Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Muon decay in an SU(2) x U(1) gauge theory with spinor and vector higgs fields and massive majorana neutrinos Hamilton, John Dwayne 1986

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

Item Metadata

Download

Media
831-UBC_1986_A1 H35.pdf [ 7.12MB ]
Metadata
JSON: 831-1.0084906.json
JSON-LD: 831-1.0084906-ld.json
RDF/XML (Pretty): 831-1.0084906-rdf.xml
RDF/JSON: 831-1.0084906-rdf.json
Turtle: 831-1.0084906-turtle.txt
N-Triples: 831-1.0084906-rdf-ntriples.txt
Original Record: 831-1.0084906-source.json
Full Text
831-1.0084906-fulltext.txt
Citation
831-1.0084906.ris

Full Text

MUON DECAY IN AN SU(2) x U ( l ) GAUGE THEORY WITH SPINOR AND VECTOR HIGGS FIELDS AND MASSIVE MAJORANA NEUTRINOS by JOHN DWAYNE HAMILTON B.E., The U n i v e r s i t y o f S a s k a t c h e w a n , 1965 M . S c , The 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 , 1967 A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE 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 c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA J a n u a r y 1986 (c) J o h n Dwayne H a m i l t o n , 1986 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 a d v a n c e d d e g r e e 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 a g r e e 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 a n d s t u d y . I f u r t h e r a g r e e 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 p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . 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 . D e p a r t m e n t o f PHYSICS  The 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 1956 Main M a l l V a n c o u v e r , C a n a d a V6T 1Y3 D a t e 20 June 1986 DE-6 (3/81) i i ABSTRACT T h i s t h e s i s i n v e s t i g a t e s t h e i m p l i c a t i o n s f o r muon d e c a y o f a gauge t h e o r y t h a t i n c o r p o r a t e s m o d i f i c a t i o n s t o t h e s t a n d a r d e l e c t r o - w e a k t h e o r y , t h e W e i n b e r g - S a l a m m o d e l . The H i g g s s e c t o r o f t h e o r i g i n a l model i s b r o a d e n e d t o i n c l u d e an i s o -v e c t o r o r t r i p l e t H i g g s f i e l d . S u c h an a d d i t i o n n a t u r a l l y p r o v i d e s f o r m a s s i v e M a j o r a n a - t y p e n e u t r i n o s and p e r m i t s t h e d e c a y modes jji -» e-tf and 11 -» 3 e w h i c h w o u l d n o t o t h e r w i s e be a l l o w e d . The p a r t i c l e s most r e s p o n s i b l e f o r t h e s e d e c a y modes a r e new, c h a r g e d , p h y s i c a l , s c a l a r H i g g s p a r t i c l e s n o t p r e s e n t i n t h e W e i n b e r g - S a l a m m o d e l . The d e c a y r a t e s f o u n d , a l t h o u g h c o n s i d e r a b l y more f a v o r a b l e t o muon d e c a y t h a n t h e s i m p l e a d d i t i o n o f n e u t r i n o mass t o t h e s t a n d a r d t h e o r y , a r e f o u n d t o be much b e l o w t h e p r e s e n t l i m i t s o f e x p e r i m e n t a l d e t e c t a b i l i t y . The l e n g t h y c a l c u l a t i o n s r e q u i r e d , i n c l u d i n g t h e d e r i v a t i o n and u t i l i z a t i o n o f t h e r e l e v a n t Feynman d i a g r a m s , a r e d i s p l a y e d i n some d e t a i l . i i i MUON DECAY IN AN SU(2) x U ( l ) GAUGE THEORY WITH SPINOR  AND VECTOR HIGGS FIELDS AND MASSIVE MAJORANA NEUTRINOS TABLE OF CONTENTS page A b s t r a c t i i L i s t o f T a b l e s v L i s t o f F i g u r e s v i A c k n o w l e d g e m e n t v i i I . I n t r o d u c t i o n and Summary 1.1 I n t r o d u c t i o n : F i f t y Y e a r s o f Weak I n t e r a c t i o n s . . . . 1 1.2 O v e r v i e w and Summary 6 I I . The SU(2) x U ( l ) Gauge T h e o r y w i t h I s o - S p i n o r and I s o - V e c t o r H i g g s F i e l d s 11.1 QED and S i g n C o n v e n t i o n s f o r C h a r g e s and F i e l d s . . 1 3 11.2 The W e i n b e r g - S a l a m M o d e l 16 11.3 The I s o - V e c t o r ( T r i p l e t ) H i g g s F i e l d 24 11.4 The P h y s i c a l and U n p h y s i c a l H i g g s F i e l d s 29 11.5 Gauge I n v a r i a n c e o f t h e T h e o r y 36 11.6 Gauge F i x i n g Terms: The G e n e r a l R^ Gauge 39 11.7 C, P, and T I n v a r i a n c e P r o p e r t i e s o f t h e T h e o r y . . 4 2 I I I . D i r a c and M a j o r a n a N e u t r i n o s 111.1 N e u t r i n o Mass 50 111.2 D i r a c N e u t r i n o s 52 111.3 M a j o r a n a N e u t r i n o s 55 IV. The Feynman R u l e s o f t h e T h e o r y IV. 1 L e p t o n - B o s o n I n t e r a c t i o n s : Summary 60 IV.2 B o s o n - B o s o n I n t e r a c t i o n s : Summary 63 I V . 3 P r o p a g a t o r s 65 I V . 4 V e r t i c e s 70 IV. 5 Changes t o W e i n b e r g - S a l a m M o d e l and t h e H a d r o n i c S e c t o r 80 V. The D e c a y p. -» etf V. l R a d i a t i v e D e c a y s i n G e n e r a l 84 V.2 The D i a g r a m s and A m p l i t u d e s f o r t h e Decay p. -• etf.87 V.3 E v a l u a t i o n o f t h e A m p l i t u d e s f o r pi -» e-rf 91 V.4 The Decay R a t e f o r p -» etf 94 i v V I . The Decay p -• 3e V I . 1 The D i a g r a m s f o r t h e D e c a y p -» 3e 97 V I . 2 E v a l u a t i o n o f t h e A m p l i t u d e s f o r p -• 3e 101 V I . 3 The D e c a y R a t e f o r p -* 3e 107 V I . 4 The C r o s s - S e c t i o n s f o r p + X -+ e + X 113 V I I . D i s c u s s i o n and C o n c l u s i o n s 117 B i b l i o g r a p h y 123 A p p e n d i c e s A. N o t a t i o n and C o n v e n t i o n s 127 B. The G e o m e t r i c A l g e b r a : C o n v e n t i o n s and A p p l i c a t i o n s 129 C. M a n i f e s t l y C o v a r i a n t C, P, and T T r a n s f o r m a t i o n s . . . 137 D. On I n v a r i a n t S p a c e - T i m e I n t e g r a l s 143 E. D e t a i l s o f t h e C a l c u l a t i o n p -• eV 147 F. D e t a i l s o f t h e C a l c u l a t i o n p -• 3e 156 G. L i s t o f V e r t i c e s f o r Feynman D i a g r a m s 164 V L I S T OF TABLES page T a b l e I : The P h y s i c a l a nd U n p h y s i c a l H i g g s F i e l d s 35 T a b l e I I : C, P and T I n v a r i a n c e o f E l e c t r o m a g n e t i s m . . . . 1 4 1 / v i L I S T OF FIGURES page F i g . 1: The W e i n b e r g a n g l e e w 20 F i g . 2: The QED v e r t e x 71 F i g . 3: The c h a r g e d l e p t o n - Z v e r t e x 72 F i g . 4: C h a r g e d l e p t o n - n e u t r i n o - W v e r t i c e s 73 F i g . 5: C h a r g e d l e p t o n - n e u t r i n o - c h a r g e d H i g g s v e r t i c e s . 7 4 F i g . 6: The c h a r g e d l e p t o n - d o u b l y c h a r g e d H i g g s v e r t e x . 7 6 F i g . 7: The W-W-A.Z v e r t e x 77 F i g . 8: The c h a r g e d H i g g s - p h o t o n v e r t e x 78 F i g . 9: The W-A-S v e r t e x 79 F i g . 10: Feynman d i a g r a m s f o r 2e -» 2p 80 F i g . 11: The r a d i a t i v e d e c a y p •+ ey 85 F i g . 12: Feynman d i a g r a m W i n t h e d e c a y p -* ey 88 F i g . 13: Feynman d i a g r a m S i n t h e d e c a y p •* ey 89 F i g . 14: Feynman d i a g r a m WS i n t h e d e c a y p •+ ey 89 F i g . 15: Feynman d i a g r a m SW i n t h e d e c a y p •+ ey .90 F i g . 16: Feynman d i a g r a m B i n t h e d e c a y p ey 90 F i g . 17: P o s s i b l e s e q u e n c e s f o r t h e d e c a y mode p -» 3e...98 F i g . 18: C o n t r i b u t i o n o f B + t o t h e d e c a y p -» 3e 99 F i g . 19: A 2x Feynman d i a g r a m f o r t h e d e c a y p -» 3e 99 F i g . 20: C o n t r i b u t i o n o f H + + t o t h e d e c a y p -* 3e 100 F i g . 21: Feynman d i a g r a m n e e d e d t o r e n o r m a l i z e F i g . 18.103 F i g . 22: C o n t r i b u t i o n o f W,S t o t h e d e c a y p -» 3e 105 F i g . 23: A p p r o x i m a t e i n t e g r a t i o n r e g i o n i n t h e x 1 - x 2 p l a n e 110 F i g . 24: Maximum and minimum e l e c t r o n e n e r g i e s 110 F i g . 25: Feynman d i a g r a m f o r p + + e -• e + e 114 F i g . 26: Feynman d i a g r a m f o r p + d e + d 115 i v i i ACKNOWLEDGMENTS The a u t h o r w o u l d l i k e t o e x p r e s s h i s a p p r e c i a t i o n t o Dr. J.N. Ng f o r h i s g u i d a n c e i n t h e c h o i c e o f t h e s i s t o p i c and h i s a s s i s t a n c e w i t h t h e l e n g t h y and c o m p l e x c a l c u l a t i o n s t h a t e n s u e d . S i n c e r e t h a n k s as w e l l a r e owed t o t h e P h y s i c s D e p a r t m e n t , p a r t i c u l a r l y t o Ms. L. H o f f m a n , f o r a s s i s t a n c e d u r i n g t h e a u t h o r ' s l o n g o f f - c a m p u s p e r i o d s d u r i n g t h e c o u r s e o f h i s employment. 1 I . INTRODUCTION AND SUMMARY 1.1 INTRODUCTION: F I F T Y YEARS OF WEAK INTERACTIONS The r e c e n t e x p l o s i v e g r o w t h i n o u r e x p e r i m e n t a l and t h e o r e t i c a l k n o w l e d g e o f weak i n t e r a c t i o n p h y s i c s c o u l d h a r d l y be i m a g i n e d f i f t y y e a r s ago when i t b e g a n w i t h t h e d i s c o v e r y o f t h e n e u t r o n a n d i t s r e l a t i v e l y s l o w B - d e c a y i n t o a p r o t o n , an e l e c t r o n and a c o n j e c t u r e d spin-™, r e l a t i v e l y m a s s l e s s , n e u t r a l p a r t i c l e , t h e n e u t r i n o . F e r m i ( 1 9 3 4 ) was t h e f i r s t t o a t t e m p t a m a t h e m a t i c a l d e s c r i p t i o n o f t h e p r o c e s s , w i t h a r a t h e r p h e n o m e n o l o g i c a l i n t e r a c t i o n L a g r a n g i a n t h a t a c c o m p l i s h e d l i t t l e more t h a n t h e s i m p l e d e s c r i p t i o n o f t h e th e n - k n o w n e x p e r i m e n t a l f a c t s . More t h a n two d e c a d e s h a d t o p a s s b e f o r e t h e d i s c o v e r y o f p a r i t y n o n - c o n s e r v a t i o n ( L e e and Yang, 1956) p r o v i d e d t h e c l u e f o r a s i g n i f i c a n t m o d i f i c a t i o n o f t h e o r i g i n a l F e r m i t h e o r y . I t was o b s e r v e d t h a t , i n t h e h i g h e n e r g y l i m i t a t l e a s t , o n l y e l e c t r o n s a n d n e u t r i n o s o f l e f t - h a n d e d h e l i c i t y ( a n d t h e i r r i g h t - h a n d e d a n t i - p a r t i c l e s ) p a r t i c i p a t e d i n weak i n t e r a c t i o n s , r i g h t - h a n d e d e l e c t r o n s n o t a t a l l . T h i s was e x p r e s s e d m a t h e m a t i c a l l y i n t h e s o - c a l l e d "V-A" ( " v e c t o r m i n u s a x i a l -v e c t o r " ) t h e o r y (Feynman and G e l l - M a n n , 1958; M a r s h a k and S u d a r s h a n , 1 9 5 8 ) , w h i c h was s h o r t l y t h e r e a f t e r e x t e n d e d t o t h e h a d r o n s , o r s t r o n g l y i n t e r a c t i n g p a r t i c l e s . E x p e r i m e n t a l s u p p o r t f o r t h e t h e o r y , a t t h e few GeV l e v e l and l e s s e r 2 e n e r g i e s , s t a r t e d t o mount. W i t h e s s e n t i a l l y t h e same c o u p l i n g c o n s t a n t s , n e u t r o n , p i o n and muon d e c a y s c o u l d be d e s c r i b e d — a q u a n t i t a t i v e s u c c e s s known as " u n i v e r s a l i t y " — b u t d r a m a t i c c o n f i r m a t i o n f o r t h e m a t h e m a t i c a l d e s c r i p t i o n was g i v e n b y p i o n d e c a y , i n w h i c h t h e r a t e o f d e c a y i n t o muons r a t h e r t h a n t h e s e e m i n g l y more l i k e l y , b u t h e l i c i t y - s u p p r e s s e d , e l e c t r o n s was q u i t e p r e c i s e l y d e s c r i b e d by t h e V-A t h e o r y , as were some o f t h e p r o p e r t i e s o f t h e much l a t e r d i s c o v e r e d T - p a r t i c l e ( P e r k i n s , 1982, S e c . 6.8, 6 . 1 4 ) . On t h e o t h e r h a n d , w e l l - k n o w n w e a k n e s s e s o f t h e t h e o r y , s u c h as i t s n o n r e n o r m a l i z a b i 1 i t y , o r m a t h e m a t i c a l i n a b i l i t y t o d e s c r i b e h i g h e r o r d e r p r o c e s s e s , and i t s v i o l a t i o n o f b a s i c c o n s e r v a t i o n o f p r o b a b i l i t y ( o r " u n i t a r i t y " ) a t h i g h e n e r g i e s ( a few h u n d r e d GeV) s t r o n g l y r e s i s t e d a t t e m p t s t o p e r f e c t t h e t h e o r y , e v e n e f f o r t s t h a t a t t e m p t e d t o i n c o r p o r a t e t h e h e a v y , c h a r g e d b o s o n s t h a t w e re p o s t u l a t e d t o m e d i a t e weak i n t e r a c t i o n s . T h e s e e f f o r t s went u n r e w a r d e d , f o r t h e most p a r t , u n t i l t h e i n v e n t i o n o f t h e n o n - a b e l i a n gauge t h e o r y now known as t h e W e i n b e r g ( 1 9 6 7 ) - S a l a m ( 1 9 6 8 ) m o d e l , w h i c h i s t h e p o i n t o f d e p a r t u r e f o r Chap. I I . The s t o r y o f t h e weak i n t e r a c t i o n t o t h e t i m e o f t h e W e i n b e r g - S a l a m t h e o r y and i t s r e l a t i o n t o t h e s t r o n g i n t e r a c t i o n i s a l o n g and f a s c i n a t i n g one, w i t h many s u c c e s s e s and f a i l u r e s on t h e t h e o r e t i c a l s i d e and enormous c h a l l e n g e s on t h e e x p e r i m e n t a l . I t has b e e n w e l l t o l d i n many p l a c e s , most r e c e n t l y i n T a y l o r ( 1 9 7 6 ) , I t z y k s o n and Z u b e r ( 1 9 8 0 ) , V e l t m a n 3 (1980), Lee (1981), Cheng and L i (1984), and many e a r l i e r accounts. (The b r i e f and s u p e r f i c i a l account given here h a r d l y begins to do i t j u s t i c e . ) The Weinberg-Salam theory became the standard model of the weak i n t e r a c t i o n s with proof of i t s r e n o r m a l i z a b i l i t y by t'Hooft (1971). Experimental support began to accumulate s h o r t l y t h e r e a f t e r . N e u t r a l c u r r e n t s , s o - c a l l e d , or i n t e r a c t i o n s mediated by a massive, n e u t r a l boson, were d i s c o v e r e d i n e l e c t r o n - n e u t r i n o and nucleon-neutrino s c a t t e r i n g , as were h e l i c i t y asymmetries i n e l e c t r o n - d e u t e r o n s c a t t e r i n g , i n which the h e l i c i t y - d e p e n d e n t weak i n t e r a c t i o n became e x p e r i m e n t a l l y n o t i c e a b l e . Even c o n t r i b u t i o n s of the weak i n t e r a c t i o n to e l e c t r o n - e l e c t r o n s c a t t e r i n g , u n t i l r e c e n t l y wholely w i t h i n the purview of quantum electrodynamics, became d e t e c t a b l e . Accounts of these experiments i n the mid to l a t e 1970's and e a r l y 1980's and t h e i r q u a n t i t a t i v e agreement with the Weinberg-Salam theory can be found i n , f o r example, Perk i n s (1982, Sec. 8.8). Now, r e c e n t l y , the long-sought p a r t i c l e s r e q u i r e d by the theory to mediate the weak i n t e r a c t i o n have been detected, with masses i n r a t h e r dramatic agreement with the theory (as to be d i s c u s s e d i n Chap. I I ) . So why, one might wonder, would anyone want to modify or extend the b a s i c Weinberg-Salam theory, the remarkable success that i t i s ? There are both experimental and t h e o r e t i c a l reasons, a few of which we now d i s c u s s . 4 Reports of measurements that i n d i c a t e a non-zero mass f o r the n e u t r i n o (Lubimov, 1980) have s e r i o u s i m p l i c a t i o n s f o r weak i n t e r a c t i o n t h e o r i e s because the Weinberg-Salam model i n p a r t i c u l a r assumed at the outset that a l l n e u t r i n o s were massless. There i s no unique and s t r a i g h t f o r w a r d way to amend the model to accommodate s p e c i e s of n e u t r i n o s with v a r i o u s masses, the s u b j e c t to which Chap. I l l i s devoted. Fur t h e r , recent r e p o r t s (e.g., Rohlf, 1985) of anomalous ± o events at the c o l l i d e r s i n which the heavy gauge bosons (W ,Z ) were f i r s t d e t e c t e d seem to i n d i c a t e that the Weinberg-Salam model cannot be the whole s t o r y , that new p a r t i c l e s might perhaps e x i s t that come as a s u r p r i s e to everyone. F i n a l l y , the time i s r a p i d l y approaching i n which the new generation of p a r t i c l e a c c e l e r a t o r s , with c e n t e r of mass energies i n the TeV range, w i l l l i k e l y t e s t a l l p a r t i c l e t h e o r i e s s e v e r e l y , and c o n c e i v a b l y show that much new p h y s i c s e x i s t s i n t h i s energy realm (Salam, 1982; Weinberg, 1984; E i c h t e n et al., 1984). I t i s not unreasonable f o r theory to attempt to p r e d i c t j u s t what new p h y s i c s t h i s might e n t a i l . On the t h e o r e t i c a l s i d e , p o s s i b l y the e x i s t e n c e of three (so f a r ) generations or f a m i l i e s of p a r t i c l e s poses the g r e a t e s t c h a l l e n g e , e s p e c i a l l y f o r the d e s c r i p t i o n of n e u t r i n o s . P o s s i b l y the mysterious n e u t r i n o s would be not q u i t e so mysterious i f they d i d possess a v a r i e t y of masses, l i k e t h e i r charged l e p t o n i c c o u s i n s , and one would l i k e to be able to d e s c r i b e such a h i e r a r c h y i n the most economical way 5 without d i s t u r b i n g too much the b a s i c Weinberg-Salam model which has come to be c o n s i d e r e d as b a s i c a l l y c o r r e c t and t h e r e f o r e r a t h e r f i x e d . A r e l a t e d concern i s with what i s c a l l e d the Higgs s e c t o r of the theory, a category of s p i n l e s s p a r t i c l e s whose e x i s t e n c e seems to be r e q u i r e d to make gauge t h e o r i e s work i n general and the Weinberg-Salam model i n p a r t i c u l a r . The new g e n e r a t i o n of p a r t i c l e a c c e l e r a t o r s and c o l l i d e r s should c r e a t e such p a r t i c l e s i f they r e a l l y e x i s t ( s t i l l an open que s t i o n , perhaps: Veltman, 1980), and i f more than the one such p a r t i c l e r e q u i r e d by the Weinberg-Salam model e x i s t s severe c o n s t r a i n t s would be p l a c e d on m o d i f i c a t i o n s to the model, and on the grand u n i f i e d t h e o r i e s that would have to i n c o r p o r a t e them i n a more general s e t t i n g . F i n a l l y , and what i s most r e l e v a n t f o r t h i s t h e s i s , new d e t e c t i o n techniques i n c r e a s e by a few orders of magnitude the p o s s i b i l i t y of o b s e r v i n g r a r e ( i f at a l l p o s s i b l e ) p a r t i c l e decay modes, i n p a r t i c u l a r the decay of a muon i n t o an e l e c t r o n and a photon (the non-observation of which i s r e s p o n s i b l e f o r the idea of separate c o n s e r v a t i o n of e l e c t r o n lepton number, muon lepton number, etc.) or i n t o an e l e c t r o n and e l e c t r o n -p o s i t r o n p a i r . It i s most i n t e r e s t i n g that such decays are not independent of the question of n e u t r i n o mass, that the former seems to imply the l a t t e r (but massive n e u t r i n o s do not i n e v i t a b l y l e a d to muon decay i n the ways mentioned). Muon decays and massive n e u t r i n o s , along with the l e a s t 6 d i s r u p t i v e changes to the Weinberg-Salam model that make them p o s s i b l e , are the primary t o p i c s of t h i s t h e s i s . 1.2 OVERVIEW AND SUMMARY There are many ways i n which the b a s i c Weinberg-Salam theory can be m o d i f i e d or extended to encompass new p h y s i c a l e f f e c t s without much changing the model's p r e d i c t i o n s i n the realms that have now been e x p e r i m e n t a l l y confirmed. Almost everyone's f i r s t p r e o ccupation would be to p r o v i d e f o r massive n e u t r i n o s and there are a number of ways i n which t h i s can be accomplished, almost always by broadening the Higgs s e c t o r of the o r i g i n a l model (see Sec. II.3 f o r examples and r e f e r e n c e s ) . A g e n e r a l i z a t i o n long known to accomplish the g e n e r a t i o n of n e u t r i n o mass i s to augment the standard doublet or i s o - s p i n o r Higgs f i e l d w ith a t r i p l e t or i s o - v e c t o r Higgs f i e l d , such a f i e l d b eing a n a t u r a l r e p r e s e n t a t i o n of the gauge group (SU(2)xU(l)) that seems most s u i t a b l e f o r the u n i f i c a t i o n of the e l e c t r o m a g n e t i c and weak i n t e r a c t i o n s . There i s one reason i n p a r t i c u l a r why the v e c t o r Higgs f i e l d i s such a n a t u r a l supplement to an i s o - s p i n o r f i e l d . The o r i g i n a l complex doublet c o n s i s t s of a charged p a i r of s c a l a r f i e l d s , and an uncharged p a i r . Although a l l but one n e u t r a l p a r t i c l e of t h i s group i s u n p h y s i c a l i n the sense that gauges 7 e x i s t i n which these p a r t i c l e s are absent, i t i s convenient f o r now, and e s p e c i a l l y l a t e r when a gauge i s chosen to f a c i l i t a t e c a l c u l a t i o n s r a t h e r than to dispose of u n p h y s i c a l p a r t i c l e s , to c o n s i d e r a l l f o u r . A t r i p l e t Higgs f i e l d i n t r o d u c e s s i x new s c a l a r p a r t i c l e s , but the p o i n t i s that they are not independent of the o r i g i n a l f o u r — they b l e n d with them, so to speak. The blend r e s u l t s i n an u n p h y s i c a l charged p a i r , an u n p h y s i c a l n e u t r a l s c a l a r p a r t i c l e , a p h y s i c a l n e u t r a l s c a l a r p a r t i c l e — a l l j u s t as i n the o r i g i n a l Weinberg-Salam model --pl u s another two n e u t r a l s c a l a r p a r t i c l e s , a new, p h y s i c a l , s i n g l y - c h a r g e d p a i r of s c a l a r p a r t i c l e s and a new, p h y s i c a l , doubly-charged p a i r . Thus the experimental prospects become enriched: r a t h e r than a s i n g l e , n e u t r a l p a r t i c l e to search f o r , one has three, and i n a d d i t i o n , two p a i r s of charged s c a l a r p a r t i c l e s although, u n f o r t u n a t e l y , a l l of unknown mass except that one s c a l a r p a r t i c l e i s massless. In g e n e r a l , the m o d i f i e d theory i s no improvement i n t h i s r e s p e c t . An independent concern i s with the g e n e r a t i o n of n e u t r i n o mass. It w i l l be shown i n Chap. I l l that the t r i p l e t Higgs f i e l d a u t o m a t i c a l l y generates n e u t r i n o mass (assuming, of course, that the a p p r o p r i a t e c o u p l i n g constants do not v a n i s h ) , of the Majorana type, thus s p a r i n g us the unpleasant task (because of i t s o b v i o u s l y a r b i t r a r y nature) of having to add ad hoc terms to the Lagrangian s p e c i f i c a l l y designed to give mass to the n e u t r i n o f i e l d s . No f u r t h e r a r b i t r a r y terms need be cons i d e r e d . 8 The S U ( 2 ) x U ( l ) gauge t h e o r y t h a t r e s u l t s seems f i r s t t o hav e been c o n s i d e r e d i n d e t a i l by P a l and W o l f e n s t e i n ( 1 9 8 2 ) who a p p l i e d i t t o t h e p r o b l e m o f m a s s i v e n e u t r i n o d e c a y . T h i s t h e s i s w i l l e m p l o y t h e m o d e l t o e x p l o r e i t s c o n s e q u e n c e s f o r muon d e c a y , t o s e e i f d e c a y modes s u c h as a r a d i a t i v e d e c a y , o r a d e c a y i n t o a t r i p l e t o f e l e c t r o n s ( u s i n g t h e t e r m " e l e c t r o n " g e n e r i c a l l y ) , may p o s s i b l y come w i t h i n t h e r a n g e o f d e t e c t a b i 1 i t y i n c o m p a r i s o n w i t h t h e u s u a l muon d e c a y i n t o an e l e c t r o n and a n e u t r i n o p a i r . One s h o u l d n o t e t h a t v e r y g e n e r a l a n a l y s e s o f v i r t u a l l y a l l s i m p l e a d d i t i o n s t o t h e H i g g s s e c t o r h a v e b e e n p u b l i s h e d i n r e c e n t y e a r s ( s e e t h e r e f e r e n c e s c i t e d e a r l y i n S e c . I I . 3 ) , w i t h r e s u l t s f o r muon d e c a y , f o r e x a m p l e , t h a t v a r y o v e r many o r d e r s o f m a g n i t u d e i n t h e i r p r e d i c t i o n s . S u c h g e n e r a l i t y , h o w e v e r , does n o t c o n s i d e r t h e many i n t e r e s t i n g r e s u l t s t h a t e x i s t w i t h i n e a c h d e t a i l e d m o d e l , a nd o v e r l o o k s t h e p o s s i b i l i t y t h a t w i t h i n a s p e c i f i c m o d el i n t e r e s t i n g p h y s i c a l p r e d i c t i o n s may be made, as i s p o s s i b l y t h e c a s e w i t h t h e model t o be e x a m i n e d i n d e t a i l h e r e , a l t h o u g h o u r r e s u l t s i n no way c o n t r a d i c t more g e n e r a l o n e s . G e n e r a l and s p e c i f i c a n a l y s e s o f gauge t h e o r e t i c m o d e l s c a n c e r t a i n l y c o - e x i s t . The n a t u r a l s t a r t i n g p o i n t f o r t h e n e x t c h a p t e r i s a b r i e f e l u c i d a t i o n o f t h e W e i n b e r g - S a l a m m o d e l . T h i s w i l l make c l e a r t h e n o t a t i o n and s i g n c o n v e n t i o n s t o be u s e d and d i s p l a y , f o r l a t e r r e f e r e n c e , t h o s e s e g m e n t s o f t h e t h e o r y t h a t w i l l be m o d i f i e d . The t r i p l e t o r i s o - v e c t o r H i g g s f i e l d w i l l be 9 i n t r o d u c e d n e x t , and i t w i l l be shown how i t augments and c h a n g es t h e o r i g i n a l W e i n b e r g - S a l a m t h e o r y ' s p r e d i c t i o n s , s u c h as, f o r example, t h e masses o f t h e gauge b o s o n s and t h e number o f new, s c a l a r H i g g s p a r t i c l e s . T hese s i x new s c a l a r p a r t i c l e s , a l o n g w i t h t h e o r i g i n a l f o u r , must be g r o u p e d i n t o p h y s i c a l and u n p h y s i c a l p a r t i c l e s . The S U ( 2 ) x U(l) gauge i n v a r i a n c e o f t h e new H i g g s f i e l d w i l l t h e n be d e m o n s t r a t e d i n d e t a i l . B e f o r e a c t u a l c a l c u l a t i o n s can p r o c e e d , however, a s p e c i f i c gauge must be c h o s e n . The p o p u l a r " r e n o r m a l i z a b l e gauge" ( o r R^ gauge, as i t i s f r e q u e n t l y d e s i g n a t e d ) w i l l be e x p l a i n e d , and u s e d t o show how i t c l e v e r l y h a n d l e s t h e p r o b l e m o f t h e u n p h y s i c a l H i g g s p a r t i c l e s and c e r t a i n o f t h e i r i n t e r a c t i o n t erms t h a t must be e l i m i n a t e d . B r i e f l y , t h e gauge g i v e s a g a u g e - d e p e n d e n t mass t o s u c h p a r t i c l e s , b u t o t h e r w i s e e n a b l e s one t o t r e a t them as i f t h e y were r e a l . A l l t r a c e o f t h e s e u n p h y s i c a l p a r t i c l e s v a n i s h e s from t h e f i n a l r e s u l t s , as r e q u i r e d , b u t t h e y a r e i n d i s p e n s a b l e d u r i n g t h e i n t e r m e d i a t e s t a g e s t o m a i n t a i n t h e t h e o r y ' s gauge i n v a r i a n c e . F i n a l l y , Chap. I I w i l l c o n c l u d e w i t h a c o n s i d e r a t i o n o f t h e C, P, and T - i n v a r i a n c e p r o p e r t i e s o f t h e t h e o r y . T h e r e w i l l be no new r e s u l t s h e r e , as i s t o be e x p e c t e d . C h a p t e r I I I w i l l d i s c u s s t h e p r o b l e m o f n e u t r i n o mass i n g e n e r a l and t h e two t y p e s o f n e u t r i n o s t h a t can e x i s t . As no new terms b e y o n d t h e i s o - v e c t o r H i g g s f i e l d and i t s i n t e r a c t i o n w i t h t h e l e p t o n s e c t o r a r e t o be added t o t h e W e i n b e r g - S a l a m 10 L a g r a n g i a n , i t w i l l be shown how a s p e c i f i c n e u t r i n o t y p e --m a s s i v e M a j o r a n a n e u t r i n o s — emerges f r o m t h e t h e o r y . The m a j o r t o o l s u s e d i n d e t a i l e d c a l c u l a t i o n s w i t h quantum f i e l d t h e o r i e s f o r t h e p a s t t h i r t y y e a r s h a v e been t h e s o -c a l l e d "Feynman r u l e s " o f t h e p a r t i c u l a r t h e o r y u n d e r i n v e s t i g a t i o n . S u c h t e c h n i q u e s d i s p l a y t h e i n t e r a c t i o n t e r m s o f a t h e o r y i n t h e i r most b a s i c and e l e m e n t a r y way, e n a b l i n g one t o t h e n s e l e c t i n a r e l a t i v e l y s i m p l e a n d s t r a i g h t f o r w a r d f a s h i o n o n l y t h o s e t e r m s n e e d e d f o r a s p e c i f i c i n t e r a c t i o n p r o c e s s , a s e l e c t i o n i m m e a s u r a b l y a i d e d by t h e "Feynman d i a g r a m s " t h a t accompany and g i v e p i c t o r a l r e p r e s e n t a t i o n t o t h e i n t e r a c t i o n t e r m s . C h a p t e r IV w i l l l i s t t h e r u l e s and d i a g r a m s t h a t f o l l o w f r o m t h e gauge t h e o r y d e s c r i b e d i n Chap. I I , I I I , b u t o n l y t h o s e t h a t w i l l be n e e d e d i n t h i s t h e s i s , a f r a c t i o n o f a l l t h a t c o u l d be l i s t e d . The r e a l c a l c u l a t i o n s o f t h i s t h e s i s w i l l b e g i n i n Chap. V, w h i c h c o n s i d e r s t h e m o d e l ' s i m p l i c a t i o n s f o r t h e r a d i a t i v e d e c a y o f t h e muon. The c a l c u l a t i o n s a r e r a t h e r l e n g t h y b u t t h e f i n a l r e s u l t i s n o t . We w i l l f i n d t h a t , w i t h t h e e x c e p t i o n o f t h e new, p h y s i c a l c h a r g e d H i g g s b o s o n s , t h e M a j o r a n a n e u t r i n o s u s e d h e r e do n o t c h a n g e t h e b a s i c r e s u l t o f Cheng and L i ( 1 9 8 0 b , 1984) c o n c e r n i n g t h i s d e c a y ( w h e r e t h e y e m p l o y e d D i r a c n e u t r i n o s ) , n a m e l y t h a t i t i s some 40 (!) o r s o o r d e r s o f m a g n i t u d e l e s s l i k e l y t h a n t h e u s u a l d e c a y mode. B u t o u r o v e r a l l r e s u l t i s q u i t e d i f f e r e n t . The v e c t o r H i g g s model c o n t a i n s an i n t e r e s t i n g p e c u l i a r i t y : t h e t e r m s i n v o l v i n g t h e 11 p h y s i c a l , charged Higgs p a r t i c l e are a c t u a l l y and remarkably independent of the n e u t r i n o masses, e n a b l i n g us to show that the r a d i a t i v e decay d i s c u s s e d here may a c t u a l l y be some 25 or more orders of magnitude g r e a t e r than the p e s s i m i s t i c r e s u l t obtained by Cheng and L i , but s t i l l , i t seems, some 10 or so orders of magnitude below c u r r e n t d e t e c t a b i l i t y . The other decay mode perm i t t e d by our gauge model i s the t h r e e - e l e c t r o n decay, to which Chap. VI i s devoted. This decay mode i s p o s s i b l e i n three d i f f e r e n t ways, a l l , i t seems, of the same order of magnitude, and, l i k e the p. -» ey decay, appear to be about 10 orders of magnitude below c u r r e n t d e t e c t a b i l i t y . I t turns out, though, that i n each case the n -* 3e decay i s more l i k e l y than the r a d i a t i v e decay. Numerical estimates of the v a r i o u s decay r a t e s are made and d i s c u s s e d i n Chap. VII. The mathematical n o t a t i o n used throughout t h i s t h e s i s i s f o r the most pa r t c o n v e n t i o n a l , but d i f f e r s from the usual i n at l e a s t one notable r e s p e c t . A m a n i f e s t l y c o v a r i a n t , matrix r e p r e s e n t a t i o n - f r e e , geometric formalism f o r the D i r a c a l g e b r a and the SU(2) or P a u l i a l g e b r a i s employed f r e e l y , although i t seems not to be w i d e l y known. It i s b r i e f l y d e s c r i b e d i n App. B, and at more le n g t h and i n g r e a t e r d e t a i l i n Hamilton (1984a,b) and r e f e r e n c e s contained t h e r e i n . With t h i s formalism one need not concern o n e s e l f with any matrix r e p r e s e n t a i o n of the D i r a c a l g e b r a , which i s e s p e c i a l l y convenient with the C, P, and T t r a n s f o r m a t i o n s which are expressed here i n a m a n i f e s t l y c o v a r i a n t way, i n marked 12 c o n t r a s t t o t h e u s u a l r e p r e s e n t a t i o n - d e p e n d e n t t r a n s f o r m a t i o n s ( a s e x p l a i n e d i n S e c . I I . 6 and App. C ) . F o r t h i s r e a s o n t h e C, P, and T t r a n s f o r m a t i o n s a r e c o n s i d e r e d i n some d e t a i l . The m a t r i x - f r e e D i r a c a l g e b r a i s e s p e c i a l l y u s e f u l i n Chap. V and App. E i n w h i c h t h e few t e r m s w i t h t h e c o r r e c t f o r m f o r t h e s p i n o r a m p l i t u d e d e s c r i b i n g a r a d i a t i v e t r a n s i t i o n c o u l d be e x t r a c t e d w i t h some e a s e f r o m among t h e m u l t i t u d e o f t e r m s a v a i l a b l e . F u r t h e r , t h e u s u a l " t r a c e " method o f e v a l u a t i n g s p i n o r a m p l i t u d e s i s r e p l a c e d b y one i n w h i c h t h e s c a l a r component o f t h e a l g e b r a i s u s e d , w h i c h p r o v i d e s , f o r t h e f i r s t t i m e , a m a t r i x and r e p r e s e n t a t i o n - f r e e method o f e v a l u a t i n g s p i n o r a m p l i t u d e s , a method t h a t i s q u i c k , s i m p l e and p o w e r f u l . The b a s i c r e s u l t s a r e e x p l a i n e d i n App. B and a r e u s e d e x t e n s i v e l y i n App. E and F i n e v a l u a t i n g t h e l e n g t h y s p i n o r a m p l i t u d e s . I n i t s 3 - v e c t o r , o r S U ( 2 ) , o r P a u l i a l g e b r a i c f o r m , t h e n o t a t i o n e n a b l e s one t o s u c c i n c t l y e x p r e s s v a r i o u s S U ( 2 ) gauge i n v a r i a n t L a g r a n g i a n t e r m s f o r b o t h t h e s p i n o r and v e c t o r H i g g s f i e l d s , e s p e c i a l l y t h e gauge f i x i n g t e r m s o f S e c . I I . 6 . The r e a d e r i s t h e r e f o r e e n c o u r a g e d t o c o n s i d e r App. A and B b e f o r e e m b a r k i n g on a r e a d i n g o f t h e m a i n t e x t , w h i c h f o l l o w s n e x t , i n w h i c h t h e 3 and 4 - v e c t o r a l g e b r a i c n o t a t i o n i s u s e d f r e e l y and w i t h o u t f u r t h e r e x p l a n a t i o n . 13 I I . THE S U ( 2 ) x U ( l ) GAUGE THEORY WITH  ISO-SPINOR AND ISO-VECTOR HIGGS FIELDS I I . 1 QED AND SIGN CONVENTIONS FOR CHARGES AND FIELDS Quantum e l e c t r o d y n a m i c s (QED) i s t h e q u i n t e s s e n t i a l gauge t h e o r y , a n d, h a v i n g b e e n d e v e l o p e d f o r o v e r t h i r t y y e a r s i n i t s modern f o r m , i s t h e quantum f i e l d t h e o r y a f t e r w h i c h one h o p e s t o m o d e l o t h e r s . The b a s i c i d e a o f QED as an a b e l i a n U ( l ) , o r p h a s e i n v a r i a n t , gauge t h e o r y i s q u i t e s i m p l e , as gauge t h e o r i e s go, p a r t i c u l a r l y when c o n f i n e d t o t h e e l e c t r o d y n a m i c i n t e r a c t i o n o f s p i n - j f e r m i o n s . The f r e e p a r t i c l e D i r a c L a g r a n g i a n f o r s p i n - j , c h a r g e d p a r t i c l e s ( w i t h t h e n o t a t i o n o f App. A ) , C = * ( i a - • ) * , ( 2 . 1 . 1 ) i s c o u p l e d m i n i m a l l y ( i t i s s a i d ) t o t h e e l e c t r o m a g n e t i c f i e l d i f t h e momentum o p e r a t o r p = i a c h a n g e s i n t h e p r e s e n c e o f an e l e c t r o m a g n e t i c f i e l d as p -+ p + eA ( 2 . 1.2a) o r a -• 3 - i e A , ( 2 . 1. 2b) where t h e c o n v e n t i o n e m p l o y e d h e r e , u n f o r t u n a t e l y n o t t h e u s u a l one ( B j o r k e n and D r e l l , 1964, Chap. 1; I t z y k s o n and Z u b e r , 1980, Chap. 2 ) , i s e > 0, s o e i s t h e c h a r g e on t h e p o s i t r o n o r p r o t o n . I n ( 2 . 1 . 2 ) A i s t h e e l e c t r o m a g n e t i c v e c t o r p o t e n t i a l , r e l a t e d t o t h e e l e c t r o m a g n e t i c b i v e c t o r f i e l d F (App. B) by 14 F = a A A , (2.1.3a) o r , i n component form, F = a A - a A . (2.1.3b) \iv n v v W i t h a k i n e t i c t e r m f o r t h e e l e c t r o m a g n e t i c f i e l d t h e L a g r a n g i a n o f QED becomes C = 4»(ia - m + e A H - j F F^" , (2.1.4) 4 JJIV w h i c h i s i n v a r i a n t u n d e r t h e change o f gauge »|/(x) -• e x p [ i e « ( x ) ] «P(x) (2.1.5a) * -» «l> e x p [ - i e « ( x ) ] (2.1.5b) A(x) -» A(x) +a«(x) , (2 . 1 . 5 c ) where «(x) i s some c o n t i n u o u s and d i f f e r e n t i a b l e f u n c t i o n o f s p a c e - t i m e . In p r a c t i c e a "gauge f i x i n g " t e r m (Nash, 1978) must be added t o t h e QED L a g r a n g i a n b e c a u s e no G r e e n f u n c t i o n e x i s t s f o r t h e e l e c t r o m a g n e t i c f i e l d e q u a t i o n s as t h e y f o l l o w f r o m ( 2 . 1 . 4 ) . T h i s w i l l be c o n s i d e r e d l a t e r i n S e c . I I . 6 i n a more g e n e r a l c o n t e x t . The i n t e r a c t i o n L a g r a n g i a n t h a t f o l l o w s f r o m (2.1.4) i s C T = e^A* = - j A P = - M , (2.1.6) I p I where t h e e l e c t r o m a g n e t i c c u r r e n t i s j P = -e i j i v % , (2.1.7) and i s t h e i n t e r a c t i o n H a m i l t o n i a n ( d e n s i t y ) . The s i g n s a r e s u c h t h a t j-A ~ J 0A Q i s a p o s i t i v e e n e r g y d e n s i t y f o r l i k e c h a r g e s . The u s u a l e x p a n s i o n o f a s p i n o r f i e l d i n terms o f F o c k -s p a c e 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 s ( B j o r k e n and D r e l l , 1965, Chap. 13) 15 * ( x ) = (2w) ^ 2 J* d 3 p J¥7€ [e 1 P ' X a (p) u (p) r r i p «x *, . . . . + e v b (p) v ( p ) ] , r r (2.1.8) where € - p Q , a and b a r e t h e c r e a t i o n o p e r a t o r s f o r e l e c t r o n s and p o s i t r o n s , r e s p e c t i v e l y , and u, v a r e b a s i s s p i n o r s . Thus t h e f i e l d «|> a n n i h i l a t e s n e g a t i v e l y c h a r g e d e l e c t r o n s , c r e a t e s p o s i t i v e l y c h a r g e d p o s i t r o n s , w h i l e 4> c r e a t e s e l e c t r o n s and a n n i h i l a t e s p o s i t r o n s . The e l e c t r i c c h a r g e t h a t r e s u l t s f r o m (2.1.7) and (2.1.8) i s (where n o r m a l o r d e r i n g o f t h e F o c k - s p a c e o p e r a t o r s has been e m p l o y e d ) . T h e s e r e s u l t s f i x o u r e l e c t r o m a g n e t i c s i g n c o n v e n t i o n s , e n a b l e us t o i d e n t i f y any c h a r g e d c u r r e n t s and a l s o p r o v i d e a c h e c k on t h e s i g n s t h a t f o l l o w when c h a r g e d s c a l a r and v e c t o r f i e l d s a r e c o n s i d e r e d . The r e m a i n i n g s i g n c o n v e n t i o n s i n t h e f i e l d L a g r a n g i a n ( s u c h as t h e n e g a t i v e s i g n i n f r o n t o f t h e e l e c t r o m a g n e t i c f i e l d term) a r e f i x e d by r e q u i r i n g t h e e n e r g y d e n s i t y o f f r e e f i e l d s t o be p o s i t i v e . Thus t h e L a g r a n g i a n f o r a r e a l , m a s s i v e s c a l a r f i e l d i s 3 0 3 ^ % Q = T d x j = -e J"d p ( a a - b b ) r r r r (2.1.9) (2.1.10) f o r a complex, m a s s i v e s c a l a r f i e l d i s C = 3<1> -a* - m* * * (2.1.11) * ( w h i c h c o u l d be w r i t t e n as two r e a l f i e l d s 4> = (1^J"2)(<|> + * ) and <|>2 = (-i'J"2) (<|> - <fr ) o f t h e same mass); f o r a r e a l , m a s s i v e v e c t o r f i e l d i s 16 C = ™ F F M V + i - m 2 A A U ; ( 2 . 1 . 1 2 ) and f o r a c o m p l e x , m a s s i v e v e c t o r f i e l d i s 1 UV 2 ^ U £ = -- F F H + m A A . ( 2 . 1 . 1 3 ) I n e a c h c a s e t h e c a n o n i c a l energy-momentum t e n s o r i s ( J a u c h and R o h r l i c h , 1976, Chap. 1) T p v = ac 3V<|>. - t^V C ( 2 . 1 . 1 4 ) a(a • ( w h e r e t h e i g e n e r i c a l l y r e p r e s e n t s f i e l d c o m p o n e n t s ) , w h i c h s a t i s f i e s t h e c o n s e r v a t i o n l a w a T p i / = 0 , ( 2 . 1 . 1 5 ) and i n e a c h c a s e t h e ( f r e e ) f i e l d e n e r g y P° = J d 3 x T 0 0 ( 2 . 1 . 1 6 ) i s p o s i t i v e . The L a g r a n g i a n s ( 2 . 1 . 1 0 ) t o ( 2 . 1 . 1 3 ) w i l l a l s o e n a b l e us t o i d e n t i f y t h e ma s s e s o f any m a s s i v e s c a l a r o r v e c t o r f i e l d s . I I . 2 THE WEINBERG-SALAM MODEL The s t a n d a r d m o d e l o f t h e e l e c t r o m a g n e t i c and weak i n t e r a c t i o n s was f o r m u l a t e d i n 1967-1968 b y W e i n b e r g ( 1 9 6 7 ) and S a l a m ( 1 9 6 8 ) as an SU(2) x U ( l ) gauge t h e o r y i n a n a l o g y t o QED and t h e n o n - a b e l i a n gauge t h e o r y o f Yang and M i l l s ( 1 9 5 4 ) . The mod e l i s r e v i e w e d i n d e t a i l i n A b e r s and Lee ( 1 9 7 3 ) , I t z y k s o n and Z u b e r ( 1 9 8 0 ) , L a n g a c k e r ( 1 9 8 1 ) , Cheng and L i ( 1 9 8 4 ) and 17 numerous o t h e r p l a c e s . The p a r t i c l e c o n t e n t o f t h e m o d e l , w h i c h i s a l m o s t a l w a y s f o r m u l a t e d f i r s t i n a gauge t h e o r y , i s as f o l l o w s : A l l p a r t i c l e s a r e i n i t i a l l y assumed t o be m a s s l e s s , so t h a t i t makes s e n s e t o c o n s i d e r t h e l e f t - h a n d e d e l e c t r o n and l e f t -h a n d e d ( e l e c t r o n ) n e u t r i n o ( a s s u m e d n o t t o h a v e a r i g h t - h a n d e d c o u n t e r p a r t ) as an S U ( 2 ) i s o - s p i n o r o r d o u b l e t * = * T = ( V ) T s Z ( v ) , ( 2 . 2 . 1 ) L e L - e w here Z + = ~[1 ± ( - i ) V 53 a r e t h e c h i r a l i t y p r o j e c t i o n o p e r a t o r s , w h i c h happen t o be t h e same as s p i n p r o j e c t i o n o p e r a t o r s i n t h e c a s e o f m a s s l e s s f i e l d s . I n ( 2 . 2 . 1 ) v and e s t a n d f o r t h e i r r e s p e c t i v e s p i n o r f i e l d s . The r e m a i n i n g r i g h t -h a n d e d e l e c t r o n (e 5 Z e) i s an SU(2) s i n g l e t . T h e r e i s an R + SU(2) gauge f i e l d A = A U * and a U ( l ) gauge f i e l d a = a P y , a n a l o g o u s t o , b u t n o t t h e same a s , t h e e l e c t r o m a g n e t i c f i e l d . The L a g r a n g i a n , i n a n a l o g y w i t h QED i n ( 2 . 1 . 4 ) , i s C ( * , A , a ) = i * Z + ( 3 - \ i g A - \ i g ' Y L a ) < P + i eZ_(a - \ i g ' Y R a ) e - \ F* . J * 1 " - i f fUV , ( 2 . 2 . 2 ) w here g, g r a r e as y e t unknown c o u p l i n g c o n s t a n t s , Y a r e R, L numbers r e f e r r e d t o as t h e "weak h y p e r c h a r g e " , and where ? = a A - a A + g A x A ( 2 . 2 . 3 a ) p.v p. v v ix p. v f = a a - a a ( 2 . 2 . 3 b ) pv p v v p i = ^ e^ . ( 2 . 2 . 3 c ) The f i r s t f a c t o r ^ i n ( 2 . 2 . 2 ) i s b e c a u s e v e c t o r s a r e A = A.a. 2 i i b u t (- a ) a r e t h e S U(2) g e n e r a t o r s ; t h e s e c o n d i s f o r l a t e r 2 l 18 c o n v e n i e n c e . The f i e l d s ( A . , a) a r e t a k e n t o be r e a l . T h i s 1 L a g r a n g i a n i s now augmented b y t e r m s f o r t h e H i g g s f i e l d * • ( M • ( 2 - 2 - 4 a > l <|>o / an SU(2) s p i n o r o r d o u b l e t t h e co m p o n e n t s o f w h i c h a r e c o m p l e x s c a l a r f i e l d s , c h a r g e d i n t h e s e n s e d e s c r i b ' e d b y t h e i r s u b s c r i p t s . One d e f i n e s = $ $ = <t> * , ( 2 . 2 . 4 b ) + o - o so t h e L a g r a n g i a n t e r m s f o r t h i s f i e l d a r e = [ ( 3 - \ i g A - \ i g ' Y ^ a ) *] • ( 2 . 2 . 5 ) x [(a - \ i g A - \ i g ' Y ^ a ) ft] - V(ft) , where t h e s e l f - i n t e r a c t i o n t e r m i s f r e q u e n t l y t a k e n t o be V(ft) = -ii ft'ft + x ( f t * f t ) 2 ; ( 2 . 2 . 6 ) h o w e v e r i t s s p e c i f i c f o r m i s n o t o f i m p o r t a n c e a t t h e p r e s e n t t i m e . I t w i l l be c o n s i d e r e d i n S e c . I I . 4 i n c o n n e c t i o n w i t h t h e i s o - v e c t o r H i g g s f i e l d . F i n a l l y , t h e L a g r a n g i a n i s augmented by a Yukawa c o u p l i n g , an i n t e r a c t i o n among t h e s c a l a r and s p i n o r f i e l d s : C ( Y u k ) = - P (ftZ $e + eZ ft + <P) . ( 2 . 2 . 7 ) e + -A d d i t i o n a l families o f f e r m i o n s a r e r e a d i l y c o n s i d e r e d --one m e r e l y adds t e r m s f o r t h e muon = Z (v /n) ] o r t a u [4> = ii - n T £ ( v / t ) ] e x a c t l y as was done f o r t h e e l e c t r o n f a m i l y . — T The c o m p l e t e L a g r a n g i a n i s U ( l ) i n v a r i a n t i f 4»L -> exp[iY L«(x)] $ L , e R •+ exp [ i Y R«( x) ] e R ft -» exp[iY«(x)] ft ( 2 . 2 . 8 ) a a + ( 2 / g ' ) a«(x) , 19 and f r o m t h e Yukawa c o u p l i n g one f i n d s —Y + Y + Y„ = 0. ( 2 . 2 . 9 ) L <|> R The weak h y p e r c h a r g e Y i s f i x e d by d e f i n i n g Q = T 3 + \ Y I , ( 2 . 2 . 1 0 ) w h e r e Q, T 3, I a r e SU(2) o p e r a t o r s ( o r m a t r i c e s ) w i t h Q b e i n g t h e c h a r g e , T 3 t h e t h i r d ( o r d i a g o n a l ) component o f t h e SU(2) g e n e r a t o r s and I i s a u n i t m a t r i x . One has Y = 2 (Q) ( 2 . 2 . 1 1 ) av where (•••) d e n o t e s a v e r a g e , so Y f o r a m u l t i p l e t i s t w i c e av t h e mean e l e c t r i c c h a r g e : Y t = - 1 ' Y „ . = + 1 » Yu = " 2 • ( 2 . 2 . 1 2 ) L <J> K I t was f o r t h i s r e a s o n t h a t we began i n i t i a l l y w i t h <t>+, r a t h e r t h a n <(>_. The SU(2) i n v a r i a n c e o f t h e L a g r a n g i a n i s c o n s i d e r e d i n d e t a i l i n S e c . I I . 5 . The m a i n t r i c k o f modern gauge t h e o r y — t h e H i g g s m e c h a n i s m — i s t o s u p p o s e t h a t t h e n e u t r a l component o f t h e H i g g s f i e l d has a n o n - z e r o vacuum e x p e c t a t i o n v a l u e ( v / J " 2 ) , d e n o t e d by <•••>: $ = I <t>+ ] ( 2 . 2 . 13) I <|>„ + v/J"2 / <$> s / 0 \ v/J/2 T h i s d e s t r o y s t h e SU(2) and U ( l ) gauge i n v a r i a n c e b u t l e a v e s a f u r t h e r U ( l ) gauge i n v a r i a n c e w h i c h c a n be i n t e r p r e t e d as t h e e l e c t r o m a g n e t i c gauge i n v a r i a n c e . F u r t h e r , and most i m p o r t a n t , many o f t h e p r e v i o u s l y m a s s l e s s p a r t i c l e s a c q u i r e a mass. The n e u t r i n o r e m a i n s m a s s l e s s b u t t h e e l e c t r o n a c q u i r e s 20 a m a s s m = 0 v / J 2 ( 2 . 2 . 1 4 ) e e f r o m ( 2 . 2 . 7 ) , a n d t h e r e a l , p h y s i c a l H i g g s f i e l d Re(<|>0) a c q u i r e s a m a s s f r o m V ( $ ) , w h i c h we n e e d n o t c o n s i d e r a t t h i s t i m e . W i t h t h e d e f i n i t i o n s W = ( 1 / J " 2 ) ( A , + i A , ) ( 2 . 2 . 1 5 a ) Z = A„ c o s e . , - a s i n e . , ( 2 . 2 . 1 5 b ) 3 W W - A = A s i n e , . + a c o s e . . , ( 2 . 2 . 1 5 c ) w h e r e t h e a n g l e e., i s d e f i n e d i n F i g . 1 , w i t h W G 2 = g 2 + g ' 2 ( 2 . 2 . 1 6 a ) e = g s i n e = g ' c o s e . , = G c o s e . , s i n e . , , ( 2 . 2 . 1 6 b ) W W W W p h y s i c a l v e c t o r f i e l d s - - t h e g a u g e b o s o n s — a r e i n t r o d u c e d , w h i c h , f r o m ( 2 . 2 . 5 ) a n d ( 2 . 2 . 1 3 ) c a n b e s h o w n t o h a v e t h e m a s s e s M 2 = \ v 2 g 2 ( 2 . 2 . 1 7 a ) M 2 = i - v 2 G 2 ( 2 . 2 . 1 7 b ) M 2 = 0 . ( 2 . 2 . 1 7 c ) A 21 The m a s s l e s s f i e l d A i s t h e u s u a l e l e c t r o m a g n e t i c v e c t o r f i e l d . From (2.2.2) and (2.2.7) t h e l e p t o n i c p a r t o f t h e L a g r a n g i a n c an be shown t o be £ ( l e p t o n ) = i v E + 3 v + e ( i a - m)e + e eAe (2.2.18) + ( g / J " 2 ) ( v E W e + e E W v) + ^GvT. Zv 2 — 1 2 2 ~ + G s i n e., e E Ze - - G ( c o s e , - s i n e ) e E Ze W - 2 w W + - - - - * - 8 ( v E • e + e E * v + e E <l> e + e E <t> e) . e + + - - + o - o The c o n n e c t i o n w i t h t h e o l d e r F e r m i f o u r - p o i n t i n t e r a c t i o n L a g r a n g i a n ( o r V-A t h e o r y ) , C T = - ( G _ / J " 2 ) [ v ( l - i V 5 ) V e] [e(l - i V j V ^ v ] , (2.2.19) I F P-i s i n t h e f o u r t h t e r m o f ( 2 . 2 . 1 8 ) , w h i c h , w i t h v o r e e n e r g i e s b e i n g much s m a l l e r t h e n M . i s e f f e c t i v e l y W G F/J/2 » g 2 / 8 M ^ , (2.2.20) U.V 2 b e c a u s e t h e W p r o p a g a t o r i s e s s e n t i a l l y (n /M„) a t low W e n e r g i e s . W i t h G „ « 1 0 _ 5 / M 2 (2.2.21) F p fr o m e x p e r i m e n t ( t h e p. l i f e t i m e , e s s e n t i a l l y ) , where M i s t h e P p r o t o n mass, t h e W mass can be e s t i m a t e d as 2 2 /40 M \ 2 M 2 = J/2 e » p . (2.2.22) 8 G F s i n 2 e w \ sine^) I t s r e c e n t l y m e a s u r e d v a l u e was ( A r n i s o n et al., 1983a,c; Banner et al., 1983) M « 83 ± 3 GeV , (2.2.23a) W w i t h ( A r n i s o n et al., 1983b; B a g n a i a et al., 1983) M » 93 ± 2 GeV , (2.2.23b) La and 22 s i n 2 e w •» .23 , ( 2 . 2 . 2 3 c ) w i t h t h e i m p o r t a n t t h e o r e t i c a l r e l a t i o n Mw / K = g 2 / ( g 2 + g ' 2 ) = c o s 2 e w ( 2 . 2 . 2 4 ) b e i n g r a t h e r w e l l v e r i f i e d ( M a r c i a n o and S i r l i n , 1984; Schewe, 1984; E i c h t e n et al., 1984; R e u t e n s et al., 1 9 8 5 ) . The o n l y unknown now r e m a i n i n g i n t h e o r i g i n a l W e i n b e r g - S a l a m m o d e l i s t h e mass o f t h e ( p h y s i c a l ) H i g g s p a r t i c l e : t h e r e i s no p r e s e n t h i n t as t o i t s v a l u e , i f , i n d e e d , s u c h a p a r t i c l e a c t u a l l y e x i s t s . The i n t e r a c t i o n s among t h e gauge b o s o n s , t h e i r s e l f -i n t e r a c t i o n s a nd t h e i n t e r a c t i o n s b e t w e e n t h e gauge b o s o n s and t h e H i g g s p a r t i c l e s a r e o b t a i n e d b y e x p a n d i n g t h e L a g r a n g i a n i n t e r m s o f t h e s e f i e l d s . U n f o r t u n a t e l y , t h e r e s u l t i n g L a g r a n g i a n i s e x t r e m e l y l e n g t h y and e x t r e m e l y c o m p l i c a t e d , a t e s t a m e n t t o t h e i n t r i n s i c a l l y c o m p l i c a t e d s t r u c t u r e o f n o n - a b e l i a n gauge t h e o r i e s ( t h e W e i n b e r g - S a l a m m o del b e i n g e s s e n t i a l l y t h e s i m p l e s t , r e a l i s t i c m o d e l ) . Among t h o s e t e r m s t h a t r e s u l t f r o m an e x p a n s i o n o f t h e H i g g s p a r t o f t h e L a g r a n g i a n a r e | i v g ( a <t> W^ - 3 4> W P) - | i v G Z l l ( l / J - 2 ) a (<J> - <t>*) ( 2 . 2 . 2 5 ) * + - | l — + * n o o * w h i c h i d e n t i f i e s <J> , (1/J"2)(<t> - $ ) as unphysical, by w h i c h i s ± o o meant t h a t s u c h f i e l d s c a n be t r a n s f o r m e d away by a s u i t a b l e gauge t r a n s f o r m a t i o n . I n p r a c t i c e , h o w e v e r , s u c h t e r m s a r e e l i m i n a t e d b y t h e gauge f i x i n g t e r m s t o be c o n s i d e r e d l a t e r , i n S e c . I I . 6 . Thus t h e f o u r - c o m p o n e n t H i g g s f i e l d c o n t a i n s o n l y one r e a l , s c a l a r f i e l d (1 /J"2) (<t> + <t> ) , o f u n d e t e r m i n e d mass as o o 23 m e n t i o n e d a b o v e . A n o t h e r t e r m i n t h e e x p a n s i o n i s - i e A ^ C * 3 * - * a * ) , ( 2 . 2 . 2 6 ) - p + + ix -w h i c h i d e n t i f i e s j = i e (4>_a<|>+ - 4>+3<t>_) ( 2 . 2 . 2 7 ) as t h e ( u n p h y s i c a l ) c u r r e n t due t o t h e c h a r g e d H i g g s f i e l d s , as ( 2 . 1 . 6 ) shows. F u r t h e r d i s c u s s i o n o f t h e H i g g s f i e l d s w i l l be p o s t p o n e d u n t i l a f t e r t h e n e x t s e c t i o n b e c a u s e t h e new H i g g s f i e l d s t o be i n t r o d u c e d t h e r e m i x i n a n a t u r a l way w i t h t h o s e o f t h i s s e c t i o n , a m i x t h a t g e n e r a t e s p h y s i c a l and u n p h y s i c a l c h a r g e d and n e u t r a l p a r t i c l e s . The i n t e r a c t i o n s among t h e gauge b o s o n s a l o n e ( t h a t i s , w i t h o u t t h e H i g g s f i e l d s ) f o l l o w f r o m t h e s e e m i n g l y compact k i n e t i c t e r m i n t h e o r i g i n a l W e i n b e r g - S a l a m L a g r a n g i a n ( 2 . 2 . 2 ) : C(W ,A,Z) = F -F^" - 7 f f P V , ( 2 . 2 . 2 8 ) * IXV * pv w i t h F and f b e i n g g i v e n i n ( 2 . 2 . 3 ) . The e x t r e m e n o n l i n e a r i t y and i n n a t e c o m p l e x i t y o f t h e s e t e r m s becomes e v i d e n t when t h e y a r e e x p a n d e d i n t e r m s o f t h e W+, A and Z f i e l d s , d e f i n e d i n ( 2 . 2 . 1 5 ) . We i g n o r e f o u r t h o r d e r t e r m s , o r t h o s e o f o r d e r g : t h e s e a r e t h e l e n g t h y and c o m p l i c a t e d t e r m s and w i l l n o t be n e e d e d i n s u b s e q u e n t c a l c u l a t i o n s . Thus t h e p r o d u c t o f t h e two (A xA ) t e r m s i n ( 2 . 2 . 3 a ) i s t o be n e g l e c t e d . The k i n e t i c t e r m s t h a t f o l l o w f r o m ( 2 . 2 . 2 8 ) a r e , as i s t o be e x p e c t e d , C(W, A , Z : k i n e t i c ) = - i - ( a W~-a W~) ( a V - a V ) ( 2 . 2 . 2 9 ) 2 pi V v p + + - j ( a z - a z ) ( a ^ z v - a v z ^ ) - j ( a A - a A ) ( a * V - a v A M ) . * p. v v p. * p v v p The r e m a i n i n g t e r m s o f ( 2 . 2 . 2 8 ) d e s c r i b e i n t e r a c t i o n s among t h e gauge b o s o n s — i n t e r a c t i o n s , b e c a u s e t h e y d e p e n d on t h e gauge 24 c o u p l i n g c o n s t a n t g. The f i n a l r e s u l t o f r e d u c i n g (2.2.28) i s , 2 e x c e p t f o r t h e terms o f o r d e r g and t h e k i n e t i c terms o f ( 2 . 2 . 2 9 ) , C ( W , A , Z : i n t ) = ^ i e [ ( A V - A V ) ( a W + - a W + ) (2.2.30) - ( A V - A V ) ( a w~-a w~) + ( a A - a A ) (wV-wV) ] + + p v v v p i / v p - + - + - J-igcose f ( z ' V - z ' V 1 ) ( a W + - a W + ) * W — - p i / v p . - (zV-z vw M)(a w~-a w+) + ( a z - a z ) (wV-wvwM) ] . + + p v v p p v v p - + - + From t h e f i r s t t erm o f t h i s i n t e r a c t i o n L a g r a n g i a n t h e c h a r g e d c u r r e n t c an be i d e n t i f i e d u s i n g t h e d e f i n i t i o n ( 2 . 1 . 6 ) : j = - i e [ ( a w+-a w+)wv - wv(a w~-a w~)] . (2.2.31) p p . v v p . — + (i v v u T h i s e x p r e s s i o n c o r r e c t l y g i v e s t h e t o t a l e l e c t r i c c h a r g e Q = -e J" d 3 k ( a _ a _ - a + a + ) (2.2.32) when expanded i n terms o f F o c k - s p a c e 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 , where a + c r e a t e s a W + . T h i s i s an i n d e p e n d e n t c h e c k on t h e c o r r e c t n e s s o f t h e W -A i n t e r a c t i o n t e r m i n ( 2 . 2 . 3 0 ) . I I . 3 THE ISO-VECTOR (TRIPLET) HIGGS FIELD The W e i n b e r g - S a l a m gauge t h e o r y c an be m o d i f i e d o r g e n e r a l i z e d i n a number o f ways, most n o t a b l y t h r o u g h an e n r i c h m e n t o f t h e H i g g s s e c t o r o f t h e t h e o r y ( L e e and S h r o c k , 1977; Cheng and L i , 1980a, 1984; L a n g a c k e r , 1981; Marshak, R i a z u d d i n and M o h a p a t r a , 1981). S i n c e t h e o r i g i n a l W e i n b e r g -Salam model i s g e n e r a l l y r e g a r d e d as b e i n g b a s i c a l l y c o r r e c t , 25 and t h e r e f o r e f i x e d , any c h a n g es one would make t o t h e t h e o r y a r e d e t e r m i n e d by f u r t h e r a s s u m p t i o n s r e g a r d i n g t h e masses o r i n t e r a c t i o n s o f t h e l e p t o n s . I t i s n o t d i f f i c u l t , f o r example, t o add s i m p l e terms t o t h e W e i n b e r g - S a l a m L a g r a n g i a n t o g i v e t h e n e u t r i n o s a mass o r t o mix t h e v a r i o u s l e p t o n f a m i l i e s . F u r t h e r , one c o u l d add SU(2) s i n g l e t H i g g s f i e l d s , a d d i t i o n a l SU(2) i s o - s p i n o r o r d o u b l e t H i g g s f i e l d s , o r H i g g s f i e l d s t h a t c o n s t i t u t e o t h e r SU(2) r e p r e s e n t a t i o n s , s u c h as t h e i s o - v e c t o r o r t r i p l e t r e p r e s e n t a t i o n t o be c o n s i d e r e d p r e s e n t l y . N e u t r i n o mass terms w i l l be c o n s i d e r e d i n d e t a i l i n t h e n e x t c h a p t e r . T h e r e a r e two b a s i c r e a s o n s f o r c o n s i d e r i n g t h e H i g g s t r i p l e t as t h e most " n a t u r a l " g e n e r a l i z a t i o n o f t h e W e i n b e r g -Salam model. F i r s t , an unwanted p r o l i f e r a t i o n o f H i g g s p a r t i c l e s i s k e p t t o a minimum b e c a u s e t h e i s o - s p i n o r and v e c t o r f i e l d s combine i n a number o f c a s e s ( n e u t r a l and s i n g l y c h a r g e d p a r t i c l e s ) t o f o r m l i n e a r c o m b i n a t i o n s o f t h e s e f i e l d s . S e c o nd, a mass term — a M a j o r a n a mass term — i s r e a d i l y c o n s t r u c t e d f r o m a Yukawa-type i n t e r a c t i o n between t h e a l r e a d y e x i s t i n g l e p t o n s and t h e H i g g s v e c t o r f i e l d w i t h o u t a d d i n g e x p l i c i t mass terms f o r t h e n e u t r i n o . T h i s i t e m , t o o , i s d i s c u s s e d i n d e t a i l i n t h e n e x t c h a p t e r . Under a g e n e r a l SU(2) t r a n s f o r m a t i o n a v e c t o r f i e l d H t r a n s f o r m s as 3 •* uihT1 , (2.3.1) w h i c h i s shown i n t h e n e x t s e c t i o n , f r o m w h i c h i t f o l l o w s t h a t t h e "gauge d e r i v a t i v e " 26 D3 s aH - i-ig(AH-HA) - ^-ig' Y ^ i f ( 2 . 3 . 2 ) t r a n s f o r m s u n d e r t h e SU(2) t r a n s f o r m a t i o n ( 2 . 3 . 1 ) as DH* U Dif U _ 1 . ( 2 . 3 . 3 ) The s e c o n d t e r m o f ( 2 . 3 . 2 ) c o u l d be w r i t t e n ( - i g A A H ) o r ( g A x i t ) , i n t h e n o t a t i o n o f App. B, s o i s j u s t an SU(2) 3-v e c t o r . Thus t h e k i n e t i c o r f i e l d p a r t t o be added t o t h e o r i g i n a l W e i n b e r g - S a l a m L a g r a n g i a n i s C(H) = S [ ( D H V ( D M i f ) ] = (D H ) . + ( D U H ) . , ( 2 . 3 . 4 ) p. p l l where S[«»»] r e f e r s t o t h e s c a l a r p a r t o f t h e SU(2) o r P a u l i a l g e b r a , as d i s c u s s e d i n App. B. I n a b a s i s ( a . ) , where x H = H. a. 1 1 t h e d e r i v a t i v e p a r t o f ( 2 . 3 . 4 ) i s s [ ( a 3)+(a^3)] = a H. a V , P i _ ( 2 . 3 . 5 ) ( 2 . 3 . 6 ) p p l l t h e c o r r e c t f o r m f o r e a c h o f t h e t h r e e c o m p l e x s c a l a r f i e l d s H^. I f t h e P a u l i m a t r i c e s a r e u s e d t o r e p r e s e n t t h e b a s i s v e c t o r s one o b t a i n s / H H, H 1 + i H 2 H l - i H 2 H J2H, >T2H ++ -H + / ( 2 . 3 . 7 ) w here t h e c h a r g e a s s i g n m e n t s h a v e b e e n made on t h e b a s i s o f h i n d s i g h t a n d t h e J"2 f a c t o r s a r e ad d e d as a r e q u i r e d n o r m a l i z a t i o n b e c a u s e ( 2 . 3 . 6 ) becomes ( 2 . 3 . 8 ) • p * p * p a H„ a H + a H a H + a H a H 0 0 i l + + LI + + + + P " p + "I" p The weak h y p e r c h a r g e a s s i g n m e n t ( 2 . 2 . 1 1 ) g i v e s ( 2 . 3 . 9 ) Now, i n a n a l o g y w i t h t h e o r i g i n a l H i g g s d o u b l e t one assumes t h a t 27 <HQ> a v H/J"2 , ( 2 . 3 . 1 0 ) so t h a t ( 2 . 3 . 4 ) makes a c o n t r i b u t i o n t o t h e W and Z masses when i t i s e x p a n d e d i n t e r m s o f t h e W, A and Z f i e l d s . U s i n g <H> i n p l a c e o f 2 i n ( 2 . 3 . 4 ) r e s u l t s i n c(<3>) = y / w ' w * ; + w y z z p , ( 2 . 3 . I D ' H \x + * H p. so t h e W and Z masses become, w i t h t h e r e s u l t ( 2 . 2 . 1 7 ) f r o m t h e d o u b l e t H i g g s f i e l d s M 2 = j v V + ^ g 2 ( 2 . 3 . 1 2 a ) M Z = ^ v 2 Q Z + V H G 2 * ( 2 . 3 . 1 2 b ) E x p e r i m e n t a l v e r i f i c a t i o n o f ( 2 . 2 . 2 4 ) , h o w e v e r , i m p l i e s t h a t v << v , ( 2 . 3 . 1 3 ) H ( A r n i s o n et al., 19 8 3 a , c ; B a g n a i a et al., 1983; M a r c i a n o and S i r l i n , 1984; R e u t e n s et al., 1985) an i n e q u a l i t y t h a t w i l l l a t e r be o f some i m p o r t a n c e . I n p r i n c i p l e and i n g e n e r a l a c o m p l i c a t e d i n t e r a c t i o n b e t w e e n t h e $ and H f i e l d s s h o u l d , i n a d d i t i o n t o V ( $ ) o f ( 2 . 2 . 6 ) , and a V(H) t e r m , be ad d e d t o t h e g e n e r a l L a g r a n g i a n a nd, when " m i n i m i z e d " , s h o u l d d e t e r m i n e t h e masses o f t h e p h y s i c a l H i g g s p a r t i c l e s and e s t a b l i s h w h i c h a r e t h e p h y s i c a l , and w h i c h t h e u n p h y s i c a l , p a r t i c l e s . T h i s i s t h e s u b j e c t o f t h e n e x t s e c t i o n . I n t h e s e q u e l we w i l l u s e ( 2 . 3 . 1 3 ) and w i l l n e e d a c h a r g e d H i g g s p a r t i c l e o f unknown mass. From ( 2 . 3 . 4 ) t h e r e f o l l o w s a t e r m (l/J-2)iv„g(W Ma H -W^a H ) - ( l / X 2 ) i v G Z ^ (H -H*) , ( 2 . 3 . 1 4 ) H - M . + + L I - H n u u w h i c h , w i t h ( 2 . 2 . 2 5 ) , i d e n t i f i e s S s <t> c o s e + H s i n e , ( 2 . 3 . 1 5 a ) + + + 28 w i t h t a n e = J/2 v /v , ( 2 . 3 . 1 5 b ) H as an u n p h y s i c a l , c h a r g e d H i g g s f i e l d , u n p h y s i c a l i n t h e s e n s e o f t h e r e m a r k s f o l l o w i n g ( 2 . 2 . 2 5 ) . The o r t h o g o n a l c o m b i n a t i o n B = s i n e + H c o s e - ( 2 . 3 . 1 6 ) ± ± ± t u r n s o u t t o be a p h y s i c a l f i e l d o f c o n s i d e r a b l e i m p o r t a n c e t o t h i s t h e s i s . From ( 2 . 2 . 2 5 ) and ( 2 . 3 . 1 4 ) one s e e s t h a t t h e c o m b i n a t i o n ( l / 4 " 2 ) [ v ( 0 0 - « > * ) + 2 v H ( H Q - H * ) ] ( 2 . 3 . 1 7 ) i s a l s o u n p h y s i c a l . T h i s i l l u s t r a t e s t h e r e m a r k made e a r l i e r , t h a t t h e v e c t o r H i g g s f i e l d h a s a n a t u r a l r e l a t i o n s h i p w i t h t h e s p i n o r H i g g s f i e l d . B e s i d e s t h e gauge f i x i n g t e r m s t o be c o n s i d e r e d l a t e r , i n S e c . I I . 5 , t h e r e r e m a i n s t h e Y u k a w a - t y p e c o u p l i n g b e t w e e n t h e l e p t o n s and t h e v e c t o r H i g g s f i e l d s . The o n l y SU(2) and U ( l ) gauge i n v a r i a n t i n t e r a c t i o n i s , as shown i n S e c . I I . 5 , £(Yuk-3) = -2. . 0 . . [ * C Z (ia , 3 )* . + ( i c j ) * * 0 ] , ( 2 . 3 . 1 8 ) I J I J I - 2 j J + 2 i Q where i , j = 1,2, ••• = e,p., <|> i s t h e c h a r g e c o n j u g a t e o f t h e s p i n o r f i e l d 4* (App. C ) , and where (ia 23) i s n e e d e d r a t h e r t h a n H a l o n e b e c a u s e i n t h e u s u a l m a t r i x r e p r e s e n t a t i o n f o r t h e ( a . ) , l a, = (° ~ 1 ) ( 2 . 3 . 1 9 a ) 2 x 0 and a 2 U* a 2 = U , ( 2 . 3 . 1 9 b ) where U i s an SU(2) t r a n s f o r m a t i o n . The f a c t o r i i s t o make (icr„) r e a l , as a r e t h e 0. . c o u p l i n g c o e f f i c i e n t s b e c a u s e o f CPT 2 n i 29 i n v a r i a n c e . When e x p a n d e d u s i n g ( 2 . 3 . 7 ) and ( 2 . 3 . 1 0 ) t h e i n t e r a c t i o n t e r m becomes C(Yuk-H) = . P . . v n ( v ° E v. + Z.X v°) ( 2 . 3 . 2 0 ) I J I J H i - j J + i -I. . P . . [ v C Z (J"2H v.-H A . ) + A ° E (-H V . - J " 2 H A . ) l j i j i - ° J + J i - + J + + J + v . E (J"2H *v°-H A ° ) + A . E (-H V C - T 2 H A ° ) ] , j + j + - l — x w h ere A , A, , • • • = e,p., w h i c h j u s t i f i e s t h e c h a r g e a s s i g n m e n t s o f t h e c o m p o n e n t s o f t h e v e c t o r H i g g s f i e l d H. T h i s i m p o r t a n t s e t o f i n t e r a c t i o n t e r m s w i l l be c o n s i d e r e d i n d e t a i l i n t h e n e x t two s e c t i o n s and i n Chap. I I I . The f i r s t t e r m i s t h e i m p o r t a n t M a j o r a n a mass t e r m . I I . 4 THE PHYSICAL AND UNPHYSICAL HIGGS FIELDS I t has a l r e a d y become a p p a r e n t , as i n ( 2 . 2 . 2 5 ) and ( 2 . 3 . 1 4 ) , t h a t n o t a l l o f t h e t e n H i g g s f i e l d s s o f a r i n t r o d u c e d , n o r a r b i t r a r y l i n e a r c o m b i n a t i o n s o f them, c a n be p h y s i c a l , o r t h e s o u r c e o f r e a l p a r t i c l e s one m i g h t e n c o u n t e r i n an a c t u a l e x p e r i m e n t . J u s t as i n t h e o r i g i n a l W e i n b e r g -S a l a m m odel t h e r e must be t h r e e u n p h y s i c a l f i e l d s t h a t c o u l d , i n p r i n c i p l e , be g a u g e - t r a n s f o r m e d away so as t o p r o v i d e f o r t h e e x t r a l o n g i t u d i n a l s t a t e s n e e d e d t o make t h e W , Z Q f i e l d s mass i v e . N ot a l l o f t h e p h y s i c a l and u n p h y s i c a l f i e l d s c a n be i d e n t i f i e d as r e a d i l y as t h o s e o b t a i n e d f r o m ( 2 . 2 . 2 5 ) and 30 (2.3.14). I n p a r t i c u l a r , t h e n e u t r a l , p h y s i c a l H i g g s f i e l d s c a n o n l y be e s t a b l i s h e d i n t h e f o l l o w i n g manner. To t h e g e n e r a l L a g r a n g i a n must be add e d an i n t e r a c t i o n t e r m V($,H) t h a t d e s c r i b e s t h e s e l f - i n t e r a c t i o n s among t h e f i e l d s and t h e o t h e r i n t e r a c t i o n s among them, and one t h a t i s S U ( 2 ) x U ( l ) gauge i n v a r i a n t . A g e n e r a l s u c h p o t e n t i a l t e r m we w i l l t a k e t o be V($,H) = P 2 * * * - x($ + $)2 + n 2 S [ H + H] - X ( S [ H * H ] ) 2 H _ H - x's[3+23+il] + a*+#s+ bs[M$*+] , (2.4.1) H where t h e g l o b a l s y mmetry $ -» 3 -» -H has b e e n i m p o s e d , as i s c o n v e n t i o n a l , t o s i m p l i f y t h e t y p e s o f t e r m s n e e d e d , where S [ * * * ] = ^ T r [ • • •] i s t h e s c a l a r p a r t o f t h e P a u l i a l g e b r a d e s c r i b e d i n App. B, and w h e r e t h e c o e f f i c i e n t s a r e g e n e r a l l y unknown. O t h e r t e r m s , s u c h as S[if + HifH*] o r S[HH*$$*] c o u l d h a v e b e e n a d d e d b u t make no c h a n g e s i n p r i n c i p l e t o t h e f i n a l r e s u l t . T h i s p o t e n t i a l and i t s i m p l i c a t i o n s seem t o h a v e b e e n f i r s t c o n s i d e r e d b y G e l m i n i a n d R o n c a d e l l i ( 1 9 8 1 ) and s u b s e q u e n t l y c o n s i d e r e d i n f u r t h e r d e t a i l b y G e o r g i , G l a s h o w and N u s s i n o v ( 1 9 8 1 ) who, i n p a r t i c u l a r , m o d i f y t h e p o t e n t i a l a b o v e s o as t o have V(<$>,<H>) = 0, b u t w h i c h i s n o t n e c e s s a r y h e r e b e c a u s e o u r i n t e n t i o n i s o n l y t o i d e n t i f y t h e p h y s i c a l s t a t e s . The i s o - s p i n o r H i g g s f i e l d $ i s w r i t t e n as $ = * + x + + (4>0+v/J/2)x_ (2.4.2) i n t e r m s o f t h e P a u l i s p i n o r s x + (App. B ) , w i t h <$ 0> = 0, and 31 t h e i s o - v e c t o r H i g g s f i e l d i s w r i t t e n as i n (2.3.5) as H = (HQ/J-2 + ^ v H ) ( a l - i a g ) + ( H + + /,T2) ( a x + i<r 2 ) + H + a 3 , (2.4.3) w i t h <HQ> = 0. Now r a t h e r t h a n " m i n i m i z e " t h e p o t e n t i a l V($,H) i t i s s u f f i c i e n t f o r t h e p u r p o s e s r e q u i r e d h e r e t o a d j u s t t h e c o e f f i c i e n t s o f (2.4.1) so as t o e l i m i n a t e l i n e a r terms i n t h e f i e l d s . On e x p a n d i n g (2.4.1) t o s e c o n d o r d e r i n t h e f i e l d s one o b t a i n s M ? v - x v 3 + j a w 2 = 0 (2.4.4) z H * as t h e c o e f f i c i e n t o f (*0+<l>0 ) / J"2 , and u 2 v - x v 3 - 2 x ' v 3 + - a v 2 v s 0 (2.4.5) H H H H . H H 2 H ^ ' as t h e c o e f f i c i e n t o f (H 0+H Q)/J"2. The c o e f f i c i e n t s o f <t>Q <|>0 and * H Q H 0 a r e , r e s p e c t i v e l y , p r o p o r t i o n a l t o (2.4.4) and (2.4.5) and t h u s a l s o v a n i s h . The c o e f f i c i e n t o f H H + + g i v e s t h e mass o f t h e d o u b l y -c h a r g e d H i g g s f i e l d : w 2 . „ . 2 2 1 2 1 , 2 ( H ± ± ) - P H - X H v H + -av + -bv = j b v 2 + 2xJjV2 . (2.4.6) What r e m a i n a r e terms f o r t h e s i n g l y c h a r g e d H i g g s f i e l d s (<|>+,H + ) and t h e r e a l p a r t s o f <|>0 and H Q , namely (<l>0 + <t>0 ) / J"2 and * (H 0+H Q)/J"2. A f t e r (2.4.4) and (2.4.5) have been employed, t h e 1 2 1 2 c o e f f i c i e n t o f <t> <l> i s -bv . o f H H i s -bv , and o f (H <t> + - 2 H + - 4 ' v + _ +H <|> ) i s - ( l / 2 J " 2 ) b v v . Thus t h e s e f i e l d s a r e l i n e a r - + H c o m b i n a t i o n s o f f i e l d s * = S c o s e - B s i n e (2.4.7) ± ± ± H = S s i n e + B c o s e , ± ± ± where 32 c o s e = v / ( v 2 + 2 v 2 ) 2 , s i n e = J"2v / ( v 2 + 2 v 2 ) ' 2 , (2.4.8) H H H w i t h mass terms - M 2 ( S + ) = 0 (2.4.9) - M 2 ( B ± ) = ( ^ b v 4 + b v 2 v 2 + b v * ) / ( v 2 + 2 v 2 ) 1 , 2 p r o v i d e d v << v, w h i c h g i v e s i n s u c h a c a s e H M(H + + ) « J/2 M ( B + ) , (2.4.10) as n o t e d by G e o r g i , Glashow and N u s s i n o v ( 1 9 8 1 ) , b u t w h i c h i s p r o b a b l y not j u s t i f i e d b e c a u s e t h e c o e f f i c i e n t s i n V($,3) a r e q u i t e unknown, even i f v ^ << v. Thus S + i s u n p h y s i c a l and B + i s p h y s i c a l : t h e s e a r e p r e c i s e l y t h e f i e l d s i d e n t i f i e d e a r l i e r , i n ( 2 . 3 . 1 5 ) . F o r t h e p h y s i c a l , n e u t r a l , s c a l a r H i g g s f i e l d s t h a t a r i s e f r o m <t>1 =(<t>0 + * * ) / J 2 and H x = ( H Q +H*) / J/2 , V($,H) g i v e s t h e terms - ( t Z + j a v 2 ) * 2 - ( M 2 + i - a v 2 ) H 2 + a v v ^ H x , (2.4.11) where (2.4.4) and (2.4.5) have been u s e d , w h i c h i s s e t e q u a l t o -JMV - j M V (2.4.12) where <t>x = <|> cosoc + w sin<x (2.4.13) H = -<J> sinoc + w cosoc , where <|> and w a r e t h e p h y s i c a l f i e l d s , and one f i n d s t a n 2 « = ( a v v H ) / [ x v 2 - ( x H + 2 x ' H ) v 2 ] (2.4.14) - ( a / x ) ( v /v) , p r o v i d e d v << v, and H 1 2 2 2 2 2 - M = xv c o s « + (x +2x' )v s i n oc + a w sinoc cosoc , (2.4.15a) 2 <j> H H H H 33 1 2 2 2 2 2 -M = xv s i n « + +2x')v cos a - a w sincx c o s « , (2.4.15b) oi H H H H so t h a t , a g a i n i f v << v, H M >> M (2.4.16) ft u and « » j ( a / x ) ( v H / v ) << 1 (2.4.17) ( p r o v i d e d a ~ x) and so ft » ftx , w » H . Note t h a t ftx was t h e o n l y p h y s i c a l f i e l d i n t h e o r i g i n a l W e i n b e r g - S a l a m model. I f we now d e f i n e <|>2 s - i ( f t Q - f t * ) / J " 2 and H 2 = - i (H Q-H*) / J"2 , t h e n (2.3.17) g i v e s ft' s <j>2 c o s e ' + H 2 s i n e ' (2.4.18) as t h e t h i r d u n p h y s i c a l f i e l d , where c o s e ' = v / ( v 2 + 4 v 2 J 1 / Z , s i n e ' = 2v / ( v 2 + 4 v 2 ) 1 , (2.4.19) H H H w h i l e t h e o r t h o g o n a l c o m b i n a t i o n w' s - f t 2 s i n e ' + H g c o s e ' (2.4.20) i s p h y s i c a l b u t massless, s i n c e f r o m V(ft,3) t h e c o e f f i c i e n t o f 2 <i>' i s 0. T h i s m a s s l e s s G o l d s t o n e b o s o n c a n n o t be a v o i d e d ( G e l m i n i and R o n c a d e l l i , 1981; G e o r g i , Glashow and N u s s i n o v , 1981): i t i s r e f e r r e d t o as a joajoron, and i s o f c o n s i d e r a b l e c u r r e n t i n t e r e s t ( G e l m i n i , N u s s i n o v and R o n c a d e l l i , 1982; Dugan et al., 1985; Glashow and Manohar, 1985). The p h y s i c a l and u n p h y s i c a l H i g g s b o s o n s , and t h e i r r e l a t i o n t o t h e o r i g i n a l H i g g s f i e l d s , a r e l i s t e d i n T a b l e I. The p r e s e n c e o f a m a s s l e s s H i g g s p a r t i c l e i s a s i g n a l t h a t a symmetry o f t h e t h e o r y has been b r o k e n by t h e p r e s e n c e o f t h e n o n - z e r o vacuum e x p e c t a t i o n v a l u e o f t h e i s o - v e c t o r H i g g s f i e l d , a c o n s e q u e n c e known as G o l d s t o n e ' s t h e o r e m ( B e r n s t e i n , 34 1974; Cheng and L i , 1984, Sec. 5 . 3 ) . The g l o b a l symmetry v i o l a t e d h e r e i s r e l a t e d t o l e p t o n - n u m b e r c o n s e r v a t i o n . As can be s e e n i n (2.2.2) and ( 2 . 2 . 7 ) , w r i t t e n out i n d e t a i l i n ( 2 . 2 . 1 8 ) , t h e o r i g i n a l W e i n b e r g - S a l a m model s e p a r a t e l y c o n s e r v e s e l e c t r o n l e p t o n number, muon l e p t o n number, e t c . , b e c a u s e o f t h e i n d e p e n d e n t g l o b a l s y m m e t r i e s ' i Q i ( e , v ) -» e ( e , v ) (2.4.21) e e / \ i e 2 ( n , v ) •* e (p., v ) , where ex , e 2 , e t c . , a r e c o n s t a n t s . From N o e t h e r ' s theorem, t h e n ( J a u c h and R o h r l i c h , 1976, Chap. 1 ) , a I, ( f d 4 x aC S<i>) = 0 , (2.4.22) v * [1 a(a ft) J v where ft r e p r e s e n t s t h e v a r i o u s f i e l d s i n t h e L a g r a n g i a n , we have t h e c o n s e r v e d e l e c t r o n l e p t o n number L = f d 3 x (e*°e + v E V ° i / ) , (2.4.23) e e + e w i t h s i m i l a r e x p r e s s i o n s f o r t h e o t h e r l e p t o n f a m i l i e s . The i s o - v e c t o r H i g g s f i e l d c hanges a l l t h i s . From (2.3.20) i t c a n be s e e n t h a t , b e f o r e s p o n t a n e o u s symmetry b r e a k i n g ( i . e . , b e f o r e H -» <H > = v /<T2), t h e r e i s t h e one g l o b a l H symmetry X 0 ( a l l l e p t o n s ) -• e ( a l l l e p t o n s ) (2.4.24) — —i©— c —i© c ( w i t h ft -> e A, Jl -»e A , e t c . ) provided one a l s o imposes - 2 i e ( H + + , H +, H Q) - e ( H + + , H +, H Q) , (2.4.25) w h i c h means t h a t t h e i s o - v e c t o r H i g g s f i e l d must a l s o be g i v e n a l e p t o n number ( w h i c h i s r e a s o n a b l e b e c a u s e we now have t h e p o s s i b i l i t y o f p r o c e s s e s s u c h as e + e -» H ). I f N(e) 35 s t a n d s f o r t h e number o f p a r t i c l e s o f t y p e e, t h e n what i s c o n s e r v e d i s N(e~) - N ( e + ) + N(p~) - N(p. + ) + ••• + N(Z v ) - N(Z + ••• - e + e - 2 N ( H + + + H + + H Q) + 2N(H__ + H_ + H*) . (2.4.26) From t h e f i r s t t erm o f ( 2 . 3 . 2 0 ) , however, i f v * 0, even t h i s H one g l o b a l symmetry i s v i o l a t e d , b e c a u s e , f o r example, c p r o c e s s e s l i k e Z v -» Z v a r e o s t e n s i b l y p e r m i t t e d . Thus t h e - e + \i p e c u l i a r v i o l a t i o n o f t h i s g l o b a l l e p t o n - n u m b e r c o n s e r v a t i o n i s by t h e M a j o r a n a n e u t r i n o mass term. TABLE I: THE PHYSICAL AND UNPHYSICAL HIGGS FIELDS O r i g i n a l F i e l d s H x , H 2 , H , H + + P h y s i c a l F i e l d s H + + B = $ s i n e + H c o s e + + + U n p h y s i c a l F i e l d s <|> c o s e - H s i n e + + ( t a n e = J"2v /v) H <!>! =(*0+<l>0)/J"2 H l S ( H 0 + H * ) / J / 2 <|> = 4> c o s a - H j sinoc w = • sinoc+Hj^ cosoc [tan2oc » ( a / x ) ( v _ / v ) ] n * 2 =-i (<J>o-<0*) /J"2 H 2 H - i ( H 0 - H * ) / J - 2 w' =-<t>2sine' +H 2cose' ( t a n e ' = [mass (co' ) =0 ] • r =<l>2 cose'+H 2 s i n e f 2 v H / v ) 36 I I . 5 GAUGE INVARIANCE OF THE THEORY The i n v a r i a n c e p r o p e r t i e s o f n o n - a b e l i a n gauge t h e o r i e s have been u n d e r s t o o d s i n c e t h e f i r s t s u c h models were f o r m u l a t e d by Yang and M i l l s (1954) and have been e x t e n s i v e l y r e v i e w e d s i n c e ( i n , f o r example, A b e r s and Lee, 1973; I t z y k s o n and Z u b e r , 1980, Chap. 1 2 ) . The s p e c i a l c a s e o f an SU(2) gauge t h e o r y c a n , however, be a p p r e c i a b l y s i m p l i f i e d by u s i n g t h e P a u l i a l g e b r a d e s c r i b e d i n App. B b e c a u s e e l e m e n t a r y 3 - v e c t o r a l g e b r a can be u s e d , r a t h e r t h a n t h e gro u p t h e o r e t i c a l t e c h n i q u e s r e q u i r e d f o r t h e g e n e r a l c a s e . The U ( l ) i n v a r i a n c e o f t h e o r i g i n a l W e i n b e r g - S a l a m L a g r a n g i a n (2.2.2) i s q u i t e r e a d i l y d e m o n s t r a t e d when t h e U ( l ) t r a n s f o r m a t i o n (2.2.8) i s a p p l i e d , w h i l e f o r t h e v e c t o r H i g g s f i e l d one needs 3 e x p [ i Y 1 T o c ( x ) ] 3 (2.5.1) and <I>C -» e x p [ - i Y « ( x ) ] *° (2.5.2a) i f * -» e x p [ i Y o c ( x ) ] * , (2.5.2b) and t h e Yukawa i n t e r a c t i o n (2.3.18) i s a l s o s e e n t o be U ( l ) i n v a r i a n t . In t h e c a s e o f t h e SU(2) t r a n s f o r m a t i o n one has 37 * T -» , e_ -» e D , $ -> U$ (2.5.3) ii L H it A -» U A U _ 1 - ( 2 i / g ) aU U _ 1 w h i c h r e s u l t s i n , as t h e r e f e r e n c e s c i t e d above show, (3 - j i g A ) * -» U(a - ^ i g A ) * (2.5.4) :* •* -* -* •* ~ l F = a A -a A + g A x A -» U F U , pv p v v p p v pv so t h e W e i n b e r g - S a l a m L a g r a n g i a n is s e e n t o be SU(2) gauge i n v a r i a n t as w e l l . T hese l a s t r e l a t i o n s a r e more r e a d i l y d e m o n s t r a t e d when axt> = -iaA t> = ~i(a5-Sa) (2.5.5) i s u s e d , as d e s c r i b e d i n App. B. The SU(2) gauge d e r i v a t i v e i n t h e v e c t o r o r t r i p l e t c a s e i s no t ( P a l and W o l f e n s t e i n , 1982, Eq. ( 4 4 ) ) Dil = (a - j i g M b u t DH = aH - j i g ( A H - H A ) = aH + gAxH , (2.5.6) as s t a t e d i n ( 2 . 3 . 2 ) , b e c a u s e a 3 - v e c t o r t r a n s f o r m s as 5 -» U 3 U _ 1 , (2.5.7) not as H - t u 3 . To show t h i s one u s e s (2.5.3) and (2.5.6) t o o b t a i n DH -» a(UHU _ 1) - i g [ U A U _ 1 - ( 2 i / g ) aUU_1 ] A UHU _ 1 (2.5.8) = uaHU~1+auu~1UHU~:i-u3u~1 a u u _ 1 - i g U A U _ 1 A u3u~l-2auu~lA UHU-1 = U ( 3 H - i g A A H ) U _ 1 = U D3 U - 1 as r e q u i r e d , where a(u _ 1) = -u" 1 au i f 1 (2.5.9) has been employed. Thus S[(D 3) t(D J 1H)] i s SU(2) i n v a r i a n t s i n c e i f 38 S[AA] = A•A = B »B (2.5.10a) B = U A U 1 . (2.5.10b) The Yukawa term (2.3.18) i s a l s o SU(2) gauge i n v a r i a n t b e c a u s e i f t h e n $ U* , * + <$U 1 ( 2 . 5 . 11a) c * * * * c ft=$^U$=U* ( 2 . 5 . l i b ) and so * C a 2 i t a -» $°(U 1 ) * a 2 U H U 1 U* (2.5.12) = * Ca 2U _ 1u3* = i°CT 2H* , as i s a l s o r e q u i r e d , where (2.3.19) has been u s e d . Thus t h e U ( l ) and SU(2) gauge i n v a r i a n c e o f t h e a d d i t i o n a l H i g g s v e c t o r terms has been d e m o n s t r a t e d . T h i s gauge i n v a r i a n c e , however, e x i s t s o n l y i n i t i a l l y . I t i s b r o k e n n o t o n l y by t h e H i g g s mechanism ( t h e vacuum i s n o t i n v a r i a n t ) b u t a l s o by t h e s p e c i f i c gauge f i x i n g terms w h i c h a r e a b o u t t o be c o n s i d e r e d . 39 I I . 6 GAUGE FIXING TERMS: THE GENERAL R. GAUGE A l t h o u g h t h e gauge i n v a r i a n c e o f t h e model t o be employed h e r e has been d e m o n s t r a t e d , i t i s n o t p o s s i b l e t o do any m e a n i n g f u l c a l c u l a t i o n s u s i n g i t . T h e r e a r e a t l e a s t two r e a s o n s f o r t h i s . No Gre e n f u n c t i o n o r p r o p a g a t o r c an be c o n s t r u c t e d f o r any m a s s l e s s v e c t o r f i e l d ( s u c h as t h e p h o t o n f i e l d ) b e c a u s e t h e d i f f e r e n t i a l o p e r a t o r on t h e f i e l d t h a t f o l l o w s f r o m t h e L a g r a n g i a n has no i n v e r s e — a d e f i n i t e gauge must be s e l e c t e d ( A b e r s and Lee, 1973; Nash, 1978; s e e a l s o Sec. I V . 3 ) . S e c o n d l y , t h e r e a r e i n t e r a c t i o n terms s u c h as (2.2.25) o r (2.3.14) and (2.3.17) t h a t a r e q u i t e a b s u r d b e c a u s e a v e c t o r p a r t i c l e c o u l d n e v e r d e c a y i n t o a s c a l a r p a r t i c l e . Such s c a l a r H i g g s f i e l d s a r e u n p h y s i c a l b e c a u s e t h e y can be e l i m i n a t e d by a j u d i c i o u s c h o i c e o f gauge. I t i s a g r e a t c o n v e n i e n c e t h a t b o t h p r o b l e m s can be e l i m i n a t e d t o g e t h e r ( t ' H o o f t , 1971), i n t h e s o - c a l l e d r e n o r m a l i z a b l e gauge, d e s i g n a t e d R^, where t i s t h e gauge p a r a m e t e r a b o u t t o be d e f i n e d . T h i s i s a c c o m p l i s h e d by a d d i n g terms t o t h e o r i g i n a l L a g r a n g i a n -- gauge f i x i n g terms -- t h a t e x a c t l y c a n c e l t h e unwanted t w o - p a r t i c l e t e r m s . A l t h o u g h t h e u n p h y s i c a l f i e l d s a r e n o t e l i m i n a t e d e n t i r e l y ( u n l e s s one u s e s what i s r e f e r r e d t o as t h e u n i t a r y gauge, i n w h i c h £-•«») t h e y can be c o n v e n i e n t l y m a n i p u l a t e d as t h o u g h t h e y were r e a l s c a l a r p a r t i c l e s , t h e masses o f w h i c h a r e dependent on t h e gauge p a r a m e t e r £. Of c o u r s e , a t t h e c o m p l e t i o n o f any c a l c u l a t i o n 40 c o n c e r n i n g r e a l , p h y s i c a l p r o c e s s e s a l l t r a c e o f t h e s e f i c t i t i o u s p a r t i c l e s must have d i s a p p e a r e d : t h e gauge p a r a m e t e r t must c a n c e l e v e r y w h e r e . The two gauge f i x i n g terms t o be added t o t h e c o m p l e t e L a g r a n g i a n a r e £ ( g a u g e f i x 1) = - ( 2 { ) _ 1 (a (2.6.1) P-- igCV[<$>$ t-$<$ t>+<H>H t-H<H t>]) 2 and C (gauge f i x 2) = -(2C) 1 (a a U (2.6.2) - igr Y t & [ < * > * * - * < * * > ] - i g ' jY i TCS[<H>H + -H<H t>] ) 2 , <p 2 H w h i c h have been w r i t t e n i n terms o f t h e P a u l i a l g e b r a o f App. B. Here V[»»»] means t h e 3 - v e c t o r p a r t o f w h a t e v e r e x i s t s i n s i d e t h e s q u a r e b r a c k e t s , and S[«««] r e f e r s t o t h e s c a l a r p a r t . The gauge f i x i n g terms have n o t been w r i t t e n t h i s way b e f o r e b u t t h e y c l e a r l y e x h i b i t a s t r i k i n g s i m i l a r i t y n o t o n l y between t h e SU(2) gauge f i x i n g t erm o f (2.6.1) and t h e U ( l ) gauge f i x i n g t erm o f ( 2 . 6 . 2 ) , b u t a l s o show a s i m i l a r i t y i n s t r u c t u r e between terms f o r t h e o r i g i n a l s p i n o r H i g g s f i e l d $ and t h e newly added H i g g s f i e l d H. W i t h t h e f i e l d $ w r i t t e n i n terms o f 2-component P a u l i s p i n o r s ( x + ) (App. B) * = <t>+x+ + * 0 x _ (2.6.3a) * + = *_*+ + + * o * _ + (2.6.3b) <$> = (v/J"2) x _ , ( 2 . 6 . 3 c ) and t h e v e c t o r H i g g s w r i t t e n e x p l i c i t l y as i n (2.3-7). if = H.a. = (1/J/2)(H+H ) a . + (-i/J"2) (H -H J o . + H a , (2.6.4a) 41 <H> = JVr (^1-i<y2) , (2.6.4b) and u s i n g t h e i m p o r t a n t s p i n o r - v e c t o r c o n n e c t i o n (App. B) V V = ± a 3 ) (2.6.5a) x _ x + + = 1-(ol - i o 2 ) (2.6.5b) x. x + = + i a 2 ) ( 2 . 6 . 5 c ) + -r e s u l t s i n , a f t e r a l e n g t h y m a n i p u l a t i o n , C ( g f l ) = - ( 2 e ) _ 1 { a A M - i g £ [ ( M w / J - 2 g ) ( a . ( S -S ) (2.6.6) p. W 1 - + - i a 2 ( S _ + S + ) ) + a 3((v/2J"2)(<t> 0-<!>*)+ ( v H / 4 " 2 ) ( H 0 - H * ) ) ] } 2 C ( g f 2 ) = - ( 2 C ) _ 1 { a a 1" + i g ' C [ (v /2J"2) (*-**) (2.6.7) p u o + ( v H / J - 2 ) ( H 0 - H * ) ] } 2 , where 7Y = Y = 1 has been u s e d . The e x p a n s i o n o f (2.6.6) and 2 H <|> (2.6.7) i s c ( g f l ) = - ( 2 C ) - 1 a A^a kv - iM„ ( w ^ a s -w^a s ) - CM 2S s p. 1 v 1 W + p - - p + W + -- igA31[(v/24-2)a^(<i>0-<i>*) + ( v H / J " 2 ) a ^ ( H 0 - H * ) ] + 2-g 2U(v/2J-2)(* 0-<t>*) + ( v H / J " 2 ) ( H 0 - H * ) ] 2 (2.6.8) and C ( g f 2 ) = - ( 2 0 _ 1 a a H a a v M v ' + ig'a^[(v/24-2)a^(<|>0-<|>*) + ( v H / X 2 ) a ^ ( H 0 - H * ) ] + \&'2t [ ( V / 2 J - 2 ) (•„-•*) + ( v H / J - 2 ) ( H 0 - H * ) ] 2 , (2.6.9) w i t h t h e f o l l o w i n g c o n s e q u e n c e s . The f i r s t t erms o f e a c h o f t h e s e e x p r e s s i o n s f i x t h e gauge f o r t h e gauge b o s o n s ; t h e s e c o n d term o f (2.6.8) and t h e c o m b i n a t i o n o f t h e f o u r t h term o f (2.6.8) w i t h t h e s e c o n d term o f (2.6.9) e x a c t l y c a n c e l t h e unwanted i n t e r a c t i o n terms (2.2.25) and ( 2 . 3 . 1 4 ) ; t h e t h i r d t e r m o f (2.6.8) g i v e s t h e mass o f t h e u n p h y s i c a l f i e l d S + as M 2 = I M 2 , (2.6.10) 42 and t h e l a s t terms o f (2.6.8) and (2.6.9) g i v e t h e mass term 2 2 f o r t h e u n p h y s i c a l , n e u t r a l H i g g s p a r t i c l e as M = CM . One <|> Z s h o u l d n o t e t h a t t h e u n p h y s i c a l masses depend on t h e ( a r b i t r a r y ) gauge p a r a m e t e r £. W i t h t h e gauge f i x i n g terms o f t h i s s e c t i o n and t h e L a g r a n g i a n o f Sec. I I . 2 , I I . 3 , we now have t h e c o m p l e t e gauge t h e o r y w i t h b o t h i s o - s p i n o r and v e c t o r H i g g s f i e l d s . B e f o r e t h e many i n t e r a c t i o n terms can be t u r n e d i n t o p r a c t i c a l "Feynman r u l e s " and a p p l i e d t o t h e p h y s i c a l p r o c e s s u n d e r i n v e s t i g a t i o n h e r e — muon d e c a y — t h e n e u t r i n o terms must be r e f o r m u l a t e d and p r o p e r l y i n t e r p r e t e d t o make s e n s e as t h o s e f o r a r e a l i s t i c p a r t i c l e . F i r s t , however, t h e i m p o r t a n t d i s c r e t e s y m m e t r i e s o f t h e t h e o r y -- t h e C, P and T t r a n s f o r m a t i o n s — w i l l be b r i e f l y c o n s i d e r e d . I I . 7 C, P AND T INVARIANCE PROPERTIES OF THE THEORY The o v e r t h r o w o f p a r i t y c o n s e r v a t i o n by Lee and Yang (1956) has d e m o n s t r a t e d t h e i m p o r t a n c e o f t h e d i s c r e t e symmetry o p e r a t i o n s o f c h a r g e c o n j u g a t i o n ( C ) , p a r i t y (P) and t i m e r e v e r s a l ( o r r e v e r s a l o f m o t i o n ) ( T ) . Not a l l t h e o r i e s , e s p e c i a l l y t h o s e c o n s t r u c t e d t o d e s c r i b e weak i n t e r a c t i o n p h y s i c s , need be i n v a r i a n t u n d e r t h e s e t r a n s f o r m a t i o n s . In t h i s s e c t i o n we w i l l see t h a t , l i k e what has come t o be known 43 about weak i n t e r a c t i o n s s i n c e Lee and Yang, t h e i n t e r a c t i o n s d e s c r i b e d i n t h i s c h a p t e r a r e n e i t h e r C n o r P i n v a r i a n t , b u t a r e CP, T and CPT i n v a r i a n t . The C, P and T t r a n s f o r m a t i o n s f o r quantum s p i n o r f i e l d s t h a t a r e t o be u s e d h e r e a r e d e s c r i b e d i n d e t a i l i n App. C In t h a t t h e y do n o t depend on any s p e c i f i c r e p r e s e n t a t i o n o f t h e D i r a c m a t r i c e s t h e y a r e somewhat n o v e l . The c o n v e n t i o n a l t r a n s f o r m a t i o n s a r e d e s c r i b e d i n , f o r example, B j o r k e n and D r e l l (1965, Chap. 1 5 ) . The t r a n s f o r m a t i o n s we use a r e ftP(t,?) = P * ( t , r ) P _ 1 (2.7.1) = tf0 M>(t,-r*) , ftC(x) 2 C 0»(x) C _ 1 (2.7.2) 3 +*(x) , ftt(t,r) E T ft(t,r) T _ 1 (2.7.3) = V 0 ft(-t,?) , ftCpt(x) = ® ft(x) o " 1 (2.7.4) s **(-x) , where P, C, T and ® S T P C a r e u n i t a r y o p e r a t o r s i n t h e F o c k - s p a c e o f 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 , and where T and © a r e a n t i - l i n e a r o p e r a t o r s . I t w i l l be e s s e n t i a l i n a p p l y i n g t h e s e t r a n s f o r m a t i o n s t o u s e t h e i m p o r t a n t d e f i n i t i o n s and i d e n t i t i e s d e s c r i b e d i n App. B, w h i c h w i l l be i d e n t i f i e d when needed. We t a k e (*) t o d e n o t e t h e complex c o n j u g a t e o f s c a l a r s and t h e h e r m i t i a n c o n j u g a t e o f F o c k - s p a c e o p e r a t o r s . A l l 4 - v e c t o r f i e l d s w i l l be assumed t o t r a n s f o r m as does t h e e l e c t r o m a g n e t i c v e c t o r p o t e n t i a l A, as d e s c r i b e d i n App. C: 44 A P ( t , ? ) s V q A ( t , - ? ) V Q (2.7.5a) o r , i n component form A ^ P ( t , ? ) = (A°(t , -2 ) , - A X ( t , - r ) ) , (2.7.5b) A ° ( x ) = - A * ( x ) (2.7.6) o r - A ( x ) , i f A i s a r e a l f i e l d , A * ( t , ? ) = V Q A ( - t , ? ) V 0 (2.7.7) A C p t ( x ) = - A * ( - x ) (2.7.8) o r - A ( - x ) , i f A i s r e a l . I t i s i m p o r t a n t t o n o t e t h a t t h e H i g g s f i e l d it i s n o t o f t h i s t y p e : i t c o n s i s t s o f a s e t o f t h r e e ( c o m p lex) s c a l a r f i e l d s . S c a l a r f i e l d s w i l l be t a k e n t o t r a n s f o r m as <D P(t,?) = 4>(t,-?) (2.7.9) <t>°(x) = * * ( x ) (2.7.10) • * ( ! , ? ) = * ( - t , ? ) (2.7.11) <D C p t(x) = <D*(-x) . (2.7.12) c * A c h a r g e d s c a l a r f i e l d , s u c h as <|>_, s a t i s f i e s <J>_ = <|> = <|>+. Of t h e k i n e t i c terms i n t h e L a g r a n g i a n o n l y t h o s e f o r t h e s p i n o r f i e l d s , s u c h as t h e f i r s t two terms o f ( 2 . 2 . 1 8 ) , a r e n o t t r i v i a l , so we c o n s i d e r t h e s e e x c l u s i v e l y . A term s u c h as C(ft) = i f t Z + 3 f t , (2.7.13) w h i c h can r e p r e s e n t e i t h e r n e u t r i n o o r e l e c t r o n ( o r muon, e t c . ) f i e l d s , has t h e f o l l o w i n g 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 P we have 45 P iftZ + 3 f t P - 1 = i f t P E + a f t P (2.7.14) = i * ( - r ) V 0 Z + a V 0 f t - ( - r ) = i * ( - r ) z _ y 0 v v v 0 a *(-?) , and as the t r a n s f o r m a t i o n r -+ - r i s per m i t t e d i n the a c t i o n i n t e g r a l , t h i s becomes C(ft) -» iftZaft . (2.7.15) Thus the c h i r a l i t y p r o j e c t i o n operators prevent such a term from being P - i n v a r i a n t ; i f absent, as they are f o r an e l e c t r o n , f o r example, the term would be P - i n v a r i a n t . Under C we have c iftz aft c - 1 = i f t C Z aft° (2.7.16) + + = ift Z v v a ft = - i a ftvcvvz v c f t = i f tv"z a ft - i a ( f t v vZ ft) = iftZ aft + s u r f a c e term , where the important and c r u c i a l i d e n t i t y (B.38) has been used, and where the s u r f a c e term disappears i n the a c t i o n i n t e g r a l . Again, we see that such a term i s not i n v a r i a n t i f the p r o j e c t i o n operators Z + are present. This term i s , however, C P - i n v a r i a n t : (CP) iftZ aft ( C P ) " 1 = ft*(-r)VnZ V VV na **(-?) ' (2.7.17) = - i a ^ f t ( - ? ) v 5 v 0 * v z + v 0 v 5 < K - ? ) = i f t ( - r ) Z V nV VV na ft(-r) + s u r f a c e term -» iftZ + 3 f t when the change r - r i s made. The CPT and T t r a n s f o r m a t i o n s are s i m i l a r : we i l l u s t r a t e with the former, remembering that both are a n t i - l i n e a r 46 t r a n s f o r m a t i o n s . We hav e ® i<PZ+a*l> © 1 = -i©^® ^ a © * ® 1 ( 2 . 7 . 1 8 ) = -i«jj (-x)z v v a <u (-x) = i a i p(-x)y * v z V - ^ K - x ) = - i i K ~ x ) £ "rfVa ^ ( - x ) + s u r f a c e t e r m + v when t h e p e r m i t t e d t r a n s f o r m a t i o n x -» -x i s made i n t h e a c t i o n i n t e g r a l . N e x t t o be c o n s i d e r e d a r e s p i n o r - b o s o n a n d b o s o n - b o s o n i n t e r a c t i o n s . The f o r m e r c o n s i s t s o f two t y p e s : f o r t h e ( v e c t o r ) gauge b o s o n s ( W +,A,Z 0) we h a v e i n t e r a c t i o n t e r m s l i k e £(*! ,*2 ' A ) ~ *iZ+k*2 + ^ 2 Z + A * * i ' ( 2 . 7 . 1 9 ) a p a r t f r o m r e a l c o n s t a n t s , and f o r t h e ( s c a l a r ) H i g g s b o s o n s t e r m s l i k e C(«l\ ,<t>) - * ! l + W 2 + * 2 • ( 2 . 7 . 2 0 ) N o t e t h a t o n l y Z i s t o be f o u n d i n t h e f i r s t , w h i l e b o t h o f Z e x i s t i n t h e s e c o n d . C o n s i d e r i n g t h e f i r s t t e r m o n l y we h a v e P Z(^,%,k) P _ 1 ~ * ? £ + A P * P ( 2 . 7 . 2 1 ) = [ ^ 1 V 0 Z + ( V 0 A V 0 ) V 0 * 2 ] (-?) = ( * l S _ A * 2 ) ( - r ) -» ( * 1 Z _ A + 2 ) ( r ) , so i s n o t i n v a r i a n t . F o r t h e C t r a n s f o r m a t i o n we h a v e C C ( * l t * 2 , A ) C _ 1 - * i £ + A ° * 2 ( 2 . 7 . 2 2 ) -* * * - * = * x Z + ( - A ) * 2 = * 2 V 5 A Z + V 5 ^ = * 2 E _ A * * 1 , wh e r e ( B . 3 8 ) has a g a i n b e e n u s e d . Thus t h e s e t e r m s a r e n o t C-i n v a r i a n t e i t h e r . They a r e , h o w e v e r , C P - i n v a r i a n t : 47 (CP) C C ^ . + g , A) ( C P ) _ 1 ~ [ * f v 0 E + ( - V 0 A V 0 ) V 0 f t * ] (-?) (2.7.23) = - ( * * E _ A % * ) (-?) = ( * 2 V s A * S _ y 5 « l ).(-?) -» (* 2Z +A** 1 ) ( ? ) . These terms a r e a l s o T - i n v a r i a n t : T £(4^ , <l>2 , A) T 1 - T ^ T 1 T E + A T ~ 1 T f t 2 T ~ 1 (2.7.24) = t * 1 V 0 Z _ ( V 0 A V 0 ) V 0 * 2 ] ( - t ) = (*1Z+h*z)(-t) + ( f t x E + A f t 2 ) ( t ) , and C P T - i n v a r i a n t : ® £ ( 4 » 1 , * 2 , A ) ® 1 - 0 ^ ® 1 ® E A® 1 ® f t 2 ® 1 (2.7.25) - [ f t f E _ ( - A * ) f t * ] ( - x ) = ( f t 2 V 5 A * E _ V 5 f t 1 ) ( - X ) - ( f t 2 E + A % x ) (x) , where t h e e s s e n t i a l a n t i - l i n e a r i t y o f T and ®=TPC s h o u l d be n o t e d . F o r t h e s c a l a r H i g g s b o s o n s we have, a g a i n c o n s i d e r i n g o n l y t h e f i r s t term, P C ( * l t * 2 , * ) P _ 1 ~ ( \ V 0 Z + 4 > V 0 * 2 ) (-?) (2.7.26) ( f t 1 E _ * f t 2 ) ( ? ) , a g a i n n ot i n v a r i a n t , C £(ft1,ft2,<l>) C _ 1 ~ * * E + * % * (2.7.27) so i s n o t C - i n v a r i a n t e i t h e r , b u t (PC) £ ( ^ ,q»2 ,•) ( P C ) " 1 - ( * f * 0 S + * * V 0 * * ) ( - ? ) (2.7.28) -» (ft 2E_0>*ft 1 ) ( ? ) , so i s C P - i n v a r i a n t . We a l s o have 48 T C(ft 1,ft 2,<l>) T _ 1 - 4>1 ( - t ) V Q T E + < l > T ~ 1 V 0 f t 2 (-t) (2.7.29) = (* 1 v 0 s_v 0 ** 2 )(-t) - ( • l E + W 2 ) ( t ) , and -1 — * — 1 * ® C(<\>1 ,ft 2 ,<t>) ® ~ ^ (-x)®£+<|>® ft2(-x) (2.7.30) = (**£_«>%*) (-x) = - ( * 2 V 5 Z _ * * * s * l ) ( - x ) - ( * 2 S _ * * * 1 ) ( x ) , so b o t h a r e i n v a r i a n t as e x p e c t e d . Of t h e b o s o n - b o s o n i n t e r a c t i o n s o n l y two t y p e s a r e o f i n t e r e s t . The A-W-Higgs s c a l a r ( H ) ( p h y s i c a l o r u n p h y s i c a l ) i n t e r a c t i o n i s C(AWH) = -eM „ ( A - W H + A-W H ) , (2.7.31) W — + + — and i s e a s i l y shown t o be C, P and T - i n v a r i a n t . The o t h e r t y p e i s t h e e l e c t r o m a g n e t i c i n t e r a c t i o n between t h e p h o t o n A and t h e W o r H i g g s c u r r e n t s . The f o r m e r c u r r e n t i s g i v e n i n (2.2.31): j p(w) = -ie [ ( a Mw v - a vw l I)w _ - w +(a^w v -a vw p) ] . (2.7.32) + + v v — — E x a m i n a t i o n o f t h e f i r s t t e r m o n l y shows t h e t r a n s f o r m a t i o n p r o p e r t i e s P j^(w) p " 1 = i e [ V 0 W _ ( - ? ) V 0 ] • a ^ [ V 0 W + ( - ? ) V 0 ] + ••• (2.7.33) = ( J . - J ) ( ~ r ) •* •* n (due t o r -» - r i n a ) , C j U ( W ) C = i e W a V + ••• (2.7.34) v + + u v = i e W a W + • • • -j"(W) , and 49 T j U ( W ) T _ 1 = - i e [ V 0 W _ ( - t ) V Q ] • a ^ f V 0 W + ( - t ) V 0 ] + ••• ( 2 . 7 . 3 5 ) = - J X ) ( - t ) (due t o t •» - t i n a H ) , a l l o f w h i c h d e m o n s t r a t e C, P and T i n v a r i a n c e . The e l e c t r i c c u r r e n t due t o t h e H i g g s b o s o n s , g i v e n i n ( 2 . 2 . 2 7 ) , j ^ O ) = i e ( < t> _ a % + - <J>+ap<t>_) , ( 2 . 7 . 3 6 ) i s r a t h e r e a s i l y shown t o have s i m i l a r t r a n s f o r m a t i o n p r o p e r t i e s , a g a i n c o n f i r m i n g C, P and T i n v a r i a n c e . Thus i n g e n e r a l we may c o n c l u d e t h a t t h e t h e o r y i s C and P i n v a r i a n t e x c e p t where t h e c h i r a l i t y p r o j e c t i o n o p e r a t o r s Z + o c c u r , i n w h i c h c a s e o n l y CP i n v a r i a n c e can be d e m o n s t r a t e d . T h e r e a r e , however, no e x c e p t i o n s t o T and CPT i n v a r i a n c e , t h e l a t t e r b e i n g r e q u i r e d q u i t e g e n e r a l l y o f any l o c a l , L o r e n t z -i n v a r i a n t f i e l d t h e o r y ( B j o r k e n and D r e l l , 1 9 6 5 , Sec. 1 5 . 1 4 ) . 50 I I I . DIRAC AND MAJORANA NEUTRINOS I I I . l NEUTRINO MASS S i n c e i t s c o n c e p t i o n a h a l f c e n t u r y ago t h e n e u t r i n o has been assumed t o be m a s s l e s s , o r a t l e a s t t o have a v e r y s m a l l mass, o f t h e o r d e r , a t most, o f a few eV. D u r i n g t h e p a s t d e c a d e , and even e a r l i e r , a number o f a n a l y s e s have been p u b l i s h e d c o n c e r n i n g t h e c o n s e q u e n c e s o f a n o n - z e r o n e u t r i n o mass ( C a s e , 1967; B i l e n k y and P o n t e c o r v o , 1978; Cheng and L i , 1980a; L a n g a c k e r , 1981; Frampton and V o g e l , 1982; L i and W i l c z e k , 1 982). At l e a s t one modern e x p e r i m e n t (Lubimov, 1980; S t o c k d a l e , 1984, R e f . 1) has c l a i m e d a mass o f t h e o r d e r o f 30 eV, w h i l e o t h e r s ( Simpson, 1982; K i r s t e n et al., 1983; A v i g n o n e et al., 1983) s t i l l c l a i m t o have o b s e r v e d no m e a s u r a b l e mass i n t h e few eV r a n g e , o r q u e s t i o n t h e measured mass (Simpson, 1984). A number o f i n t e r e s t i n g p h y s i c a l c o n s e q u e n c e s f o l l o w f r o m t h e c o n j e c t u r e d e x i s t e n c e o f m a s s i v e n e u t r i n o s : n e u t r i n o o s c i l l a t i o n s ( B i l e n k y and P o n t e c o r v o , 1978; Cheng and L i , 1980a; B a r g e r , L a n g a c k e r and L e v e i l l e , 1980; Frampton and V o g e l , 1982), o r t h e s p o n t a n e o u s change o f n e u t r i n o t y p e , s u c h as t o e l e c t r o n t y p e f r o m muon t y p e , and so on; n e u t r i n o d e c a y (Cheng and L i , 1980b; P a l and W o l f e n s t e i n , 1982), i n w h i c h t h e h e a v i e r n e u t r i n o s w o u l d d e c a y i n t o t h e l i g h t e r v a r i e t i e s , s u c h as , f o r example, t h e r a d i a t i v e d e c a y o f a muon n e u t r i n o i n t o an 51 e l e c t r o n n e u t r i n o ; a t o t a l n e u t r i n o mass o f c o s m o l o g i c a l s i g n i f i c a n c e ( F r ampton and V o g e l , 1982), as t h e t o t a l mass o f t h e n e u t r i n o s i n t h e u n i v e r s e would, i n a l l l i k e l i h o o d , have an i m p a c t on t h e " m i s s i n g mass p r o b l e m " and t h e o v e r a l l t o p o l o g y o f t h e u n i v e r s e ; new ways i n w h i c h t h e v a r i o u s l e p t o n s i n t e r a c t w i t h n e u t r i n o s o f o t h e r f a m i l i e s , one o f t h e c o n c e r n s o f t h i s t h e s i s . Such p h y s i c a l c o n s e q u e n c e s a r e , a t b e s t , a t t h e v e r y edge o f d e t e c t a b i l i t y , and o n l y t h r o u g h v a r i o u s a s s u m p t i o n s and m odels can t h e p o s s i b i l i t i e s be e x p l o r e d and e x p e r i m e n t a l l i m i t s a p p r o a c h e d . Indeed, i t was t h e v e r y u n d e t e c t a b i l i t y o f c e r t a i n o f t h e s e p r o c e s s ( e s p e c i a l l y t h e r a d i a t i v e d e c a y p.-»eV) t h a t gave r i s e t o t h e c o n c e p t o f n o t o n l y c o n s e r v a t i o n o f l e p t o n number, b u t s e p a r a t e l y o f e l e c t r o n l e p t o n number, muon l e p t o n number, e t c . I t must be remembered, however, t h a t a t t h e p r e s e n t t i m e t h e r e i s no c o n c l u s i v e e v i d e n c e t h a t any o f t h e s e c o n s e r v a t i o n laws i s n o t r e a l l y v a l i d . I t so happens t h a t t h e r e a r e a t l e a s t two t h e o r e t i c a l d e s c r i p t i o n s o f m a s s i v e n e u t r i n o s t o be c o n s i d e r e d , w h i c h r e s u l t , i n a number o f c a s e s , i n q u i t e d i f f e r e n t p r e d i c t i o n s b e i n g made. F i r s t , t h e D i r a c n e u t r i n o , a s o r t o f l i g h t , n e u t r a l e l e c t r o n w i t h two s p i n s t a t e s f o r e a c h p a r t i c l e and a n t i - p a r t i c l e , and t h e b e s t u n d e r s t o o d o f t h e two b e c a u s e i t s m a t h e m a t i c a l d e s c r i p t i o n so r e s e m b l e s t h a t o f t h e e l e c t r o n . S e c ond, a s i n g l e , t w o - s t a t e n e u t r a l p a r t i c l e r e f e r r e d t o as a M a j o r a n a n e u t r i n o , one t h a t i s , i n d i f f e r e n t l y , i t s own a n t i -p a r t i c l e o r t h a t has no a n t i - p a r t i c l e c o u n t e r p a r t , l i k e t h e 52 p h o t o n . T h e s e p a r t i c l e s , t h e i r d e s c r i p t i o n s and i n t e r a c t i o n s , t h e i r r o l e i n t h e gauge t h e o r y o f t h e p r e v i o u s c h a p t e r , a r e , i n t u r n , t h e s u b j e c t o f t h e r e s t o f t h i s c h a p t e r . s I I I . 2 DIRAC NEUTRINOS To a s s u r e t h e e x i s t e n c e o f a D i r a c - t y p e n e u t r i n o mass one must a r r a n g e t h a t t h e L a g r a n g i a n c o n t a i n t h e terms n e c e s s a r y t o r e s e m b l e t h o s e o f a g e n e r a l D i r a c - t y p e p a r t i c l e : C = ^ ( i a - m ) q ; (3.2.1) = i«l>Z+a»l> + i«l>Z_aiP - m » K E + + E _ ) * . The l e p t o n i c p a r t o f t h e W e i n b e r g - S a l a m L a g r a n g i a n (2.2.2) c o n t a i n s , as f a r as t h e n e u t r i n o i s c o n c e r n e d , o n l y t h e f i r s t t e r m o f ( 3 . 2 . 1 ) , o r i v E 3v. T h e r e f o r e an SU(2) s i n g l e t v =E v •+* R + must be added t o t h e o r i g i n a l p a r t i c l e c o n t e n t , w i t h t h e f o l l o w i n g a e s t h e t i c a l l y d i s p l e a s i n g a s p e c t . I t s weak h y p e r c h a r g e , a c c o r d i n g t o t h e r u l e ( 2 . 2 . 1 1 ) , must be z e r o , so t h a t t h i s p a r t i c l e , a t l e a s t i n t h e p r e - H i g g s mechanism s t a g e , p a r t i c i p a t e s i n no i n t e r a c t i o n s w h a t e v e r w i t h t h e gauge b o s o n s , and o n l y w e a k l y w i t h t h e gauge b o s o n s and o t h e r f e r m i o n s --much more w e a k l y t h a n i t s l e f t - h a n d e d p a r t n e r — a f t e r symmetry b r e a k i n g . An S U ( 2 ) x U ( l ) gauge i n v a r i a n t Yukawa-type i n t e r a c t i o n t e r m t h a t can be added t o t h e o r i g i n a l L a g r a n g i a n , i n a n a l o g y w i t h ( 2 . 2 . 7 ) , i s , summing o v e r l e p t o n f a m i l i e s , 53 -2 f t 0 « + * ' • * „ ) , (3.2.2) A vH A + A A - A where *' = i a 2 $ = j 0>o j , (3.2.3) w h i c h i s needed r a t h e r t h a n <$ t o have t h e h y p e r c h a r g e i n (3.2.2) sum t o z e r o , w h i c h e x p r e s s e s t h e U ( l ) gauge i n v a r i a n c e as i n ( 2 . 2 . 9 ) . A f t e r symmetry b r e a k i n g where * ' "» j *o + v / J " 2 J (3.2.4a) <*' > = jv/J"2j , (3.2.4b) one o b t a i n s a term o f t h e f o r m ~In 0 n (v/J"2) ( v f t E v n + ZaZ v f t) , (3.2.5) A v A A + A A - A so t h a t t h e t y p e A n e u t r i n o has a mass m ( v A ) = BvH V / T 2 ' (3.2.6) t h a t , l i k e t h e c o u p l i n g c o n s t a n t s 0 , i s t o t a l l y a r b i t r a r y . v A T h e r e i s no c o n n e c t i o n w h a t e v e r between t h e mass o f a l e p t o n A and i t s a s s o c i a t e d n e u t r i n o v^, a n o t h e r a e s t h e t i c s h o r t c o m i n g o f t h i s c o n s t r u c t i o n . I m p l i c i t i n t h e model so f a r has been t h e a s s u m p t i o n t h a t t h e n e u t r i n o s t a t e s (v = v ) i n t h e L a g r a n g i a n e, p., • • • t h e s o - c a l l e d weak e i g e n s t a t e s -- a r e a l s o mass e i g e n s t a t e s ; t h a t i s , e a c h o f t h e f i e l d s has been s u p p o s e d t o have a d e f i n i t e mass as o p p o s e d t o b e i n g l i n e a r s u p e r p o s i t i o n s o f f i e l d s o f v a r i o u s b u t d e f i n i t e masses. The most g e n e r a l Yukawa term t h a t g e n e r a t e s a ( D i r a c ) mass f o r n e u t r i n o s i s a g e n e r a l i z a t i o n o f ( 3 . 2 . 2 ) : 54 -I. . P. . ( * . Z $ r v . + v . Z ) , (3.2.7) i j i j i + J J - i w h i c h , a f t e r symmetry b r e a k i n g , c o n t a i n s t h e mass t e r m -I. . P. . (v/J/2) ( v . Z v . + v . Z v . ) (3.2.8) i j i j i + J J - i = -S. . P. . (v/J/2) ( v . „ . ) , i j i j i J where P. . = 0 . . ( s o t h a t b o t h c h i r a l i t y s t a t e s a r e t o have t h e same ma s s ) , w h i c h can be d i a g o n a l i z e d by an o r t h o g o n a l t r a n s f o r m a t o n t o m. v . v . , (3.2.9) 1 1 1 1 where t h e m. a r e t h e e i g e n v a l u e s o f t h e mass m a t r i x (P. .v/J/2), i i j assumed p o s i t i v e (Cheng and L i , 1980a). I f any m^ were t o be n e g a t i v e a t r a n s f o r m a t i o n m. -» -m. , v. •* V c v . would be l l l 5 l p e r f o r m e d , b e c a u s e v3v •* v3v w h i l e vv -* -vv. The o r t h o g o n a l t r a n s f o r m a t i o n i s v. = 1 U v (3.2.10) k ex ock oc where w i t h Greek s u b s c r i p t s d e n o t i n g mass e i g e n s t a t e s , so t h a t t h e c h a r g e d l e p t o n - n e u t r i n o terms i n t h e L a g r a n g i a n t h a t f o l l o w f r o m (2.2.18) and (3.2.7) a r e , r e s p e c t i v e l y , I n [ ( g / J " 2 ) U ( v Z W A + A Z W v ) (3.2.12) ocA OCX. oc + + + - oc + P n u n ( v E 4> A + A Z • v ) ] A ocA oc + + — - oc ( w i t h p^v/J/2 = M^) and w i t h 2„ . P„ -U . ( A Z <t> v + v Z <l> A ) , (3.2.13) A Joe A j ocj + - a oc ~ + 2 . P „ .U .v/J/2 = m U . (3.2. 14) J A j ocj oc ocA 55 f r o m ( 3 . 2 . 1 1 ) . Thus i f t h e o f f - d i a g o n a l e l e m e n t s o f t h e o r t h o g o n a l m a t r i x do n o t v a n i s h t h e r e i s a m i x i n g among t h e v a r i o u s n e u t r i n o t y p e s ; a g i v e n n e u t r i n o o f f i x e d mass i n t e r a c t s w i t h s e v e r a l l e p t o n s (and v i c e v e r s a ) , and t h e p r o b a b i l i t y o f p r o c e s s e s s u c h as n e u t r i n o d e c a y a r i s e s . No l o n g e r w o u l d t h e r e be any s e p a r a t e c o n s e r v a t i o n o f e l e c t r o n l e p t o n number, muon l e p t o n number, e t c . I I I . 3 MAJORANA NEUTRINOS One can u n d e r s t a n d from t h e p r e c e d i n g s e c t i o n t h a t t h e g e n e r a l i z a t i o n o f t h e o r i g i n a l W e i n b e r g - S a l a m L a g r a n g i a n t o i n c o r p o r a t e m a s s i v e n e u t r i n o s o f t h e D i r a c t y p e i s n o t a l t o g e t h e r s a t i s f a c t o r y b e c a u s e o f t h e number and t y p e o f new terms t h a t must be added and t h e l a c k o f r e s t r i c t i o n on any o f t h e new p a r a m e t e r s . I t so happens t h a t n e u t r i n o mass can be g e n e r a t e d w i t h o u t t h e a d d i t i o n o f any o f t h e s e o b j e c t i o n a b l e terms p r o v i d e d an i s o - v e c t o r H i g g s f i e l d , w i t h i t s Yukawa i n t e r a c t i o n ( 2 . 3 . 1 8 ) , i s i n t r o d u c e d (Cheng and L i , 1980a, 1984; Marshak, R i a z u d d i n and M o h a p a t r a , 1981; S c h e c h t e r and V a l l e , 1980). The p a r t i c l e t h a t r e s u l t s — r e f e r r e d t o as a M a j o r a n a n e u t r i n o — i s a s i n g l e t w o - s t a t e p a r t i c l e w i t h o u t a n t i -p a r t i c l e ( o r , i f p r e f e r r e d , i s i t s own a n t i - p a r t i c l e ) . A f t e r symmetry b r e a k i n g t h e r e r e s u l t s t h e f i r s t t erm o f (2.3.20) ( i n t h i s s e c t i o n one i s t o sum o v e r r e p e a t e d 56 s u b s c r i p t s ) : -B. .v T T(i/°E v. + v.E v°) , (3.3.1) I J H I - J j + i where, by c o n s t r u c t i o n , 0.. = B..» and th e 0's a r e r e a l , by CPT i n v a r i a n c e . The mass m a t r i x , v 0. . = ;-M. ., c a n , as i n t h e H i j 2 i j p r e v i o u s s e c t i o n , be d i a g o n a l i z e d by an o r t h o g o n a l t r a n s f o r m a t i o n : U M U T = M J. , (3.3.2a) d i a g o r U., M U = S. .m.-n. , (3.3.2b) l k kJl jJL i j l l where t h e m. a r e a l l p o s i t i v e and -n. = ±1, where t h e l l e i g e n v a l u e s o f M a r e m. <n. . In t h e c a s e o f two f a m i l i e s , f o r I l T -1 example, i t f o l l o w s from (3.3.2a) t h a t , s i n c e U = U , Tr(UMU T) = Tr(M) = + M 2 2 (3.3.3) = T r ( M d i a g } = " i - i + ^ m 2 and Det(UMU T) = Det(M) = M 1 X M 2 2 - M 1 2 M 2 1 (3.3.4) = D e t ( M d i a g ) = U i , ° 1 ) ( V 2 ) • w h i c h i s r e a d i l y s o l v e d f o r t h e masses ml, m2 i n terms o f t h e o r i g i n a l 0 c o e f f i c i e n t s : mx = j | T r ( M ) ± [ T r 2 ( M ) - 4 D e t ( M ) ] 1 / 2 | (3.3.5) I t i s n e i t h e r c o n v e n i e n t n o r n e c e s s a r y t o c o n s i d e r more g e n e r a l e x a m p l e s . One d e f i n e s t h e M a j o r a n a f i e l d x ( P a l and W o l f e n s t e i n , 1982) by x. = (1/J/2) U. . ( Z v. + n.Z v.) (3.3.6a) i i j - J i + J 57 x. = (1/J"2) U. . (v .E + n . v ° E ) , (3.3.6b) i I J j + l j -b u t where t h e r e f e r e n c e j u s t c i t e d does n o t have t h e ( n e c e s s a r y ) (1/J"2) f a c t o r . We have — i — c — c m x x = - U . U .m n ( v . E v. + v . E v.) a a a 2 a i a j a a i + j i _ J = 7 M. . ( v . E v° + v C E v.) (3.3.7) 2 1 J J + i i - J w h i c h i s ( 3 . 3 . 1 ) , t h e mass term, where, from (3.3.2b) f o l l o w s M , = U. U m.-n. (3.3.8) ab i a l b l l and — c — * — * — c v. E v. = v . E v. - - v . V c E V c v. - v . E v. , (3.3.9) i + j i + J j 5 + 5 i J + i where (B.38) has been a p p l i e d . We a l s o have — I — — c c i x ax = - i U .U . ( v . E 3v . + TI -n v . E 3v .) a a 2 a i a j i + j a a i - j = 7 i ( v . E av. + v ° E a v C ) (3.3.10) 2 1 + 1 1 - 1 b u t t h e l a s t t erm becomes, a g a i n u s i n g ( B . 3 8 ) , — c c — v — v E 3v. = - a v.V„V E V„v. = v . E 3v. + s u r f a c e term, (3.3.11) l - i v i 5 - 5 i 1 + 1 so t h a t i x 3 x = i v . E 3 v . (3.3.12) a a 1 + 1 i n t h e a c t i o n i n t e g r a l . Thus t h e p u r e n e u t r i n o p a r t o f t h e L a g r a n g i a n becomes C(x ) = i x 3 x - m x x , (3.3.13) a a a a a as r e q u i r e d . What r e m a i n s now i s t o r e p l a c e t h e v - f i e l d s i n t h e i n t e r a c t i o n terms w i t h t h e x - f i e l d s . I t i s i m p o r t a n t t o n o t e t h a t , t o w i t h i n a phase, t h e M a j o r a n a f i e l d x i s s e l f - c o n j u g a t e : x° = x* = n. x. , (3.3.14) l I i i a f a c t t h a t f o l l o w s e a s i l y f r o m i t s d e f i n i t i o n ( 3 . 3 . 6 a ) . The u s u a l e x p a n s i o n o f a f r e e s p i n o r f i e l d (2.1.8) 58 — 3 / 2 a - i n 'X 1D*X * ft = (2ir) I d p Jm/e ( e P a u + e P b v ) (3.3.15) r r r r y i e l d s , when (3.3.14) i s a p p l i e d , b = a , (3.3.16) r r t o w i t h i n a phase, so t h a t t h e f i e l d ft d e s c r i b e s b u t one two-s t a t e p a r t i c l e , as c l a i m e d . In o r d e r t o o b t a i n t h e c o r r e c t f r e e - f i e l d energy-momentum P P = J d 3 p p M a * ( p ) a (p) (3.3.17) r r from t h e c a n o n i c a l p r e s c r i p t i o n s (2.1.14) and (2.1.16) a f a c t o r o f ( 1 / J 2 ) must be added t o ( 3 . 3 . 1 5 ) : ( o . o . l o ) — 3 ' 2 3 — i D ' X i D * x • x. = (1/4"2)(2TT) / d p Jm./€ (e P a u + „ . e p a v ). l l r r l r r T h i s f a c t o r o f (l/J"2) means t h a t t h e x - f i e l d p r o p a g a t o r ( S e c . IV.3) i s o n e - h a l f o f t h e u s u a l f o r s p i n - j p a r t i c l e s . I t i s v e r y i m p o r t a n t t o n o t e t h a t t h e x - f i e l d b o t h c r e a t e s and a n n i h i l a t e s i t s a p p r o p r i a t e M a j o r a n a p a r t i c l e . From t h e d e f i n i t i o n s (3.3.6a) and (3.3.6b) f o l l o w (3.3.19a) (3.3.19b) (3.3.19c) (3.3.19d) w h i c h can now be u s e d t o e l i m i n a t e a l l r e f e r e n c e t o t h e v-f i e l d s i n t h e o r i g i n a l i n t e r a c t i o n L a g r a n g i a n . The c h a r g e d l e p t o n - n e u t r i n o - W i n t e r a c t i o n f r o m (2.2.18) becomes CUxW) = gU ( x . Z W A + Iz W x.) , (3.3.20) l H l + + + - l w h i l e t h e c h a r g e d l e p t o n - n e u t r i n o - c h a r g e d H i g g s i n t e r a c t i o n becomes C ( i l x * ) = -J"2 U. R ( x . Z <l> Jl + Iz ft x. ) (3.3.21) ± 1 A J I 1 + + - - 1 Z v . --- J • = J"2 U. .Z x. i j - 1 v .Z = J + = J2 U. . x.Z i j 1 + c Z v . -+ J J"2 U. .n.Z i j 1 + 1 - c v .Z = J -= J"2 U..n. x.Z i j 1 1 -59 and, from ( 2 . 3 . 2 0 ) , C(AxH) = 2J/2 U .B.nn ( x E H A + AE H x ) . (3.3.22) a i i A a a - + + - a These l a s t two terms can be combined i n t o i n t e r a c t i o n s w i t h t h e u n p h y s i c a l and p h y s i c a l H i g g s f i e l d s , S and B o f ( 2 . 3 . 1 5 ) , ( 2 . 3 . 1 6 ) , r e s p e c t i v e l y , i n t o t h e f i n a l form C(AxS) = _ ( g / M ) U . n [ x . ( M E -m.E )S A + A(M E -m.E )S x.] W 1 A 1 A + 1 - + A - 1 + - 1 and C(AxB) = ( g / M M ) U . n [ x . ( M t a n e E +m.coteE )B A W 1 A 1 A + 1 - + + A(M t a n e E +m.coteE )B x.] , (3.3.24) A - 1 + - 1 where t h e P., P. ., v, v „ f a c t o r s have been w r i t t e n i n terms o f 1 1 j H t h e c h a r g e d l e p t o n and n e u t r i n o masses (M. and m., 1 1 r e s p e c t i v e l y , f r o m (2.2.14) and ( 3 . 3 . 2 b ) ) and t h e c o u p l i n g c o n s t a n t g f r om ( 2 . 3 . 1 2 a ) . These i n t e r a c t i o n terms w i l l be t h e b a s i s o f a l l f u t u r e work. 60 IV. THE FEYNMAN RULES OF THE THEORY IV.1 LEPTON-BOSQN INTERACTIONS: SUMMARY In t h i s s e c t i o n a l l o f t h e terms i n t h e L a g r a n g i a n d e s c r i b i n g t h e i n t e r a c t i o n s between t h e l e p t o n s and t h e gauge b o s o n s and between t h e l e p t o n s and t h e H i g g s b o s o n s w i l l be l i s t e d and summarized. Such terms w i l l be n eeded t o c o n s t r u c t t h e "Feynman r u l e s " f o r t h e v e r t i c e s o f t h e t h e o r y , o r a t l e a s t t h o s e c o n s i s t i n g o f two l e p t o n s and one gauge b o s o n o r H i g g s b o s o n r e l e v a n t f o r t h e c e n t r a l p r o b l e m o f t h i s t h e s i s (Chap. V, V I ) , w h i c h i s t h e c a l c u l a t i o n o f t h e r a t e o f d e c a y o f a muon i n t o an e l e c t r o n p l u s a p h o t o n , o r an e l e c t r o n p l u s an e l e c t r o n - p o s i t r o n p a i r . T h e r e a r e a t l e a s t two good r e a s o n s why s u c h a c o n s t r u c t i o n must be done f r o m t h e b e g i n n i n g . E a c h g a u g e - t h e o r e t i c model has i t s own p a r t i c u l a r i n t e r a c t i o n terms and t h o s e f o r t h e s p e c i f i c model u s e d h e r e , namely t h e model w i t h a v e c t o r H i g g s f i e l d and m a s s i v e M a j o r a n a n e u t r i n o s , have n o t been l i s t e d b e f o r e . S econd, i n c o n s t r u c t i n g a l l o f t h e r u l e s t o be u s e d h e r e a c o n s i s t e n t s i g n c o n v e n t i o n can be u s e d t h r o u g h o u t , t h e same f o r a l l i n t e r a c t i o n s , so t h a t any a m b i g u i t y o f p h a s e w i t h v e r t i c e s p u b l i s h e d e l s e w h e r e can be a v o i d e d . The i n t e r a c t i o n s among t h e l e p t o n s and t h e gauge bosons (W +, A, Z) a r e g i v e n i n (2.2.18) w h i c h i s m o d i f i e d t o i n c l u d e a 61 sum o v e r t h e l e p t o n f a m i l i e s , and t h e n e u t r i n o f i e l d s g i v e n i n Chap. I I a r e m o d i f i e d t o become M a j o r a n a n e u t r i n o s , as g i v e n i n ( 3 . 3 . 1 9 ) . T h e s e c hanges r e s u l t i n t h e i n t e r a c t i o n terms CU,x;W,A,Z) = S A i g U . J l ( x i E + W + J l + iz +W_x.) (4.1.1) + Z j J e i U A + G s i n 2 e w J l Z _ Z J l - ±G( c o s 2 e ^ s i n 2 e t f) Jl£ +ZJl] + ln • .GU • U x . Z Z E x. . JU j i i j J l l + - j The o r t h o g o n a l i t y o f t h e m a t r i x U can be e x p l o i t e d t o r e w r i t e t h e l a s t t e r m as 2 i G x i Z + Z x i , (4.1.2) b u t t h i s t e r m w i l l n o t be needed i n what f o l l o w s . The most i m p o r t a n t term o f (4.1.1) f o r t h e needs o f t h i s t h e s i s , e x c e p t f o r t h e e l e c t r o m a g n e t i c i n t e r a c t i o n , i s t h e f i r s t , w h i c h shows t h e c o u p l i n g o f t h e v a r i o u s c h a r g e d l e p t o n s (e, p., •••) t o t h e v a r i e t i e s o f M a j o r a n a n e u t r i n o s x. l (i=l,2,'««)» a c o u p l i n g g o v e r n e d by t h e unknown components (U. .) o f t h e o r t h o g o n a l t r a n s f o r m a t i o n t h a t d i a g o n a l i z e s t h e Yukawa c o u p l i n g m a t r i x o f ( 2 . 3 . 1 8 ) . One i s tempted t o presume t h a t t h e v a l u e s o f U.. a r e g r e a t e s t when i = j and d e c r e a s e as t h e d i f f e r e n c e between i and j i n c r e a s e s b u t , u n f o r t u n a t e l y , t h e r e i s no c o n c r e t e e v i d e n c e f o r s u c h an a s s u m p t i o n . The i n t e r a c t i o n s among t h e l e p t o n s and t h e H i g g s f i e l d s a r e a l l t h a t r e m a i n t o be c o n s i d e r e d . The b a s i c i n t e r a c t i o n terms f o r t h e i s o - s p i n o r H i g g s f i e l d were g i v e n i n ( 2 . 2 . 1 8 ) , o r , when g e n e r a l i z e d t o i n c l u d e a sum o v e r l e p t o n f a m i l i e s , C(v,A,<D) = ~2 J lB i l(^ AS +<t> +Jl + lz_+_Vji (4.1.3) 62 a l o n g w i t h (2.3.20) f o r t h e v e c t o r H i g g s f i e l d . As m e n t i o n e d e a r l i e r i n Chap. I I t h e f i e l d s <|>+ and H + a r e r e p l a c e d by a p h y s i c a l H i g g s f i e l d B + and an u n p h y s i c a l f i e l d S + d e f i n e d i n (2.3.15) and ( 2 . 3 . 1 6 ) , and t h e n e u t r i n o v - f i e l d s a r e t o be r e p l a c e d by t h e M a j o r a n a x ~ f i e l d s o f t h e p r e v i o u s c h a p t e r . The r e s u l t was d i s p l a y e d e a r l i e r , i n (3.3.23) and ( 3 . 3 . 2 4 ) : £ ( A , x , S ) = - Z . n ( g / M w ) U . n [ x . ( M n Z -m.Z )S A (4.1.4) l A W 1 A 1 A + 1 - + + M M ^ E - m . E ^ S x . ] , C(A,x,B) = „ ( g / M w ) U . n [ x . ( M t a n e E +m.coteE )B A (4.1.5) l A W 1 A 1 A + 1 - + +A(M t a n e E + m . c o t e £ )B x.] • A - 1 + - 1 The d o u b l y c h a r g e d H i g g s f i e l d makes a c o n t r i b u t i o n g i v e n i n ( 2 . 3 . 2 0 ) : C(A,H ) = 2. .J/2B. . ( A ° E H A. + A.E H A°) . (4.1.6) ±± i j i j 1 - ++ j j + — 1 B o t h (4.1.3) and (2.3.20) c o n t a i n i n t e r a c t i o n terms between t h e l e p t o n s and n e u t r a l H i g g s p a r t i c l e s , w h i c h , l i k e t h e i r s i n g l y c h a r g e d c o u n t e r p a r t s , must f i r s t be r e c a s t i n t o p h y s i c a l and u n p h y s i c a l f i e l d s , as was done i n Se c . I I . 4 . From (4.1.3) and u s i n g <t>0 = <J>1+i<t>2 we have " ^ A V ^ O * + i s _ * o * A ) = - 2 A V ( * i + * 5 « > 2 ) A (4.1.7) " "VA*** from T a b l e I, w h i c h i s by f a r t h e l e a d i n g c o n t r i b u t i o n . The l e p t o n - s i n g l y c h a r g e d , p h y s i c a l H i g g s b o s o n i n t e r a c t i o n o f (4.1.5) c o n t a i n s s o m e t h i n g v e r y i n t e r e s t i n g , s o m e t h i n g n o t c o n s i d e r e d b e f o r e i n i t s p o s s i b l e c o n t r i b u t i o n t o muon decay: th e f a c t o r m.cote. From (2.3.15b) and (3.3.2) one s e e s t h a t 1 ( m . X c o t e ) - (v T I B . .) ( v / v t T ) (4.1.8) 63 so t h e f a c t o r i s a c t u a l l y independent o f v and t h e r e f o r e o f H t h e n e u t r i n o masses as w e l l . I n d e e d , i f t h e Yukawa c o u p l i n g c o n s t a n t s P. . and 0^ a r e o f t h e same o r d e r o f m a g n i t u d e , t h i s i J SL f a c t o r i s o f t h e o r d e r o f m a g n i t u d e o f a c h a r g e d l e p t o n mass, and makes, t h e r e f o r e , a c o n t r i b u t i o n p o t e n t i a l l y much g r e a t e r t h a n any t h a t depend on n e u t r i n o masses. T h i s i n t e r a c t i o n term w i l l , i n what f o l l o w s , make a most i n t e r e s t i n g c o n t r i b u t i o n t o muon de c a y , b o t h t h e r a d i a t i v e d e c a y o f Chap. V and t h e t r i p l e e l e c t r o n d e c a y o f Chap. VI. IV.2 BOSON-BOSON INTERACTIONS: SUMMARY The l a s t t o p i c f o r c o n s i d e r a t i o n b e f o r e we can c o n s t r u c t t h e Feynman r u l e s n e e d ed i n t h e s e q u e l i s a l i s t o f t h e r e l e v a n t i n t e r a c t i o n terms among t h e gauge b o s o n s , and among t h e H i g g s and gauge b o s o n s . The l i s t t o be g i v e n h e r e w i l l o n l y be t o t h e o r d e r o f p e r t u r b a t i o n t h e o r y needed i n s u b s e q u e n t c a l c u l a t i o n s w h i c h , g e n e r a l l y , i s t r i l i n e a r i n t h e f i e l d s . The i n t e r a c t i o n s among t h e gauge b o s o n s a r e an i n t e g r a l p a r t o f t h e o r i g i n a l W e i n b e r g - S a l a m model, and were g i v e n and e x p l a i n e d i n Sec. I I . 2 . T h e s e terms a r e i n d e p e n d e n t o f t h e H i g g s f i e l d s . Thus, t o an o r d e r t r i l i n e a r i n t h e f i e l d s , we have t h e i n t e r a c t i o n term o f (2.2.30)': 64 C(W,A,Z) = ( i / 2 ) e [ ( A V - A V ) ( a W+-a W +) (4.2.1) - - p v v p - (A^WV-AVWU)(a w~-a w~) + (a A -a A )(w^ wv-wvw^ )] + + \X V V \X \X V V \x — + - + - ( i / 2 ) g c o s e r , [ (zMwv-zvwM) (a w+-a w+) W - - ix v v \x - ( z V - z V)(a w~-a w~) + (a z -a z MwV-wV1)] . + + IX V V IX IX V V ix — + — + The e l e c t r o m a g n e t i c i n t e r a c t i o n o f t h e H i g g s b o s o n s was d e s c r i b e d by (2.2.26) f o r t h e s p i n o r H i g g s f i e l d , w i t h s i m i l a r t erms f o l l o w i n g f r o m t h e v e c t o r H i g g s L a g r a n g i a n ( 2 . 3 . 4 ) : C(A,<D,H) = -ieA^(<|> a <|> -<t>a <|> ) (4.2.2) - p + + p -- i e A M ( H a H -H a H ) - i e A ^ ( H a H -H a H ) . - n + + p - — n ++ ++ p. — In terms o f t h e p h y s i c a l f i e l d (B) and u n p h y s i c a l f i e l d (S) o f (2.3.16) and ( 2 . 3 . 1 5 ) , r e s p e c t i v e l y , t h e f i r s t two terms o f (4.2.2) become, not s u r p r i s i n g l y , C(A,B) = - i e A ^ ( B a B - B a B ) (4.2.3a) - \x + + ix -C(A,S) = - i e A U ( S a S - S a S ) . (4.2.3b) - p. + + p -F i n a l l y , f r o m t h e s p i n o r H i g g s L a g r a n g i a n (2.2.5) and t h e v e c t o r H i g g s L a g r a n g i a n (2.3.4) a r i s e i n t e r a c t i o n s among t h e c h a r g e d H i g g s , t h e W and t h e e l e c t r o m a g n e t i c f i e l d : C(W,A,* .H ) = ( 1 / J 2 ) g g ' a [ W*1( v H + ( v / J"2) <t> ) (4.2.4) ± ± p - H + + + W^(v_H +(v/,T2)* )] , ' n — ~ w h i c h , w i t h t h e d e f i n i t i o n o f t h e u n p h y s i c a l f i e l d S o f ( 2 . 3 . 1 5 ) , becomes £ ( W , A , S ) = - J ^ e A (W^S + + W^S) , (4.2.5) where (2.3.12a) was u s e d , and (2.2.15c) was u s e d t o o b t a i n A from t h e U ( l ) gauge f i e l d a . S i m i l a r i n t e r a c t i o n s w i t h t h e n e u t r a l gauge b o s o n Z have been i g n o r e d , n ot b e i n g needed i n t h e s e q u e l . 65 I t i s i m p o r t a n t t o n o t e t h a t t h e r e i s no i n t e r a c t i o n s i m i l a r t o (4.2.5) f o r t h e p h y s i c a l H i g g s b o s o n B + : t h i s i n t e r a c t i o n i s a r e s u l t o f t h e gauge f i x i n g t erm (2.6.8) and i s c r u c i a l f o r t h e gauge i n v a r i a n c e o f t h e t h e o r y , as t h e Feynman d i a g r a m s o f S e c . V.2 w i l l d i s p l a y . T h e r e a r e many more i n t e r a c t i o n terms w i t h i n t h e gauge t h e o r y o f C h a p . I I t h a n have been l i s t e d i n t h i s and t h e p r e v i o u s s e c t i o n , i n t e r a c t i o n s w i t h t h e n e u t r a l p h y s i c a l H i g g s f i e l d s and, f o r example, i n t e r a c t i o n s among n e u t r i n o s . But we have l i s t e d a l l t h o s e t h a t w i l l be needed i n what f o l l o w s and t h e y a r e now t o be u s e d t o c o n s t r u c t t h e Feynman r u l e s o f t h e t h e o r y . IV.3 PROPAGATORS The Feynman r u l e s o f t h e t h e o r y , c o n s i d e r e d i n t h i s and t h e s u b s e q u e n t s e c t i o n , c o m p r i s e a c o n s i s t e n t s e t o f " d i a g r a m s " r e p r e s e n t i n g e l e m e n t a r y i n t e r a c t i o n p r o c e s s e s t h a t a r e p i e c e d t o g e t h e r i n s u c h a way as t o g e n e r a t e w i t h some e a s e , t o t h e o r d e r o f p e r t u r b a t i o n t h e o r y r e q u i r e d , t h e e x a c t i n t e r a c t i o n t erms n eeded f o r a s p e c i f i c p r o c e s s i n v o l v i n g g i v e n i n i t i a l and f i n a l p a r t i c l e s t a t e s . The d i a g r a m s c o n s i s t o f two b a s i c t y p e s : propagators, so c a l l e d , w h i c h a r e e s s e n t i a l l y Green f u n c t i o n s f o r t h e v i r t u a l m o t i o n o f a p a r t i c l e from one s p a c e -66 t i m e e v e n t t o a n o t h e r , and vert ices, w h i c h r e p r e s e n t s p e c i f i c i n t e r a c t i o n terms when t h r e e o r more p a r t i c l e s a r e i n v o l v e d a t , e i t h e r e n t e r i n g o r l e a v i n g , some e v e n t . T h i s s e c t i o n w i l l l i s t t h e v a r i o u s p r o p a g a t o r s r e q u i r e d . T h e r e i s a b s o l u t e l y n o t h i n g new i n t h i s -- a s t a n d a r d t e c h n i q u e s i n c e t h e i n c e p t i o n o f modern quantum e l e c t r o d y n a m i c s more t h a n t h i r t y y e a r s ago — b u t by c a l c u l a t i n g them a l l a c c o r d i n g t o a d e f i n i t e scheme we can be a s s u r e d t h a t t h e r e l a t i v e s i g n s among them w i l l be c o r r e c t . S i n c e a common phase f a c t o r f o r them i s i r r e l e v a n t we need n o t be c o n c e r n e d h e r e , n o r i n t h e n e x t s e c t i o n , a b o u t w h e t h e r t h e s i g n s o b t a i n e d a r e t h e same as t h o s e l i s t e d e l s e w h e r e . F o r t h e p r o p a g a t o r s t h e y w i l l be, f o r t h e v e r t i c e s t h e y w i l l n o t . P r o p a g a t o r s a r e c o n s i d e r e d i n d e t a i l i n , f o r example, B j o r k e n and D r e l l (1964, 1965), I t z y k s o n and Z u b e r (1980) and Cheng and L i ( 1 9 8 4 ) . I t i s e x t r e m e l y q u i c k and c o n v e n i e n t t o use t h e p r o c e d u r e d e v e l o p e d i n t h e l a s t work c i t e d : a p r o p a g a t o r i s i t i m e s t h e F o u r i e r t r a n s f o r m o f t h e i n v e r s e o f t h e d i f f e r e n t i a l o p e r a t o r o b t a i n e d f r o m t h e k i n e t i c p a r t o f t h e f r e e f i e l d L a g r a n g i a n . I f , f o r example, C ( f r e e ) = | <D.(x) A . . ( x ) <D.(x) , (4.3.1) where t h e f a c t o r j a p p e a r s o n l y f o r r e a l s c a l a r and r e a l v e c t o r f i e l d s , one t h e n d e f i n e s A . . ( x ) A T ' ( x - y ) — S., S 4 ( x - y ) (4.3.2) i j j k i k and A T ^ X ) = ( 2 T T ) ~ 4 r d 4 k e ~ l k ' X AT 1 , (k) , (4.3.3) 67 and t h e n t h e p r o p a g a t o r i s s a i d t o be A. .(k) = i A7\(k) , (4.3.4) i J i J where k i s t o be i n t e r p r e t e d as t h e p a r t i c l e ' s 4-momentum. Fo r a s c a l a r f i e l d <J> we have C ( f r e e ) = -^a * a % - \m Z <t>2 (4.3.5) 2 p. 2 1 2 2 = --<l>(a + m ) $ + s u r f a c e term and - ( a 2 + m 2)(2w) 4 / d 4 k e l k ' ( X y ) A ' ( k ) = S 4 ( x - y ) , (4.3.6) from w h i c h we o b t a i n - ( - k 2 + m 2) A _ 1 ( k ) = 1 , (4.3.7) b e c a u s e o f t h e r e p r e s e n t a t i o n o f t h e d e l t a f u n c t i o n S 4 ( x ) = ( 2 T T ) " 4 J"d 4k e " l k ' X , (4.3.8) so t h e p r o p a g a t o r f o r a r e a l o r complex s c a l a r f i e l d ( t h e f a c t o r i i n (4.3.5) i s i r r e l e v a n t ) i s * ( k ) = i / ( k 2 - m 2) , (4.3.9) as i s l i s t e d i n e v e r y t e x t b o o k on quantum f i e l d t h e o r y . F o r t h e s p i n o r f i e l d we have C ( f r e e ) = i ( i a - m)<|> (4.3.10) so t h a t (ia-m) ( 2 T T ) ~ 4 f d 4 k e - l k , ( x ~ v ) A - 1 ( k ) = S 4 ( x - y ) (4.3.11) /o \ - 4 P J 4 I - i k - ( x - y ) - i = (2ir) J d k e (k-m) A (k) , so t h e p r o p a g a t o r f o r s p i n - j f e r m i o n s ( e l e c t r o n s , muons, D i r a c n e u t r i n o s , e t c . , b u t not M a j o r a n a n e u t r i n o s — see below) i s ^ ( k ) = i / ( k - m) , (4.3.12) t h e s t a n d a r d r e s u l t . F o r a m a s s i v e , complex ( o r r e a l , f o r w h i c h a f a c t o r o f j 68 must be added) v e c t o r f i e l d , w i t h t h e i m p o r t a n t and n e c e s s a r y gauge f i x i n g t e r m o f (2.6.8) o r ( 2 . 6 . 9 ) , we have 1 % U V 2 U — 1 U £ V C ( f r e e ) = --F F H + m A A — £ a^A a A , - (4.3.13a) A uv U u v where F s a A - 3 A (4.3.13b) p.1/ (l 1/ v M as u s u a l , and we o b t a i n L l i / 2 2 —1 LI V C = A [n ( a +m ) + (£ - l ) a a ]A + s u r f a c e term , (4.3.14) \x v w h i c h r e s u l t s i n [^ V(a a+« a) + ( r 1 - l ) 3 J J 3 V ] ( 2 T r ) " 4 Pd 4k e " i k * ( X " y ) A _ 1 ( k ) (4.3.15) vX * S 4 ( x - y ) m \ - 4 P J 4 I - i k « ( x - y ) U V 2 2 -1 , x, n, v. - i . = (2w) |d k e [-n (-k +m ) - (C - l ) k k ]A (k) . vX The s o l u t i o n f o r i A 1 i s t h e p r o p a g a t o r A: a (k) = ( k 2 - m 2 ) 1 p.v p.v + i ( C _ 1 - l ) k k ( k 2 - m 2 ) _ 1 [ ( C _ 1 - l ) k 2 + ( k 2 - m 2 ) ] _ 1 . (4.3.16) IX V Note t h e l i m i t s : i n t h e " u n i t a r y gauge" (£ 1 = 0 ) we have A (k) = -i-n ( k 2 - m 2 ) _ 1 + i ( k k /m2 ) ( k 2 - m 2 ) _ 1 , (4.3.17) \xv uv U v and i n t h e m a s s l e s s c a s e ( p h o t o n p r o p a g a t o r ) we have A (k) = -i-n ( k 2 ) " 1 + i ( l - O k k ( k 2 ) " 2 , (4.3.18) Uv \xv U v where one can not now have £ 1 = 0. The Feynman ( o r t ' H o o f t -Feynman) gauge has £ = 1. I t i s v e r y u s e f u l t o n o t e t h a t (4.3.16) can be w r i t t e n A (k) = - i i , ( k 2 - m 2 ) _ 1 + i ( k k /m ) ( k 2 -m2 ) _ 1 * (4.3.19) \xv \xv u v - i ( k k / m 2 ) ( k 2 - ? m 2 ) _ 1 . U v The p r e s c r i p t i o n s u s e d above c o n t a i n no h i n t o f t h e o r i g i n o f p r o p a g a t o r s and t h e i r u s e . They a c t u a l l y a r i s e f r o m an i n t e g r a t i o n o v e r i n t e r n a l ( o r v i r t u a l ) l i n e s i n Feynman 69 d i a g r a m s , an i n t e g r a l t h a t c o n t a i n s a vacuum e x p e c t a t i o n v a l u e o f a t i m e - o r d e r e d - s e r i e s o f f i e l d o p e r a t o r s . F o r example, i n t h e s c a l a r f i e l d c a s e one has ( B j o r k e n and D r e l l , 1965) A ( k ) = T d 4 k e ~ l k ' X < 0 | T [ O ( x ) * ( 0 ) ] | 0 > (4.3.20) = i / ( k 2 - m2 + i e ) , where |0> i s t h e vacuum s t a t e , T t h e t i m e o r d e r i n g o p e r a t o r , and €>0 a p r e s c r i p t i o n d e s c r i b i n g how t h e i n t e g r a l i s t o be c a r r i e d out t o a v o i d p o l e s i n t h e complex t o r x Q - p l a n e , w h i c h a r i s e s from e ( t ) a ( *' t > 0 (4.3.21) 1 0, t<0 = (2wi) 1 Td« e 1 < x t (« - i € ) 1 , and w h i c h g i v e s meaning t o t h e p r o p a g a t o r as i t i s i n t e g r a t e d a r o u n d i t s p o l e s i n t h e complex k Q - p l a n e . A l t h o u g h t h e p r o c e d u r e (4.3.20) g i v e s t h e p r o p a g a t o r f o r p r o p e r l y n o r m a l i z e d f i e l d s ( w h i c h d e s t r o y t h e p a r t i c l e and c r e a t e t h e a n t i - p a r t i c l e ) t h e M a j o r a n a f i e l d i s u n i q u e among s p i n o r f i e l d s — i t b o t h c r e a t e s and d e s t r o y s i t s p a r t i c l e . As a r e s u l t an e x t r a f a c t o r o f 1/J"2 must be added t o t h e f i e l d ' s momentum e x p a n s i o n as e x p l a i n e d i n ( 3 . 3 . 1 8 ) , so t h a t (4.3.20) i m p l i e s t h a t , i n s u c h a c a s e , t h e M a j o r a n a f e r m i o n p r o p a g a t o r i s A ( k ) = 2 - 1 / (k-m) , (4.3.22) t h e l a s t o f t h e p r o p a g a t o r s t h a t w i l l be n e e ded. 70 IV.4 VERTICES W h i l e t h e p a r t i c l e p r o p a g a t o r s depend o n l y on t h e p a r t i c l e t y p e ( s c a l a r , s p i n o r , e t c . ) and a r e t h e same f o r a l l t h e o r i e s t h e r u l e s f o r t h e v e r t i c e s depend t o t a l l y on t h e s p e c i f i c i n t e r a c t i o n s p r o s c r i b e d by t h e t h e o r y . Indeed, i n a p r a c t i c a l s e n s e , t h e c o l l e c t i o n o f v e r t i c e s is t h e t h e o r y . T h e r e b e i n g , i t seems, no s t a n d a r d method o f e s t a b l i s h i n g t h e v e r t i c e s t h e a p p r o a c h t o be employed h e r e w i l l c o n s t r u c t t h e t r a n s i t i o n a m p l i t u d e d i r e c t l y f r o m t h e i n t e r a c t i o n L a g r a n g i a n , f r o m w h i c h t h e v e r t i c e s w i l l be e x t r a c t e d . F o r an i n i t i a l s t a t e |i> and f i n a l s t a t e |f> t h e c o m p l e t e s c a t t e r i n g a m p l i t u d e i n terms o f t h e i n t e r a c t i o n L a g r a n g i a n C^ . i s ( I t z y k s o n and Z u b e r , 1980, Chap. 4) S i - f B < f ' s l i > = i f * * * < f l ^ I l i > 5 A C 2 " ) 4 S 4 ( 2 P f - I P A ) (4.4.1) where S » 1 - i / d 4 x (4.4.2) ( i n t h e f i r s t a p p r o x i m a t i o n ) i s t h e " s c a t t e r i n g o p e r a t o r " , A t h e i n v a r i a n t a m p l i t u d e , = - C^ . i s t h e i n t e r a c t i o n H a m i l t o n i a n d e n s i t y , where |f> * |i> has been assumed, and where t h e energy-momentum c o n s e r v i n g S - f u n c t i o n has been s e p a r a t e d o u t . F o r example, t h e one v e r t e x o f s p i n o r QED ( F i g . 2) f o l l o w s from ( 2 . 1 . 6 ) : C j X e e t f ) = e *A<|> . (4.4.3) 71 The u n r e n o r m a l i z e d f i e l d s a r e ( t r a d i t i o n a l l y one r e p l a c e s t h e n o r m a l i z a t i o n c o n s t a n t s when t h e t r a n s i t i o n r a t e o r c r o s s -s e c t i o n i s c a l c u l a t e d ) * = .fd 3p e ~ l p ' X a (p) u (p) (4.4.4) r r ( f o r e l e c t r o n s o n l y ) and . , 3 . (oc) , -ik«x ik«x * . , , „ „ _. A = Td k e [e c . (k) + e c, ( k ) ] (4.4.5) P P (oc) (oc) f o r t h e e l e c t r o m a g n e t i c f i e l d , and f o r t h e i n i t i a l and f i n a l s t a t e s |i> = a *(P) |0> (4.4.6) r |f> = a* r (P' ) c* (K) |0> , r {a) w i t h t h e commutation r e l a t i o n s {a ( p ) , a * r ( p ' ) } = S , S 3 ( p - p ' ) (4.4.7) r r r r [c ( k ) , c * (k' )] = 8 , S3(Z-P ) , K<x) (. oc ; oeoc one f i n d s A ( 2 w ) 4 S 4 (P'+K-P) = i e ( 2 w ) 4 S 4 (P r+K-P) (4.4.8) x u , (P' ) V ^ u (P) € ( o c ) (K) . r' r p. The QED v e r t e x V i s A w i t h t h e e x t e r n a l s p i n o r s and v e c t o r s d e l e t e d : V(QED) = i e V . (4.4.9) P 7 2 T h i s i s t h e n e g a t i v e o f t h a t u s u a l l y l i s t e d ( i n , f o r example, B j o r k e n and D r e l l , 1964, App. B ) , b u t we have d e c r e e d e a r l i e r ( S e c . I I . 1 ) t h a t e>0, t h e o p p o s i t e o f t h e u s u a l c o n v e n t i o n . The r e s u l t (4.4.9) f o l l o w s f o r any c o m b i n a t i o n o f d i r e c t i o n s , b u t t h i s i s n o t a g e n e r a l f e a t u r e o f gauge t h e o r i e s : t h e d i r e c t i o n s do m a t t e r on o c c a s i o n . Our t a s k i s now s t r a i g h t f o r w a r d : we must p r o c e e d s y s t e m a t i c a l l y t h r o u g h t h e l e p t o n - b o s o n and b o s o n - b o s o n i n t e r a c t i o n s l i s t e d i n t h e f i r s t two s e c t i o n s o f t h i s c h a p t e r t o e x t r a c t t h e v e r t i c e s as was done above f o r quantum e l e c t r o d y n a m i c s . F i g . 3: The c h a r g e d l e p t o n - Z v e r t e x The c h a r g e d l e p t o n ( A ) - Z b o s o n i n t e r a c t i o n i s c o n t a i n e d i n ( 4 . 1 . 1 ) : C j O U Z ) = 2 a [ G s i n 2 e w A E Z A - j G ( c o s z e ^ s i n 2 e w ) A E + Z A ] (4.4.10) = 7 ( G s i n 2 e „ Iza - -G AZE A) . A W 2 — The Z b o s o n can be w r i t t e n e x a c t l y as was t h e p h o t o n i n (4.4.5) and e x a c t l y as above one f i n d s t h e v e r t e x t o be ( F i g . 3) ) 73 F i g . 4: C h a r g e d l e p t o n - n e u t r i n o - W v e r t i c e s V(AAZ) = i G * ( s i n 2 e „ - |z ) (4.4.11) p W 2 — f o r a l l d i r e c t i o n s . The c h a r g e d l e p t o n ( A ) - M a j o r a n a n e u t r i n o ( x ) - W b o s o n i n t e r a c t i o n i s d e s c r i b e d by t h e f i r s t t erm o f ( 4 . 1 . 1 ) : C T(AxW) = 2. ngU ( x . Z W A + I I W x.) . (4.4.12) 1 i i i i ) l l + + + — l U n f o r u n a t e l y , t h i s i s somewhat c o m p l i c a t e d by t h e phase a m b i g u i t y i n ( 3 . 3 . 1 8 ) , as I K = ±1. F o r c a s e s ( a ) - ( d ) i n F i g . 4 one f i n d s V(AxW) = i g U . n V Z , (4.4.13) l A p. -w i t h t h e same, t i m e s IK » f o r c a s e s ( e ) - ( h ) i n F i g . 4. In j i d e c a y d i a g r a m s (a) and (d) o f F i g . 4 a r e needed, w h i c h have no TK f a c t o r , w h i l e f o r p d e c a y d i a g r a m s (e) and (h) a r e needed, 2 w i t h two s u c h f a c t o r s , b u t w h i c h r e s u l t s i n (t\.) =+1. Thus t h e 74 f a c t o r c a u s e s no p a r t i c u l a r d i f f i c u l t y . The l a s t i n t e r a c t i o n d e s c r i b e d i n (4.1.1) o r (4.1.2) i s t h e x-Z i n t e r a c t i o n , b u t as i t w i l l n ot be needed i n what f o l l o w s i t w i l l n o t be c o n s i d e r e d f u r t h e r . The l e p t o n - H i g g s b o s o n i n t e r a c t i o n s were g i v e n i n (4.1.3) t o ( 4 . 1 . 6 ) . The i n t e r a c t i o n s w i t h t h e n e u t r a l H i g g s p a r t i c l e a r e n o t o f m a j o r i m p o r t a n c e h e r e and so w i l l n o t be p u r s u e d b e y o n d ( 4 . 1 . 7 ) . However, t h e i n t e r a c t i o n s w i t h t h e c h a r g e d H i g g s b o s o n s -- t h e u n p h y s i c a l S + , t h e p h y s i c a l B + and t h e d o u b l y c h a r g e d H + + — a r e c r u c i a l f o r t h e t h e o r y . Here t h e s e q u e n c e o f e v e n t s makes a d i f f e r e n c e : t h e v e r t i c e s f o r an i n i t i a l e l e c t r o n and f i n a l n e u t r i n o a r e not t h e same as t h o s e f o r as i n i t i a l n e u t r i n o and f i n a l e l e c t r o n . F i g . 5: C h a r g e d l e p t o n - n e u t r i n o - c h a r g e d H i g g s v e r t i c e s F o r t h e u n p h y s i c a l H i g g s p a r t i c l e we u s e ( 4 . 1 . 4 ) : 75 C j U x S ) = - ( g / M „ ) 2 i j l U i i i [ x i ( M A Z + - m i Z _ ) S + Jl (4.4.14) + Jl(M Z -m.Z )S x.] , a - i + - i where, a g a i n , M„ and m. a r e t h e c h a r g e d l e p t o n and n e u t r i n o Jl l masses, r e s p e c t i v e l y . The f i r s t t e r m o f (4.4.14) g i v e s d i a g r a m s (a) and (b) o f F i g . 5, and one has V ( A % x S ) = - i ( g / M w ) U . n ( M n Z -m.Z ) (4.4.15) W lft, Jl + l -( w i t h t h e f a c t o r n. i f F i g . 5 ( b ) i s b e i n g u s e d ) , and s i m i l a r l y , l t h e s e c o n d t e r m o f (4.4.14) g i v e s d i a g r a m s ( c ) and (d) o f F i g . 5, w i t h V(x-»A~S) = -i(g/M w)U. n ( M Z -m.Z) (4.4.16) ( w i t h n. a g a i n , f o r F i g . 5 ( d ) ) , where t h e momentum e x p a n s i o n l <D_ = J/d 3k ( e ~ l k ' X a _ ( k ) + e l k * X a * ( k ) ) , (4.4.17) w i t h <l>+=<l> , i s u s e d f o r a l l c h a r g e d s c a l a r f i e l d s . F o r t h e p h y s i c a l H i g g s b o s o n we use ( 4 . 1 . 5 ) : C T U x B ) = (g/M t I )£. U n [ x . ( M t a n e Z +m.coteZ )B Jl (4.4.18) i w i i i i I i + i - + + Jl(M t a n e Z +m.coteZ )B x. ] . Jl - l + - l F i g u r e 5 can be u s e d f o r B + as r e a d i l y as f o r S + . The f i r s t t e r m o f (4.4.18) g i v e s d i a g r a m s (a) and (b) o f F i g . 5 and t h e v e r t e x V ( jT*xB) = i ( g / M w ) U " i j i ( M a t a n e Z + + m . c o t e Z ) (4.4.19) ( w i t h a f a c t o r i\. f o r d i a g r a m ( b ) ) and t h e s e c o n d term o f l ( 4.4.18) g i v e s d i a g r a m s ( c ) and (d) and t h e v e r t e x V(x-»Jl~B) = i (g/M ) U. (M t a n e Z + m . c o t e Z ) (4.4.20) W l J l Jl - l + ( w i t h a f a c t o r TI. f o r d i a g r a m ( d ) ) . l We w i l l a l s o n eed t h e l e p t o n - d o u b l y c h a r g e d H i g g s v e r t i c e s , t h e i n t e r a c t i o n b e i n g g i v e n by ( 4 . 1 . 6 ) : 76 C (JLH ) = £ . .J"20. . (A°Z H H. + A.Z H A°) I ±± i j i j 1 - ++ j j + — 1 (4.4.21) H Jl 7 F i g . 6: The c h a r g e d l e p t o n - d o u b l y c h a r g e d H i g g s v e r t e x The v e r t e x i s ( F i g . 6) V U . J l .H ) = iJ - 20 . .S , i j ±± i j ± (4.4.22) where Z i s u s e d i f an H i s d e s t r o y e d ( o r H c r e a t e d ) and Z + — ++ -i s u s e d i f an H__ i s c r e a t e d ( o r H + + d e s t r o y e d ) as (4.4.21) i n d i c a t e s . T h i s c o m p l e t e s t h e l i s t o f r e l e v a n t l e p t o n v e r t i c e s — t h a t i s , t h o s e t h a t w i l l be needed i n t h e c a l c u l a t i o n o f muon d e c a y — and we must now l i s t t h e r a t h e r more c o m p l i c a t e d t r i p l e -b o s o n v e r t i c e s . The c o u p l i n g between t h e W+ gauge b o s o n s and t h e e l e c t r o m a g n e t i c f i e l d A i s g i v e n i n t h e f i r s t t erm o f ( 4 . 2 . 1 ) : (4.4.23) n ( w w A ) = l i e f ( A V - A V ) ( a w + - a w + ) I 2 — — p. v v p. - (AV-AV 1) ( a w~-a w~) + ( a A - a A )(W^WV-WVWU) . + + L t v v p \x v v \x - + - + The c a s e o f a W , f o r example, r a d i a t i n g a p h o t o n would be d e s c r i b e d by A* 1 = Jd3k e i k * X c*(k) e V ) = fd3p e _ i p " X a ( P ) E ^ p ) (4.4.24a) (4.4.24b) 77 = W J = . f d V e i p ' * X a V ) E^(p') , where € and E a r e p o l a r i z a t i o n v e c t o r s , and s t a t e s ( 4 .4.24c) |i> = a (P) |0> |f> = a * ( P r ) c * ( K ) |0> , (4.4.25a) (4.4.25b) ( i g n o r i n g p o l a r i z a t i o n i n d i c e s ) and a l l t h i s , when p u t i n t o (4.4.1) r e s u l t s i n ( 2 i r ) 4 S4 (PF +K-P)A = - 2 - i e ( 2 T t ) 4 S4 (K+P R-P)e° C(K)E 0(P)E V(P' ) (4.4.26) * [(s^V-s^MP'-r, -P'H ) + (siXsv-svsiX)(p i. -P -n ) ( x p o c p L i v V v p - V a V o c t f L i R v v p p . + ( S Q S ^ - S ^ S ^ ) ( K T, — K TJ ) ] . p V p V p. ocv v ocp. F i g . 7: The W-W-A.Z v e r t e x When t h e p o l a r i z a t i o n v e c t o r s a r e d r o p p e d , t h e WWA v e r t e x i s o b t a i n e d : V(WWA) = - i e [ ( P ' + P )n + (K - P ' ) n o + (-P -K )n ] , (4.4.27) ot oc B V P 0 ocV V V OC0 where, as i n F i g . 7, t h e momenta a r e p o s i t i v e i n t h e d i r e c t i o n s i n d i c a t e d . T h i s r e s u l t i s i n agreement w i t h Cheng and L i (1984, App. B ) . The WWZ v e r t e x , d e s c r i b e d by t h e s e c o n d term o f ( 4 . 2 . 1 ) , i s v i r t u a l l y i d e n t i c a l ( F i g . 7 ) : V(WWZ) = i g c o s e w [ ( P ' + P ) n _ , + ( K Q - P r Q ) n „ + ) i p ] ( 4 . 4 . 2 8 ) W ot ot p V P P otV 8 V otp 78 F i g . 8: The c h a r g e d H i g g s - p h o t o n v e r t e x The c h a r g e d H i g g s - e l e c t r o m a g n e t i c c o u p l i n g i s g i v e n by (4.2.3) as C (AB) = - i e A ^ ( B a B -B a B ) (4.4.29) I - p + + p -f o r t h e p h y s i c a l H i g g s f i e l d , and C f A S ) = - i e A P ( S a S -S a S ) (4.4.30) I - p + + p -f o r t h e u n p h y s i c a l H i g g s f i e l d . W i t h t h e f i e l d B b e i n g g i v e n as i n (4.4.17) and t h e e l e c t r o m a g n e t i c f i e l d as i n (4.4.5) we have |i> = a*(P) |0> (4.4.31a) |f> = a*(P') c * ,(K) |0> , (4.4.31b) - ( oc) and one o b t a i n s ( 2 T T ) 4 S 4 (P r +K-P)A = i e ( 2 i r ) 4 S 4 (P F+K-P) . (K) (P r+P ), (4.4.32) (oc) p p so t h e v e r t e x , f o r t h e d i r e c t i o n s o f F i g . 8, i s V(AB) = i e ( P r + P ) , (4.4.33) and t h e same a p p l i e s t o t h e S f i e l d : V(AS) = i e ( P r + P ) . (4.4.34) 79 F i g . 9: The W-A-S v e r t e x F i n a l l y , f o r t h e W-A-S v e r t e x ( F i g . 9) d e s c r i b e d by (4.2.5) C].(WAS) = -eM w A^(W^S + + W^S_) , (4.4.35) we have, s i m p l y , V(WAS) = -ieMr,-n (4.4.36) W \xv r e g a r d l e s s o f momentum d i r e c t i o n . The v e r t i c e s c a l c u l a t e d h e r e t h a t a r e r e l e v a n t f o r muon d e c a y a r e c o l l e c t e d and d i s p l a y e d i n App. G. Now, f i n a l l y , t h e t a s k o f e x p l o r i n g t h e i m p l i c a t i o n s o f o u r gauge t h e o r y f o r muon de c a y can b e g i n . 80 IV.5 CHANGES TO WEINBERG-SALAM MODEL AND THE HADRONIC SECTOR I t i s i m p o r t a n t t o know t o what e x t e n t t h e m o d i f i c a t i o n s t o t h e o r i g i n a l W e i n b e r g - S a l a m t h e o r y i n t r o d u c e d above change t h e t h e o r y ' s p r e d i c t i o n s . W i t h t h e Feynman r u l e s o f t h e p r e v i o u s two s e c t i o n s we a r e i n a p o s i t i o n t o answer t h i s q u e s t i o n , i n so f a r as we a r e a b l e t o e s t i m a t e t h e many unknown p a r a m e t e r s i n t h e changes b r o u g h t about by t h e i s o - v e c t o r H i g g s f i e l d . F i g . 10: Feynman d i a g r a m s f o r 2e -» 2p. As an example we w i l l c o n s i d e r t h e r e a c t i o n 2e -» 2p w h i c h can happen i n a number o f ways ( F i g . 1 0 ) . D i a g r a m (a) i s p o s s i b l e i n t h e W e i n b e r g - S a l a m t h e o r y , w h i l e (b) and ( c ) a r e o n l y made p o s s i b l e by t h e changes t o t h e t h e o r y made e a r l i e r . In F i g , 10, <J>Q and H + s t a n d f o r n e u t r a l and s i n g l y c h a r g e d 81 H i g g s p a r t i c l e s , w h e t h e r p h y s i c a l o r u n p h y s i c a l . The b a s i c a m p l i t u d e f r o m QED f o r F i g . 10(a) w i t h t h e i n t e r m e d i a t e V i s , i g n o r i n g p h a s e s , A(2e-»*-»2p) - ( e 2 / q 2 ) (v * u ) (u * % ) . (4.5.1) e v e p. p. From ( 4 . 4 . 1 1 ) , t h e Z's c o n t r i b u t i o n i s A(2e-Z-2p) ~ [ G 2 / ( q 2 - M 2 ) ) [v V ( s i n 2 e w - j Z )u ] (4.5.2) Z e v W 2 - e x [u V V ( s i n 2 e U J - ;-Z )v ] , p W 2 - p w h i c h i s c o m p a r a b l e t o (4.5.1) o n l y a t v e r y h i g h e n e r g i e s , s u c h 2 2 as q ~ M , o r h i g h e r . The n e u t r a l H i g g s c o n t r i b u t i o n i s , f rom ( 4 . 1 . 7 ) , A(2e-><|> -*2p) ~ [0 0 / ( q 2 - M 2 ) ] v Z u u Z v (4.5.3) 0 e p <j> e - e p + p and as 0 0 = g 2 m M /M2 ~ 1 0 ~ 8 g 2 , (4.5.4) e p e p W t h i s i s s e e n t o be q u i t e n e g l i g i b l e a t a l l e n e r g i e s . The f i r s t new term a r i s e s f r o m F i g . 1 0 ( b ) , w h i c h f o r t h e W i s , f r o m ( 4 . 4 . 1 2 ) , A(2e-»Wx-»2p) - 2. ./d 4k g 2U..U.. * 2U..U._ v I 1 l l j v i j i i i 2 j l j 2 e + pi I + v p 1p.-k-m.' l l x( \ l y V P \ u Z^V f 1 | Z y u (4.5.5) l V - M 2 M k ' 2 - M 2 l M + p l p . - k ' _ f f l . l + CT e W W J J w h i c h i s t o t a l l y n e g l i g i b l e b e c a u s e , n o t o n l y a r e t h e r e two W p r o p a g a t o r s , a t l e a s t two o f (U.,U.„U .,U . _) w o u l d be e x p e c t e d i i i 2 j l j 2 t o be s m a l l , b e i n g o f f - d i a g o n a l . F o r t h e H i g g s p a r t i c l e s one has g -» g(M /M ) , where M i s a l e p t o n mass, b u t i n g e n e r a l t h e Jl W Jl a m p l i t u d e s would be e x p e c t e d t o be c o m p a r a b l e t o t h a t f o r t h e W p a r t i c l e b e c a u s e u n p h y s i c a l H i g g s p a r t i c l e s a r e needed t o keep t h e o v e r a l l r e s u l t gauge i n v a r i a n t . A f t e r a l l , F i g . 10(b) i s a 82 f o u r t h o r d e r p r o c e s s , w h i l e (a) and ( c ) a r e o f s e c o n d o r d e r . F o r F i g . 1 0 ( c ) we have, f r o m ( 4 . 4 . 2 1 ) , A(2e-H + +-2p.) ~ [ ( B 1 2 ) 2 / ( k 2 - M 2 ) ] v £ v v E v (4.5.6) ±± 1 2 H e + p e - p and, i n a s p e c u l a t i v e and p r o b a b l y g e n e r o u s a p p r o x i m a t i o n t o be made and d i s c u s s e d l a t e r , we i m a g i n e t h e Yukawa c o u p l i n g c o n s t a n t s t o be c o m p a r a b l e , so 0 .. ~ B. » g(M /M ), i n w h i c h i j l A W 2 — Q c a s e t h i s i s o f o r d e r (m M /M ) " 1 0 t i m e s t h e a m p l i t u d e f o r e p. W t h e Z p a r t i c l e , a s s u m i n g M ~ M . H W, Z Thus i n t h i s example, and o t h e r s , t h e new e f f e c t s i n t r o d u c e d • a l o n g w i t h t h e v e c t o r H i g g s f i e l d s a r e c o m p a r a b l e t o t h e e f f e c t s i n d u c e d by t h e o r i g i n a l H i g g s p a r t i c l e s , w h i c h i s t o s a y , v e r y s m a l l i n d e e d . The h a d r o n i c s e c t o r c a n n o t be t o t a l l y i g n o r e d h e r e as i t might b e a r on weak i n t e r a c t i o n p r o c e s s e s t h a t a r e r e l a t e d t o t h o s e o f t h i s t h e s i s , p a r t i c u l a r l y t h e p. -* 3e d e c a y o f Chap. VI. I t s p r i m a r y c o n n e c t i o n t o t h e W e i n b e r g - S a l a m model i s t h r o u g h t h e SU(2) s p i n o r o r d o u b l e t <P = ( " ) , ~ E (,), and q d L - d s i n g l e t s u^ = £ + u , d^, where t h e u and d q u a r k s have e l e c t r i c 2 1 c h a r g e s q =-e and q =--e, a n a l o g o u s t o t h e e l e c t r o n f a m i l y ' s u 3 d 3 E ( V ) S ( V ) , , e . T h e i r weak h y p e r c h a r g e s , a c c o r d i n g t o t h e r u l e - e e L R ( 2 . 2 . 1 1 ) , a r e Y ( * ) = j , Y ( u D ) = | and Y ( d ) =-|. The q u a r k s ' i n t e r a c t i o n w i t h t h e H i g g s s e c t o r i s t h r o u g h t h e Yukawa c o u p l i n g C(Yuk-q) = - B J ( * E $d + dE $ + * ) (4.5.7) d q + q - B (*E $' u + uE ** ) , u + - q 83 w i t h $ and $ r = i o - 2 $ as i n S e c . I I . 2 , I I I . 2 . B e c a u s e o f t h e q u a r k s ' f r a c t i o n a l h y p e r c h a r g e t h e r e can be no Yukawa c o u p l i n g w i t h t h e v e c t o r H i g g s f i e l d H. Thus t h e o n l y q u a r k - H i g g s c o u p l i n g f o l l o w s f r o m ( 4 . 5 . 7 ) , w h i c h , expanded i n terms o f t h e b a s i c f i e l d s , i s C(Yuk-q) = - B J ( v / J " 2 ) d d - 0 (v/4"2)uu (4.5.8) d u - 0 J ( u E <t> d + dE 4> u + dE <|> d + dE 4>*d) - * - - -- 0 ( u E <t>„u + uE *„u - dE $ u - uE * d) . u + ° - 0 + - - + Thus t h e q u a r k masses a r e m J = 0_, v/J"2 , m = 0 v/J"2 , (4.5.9) d d u u so t h a t 0 - g (m / J"2Mw) (4.5.10) u, d u, d W and t h e r e s t o f (4.5.8) i s u s e d f o r t h e q u a r k c o u p l i n g s t o t h e p h y s i c a l and u n p h y s i c a l H i g g s f i e l d s , and, t h r o u g h them, t o t h e l e p t o n s e c t o r . Thus t h e q u a r k s a r e c o u p l e d t o a l l o f t h e H i g g s f i e l d s except t h e d o u b l y c h a r g e d f i e l d H + + . These i n t e r a c t i o n s w i l l be u t i l i z e d l a t e r (Chap. VI) when t h e q u e s t i o n o f p r o c e s s e s s i m i l a r t o u -» 3e i s c o n s i d e r e d . 84 V. THE DECAY u -> e V V.1 RADIATIVE DECAYS IN GENERAL A c o m p l e t e l i s t o f t h e Feynman d i a g r a m s a l l o w e d by t h e gauge t h e o r y i n v e s t i g a t e d h e r e w i t h one muon i n t h e i n i t i a l s t a t e and an e l e c t r o n and a p h o t o n i n t h e f i n a l s t a t e would be r a t h e r l e n g t h y . The d i a g r a m s w o u l d be o f d i v e r s e m a g n i t u d e s ( b e c a u s e , f o r example, M >> m ), and, i n d e e d , many wou l d p. e d i v e r g e , a c h a r a c t e r i s t i c o f p e r t u r b a t i o n methods i n quantum f i e l d t h e o r y . A l l d i v e r g e n t d i a g r a m s must, however, c a n c e l b e c a u s e o f t h e gauge t h e o r y ' s g u a r a n t e e d r e n o r m a l i z a b i l i t y ( I t y z k s o n and Z u b e r , 1980, Chap. 8; Cheng and L i , 1984, Chap. 6 ) . In t h e p a r t i c u l a r c a s e o f a r a d i a t i v e t r a n s i t i o n i t i s f o r t u n a t e l y p o s s i b l e t o e x t r a c t t h o s e f i n i t e t e r m s , a l l o f t h e same o r d e r o f m a g n i t u d e , r a t h e r d i r e c t l y , w i t h o u t b o t h e r i n g a t a l l w i t h t h e m u l t i t u d e o f terms t h a t become u l t i m a t e l y i r r e l e v a n t ( N i e v e s , 1982; Cheng and L i , 1984, Sec. 1 3 . 3 ) . The e l e c t r o m a g n e t i c t r a n s i t i o n a m p l i t u d e f o r p -• ey ( F i g . 11) must have t h e f o r m A ~ .(q) u ( p r ) J u (p) (5.1.1) (. oc) e v p where e i s t h e p o l a r i z a t i o n v e c t o r o f t h e p h o t o n o f 4-momentum 2 q ( q =0 f o r r e a l p h o t o n s ) and p o l a r i z a t i o n oc ( = 1,2), and where t h e o p e r a t o r o r m u l t i v e c t o r J (App. B) i s unknown. As t h e e l c t r o m g n e t i c i n t e r a c t i n f o r an e l e c t r i c c u r r e t j i s j «A t h e c u r r e n t has t h e f o r m 85 M F i g . 11: The r a d i a t i v e d e c a y p. -» etf j - u J u , (5.1.2) v e v u and c u r r e n t c o n s e r v a t i o n 3 - j = 3 j " = 0 , (5.1.3) v o r , i n momentum s p a c e q - j = 0 , (5.1.4) r e q u i r e s t h a t J have t h e form v J = q ( a + a'V 5) + q A * (b + b r V 5 ) (5.1.5) V V 5 V 5 2 s i n c e a r e a l p h o t o n s a t i s f i e s q = 0 and € . q = e v q = 0 = _-(€q + q€) , (5.1.6) v * where t h e v e c t o r p r o d u c t s d e s c r i b e d i n App. B have been employed. The r e s u l t (5.1.6) makes t h e f i r s t term o f (5.1.5) u n n e c e s s a r y . Thus t h e t r a n s i t i o n a m p l i t u d e has t h e form A ~ £V (q) u ( p r ) q A V ( a + b * 5 ) u (p) . (5.1.7) In what f o l l o w s t h e e l e c t r o n ' s mass w i l l be i g n o r e d b e c a u s e M >>m . n e The b a s i s s p i n o r s s a t i s f y p u (p) = M u (p) (5.1.8a) p' u (p' ) = m u ( p r ) , (5.1.8b) e e e 86 and as v s ( q A * ) = U A * ) x s (5.1.9) we need o n l y l o o k f o r t h e g e n e r a l f o r m , w i t h o r w i t h o u t a V s> C V u (p r ) q/\V u (p) = u q A € u = u q€u . (5.1.10) e v p e p e p T h i s has a number o f e q u i v a l e n t f o r m s , u s i n g q = p-p' and (5.1.8) u qeu = u peu - m u €u (5.1.11) e p e p e e p = -u equ = u €p' u - M u eu , e p e p p e p and as p€ = -€p + 2p«€ (5.1.12) we have u q€u = (2p«€) u u + ••• (5.1.13) e p e p = (2p r •€) u u + • • • , e p. where t h e d r o p p e d t e r m s , s u c h as t h e u €u terms o f ( 5 . 1 . 1 1 ) , e p have e i t h e r t h e wrong s t r u c t u r e f o r a r a d i a t i v e t r a n s i t i o n o r a r e n e g l i g i b l e ( i f a f a c t o r m a p p e a r s ) . e To summarize, a l l terms i n t h e t r a n s i t i o n a m p l i t u d e s t o f o l l o w e x c e p t t h o s e t h a t c o n t a i n one o f t h e e q u i v a l e n t forms u q€(a+b"rf)u ~ u p € ( a + b V c ) u (5.1.14) e 5 p e 5 p ~ u €p' ( a + b V c ) u - (2p«€)u ( a + b v j u - (2p r «e)u ( a + b * c ) u e p e 5 p . e 5 p a r e t o be i g n o r e d as h a v i n g t h e wrong s t r u c t u r e f o r an e l e c t r o m a g n e t i c t r a n s i t i o n . T h i s s h o r t l i s t w i l l be a c o n s i d e r a b l e a i d i n s i m p l i f y i n g t h e complex d i a g r a m s t o f o l l o w . I t i s e s p e c i a l l y g r a t i f y i n g t o d i s c o v e r t h a t no d i v e r g e n t t e r m has one o f t h e wanted f o r m s . 87 V. 2 THE DIAGRAMS AND AMPLITUDES FOR THE DECAY u -> e V I t i s a p p r o p r i a t e now t o c o n s i d e r and l i s t t h e f i v e Feynman d i a g r a m s t h a t a r e r e s p o n s i b l e f o r t h e r a d i a t i v e d e c a y s o f muons as p e r m i t t e d by t h e S U ( 2 ) x U ( l ) gauge t h e o r y o u t l i n e d t o t h i s p o i n t . The p r o p a g a t o r s and v e r t i c e s o f t h e p r e v i o u s c h a p t e r p r o v i d e f o r an unambiguous t r a n s l a t i o n f r o m t h e s e d i a g r a m s t o t h e i r m a t h e m a t i c a l r e p r e s e n t a t i o n . The W p r o p a g a t o r i s r a t h e r complex and i t w i l l h e l p t o f i r s t r e d e f i n e i t , g i v e n i n ( 4 . 3 . 1 9 ) , by f a c t o r i n g o u t t h e i : - i * (k) = - i { „ / ( k 2 - M f ) - k k / [ M 2 ( k 2 - M 2 ) ] pv pv W p. v W W + k k / [ M 2 ( k 2 - C M 2 ) ] } p v n W = - i [ a . ( k ) + <v (k) + A 3 ( k ) ] (5.2.1) 1 2 J pv The a m p l i t u d e f o r F i g . 12 ( o r D i a g . W) i s , f r o m t h e Feynman r u l e s o f Chap. IV, A(W) = £. r d " k « ( P r . i g V Z )( 1 ' Z i \ ( U i g y Z )u ( P ) X J ( 2 W ) 4 6 1 1 " " \ ( p - k ) - m j 1 2 v ~ » l x ( - i ) ^ P ( k - q ) ( - i ) A ° V ( k ) ( - i e € X P ) , (5.2.2) Xap where r « <-q-k) i\ + [ k - ( - k + q ) ] it + [ ( - k + q ) - ( - q ) ] -n (5.2.3) Xap p Xcr X ap a Xp a r i s e s f r o m t h e W-W-A v e r t e x , Eq. ( 4 . 4 . 2 7 ) , where i = l = e l e c t r o n , i=2= muon have been used, and where t h e o t h e r v e r t i c e s and p r o p a g a t o r s a r e as g i v e n i n Sec. IV.3 and IV.4. The momentum and o t h e r symbols employed i n A(W) a r e g i v e n i n 88 ^ 'P X, 'P-l< P' F i g . 12: Feynman d i a g r a m W i n t h e dec a y p. •+ etf t h e a p p r o p r i a t e p l a c e i n F i g . 12. The i n t e g r a t i o n i s o v e r t h e W l o o p : as t h e 4-momentum k i s u n d e t e r m i n e d by energy-momentum c o n s e r v a t i o n ( w h i c h g u a r a n t e e s o n l y p = p'+q) a l l v a l u e s o f t h e W momentum make a c o n t r i b u t i o n ( B j o r k e n and D r e l l , 1964, App. B) . The a m p l i t u d e f o r F i g . 13 ( D i a g . S) i s (2*r)' ACS) = L f J _ J L _ u (pr )/z±gJJ (m Z -m.Z )W 1 ^2 i \ i j ( 2 w ) 4 e i M w 1 1 e " 1 + y ( ( p - k ) - m J w 1 f = ±gU (M Z +-m Z_)\ u ( p ) / i W ^ J 1 I " / \ ( k - q ) 2 - £ M 2 j / l k 2 - C M 2 j / W x i e [k + ( k - q ) ] ^ e , (5.2.4) where t h e v e r t i c e s ( 4 . 4 . 1 5 ) , (4.4.16) and (4.4.34) have been 2 2 u s e d , as has t h e mass M = ?M o f (2.6.10) f o r t h e s c a l a r f i e l d S W p r o p a g a t o r , g i v e n i n ( 4 . 3 . 9 ) . The a m p l i t u d e f o r F i g . 14 ( D i a g . WS) i s 8 9 / k / / k-q F i g . 13: Feynman d i a g r a m S i n t h e d e c a y p -» ey F i g . 14: Feynman d i a g r a m WS i n t h e d e c a y p -* ey A(WS) = 2. r d k u (P r )/-i<tU. , (m £ -m.Z )\/ 1 ^2 i r d k u ( P' )/• ... „ _ 1 J ( 2 W ) 4 6 U M 1 1 6 - 1 + / l ( p - k ) - m \ ( k - q ) 2 - C M 2 / x (u i g y z )u (P)I i y - i ) * " " ( k ) i 2 v - p. I . 2 X x ( - i e M ^ ^ ) € , (5.2.5) where t h e one new v e r t e x , Eq. ( 4 . 4 . 3 5 ) , has been employed. The a m p l i t u d e f o r F i g . 15 ( D i a g . SW) i s , s i m i l a r l y , A(SW) = 7. f d 4 k u ( p ' ) ( U . i g y Z )/ 1 ^2 i ^ (5.2.6) i j T T ^ T e i l ( 2 T T ) f S_>/ 1  >M \ ( p - k ) - m . / i J ^ U . (M Z -m.Z )\u ( P ) / i ^ ( - i ) A C T t I ( k - q ) x ( " i e M w \ x ) e ' F i n a l l y , t h e d e c a y m e d i a t e d by t h e p h y s i c a l H i g g s b o s o n (B) i s , f r o m F i g . 16 ( D i a g . B ) , A(B) = 2. r <i4k u (p' )/ i£U. , (m Z tane+m.Z c o t e ) ) / 1 ^2 i \ 1 J ( 2 W ) 4 6 U 1 1 6 - 1 + M(p-k)-m. ; W 1 x / lg-U. „(M Z tane+m. Z c o t e ) ) u (p) / i \ / i \ U w 1 2 M + 1 - ) » ((k-q)2-M;J(k2-M;i x i e f k + ( k - q ) ] € X , (5.2.7) where t h e v e r t i c e s o f ( 4 . 4 . 1 9 ) , (4.4.20) and (4.4.33) have been u s e d . 91 V. 3 EVALUATION OF THE AMPLITUDES FOR u -» e V The f i v e i n t e g r a l s o f t h e p r e v i o u s s e c t i o n , i n t h e i r p r e s e n t form, pose a r a t h e r f o r m i d a b l e m a t h e m a t i c a l p r o b l e m . I t i s f o r t u n a t e l y p o s s i b l e t o c o n s i d e r a b l y r e d u c e t h e i r c o m p l e x i t y by t h e a p p r o p r i a t e use o f a p p r o x i m a t i o n s , s u c h as m.,m << M << M .M . and by k e e p i n g o n l y t h e l o w e s t p o s s i b l e 1 e u W B o r d e r o f t h e n e u t r i n o masses. The n e u t r i n o p r o p a g a t o r (p-m.) 1 i s a p p r o x i m a t e d by l l /(p-m) = (p+m)/(p 2-m 2) = [(p+m)/p 2] (1 - m 2 / p 2 ) _ 1 (5.3.1) 2 2 2 4 3 4 = P/P + m/P + m P/P + m /p + • • • and o n l y t h e f i r s t n o n - v a n i s h i n g term w i l l be k e p t . From A(W) ( E q . ( 5 . 2 . 2 ) ) we have a f a c t o r ( u s i n g p-k - f p) Z . U U . - Y Z ( p / p 2 + m./p 2 + m 2p/p 4 + m 3/p 4 + • • • ) * Z . (5.3.2) 1 X1 l 2 p. - 1 1 1 v -The f i r s t t e r m v a n i s h e s b e c a u s e U i s an o r t h o g o n a l m a t r i x : 1± U u U i 2 = S 1 2 = 0 ; (5.3.3) th e s e c o n d v a n i s h e s b e c a u s e Z m.V Z = m.V Z Z = 0 . (5.3.4) - 1 i/ - i v + -The n e x t t e r m — t h e f i r s t n o n - v a n i s h i n g one — r e d u c e s t h e f a c t o r (5.3.2) t o , a p p r o x i m a t e l y , 2.U. . U . V z * PV / P 4 . (5.3.5) l l l i 2 l + u v S i m i l a r l y , t h e f a c t o r S.U..U._(m Z -m.E)[(p+m.)/p2 + m 2p/p 4 + ...](M Z^-m.Z ) ( 5 . 3 . 6 ) l x l i 2 e - I + x x U + i -i n A (S) from (5.2.4) becomes ( s e t t i n g m = 0 : t h e r e w o u l d be no e m m. term anyway) e x 92 _.U U m 2E (p-M ) / p 2 ; (5.3.7) t h e f a c t o r _.U U. (m E -m.E ) [ ( p + m . ) / p 2 + _ 2 p / p 4 + •• • ] £ _ * (5.3.8) l i i i ^ J e — i + l I + v i n A(WS) from (5.2.5) becomes -I.U..U m 2 £ V / p 2 ; (5.3.9) l i l i 2 l + v t h e f a c t o r _.U. U . 0 E V [(p+m.)/p 2 + m 2p/p 4 + ...](M E -m.E ) (5.3.10) l i l i _ + p l l p + I -i n A(SW) from (5.2.6) becomes _.U U. 0m 2E V (-1/P 2 + M p / p 4 ) ; (5.3.11) i i l i _ i + p p and, f i n a l l y , t h e f a c t o r _.U.,U._(m E t a n e + m.E c o t e ) [ ( p + m . ) / p 2 + m 2p/p 4 + •••] l 11 i<2 e — i + l I x(M E t a n e + m.E c o t e ) (5.3.12) p + I -i n A(B) from (5.2.7) becomes _.U. U. 0m 2E (M / p 2 + p c o t 2 e / p 2 ) . (5.3.13) I i l i _ l + p The a m p l i t u d e s a r e now A(W) = - i 6 _ g l e _ j _ . U U . r n 2 f d 4 ku E V (p-k) V u .4 2 l i l i 2 l e + pi 4 v p (2w) J ( p - k ) x _ M P ( k - q ) _ v < x ( k ) e x r (5.3.14) X a p - 6 2 i r U. H. n 2 .4 - _ (p-k-M ) A(S) = +i g e r _ . _ _ _ _ _ _ . - . fd ku E p u (2w) M w J (p-k) x 2k •€ (5.3.15) ( k 2 - C M 2 ) [ ( k - q ) 2 - S M 2 ] A(WS) = + i 6 g 2 e ^_.U. U. om 2 f d 4 ku E V u ( 2 i r ) _ J2. . , m  4 2 i i l i 2 i e + v p _ V q ( k ) € (5.3.16) (p-k) 2[(k-q) 2-eM 2] w 93 A(SW) = - i g e r l . U U . m rd ku E V -1 u ( P } u 4" i l 12 i e + v h 7-T + H ,4 u J 1 P-k (p-k) ' ( 2 * ) ' J l ( p - k ) (P-k) x A V < T ( k - g ) e (5.3.17) <k 2 -tM;) -A(B) = + i 6 g 2 e r?!. U i l U i 2 m2 r d 4 k u E / M p (p-k) c o t 2 e\ u / o \ 4 1 »«2 1 e + W i \ 2 + /• ^ 2 M (2ir) M J HP-It) (p-k) ' w x 2 k-e . (5.3.18) ( k 2 - M 2 ) [ ( k - q ) 2 - M 2 ] F u r t h e r r e d u c t i o n o f t h e i n t e g r a n d s , and t h e a c t u a l i n t e g r a t i o n s , a r e much t o o l e n g t h y t o d i s p l a y h e r e , and so have been r e l e g a t e d t o App. E. W i t h t h e n o t a t i o n s A = / J. g 2 e U..U..m2| ( i r r l M \ (u E q€u ) , ( 5 . 3 . 19) I 1 2 ( 2 w ) 4 1 1 1 2 ^U; »I E + ^ f(£) = ( I n € ) / < € - l ) (5.3.20a) g(C) = (£ i n O / U - l ) 2 - 1 / ( C - D , (5.3.20b) f o l l o w i n g Cheng and L i (1984, S e c . 13.3), t h e f i n a l r e s u l t s f o r th e a m p l i t u d e s a r e A(W) = A [ l / 4 - f ( C ) / 1 2 + g(£)/2] (5.3.21a) A(S) = A ( - 5 / 1 2 £ ) (5.3.21b) A(WS) = \ A[5/6£ - f ( C ) + g ( C ) / 3 ] ( 5 . 3 . 2 1 c ) A(SW) = \ A[5/6C + 4 f ( € ) / 3 - 7 g ( £ ) / 3 ] (5.3.21d) A(B) = \ A ( M 2 / M 2 ) ( l + \ c o t 2 e ) (5.3.21e) 2 W B 6 w i t h t h e sum A ( u - e * ) = \ A [ j + ( M 2 / M 2 ) ( l + \ c o t 2 e ) ] , (5.3.22) w h i c h i s i n d e p e n d e n t o f t h e gauge p a r a m e t e r £ as i t must be. T h i s r e s u l t , w i t h t h e o m i s s i o n o f t h e l a s t t e r m w h i c h i s n o t p r e s e n t i n t h e c a s e o f p u r e l y D i r a c n e u t r i n o s , i s a l m o s t i d e n t i c a l t o Cheng and L i ' s r e s u l t (1984, S e c . 13.3) ( o u r s 94 c o n t a i n s t h e s p i n o r a m p l i t u d e t h a t Cheng and L i seem t o have i n a d v e r t a n t l y d r o p p e d ) . Thus, w i t h t h e e x c e p t i o n o f t h e terms t h a t depend on t h e p h y s i c a l , c h a r g e d H i g g s b o s o n B, t h e d e c a y p 2 -» eV i s i n d i f f e r e n t t o t h e n e u t r i n o t y p e -- f a c t o r s o f (g/J"2) i n t h e c o u p l i n g o f t h e D i r a c n e u t r i n o s t o t h e c h a r g e d l e p t o n s and t h e b o s o n s become a f a c t o r j i n t h e M a j o r a n a n e u t r i n o p r o p a g a t o r . But w i t h t h e c h a r g e d H i g g s p a r t i c l e s and t h e p o t e n t i a l f o r c o t e t o be l a r g e ( o r more a c c u r a t e l y , f o r m^cote t o be i n d e p e n d e n t o f n e u t r i n o mass) t h e r a d i a t i v e d e c a y r a t e o f t h e muon has t h e p o t e n t i a l o f b e c o m i n g s i g n i f i c a n t l y e n h a n c e d i n t h e c a s e o f t h e v e c t o r H i g g s f i e l d . T h i s i s d i s c u s s e d i n Chap. V I I . S t a n d a r d t e c h n i q u e s can now be employed t o c a l c u l a t e t h e t r a n s i t i o n r a t e f o r t h e d e c a y p. -* eV f r o m t h e i n v a r i a n t a m p l i t u d e A(p-*etf) o f t h e p r e v i o u s s e c t i o n ( B j o r k e n and D r e l l , 1964, App. B; I t y z k s o n and Z u b e r , 1980, App. A 3 ) . The t r a n s i t i o n r a t e i s r ( p - e * ) = rlA(u->eV)| 2 d3 q "e d V ( 2 T T ) 4 g 4 (p-q-p' ) (5.4.1) w h i c h has t h e meaning o f t h e p r o b a b i l i t y o f d e c a y p e r u n i t t i m e p e r p a r t i c l e , o r T = 1/T, where T i s t h e p a r t i c l e ' s ( t h e V.4 THE DECAY RATE FOR u •* e V 95 union's) mean l i f e t i m e f o r t h i s d e c a y mode. W i t h (5.4.2) 2M 1 1 .1 2 , r - T T T T 2 . / 2 M 1 1 R 1 . 2 , „2 . . , 1 , 2 . . ^ 4 - ( S i U i l U i 2 m i ) r - ? * ' [ » + ( M W / M B ) U + e c o t e>i 2 t r ) M ' 2 ( 2 t „ w we have, from (5.3.22) | A ( u - e V ) | 2 = C 2 |u ( p ' ) Z qeu ( p ) | 2 . (5.4.3) e + p One must now a v e r a g e o v e r t h e i n i t i a l muon s p i n and sum o v e r t h e f i n a l e l e c t r o n and p h o t o n s p i n s ( d e s i g n a t e d by 5" ). s p i n s One t h e r e b y o b t a i n s 2 . |A(p-»eV)| 2 = \ C 2 I . |u Z q€u | 2 (5.4.4a) s p i n s 2 s p i n s e + p and w i t h ( B j o r k e n and D r e l l , 1964, Chap. 7) 2 € ? . € " = (5.4.4b) ex (<x) (a) we have 2 . |u Z q€u | 2 (5.4.5) s p i n s e + p = 4 S [ Z q* _-<l+p/M )V qZ jd+p'/m )] + ix p v — * e = 2 (p.q) (p' -q) / (m M ) , e p w h i c h i s e v a l u a t e d i n App. E and where (App. B) 4S[ • • • ]=Tr[ • • • ] ( i n t h e n o t a t i o n o f t h e m a t r i x r e p r e s e n t a t i o n o f t h e D i r a c a l g e b r a ) . We now have T(p-»ey) = C 2 m e F d 3 q d 3 p f (p«q) (p'«q) S 4 ( p - q - p ' ) . (5.4.6) m M ( 2 f f ) 2 J 2 q Q p' e p 0 0 S i n c e q 2 = (P-P' ) 2 = 0 = M 2 - 2p.p' + m2 « M 2 - 2p-p' (5.4.7) p e p we have 1 2 p.q = p«(p-p' ) « - M (5.4.8a) 2 U P'-q = P' «(P-P') - i M 2 , (5.4.8b) so t h a t 96 .2 M ,3 ,3 , <1 r(p.-e*) = <T u f d g d p ' S ( p - q - p ' ) . (5.4.9) \ 4(2T T ) ~ J 2 q Q p^ As t h e i n t e g r a l has t h e v a l u e w (App. E) we h a v e r(p.->ey) = C 2 M3 / 16TT . (5.4.10) To compare w i t h t h e u s u a l r a t e f o r muon d e c a y ( C h e n g and L i , 1984, S e c . 13.3) r ( ^ e v v ) = G 2 M5 / 192ir 3 , (5.4.11) F p. where ( f r o m (2.2.20)) G_ / J2 = g 2 / (8M 2) , (5.4.12) we h a v e F(u->ev) _ 12 TT 2 C 2 (5.4.13) r(p->ew) ~ G 2 M2 F p. = (3<x/8 w ) ( S.U i lU. 2m 2/M 2) 2[ 2- + (M w/M B) 2(1 + J - c o t ' e ) ] 2 2 w h e r e « B e /4TT » 1/137 i s t h e f i n e s t r u c t u r e c o n s t a n t . One c a n g a t h e r f r o m (5.4.13) t h a t t h e p -» ey d e c a y mode a p p e a r s t o be s u p p r e s s e d ( a n d i s , i n t h e c a s e o f D i r a c n e u t r i n o s , w here 1/M = 0) by t h e ( p r o b a b l y ) e x t r e m e l y s m a l l B 4 —4 O f a c t o r (m./M ) - 10 , f o r m. ~ lOeV ( a s s u m i n g t h e D,, , U, , 1 W 1 1 1 1 2 v a l u e s a r e n o t n e g l i g i b l e — t h e y a r e q u i t e u n k nown). B u t i f 4 M ~ M t h i s i s n o t t h e c a s e , s i n c e ( c o t e ) i s q u i t e l a r g e . B W T h i s p o i n t , and i t s i m p l i c a t i o n s , w i l l be d i s c u s s e d i n d e t a i l i n Chap. V I I . 97 VI. THE DECAY u -• 3e VI. 1 THE DIAGRAMS FOR THE DECAY p 3e I t so happened t h a t t h e d i a g r a m s o f t h e p r e v i o u s c h a p t e r f o r t h e r a d i a t i v e d e c a y o f t h e muon were e a c h o f t h e same o r d e r o f m a g n i t u d e i n t h e f i r s t n o n - v a n i s h i n g a p p r o x i m a t i o n , a f a c t r e q u i r e d by t h e gauge i n v a r i a n c e o f t h e t h e o r y . A p o s s i b l e e x c e p t i o n was t h e d e c a y s e q u e n c e p. -• Bx -» Ve ( n o t dep e n d e n t on t h e a r b i t r a r y gauge p a r a m e t e r ) b e c a u s e o f t h e v a l u e s o f t h e t h e o r y ' s p a r a m e t e r s . In t h i s c h a p t e r t h e d e c a y mode p -» 3e w i l l be c o n s i d e r e d . T h i s d e c a y mode, n o t b e i n g a r a d i a t i v e t r a n s i t i o n , w i l l r e s u l t i n a much g r e a t e r v a r i e t y o f p o s s i b l e d e c a y s e q u e n c e s , w i t h a c o r r e s p o n d i n g l y g r e a t e r v a r i e t y i n t h e i r r e l a t i v e c o n t r i b u t i o n s t o t h e i n v a r i a n t a m p l i t u d e . The p r i m a r y t a s k , t h e r e f o r e , i s t o i s o l a t e t h o s e d i a g r a m s t h a t y i e l d t h e f i r s t n o n - v a n i s h i n g a p p r o x i m a t i o n t o t h e d e c a y mode p. -» 3e. When d i a g r a m s s i m i l a r t o t h o s e u s e d i n t h e p r e v i o u s c h a p t e r a r e c o n s i d e r e d one o b s e r v e s t h a t as t h e p h o t o n i s " o f f - s h e l l " ( o r v i r t u a l , o r has a 4-momentum q t h a t s a t i s f i e s q >0) i t s r o l e can a l s o be p l a y e d by t h e n e u t r a l Z p a r t i c l e , o r even by n e u t r a l H i g g s p a r t i c l e s . Thus i n F i g . 17, f o r example, one has t w e l v e s e p a r a t e d i a g r a m s : e i t h e r a W o r an u n p h y s i c a l , i n t e r n a l S (one o f e a c h p o s s i b l e c a s e , o r f o u r c h o i c e s ) , and f o r e a c h s u c h c h o i c e a V o r Z o r q>Q ( r e p r e s e n t i n g n e u t r a l H i g g s F i g . 17: P o s s i b l e s e q u e n c e s f o r t h e d e c a y mode p -» 3e f i e l d s ) c o u l d g i v e r i s e t o t h e e -e p a i r . The n e u t r a l H i g g s c o u p l i n g t o t h e e l e c t r o n i s p r o p o r t i o n a l t o m (as i n Eq. (2.2.7) and ( 2 . 2 . 1 4 ) ) , and t h e y a r e i n a l l e c a s e s much, much s m a l l e r t h a n y-e c o u p l i n g s , b e c a u s e , from (2.2.14) and ( 2 . 2 . 1 7 a ) , 0 ~ m /v ~ g (m /M ) - e (m /M ) (6.1.1) e e e W e W — 5 t h a t i s , o f o r d e r (m /M ) ~ 10 t i m e s s m a l l e r , even w i t h o u t e W t a k i n g i n t o a c c o u n t t h e ( p r o b a b l y ) h e a v y mass o f t h e n e u t r a l H i g g s p a r t i c l e . F u r t h e r , a t t h e e n e r g i e s i n v o l v e d h e r e (~M ), t h e p h o t o n 2 2 p r o p a g a t o r ( ~ l / q ~ 1/M ) i s v e r y much g r e a t e r i n m a g n i t u d e M 2 t h a n t h a t o f t h e Z - p a r t i c l e ( p r o p a g a t o r ~ 1/M ) . Thus one s h o u l d a l s o i g n o r e t h e p o s s i b l e r e p l a c e m e n t o f a v i r t u a l p h o t o n w i t h a v i r t u a l Z. The d e t a i l e d c a l c u l a t i o n o f t h e f o u r Feynman d i a g r a m s o f F i g . 17 w i l l be done i n t h e n e x t s e c t i o n ( a n d App. F ) , b u t we q u o t e h e r e t h e i r combined o r d e r o f m a g n i t u d e : A(W,S) - g 2 e 2 (Z.U..U. m 2 ) / ( M 2 q 2 ) . (6.1.2) l i i i 2 l W F i g . 19: A 2* Feynman d i a g r a m f o r t h e d e c a y p -» 3e The p h y s i c a l H i g g s p a r t i c l e B ( F i g . 18) w i l l be shown t o make t h e c o n t r i b u t i o n A(B) ~ g % 2 (_.U..U m 2 c o t 2 e ) M 2 / ( M 2 q 2 M 2 ) , (6.1.3) l i l i 2 l p. W B w h i c h i s ( p o s s i b l y ) o f t h e same o r d e r o f m a g n i t u d e o f (6.1.2) 2 b e c a u s e o f t h e c o t e f a c t o r . T h e r e r e m a i n two f u r t h e r d i a g r a m s t o c o n s i d e r . F i r s t , i n F i g 19, i s a d i a g r a m t h a t i n v o l v e s two i n t e r n a l M a j o r a n a n e u t r i n o s . I t s a m p l i t u d e i s , r o u g h l y , A(2B,2x) ~ g 4 (_.U U m 2 c o t 2 e ) 2 / ( M 4 M 2 ) . (6.1.4) 1 1 1 1Z 1 Wt) 100 F i g . 20: C o n t r i b u t i o n o f H + + t o t h e d e c a y p 3e 2 2 w h i c h , compared t o A(B) i s ( a s q - M ) P A ( 2 B , 2 * ) / A ( B ) ~ (_.U. _U. m 2 c o t 2 e ) / M 2 << 1 l l 1 i 2 l W (6.1.5) b e c a u s e a g e n e r o u s v a l u e f o r m.cote i s o f t h e o r d e r o f a l e p t o n mass. F i n a l l y , t h e r e i s t h e d i a g r a m o f F i g . 20 w h i c h i n v o l v e s t h e d o u b l y c h a r g e d H i g g s p a r t i c l e , a c o n t r i b u t i o n t h a t f o l l o w s f r o m t h e i n t e r a c t i o n t erm (4.1.6) w h i c h i s n e u t r i n o i n d e p e n d e n t . T h i s i s n ' t even a l o o p d i a g r a m and i s t h e r e f o r e r e a d i l y w r i t t e n down. I t s a m p l i t u d e i s , r o u g h l y , a s s u m i n g M >>M , H \x A(H ) ~ 0 0 /M_ 1 1 1 2 H so t h a t A(H) A(B) 0 0 1 1 1 2 2 2 W B 2 2 2 2 M „ g e (m.cote) H I (6.1.6) (6.1.7) From (4.1.8) f o l l o w s m.cote - ( v „ 0 . . ) ( v / v ) = v 0 . . l H i j H i j (6.1.8) 2 2 2 and as g v ~ M , we have W A ( H ) / A ( B ) ~ M 2 / ( e 2 M 2 ) , (6.1.9) w h i c h c o u l d have any v a l u e b e c a u s e t h e H i g g s b o s o n masses a r e unknown. 101 Thus we s e e t h a t any one o f t h r e e d i a g r a m s — F i g s . 17,18,20 — c o u l d make t h e l a r g e s t c o n t r i b u t i o n t o t h e d e c a y p. -• 3e. B e c a u s e t h e masses m., , M „ and t h e p a r a m e t e r c o t e a r e unknown t h e r e i s no way o f d e c i d i n g w h i c h makes t h e g r e a t e s t c o n t r i b u t i o n : t h e r e i s even t h e p o s s i b i l i t y t h a t t h e y a r e a l l o f t h e same o r d e r o f m a g n i t u d e . U n f o r t u n a t e l y , t h e r e a r e d i v e r g e n c e s i n v o l v e d i n t h e i n t e g r a l s a s s o c i a t e d w i t h t h e s e d i a g r a m s w h i c h have t o be c a r e f u l l y h a n d l e d , a t a s k t o w h i c h we now t u r n o u r a t t e n t i o n . V I . 2 EVALUATION OF THE AMPLITUDES FOR u -» 3e We b e g i n t h e d e t a i l e d e v a l u a t i o n o f t h e a m p l i t u d e s f o r t h e d e c a y p -* 3e by c o n s i d e r i n g t h e r o l e p l a y e d by t h e p h y s i c a l H i g g s boson B. The r e a s o n f o r t h i s i s t h a t d i v e r g e n t i n t e g r a l s soon a p p e a r t h a t have t o be c a r e f u l l y m a n i p u l a t e d , i n t e g r a l s t h a t a p p e a r as w e l l i n t h e a m p l i t u d e s i n w h i c h th e u n p h y s i c a l H i g g s boson S p l a y s an i m p o r t a n t r o l e i n k e e p i n g t h e o v e r a l l r e s u l t gauge i n v a r i a n t . In t h i s s e c t i o n t h e r e i s t r u l y an o v e r w h e l m i n g a d v a n t a g e t o be g a i n e d by s e l e c t i n g t h e F e y n m a n - t ' H o o f t gauge (£ = 1 ) , so s u c h a c h o i c e w i l l n o t be r e s i s t e d , i n c o n t r a s t w i t h t h e p r e v i o u s s e c t i o n where i t was p o s s i b l e t o keep £ a r b i t r a r y . In t h i s s e c t i o n i t i s n o t p o s s i b l e t o i d e n t i f y t h e r e l e v a n t , 102 f i n i t e c o n t r i b u t i o n s d i r e c t l y , and p r o p a g a t o r t e r m s l i k e (j v 2 2 k k / ( k -M ) c o n s i d e r a b l y e x a c e r b a t e t h e d i v e r g e n c e d i f f i c u l t i e s . The c h o i c e £ = 1 e l i m i n a t e s s u c h t e r m s . M o s t o f t h e p r o p a g a t o r s and v e r t i c e s f o r t h e d e c a y s e q u e n c e o f F i g . 18 were u s e d e a r l i e r , i n ( 5 . 2 . 7 ) , f o r a c o n t r i b u t i o n t o t h e d e c a y p. -• etf. H e r e , h o w e v e r , o n l y t h e t e r m s w i t h c o t e w i l l be r e t a i n e d , r e f l e c t i n g t h e f a c t t h a t c o t e >> 1, and t h a t s u c h t e r m s make b y f a r t h e l a r g e s t c o n t r i b u t i o n . I n t h i s c a s e t h e a m p l i t u d e f o r F i g . 18 i s A(B) = y. r d k u (p. U i g U . .m.coteZ V 1 ^ 2 i \ igU.„m.coteZ u ( p ) ' J T i ^ e » 1 1 1 1 (p-k)-. » 1 2 1 W 1 W i U e ( 2 k - g ) f - i n ^ W ( P , ) ( i e * ) v ( p r ) ( k - q ) 2 - M ; A k 2 - M 2 B ; " x \ q 2 j e 2 * e - ( P , P 2 ) (6.2.1) w h e r e t h e gauge p a r a m e t e r £ = 1 has b e e n u s e d i n t h e p h o t o n p r o p a g a t o r (4.3.18) and where t h e l a s t t e r m i n d i c a t e s t h a t an a n t i s y m m e t r i z a t i o n o v e r t h e two f i n a l i d e n t i c a l e l e c t r o n s must be p e r f o r m e d . T h i s e x p r e s s i o n c a n be r e w r i t t e n A(B) = -1 g e 2. U. . U. 0m. c o t e j d k e + \x (6.2.2) 0/0 \ 4 u 2 1 i l i Z 1 2 . . 2 2(2w) M q ( p - k ) w u ( 2 k - q ) v x e e - ( p x P 2 ) • ( k 2 - M 2 ) [ ( k - q ) 2 - M 2 ] C a l l t h e f a c t o r i n f r o n t o f t h e i n t e g r a l s i g n C. When an i n t e g r a t i o n i s p e r f o r m e d (App. F) a f t e r d i m e n s i o n a l 4 n r e g u l a r i z a t i o n ( d k -* d k, n<4) has b e e n i m p o s e d on t h e i n t e g r a l , one o b t a i n s 103 k y -/ \ G V Xi P - k ^ F i g . 21: Feynman d i a g r a m needed t o r e n o r m a l i z e F i g . 18 A(B) = C ( - u E V u ) ( u * \ ) rdcxd nK K 2 (!-«) (6.2.3) J q [K -(1-«)M ] - JC . T 2 M (u E u ) ( u pv ) - (p *~> p ) . 2 2 p. e + p. e e 1 2 6q M B The i n t e g r a l , f o r n = 4, i s n o t f i n i t e . T h i s d e c a y s e q u e n c e , or d i a g r a m , has t o be r e n o r m a l i z e d , o r augmented by t h e d i a g r a m o f F i g . 21 i n w h i c h t h e v i r t u a l p h o t o n emerges from t h e e m e r g i n g e l e c t r o n l i n e . A v i r t u a l p h o t o n e m e r g i n g from t h e e n t e r i n g muon l i n e i s o f o r d e r (m /M ) r e l a t i v e t o F i g . 21 e p (App. F ) , and i s t h e r e f o r e t o be i g n o r e d . The a m p l i t u d e f o r F i g . 21 i s A ( r e n o r m ) = 7. f d ^ k u (p, ) i e t f / i \ i g U . . m . c o t e E / 1 ^2 i \ O c i ^ e H P ^ 1 M w 1 1 1 + ( ( p - k ) - B . e w 1 x i£U. _m. c o t e E u ( p ) / i W-in^^u ( p _ ) i e t f v (p' ) M „ 1 2 1 ~ » \k2-M2A q 2 / 6 1 1 6 W B - (Pj_ P 2 ) (6.2.4) w i t h o n l y t h e h i g h e s t o r d e r terms k e p t , as i n ( 6 . 2 . 1 ) , w h i c h becomes, w i t h C d e f i n e d as i n ( 6 . 2 . 2 ) , and a f t e r 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 has been imposed, J q ' [K'-(1-«)M; (6.2.5) B A ( r e n o r m ) = C ( u E V u ) ( u V v ) fdcxd K (!-«) - (p. <--»P, ) e + x p e e I 2 , „ 2 . . . 2 , 2 1 z 104 w h i c h , t o o , i s d i v e r g e n t f o r n = 4. We now employ s p e c i a l f o r m u l a e f r o m 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 (Nash, 1978, A p p e n d i x ) t o e v a l u l a t e (6.2.3) as A (B) = _C_ u £ V u u V X v r(l-«)d« i T T " / 2 2 F ( 3 - l - n / 2 )  q 2 e + x p e e J r ( 3 ) B 3 1 ~ n / a - Ci-n-^ M u £ u u pv - (p p ) (6.2.6) 2 2 p e + p e e 1 2 6q MT B 2 where B = -(l- o c ) M , F i s t h e gamma f u n c t i o n and (6.2.5) as B A ( r e n o r m ) = _C u £ * u u y X v r ( l - « ) d a J T r " / 2 F ( 2 - n / 2 ) (6.2.7) q 2 6 + X 11 6 6 J F ( 2 ) fl2""'2 - ( P x P 2 ) . w h i c h e x a c t l y c a n c e l s t h e f i r s t t e r m o f ( 6 . 2 . 6 ) . Thus t h e a m p l i t u d e f o r p. •+ 3e v i a B i s A (B) = - J O 2 M u £ u u pv - (p, «--» p, ) . (6.2.8) 2 2 p e + p e e 1 2 6q M B The a m p l i t u d e f o r F i g . 22, w i t h i n t e r n a l W's, i s , s i m i l a r t o i t s p-»eV d i a g r a m d e s c r i b e d i n ( 5 . 2 . 2 ) , ( P ) I ~L£ V ~ p i A(W) =2. f d 4 k u (p )U i g V £ f W 2 i \ U i g y £ u ( 1 4 e 1 i l ^ i 2 v - p J(2w) ' ( p - k ) - - . ' K k - q ^ - M ^ - M ^ A q 2 / - i t i U - i i i \ ( - i e F ) u (p.)iev v ( p r ) X a p e 2 « e - ( P l «~> p 2 ) ( 6 . 2 . 9 ) i n t h e I - 1 gauge, where r = ~ ( q + k ) <n + ( 2 k - q ) -n - ( k - 2 q ) -n . ( 6 . 2 . 1 0 ) X a p p X a X a p p X a - \ ,4. - „ p . . . a T u V V T h i s i s e a s i l y r e d u c e d , u s i n g ( 5 . 3 . 5 ) , t o I where (6.2.11) A(W) = C fd k u £ -j (p-k) -j u X a p e e - (p 1<-»p 2) I 2 6 + ( p - k ) 4 M [ ( k - q ) 2 - M 2 ] ( k 2 - M 2 ) C a i 8 j g 2 e 2 Z . U . n U . 0 m 2 / ( 2 T T ) 4 . (6.2.12) 2 l i l i 2 l F i g . 22: C o n t r i b u t i o n o f W,S t o t h e d e c a y p. -» 3e T h i s i n t e g r a l i s e v a l u a t e d i n App. F. The r e s u l t i s , u s i n g 2 o n l y t h e l e a d i n g k terms i n t h e n u m e r a t o r , A(W) = - J T T 2 C 3u £ y u u y \ - (p, «--• p„ ) . (6.2.13) 2 2 e + X p e e 1 2 M w q The a m p l i t u d e f o r F i g . 22 w i t h t h e s e c o n d W r e p l a c e d by an S i s ( s e t t i n g m = 0 ) V 8 e ; (6.2.14) A(WS) = 7. f d 4 k u f-igU. (-m,)E V W 2 i \ u i g y £ u i J ( 2 w ) 4 e \ M w 1 1 1 + y ! ( P - k ) - m J 1 2 V ~ » W l x / i V-i'n 1 / g V-ieM TI )/ - i t i X o c \ u i e y v - (p.*-*p 2) l ( k - q ) 2 - M 2 / k 2 - M ; / I q 2 / « 6 w h i c h i s e v a l u a t e d (App. F) t o be A(WS) = - i i r 2 C r u £ y u u y \ - (p, p„) . (6.2.15) 2 . . 2 e + x p e e 1 2 q Mw The a m p l i t u d e f o r F i g 22 w i t h t h e f i r s t W r e p l a c e d by an S i s A(SW) = 7 f d 4 k u_U.,igV £ r    i g y / 1 ^2 i \/-igU. _(M £ -m.E )\u J ^ T " 6 1 1 " " (p-k)-m.A M w 1 2 ^ + 1 - y ^ / - J T i M g V i \ ( - i e M -n ) / - J T I X O C \ U i e y v - ( p «--• p ) ( ( k - q ^ - M ^ - M 2 ] W ^ i q 2 j e « e 1 (6.2.16 W W w h i c h i s s i m i l a r l y e v a l u a t e d (App. F) t o be 106 A(SW) = - J T T 2 C r u E * u u y \ - (p, ••-» p ,) . (6.2.17) 2 w 2 e + x u e e 1 2 q Mw The a m p l i t u d e f o r F i g . 22 w i t h b o t h W's r e p l a c e d by S's i s (6.2.18) A(S) = >. r d 4 k u / - i g U . (-m, ) E J f W 2 i | i g U (M E -m. E ) u J (2TT) \ M n (p-k)-m.' M W 1 w i } i e ( 2 k - q ) ( - i n X o c ) U i e * v - (p «--» p ) / i \ 2 » « 2 H . 2 m 2 1 M 2 e <x e (k-q) -M 'Ik -M / » q w w T h i s i n t e g r a l d i v e r g e s and must be h a n d l e d e x a c t l y as was done f o r t h e s i m i l a r H i g g s B p a r t i c l e , w h i c h i s t o s a y , i t must be c o n s i d e r e d i n a d d i t i o n t o a d i a g r a m s i m i l a r t o F i g 21. The o n l y d i f f e r e n c e i s i n t h e f a c t o r s m u l t i p l y i n g t h e i n t e g r a l , so t h e r e s u l t i s A(S) = I J T T 2 C f M \u E u u pv - (p, p,) (6.2.19) U 2 4 uj e + u e e 1 2 »6q Mtf ' w h i c h i s o f o r d e r (M 2/M 2) l e s s t h a n ( 6 . 2 . 1 3 ) , (6.2.15) and u W (6.2.17) and so i s i g n o r e d . Thus t h e f i n a l r e s u l t f o r t h e u -» 3e d e c a y m e d i a t e d by t h e W w i t h i t s c o - r e q u i s i t e gauge p a r t i c l e s , t h e S's, d e s i g n a t e d c o l l e c t i v e l y as A(W), i s A(W) = / - i t r 2 C f \ 5u E V u u V X v - (p, •--» p, ) . (6.2.20) I 2 „ 2 e + x u e e 1 2 *q M W / F i n a l l y , t h e a m p l i t u d e f o r F i g . 20 i s ( f r o m (4.1.6) and C.7)) A ( H ) = (iJ-20 )(iJ"2B ) V u ( p ) V e ( p f } [u ( p j v ( P 2 ) - u ( P 2 ) V (p )] 1 1 1 2 t f \ 2 M 2 e 1 e 2 e z , c e i (P-P ) ~ M (6.2.21) i i 2 2 2 and as (p-p') ~ M << M ( p r e s u m a b l y ) , we have u H A(H) « 2 P n P i 2 v ( p ) v (p' ) [u (p )v ( P 2 )-u ( p 2 )v (p ) ] . ~~M~ 2 »* 6 6 (6.2.22) H 107 As m e n t i o n e d , we have no way o f knowing w h i c h o f t h e t h r e e a m p l i t u d e s ( 6 . 2 . 8 ) , (6.2.20) o r (6.2.22) makes t h e g r e a t e s t c o n t r i b u t i o n t o t h e d e c a y p. •+ 3e, b e c a u s e e a c h c o n t a i n s unknown masses. We now c o n s i d e r t h e p. -» 3e d e c a y r a t e s as i m p l i e d by t h e s e a m p l i t u d e s . W i t h t h e f i n a l f o r m f o r t h e p -» 3e d e c a y a m p l i t u d e s a t hand s t a n d a r d t e c h n i q u e s , s u c h as t h o s e a p p l i e d i n t h e p r e v i o u s c h a p t e r , w i l l now be u s e d t o o b t a i n t h e t r a n s i t i o n r a t e . As we a r e m a i n l y c o n c e r n e d w i t h e s t a b l i s h i n g t h e a p p r o x i m a t e p l i f e t i m e f o r t h i s d e c a y we w i l l o n l y e v a l u a t e them f o r e a c h o f t h e t h r e e d e c a y modes s e p a r a t e l y , and n o t c o n s i d e r t h e i n t e r f e r e n c e e f f e c t s . The r e a s o n f o r t h i s i s t h a t e a c h a m p l i t u d e has a d i f f e r e n t unknown p a r a m e t e r i n i t ( s u c h as a mass o r Yukawa c o u p l i n g c o n s t a n t o r t h e p a r a m e t e r e ) . The s p i n o r a m p l i t u d e s t o be e v a l u a t e d a r e , f i r s t , f r o m A(W) i n ( 6 . 2 . 2 0 ) , where t h e o v e r b a r i n t h e summation s i g n once a g a i n r e f e r s t o an VI. 3 THE DECAY RATE FOR p -» 3e (6.3.1) a v e r a g e o v e r t h e i n i t i a l muon s p i n ( a f a c t o r j , e s s e n t i a l l y ) 108 and where, u s i n g p = P1+P2+p' , q = P'+ P = P ~ P , (6.3.2) 1 , 2 1 , 2 2 , 1 from A(B) i n ( 6 . 2 . 8 ) , Sf, s S . |u ( p , ) Z u (p) u ( P 2 ) p v (p') / q 2 (6.3.3) B s p i n s e 1 + p. e 2 e 2 - u ( P 2 ) T u (p) u ( p . ) p v (p' ) / q 2 | 2 , e 2 + ix e 1 e 1 and f i n a l l y , f r o m A(H) i n ( 6 . 2 . 2 2 ) , S 2 = I . v ( P ) v (p') [u (p )v ( p . ) - u (p )v ( p , ) ] f . (6.3.4) H s p i n s u e e 1 e 2 e 2 e 1 The r e s u l t s o f t h e s e r a t h e r l e n g t h y c a l c u l a t i o n s a r e (App. F ) , u s i n g M = M, m = m i n t h i s s e c t i o n , p. e S 2 = (l/2Mm 3 ) [ ( m 2 P - p 1 + p . p 2 p 1 ^ p'+p ^ p'p x -p 2 ) / ( p - p x ) 4 (6.3.5) + (m 2p 'P 2+P * P 1 P 2 * P f + P *P'P x * P 2 ) / ( P ~ P 2 ^ + (m P'Pj^+m P'P 2-m P 'Pr +2p «p' P x «P 2 ) / ( p ~ P x ) ( p ~ P 2 ) ] , = M / 4 T v S 2  ( l / 4 M m 3 ) { p - P l ( 2 p . p ' p . p 2 - M 2 p ' . p 2 - M 2 m 2 ) / ( p - P l ) 4 (6.3.6) + P*P 2 (2p«p r P'Pj^-M p' -p^M m ) / ( p - p 2 ) ' .2 - [ 4 p « p i p « p 2 p « p ' -M ( p ' P 2 p ' « P X + P ' P x p' •P 2+P 1 « P 2 P *p' ) +M 2m 2(p «p'-p » p x - p « p 2 ) ] / 2 ( p - p x ) 2 ( p - p 2 ) 2 } and, S 2 = 2 ( l + P - p ' / m M ) ( P l . p 2 / m 2 - l ) (6.3.7) In t h e r e s t frame o f t h e muon where p = Mtf 0, we have p-p. = ME. ( i = l , 2 ) (6.3.8) l l p.p' = P - ( P - P 1 - P 2 ) = M 2 - M ( E X + E 2 ) , and p1 «p 2 f o l l o w s f r o m P' 2 = (P —Pj^ — P 2 ) 2 = m2 = M 2+2m 2-2M(E 1+E 2)+2p 1 -p 2 . (6.3.9) I f we now d e f i n e E. 3 -Mx. ( i = l , 2 ) (6.3.10) l 2 l and s e t m = 0 w h e r e v e r i t o c c u r s i n t h e n u m e r a t o r o n l y , t h e 109 s p i n o r a m p l i t u d e s s q u a r e d a r e W 8Mm' ( 2 - x 1 - x 2 ) ( x l + x 2 - l ) + x 2 ( l - x 2 ) ( l - x x + m 2 / M 2 ) 2  + ( 2 - x l - x a ) - ' ( x 1 + x 2 - l ) + x ^ l - X j ) ( l - x 2 + m 2 / M 2 ) 2  + 2 ( 2 - x 1 - x 2 ) ( x 1 + x 2 - l ) ( l - x x +m2 /M2 ) ( l - x 2 +m2 /M2 ) (6.3.11) M 16m" x 1 x g ( 2 - x 1 - x 8 ) + x ^ - 1 ) (6.3.12) ( l - x 1 + m 2 / M 2 ) 2 + X l X 2 ( 2 ~ X l " X 2 ) + X 2 ( X 2 ~ 1 ) ( l - x 2 + m 2 / M 2 ) 2  + ( 2 - x l - x 2 ) ( 2 x 1 x 2 - x l - x 2 + l ) + x ^ x ^ l ) + x 2 ( x 2 - l ) ( l - x 1 + m 2 / M 2 ) ( l - x 2 + m 2 / M 2 ) and H ( M 3 / 2 m 3 ) ( 2 - x 1 - x 2 ) ( x 1 + x 2 - l ) (6.3.13) The t r a n s i t i o n r a t e , as i n t h e p r e v i o u s c h a p t e r , i s r(u-»3e) = j J" |A(p.-*3e) | 2 (2t r ) 4 S 4 (p-p, -p 2-p' ) (6.3.14) d 3 P i m d3 p . x m_ d p m E' ( 2 T T ) 3 E1 ( 2 I T ) ° E 2 (2ir) ' w i t h t h e r e q u i r e d s t a t i s t i c a l f a c t o r ( j ) b e c a u s e o f t h e two i d e n t i c a l e l e c t r o n s i n t h e f i n a l s t a t e . A f i r s t i n t e g r a t i o n o v e r p' ( t h a t i s , o v e r t h e e s t a t e s ) r e s u l t s i n (App. F) T = m" LAI f_ A J (2ir 2 , 1 d p x d p 2 S [ ( P - P 1 - P 2 ) -m ] - [ l + €(p-p 1-p 2 )] (2ir) E^. (6.3.15) 2 and b e c a u s e |A| depends o n l y on and E 2 we o b t a i n (App. F) i = [ m 3M 2/4(2ir) 3 ] T|A| 2 dx x d x 2 (6.3.16) w h i c h i s e x a c t . F i g . 23: A p p r o x i m a t e i n t e g r a t i o n r e g i o n i n t h e -x p l a n e l^ min < 3 — o o O O O o O c> , Emax ( a ) ( b ) F i g . 24: Maximum and minimum e l e c t r o n e n e r g i e s Now, t h e x - v a l u e s v a r y f r o m 0 t o 1, a p p r o x i m a t e l y , as i n d i c a t e d i n F i g . 23, b u t i t i s i m p o r t a n t t o be more p r e c i s e so as t o a v o i d (6.3.11) and (6.3.12) d i v e r g i n g i n t h e m -» 0 l i m i t . One e l e c t r o n c o u l d be a t r e s t ( F i g . 24a) so t h e minimum x - v a l u e from (6.3.10) i s x . = 2E . /M = 2m/M , (6.3.17) min min and t h e maximum x - v a l u e i s ( F i g . 24b) x = 1 - 3m 2/M 2 (6.3.18) max w h i c h f o l l o w s f r o m e l e m e n t a r y energy-momentum c o n s e r v a t i o n 2 2 p r i n c i p l e s . From ( p r ) = ( p - p ^ P g ) f o l l o w s M 2 ( X l + x 2 ) = M 2 + m2 + 2 P l -p 2 (6.3.19) w h i c h g i v e s t h e e x a c t x x ~ x 2 i n t e g r a t i o n r e g i o n , and I l l ( x , + x j = 2 - 2m/M , (6.3.20) 1 2 max w h i c h f o l l o w s f r o m F i g . 24a w i t h t h e p o s i t r o n a t r e s t , and (x +x,) . = 1 + 3m 2/M 2 , (6.3.21) 1 2 min 2 b e c a u s e t h e minimum v a l u e o f px «p 2 i s m , when t h e e l e c t r o n s a r e a t r e s t r e l a t i v e t o ea c h o t h e r . We must now i n t e g r a t e (6.3.11) o v e r t h e a p p r o p r i a t e X j - x 2 r e g i o n . The f i r s t two terms o f (6.3.11) w i l l g i v e t h e same i n t e g r a l , namely r 1 _ € 2 d x r 1 _ € 2 dx ( 2 - ^ 1 - x 2 ) ( x i + x 2 - 1 ) + x 2 ( 1 - x 2 ) € l 1 1 + € 2 _ X 1 2 ( l - x 1 + m 2 / M 2 ) 2 (6.3.22) 2 2 where € x s 2m/M and £ 2 = 3m /M . The i n t e g r a l i s , w i t h x = x1, r 1 _ € 2 d x (-2x 3/3 + x 2 ) (6.3.23) € 1 , , 2 ,.,2 . 2 (1-x+m /M ) where t h e € terms f r o m t h e x 2 i n t e g r a t i o n c a n be i g n o r e d a t t h i s s t a g e . T h i s i n t e g r a l i s 2/ i-( 1-x+m2/M2 ) 2 - 3(l+m 2/M 2 ) (l-x+m 2/M 2 ) + (l+m 2/M 2 ) 3 (6.3.24) (1-x+m 2/M 2) + 3(l+m 2/M 2 ) In ( 1-x+m2/M2 )) + (-(1-x+m 2/M 2 ) + q+m 2/M 2 ) 2 + 2 (1+m 2/M 2 ) In (1-x+m 2/M 2 ) (1-x+m 2/M 2) -2 , „ 2 „ \ . 1~ € 2 € l 2 2 i n w h i c h t h e dominant terms a r e t h e (1-x+m /M ) f a c t o r s i n t h e d e n o m i n a t o r e v a l u a t e d a t t h e up p e r l i m i t . The v a l u e o f t h i s i n t e g r a l i s , a p p r o x i m a t e l y , - | [ l / ( 4 m 2 / M 2 ) ] + [ l / ( 4 m 2 / M 2 ) ] = M 2/12m 2 . (6.3.25) The s e c o n d i n t e g r a l f r o m (6.3.11) i s , a p p r o x i m a t e l y , r l - € 2 , .1 2(2-x -x ) ( x +x -1) S dx, r dx. 1 2 1 2 (6.3.26) (1-Xj^+m /M ) ( l - x 2 + m /M ) and i t s h i g h e s t o r d e r term i s a l o g a r i t h m i c t e r m l i k e t h o s e i n 112 ( 6 . 3 . 2 4 ) , w h i c h , compared t o ( 6 . 3 . 2 5 ) , i s n e g l i g i b l e . Thus t h e f i n a l v a l u e f o r t h e W-mediated d e c a y r a t e i s 3 ..2 2 „ , , 2 , ^ w „.,2 I M m 3 ' ' r (u-»3e) = m M |5TT C \ I 1 1/2M 1 (6.3.27) W „ 2 , „ .3 ..2 3/ , „ 2 2 (2w) | M l \8Mm n12m w _ 2 4 4 . „ T T T r 2 . 2 / .,3 i = 5 g e (Z.U U m ) / _M 3 ( 2 1 2 ) ( 2 W ) 7 i H 12 i (M2M*J W _ 2 2 4 3 , „ 2 . 2 = 5 G", e M ( S.U.,U.„m ) 3 ( 2 7 ) ( 2 W ) 7 \7 1 1 1 1 2 1 2 2 where, a g a i n , G /J"2 = g / ( 8 M ) . In c o m p a r i s o n w i t h t h e p.-»ew F W t r a n s i t i o n r a t e (5.4.11) we have rw U" 3 e ) 5 2 e 4 ( y . U . U , m 2 ) 2 . (6.3.28) w N = „ 4 . „ .4 2„,2 l l l l 2 l r(pi-»evv) 2 (2w) m M F o r t h e B - m e d i a t e d d e c a y s e q u e n c e we have f o r t h e f i r s t t e rm o f (6.3.12) J d x 1 d x 2 1 2 v 1  z' 1 v 1 ' f x dx (6.3.29) (1-x +m 2/M 2) 2 " J 6 ( l - x + m 2 / M 2 ) 2 » M 2 / 24m 2 , o b t a i n e d i n a f a s h i o n s i m i l a r t o t h e i n t e g r a l ( 6 . 3 . 2 2 ) . The s e c o n d term w i l l g i v e t h e same r e s u l t , w h i l e t h e t h i r d t e r m i s n e g l i g i b l e by c o m p a r i s o n , as i t was i n t h e p r e v i o u s c a s e , and f o r t h e same r e a s o n . The f i n a l f o r m f o r t h i s d e c a y r a t e i s t h e n 113 I (ifm^L2) r (u-+3e) = m3 M 2 [ C T T 2 M | 2 ( M V2M" ) (6.3.30) 2 2 ( 2 t r ) 3 |6M2 4 4 7 . „ 2 2 ^ 2 = g e M (Z.U..U._m.cot e) 3 12 2 M 4 M 4 1 l l 1 2 1 3 2 (2IT) m M M W B = e 4 M 7 G 2 ( 2 . U . , U . „ m 2 c o t 2 e ) 2 „ 3 „ 7 / i r , . 7 2 „ 4 F 1 l l l 2 1 3 2 (2TT) m M B and i n c o m p a r i s o n t o t h e \x-*evv r a t e F B ( t I " > 3 e ) e 4 M 2 ( J . U . U . 0 m 2 c o t 2 e ) 2 (6.3.31) ,w \ ~ " o 2 o 4 / » . \ 4 2 . . 4 I l l l 2 l r(p.-»ew) 3 2 (2TT) m M B and t h e W-mediated r a t e r B(u-»3e) ^ ( M c o t e ) 4 (6.3.32.) r w(u-»3e) " ( 3 2 ) M 4 w h i c h may i n d i c a t e t h a t t h e r a t e s a r e c o m p a r a b l e . F o r t h e d o u b l y c h a r g e d H i g g s p a r t i c l e we have t h e i n t e g r a l , f r o m ( 6 . 3 . 1 3 ) , J - d X l d x 2 ( 2 - x l - x 2 ) ( x l + x 2 - l ) = 1 / 1 2 (6.3.33) and t h e d e c a y r a t e becomes T (u-»3e) = ( B 1 1 B 1 2 ) 2 M 5 , (6.3.34) 24 ( 2 T T ) 3 M 4 w h i c h , i n c o m p a r i s o n t o t h e p.-»ei/v r a t e , i s r H U " 3 e ) = ( B u B i z ) 2 . (6.3.35) r(u-»ei/V) G 2 M 4 F H H a v i n g no n e u t r i n o mass o r e - p a r a m e t e r i n i t , t h e r e seems t o be no p o i n t i n c o m p a r i n g T* w i t h T o r T a t t h i s t i m e . H W B 114 VI. 4 THE CROSS-SECTIONS FOR u + X -» e + X An e x p e r i m e n t s e a r c h i n g f o r t h e dec a y n -» 3e c o u l d c o n c e i v a b l y be c o m p l i c a t e d by o t h e r p r o c e s s e s and r e a c t i o n s i n w h i c h muons d i s a p p e a r w i t h t h e p r o d u c t i o n o f e l e c t r o n s , p r o c e s s e s t h a t , l i k e t h e j i -• 3e decay, v i o l a t e e l e c t r o n and muon number c o n s e r v a t i o n . We n o t e h e r e two p o s s i b i l i t i e s i n + w h i c h an i n c o m i n g p beam i s s u b j e c t t o t h e p o s s i b i l i t y , t h r o u g h p h y s i c a l , b u t v i r t u a l , c h a r g e d H i g g s p a r t i c l e s , t h a t a + + n i s c o n v e r t e d t o an e . I t i s i m p o r t a n t t o e s t a b l i s h w h e t h e r s u c h r e a c t i o n r a t e s would be l a r g e enough t o i n t e r f e r e w i t h a \s -» 3e e x p e r i m e n t . In F i g . 25 t h e d o u b l y c h a r g e d H i g g s f i e l d c o u l d p l a y t h i s r o l e . The a m p l i t u d e f o r t h e Feynman d i a g r a m o f F i g . 25 i s A ( n + , e % e + , e " ) = ( l J 2 ^ * } ( l J " 2 P i 2 ) v ( p ) E v ( p J v ( P ) E v ( p ) 2 2 — + l l — — k - M* ^ (6.4.1) H » ( 2 0 X 1 B12/M ZE) v ( p ) E _ v ( P + ) v ^ ( P ) E _ v ( p _ ) , 2 2 2 s i n c e M >> k ~ M . T h i s a m p l i t u d e f o l l o w s from t h e H \s 115 i n t e r a c t i o n t erm ( 4 . 1 . 6 ) : £ ( « . . , A. ,H ) = J"2B. . ( A C £ H x,. + iX.E H it°) (6.4.2) l j ±± i j l - ++ j j + — l and ip , f r o m ( C . 7 ) , where t h e f i r s t t erm a p p l i e s t o t h e l o w e r v e r t e x i n F i g . 25, and t h e s e c o n d t o t h e up p e r v e r t e x . T h i s i s summed and a v e r a g e d o v e r s p i n s t o become 1 . | A | 2 * [ ( P 1 1 0 1 2 ) 2 / 4 M 4 J ] (l+E/m )(1+E /m ) . (6.4.3) s p i n s i i 12 H + e - e T h e r e w o u l d be no need f o r t h e muon beam t o have a h i g h k i n e t i c e n e r g y so t h a t E « E » . The c r o s s - s e c t i o n (a) i s + 2 p e v a l u a t e d u s i n g s t a n d a r d t e c h n i q u e s ( B j o r k e n and D r e l l , 1964, App. B) t o be, a s s u m i n g |v | ~ 1, o- ~ [ ( B ^ P ^ ) 2 / ^ * ] (M 2/M 4) ( * 2 / c 2 ) , (6.4.4) where t h e l a s t f a c t o r was p ut i n f o r u n i t r e a s o n s . I f as b e f o r e , one g e n e r o u s l y assumes IB. . I ~ IB. I ~ S (M /M ) , (6.4.5) 1 J 1 X, n where M i s a l e p t o n mass (~J"m M ), one f i n d s Jl e p o - (g 4/16Tt) (m 2M 4/M^) ( M / M ) 4 ( * 2 / c 2 ) (6.4.6) e \i W W H - I O " 5 6 ( M W / M H ) 4 cm 2 , w i t h a mean " s c a t t e r i n g l e n g t h " i n m a t t e r o f Jl = 1/no- - 1 0 1 8 I t y r , (6.4.7) 2 0 — 3 f o r an e l e c t r o n d e n s i t y n ~ 10 cm , M ~ M , w h i c h shows H W j u s t how n e g l i g i b l e s u c h a r e a c t i o n i s . W i t h s i m i l a r a s s u m p t i o n s t h e qua r k i n d u c e d p r o c e s s shown i n F i g . 26 would have t h e v e r y r o u g h a m p l i t u d e ( a f t e r a r a t h e r c r u d e l o o p i n t e g r a t i o n ) A(u +,q->e +,q) - (gm /M J ( g M /M )(gM /M ) 2 ( 1 / M 2 ) (6.4.8) e W p W q W n from (4.5.8) and (4.5.10) where M i s a qua r k mass, w h i c h , when q 116 compared t o ( 6 . 4 . 1 ) , i s A(p +,q->e +,q) / A ( p + , e ~ - e + , e ~ ) ~ g 2 (M 2/M 2) << 1 , (6.4.9) q W so t h a t s u c h a p r o c e s s would be even more n e g l i g i b l e . d + + F i g . 26: Feynman d i a g r a m f o r p + d -» e + d We may c o n c l u d e , t h e n , t h a t a l t h o u g h t h e d e c a y p -* 3e i s , a t most, v e r y r a r e , o t h e r muon l e p t o n number n o n - c o n s e r v i n g p r o c e s s e s , s u c h as t h e d i r e c t c o n v e r s i o n o f a p. i n t o an e, a r e p r o b a b l y v e r y much r a r e r by c o m p a r i s o n , and do n o t mimic p r o c e s s e s by w h i c h one would a t t e m p t t o measure t h e d e c a y r a t e p -> 3 e. 117 V I I . DISCUSSION AND CONCLUSIONS I t i s a p p r o p r i a t e , now, t o c o n s i d e r t h e q u a n t i t a t i v e p r e d i c t i o n s t h a t f o l l o w from t h e d e c a y r a t e s c a l c u l a t e d i n t h e p r e v i o u s two c h a p t e r s . U n f o r t u n a t e l y most o f t h e p a r a m e t e r s t h a t went i n t o t h e model's c o n s t r u c t i o n a r e unknown. But some, however, c a n c e l , and o t h e r s can p e r h a p s be e s t i m a t e d on t h e b a s i s o f p r e v i o u s e x p e r i m e n t a l work o r on t h e b a s i s o f o t h e r s ' work on l i m i t a t i o n s f r o m a s t r o p h y s i c a l c o n s i d e r a t i o n s . From (2.3.12) one has P s M 2 v/(M 2,cos 2e w) = ( V 2 + 2 V 2 I ) / ( V 2 + 4 V 2 I ) (7.1) » 1 - 2 v 2 / v 2 = 1 - t a n 2 e , H and f r o m (2.3.13) (and r e f e r e n c e s m e n t i o n e d t h e r e ) one can c o n s i d e r as e x p e r i m e n t a l l y e s t a b l i s h e d t h a t p « 1 and v <<v. H C u r r e n t e x p e r i m e n t s a r e c o n s i s t e n t w i t h p = 1, and an e s t i m a t e o f p = .985±.015 (Marshak, R i a z u d d i n and M o h a p a t r a , 1981) l e a d s t o c o t 2 e > 10 2 . (7.2) An i n d e p e n d e n t e s t i m a t e b a s e d on a s t r o p h y s i c a l e v o l u t i o n ( G e l m i n i , N u s s i n o v and R o n c a d e l l i , 1982; Dugan et al., 1985; Glashow and Manohar, 1985) g i v e s v ~ 100 keV ( w h i c h i s v e r y H r o ugh) so t h a t w i t h v = (1/J2G)1/Z » 250 GeV (7.3) F (Cheng and L i , 1984, p. 3 5 3 ) , we have c o t e = v/J"2v„ - 1 0 6 , (7.4) 118 w h i c h , a l t h o u g h a c r u d e e s t i m a t e , i s a more s t r i n g e n t bound t h a n (7.2). Of c o u r s e , v = 0 i s n o t r u l e d o u t . H As p o i n t e d o ut e a r l i e r , i n (4.1.8), t h e p r o d u c t m c o t e i s l a c t u a l l y i n d e p e n d e n t o f v , and t h e r e f o r e o f t h e n e u t r i n o H masses, however s m a l l : m. c o t e ~ |v p..| ( v / v ) = |vB..| , (7.5) 1 n i l n 1 1 b e c a u s e f r o m (3.3.2b) we have • i = | 2 Z j k D l J D l k B J k ' H l " ' " i i V < 7- 6 > i f , i n t h e f i r s t a p p r o x i m a t i o n U. . » S. ., as seems l i k e l y ( s e e i j i j t h e r e f e r e n c e s b e f o r e ( 7 . 3 ) ) . An a s s u m p t i o n we w i l l make now, a l t h o u g h t h e r e r e a l l y i s no e v i d e n c e f o r i t , i s t h a t t h e Yukawa c o u p l i n g c o n s t a n t s f o r t h e i s o - s p i n o r H i g g s f i e l d and t h o s e f o r th e i s o - v e c t o r H i g g s f i e l d a r e o f t h e same o r d e r o f m a g n i t u d e . In s u c h a c a s e |v0..| - v8 ~ , (7.7) 1 '2 where M„ i s a l e p t o n mass, s a y M. ~ (M m ) . These r o u g h Jl Jl u e e s t i m a t e s w i l l now be a p p l i e d t o t h e d e c a y r a t e s c a l c u l a t e d e a r l i e r . I t i s s u r p r i s i n g t h a t (7.4) and (7.7) t u r n o ut t o be o f t h e same o r d e r o f m a g n i t u d e . The b a s i c r a d i a t i v e d e c a y r a t e f o r t h e muon i s g i v e n i n (5.4.10). I g n o r i n g , f o r now, t h e p h y s i c a l , c h a r g e d H i g g s b o s o n B, t h e r a t e as a f r a c t i o n o f t h e u s u a l d e c a y r a t e i s , from (5.4.13), r(u-»W-»ey) _ 3_oc (2.U..U._ m 2 / M 2 ) 2 (7.8) _ . — 1 1 1 1 £ 1 w r(u-»ew) 32ir i n agreement w i t h Cheng and L i (1984, p. 427), and w i t h a 119 n e u t r i n o mass i n t h e 10 eV r a n g e one has r(u-»W-»eV) ~ 6 x l Q ~ 4 4 [£.U U ( m . / 1 0 e V ) 2 ] 2 , (7.9) F( M - » e w ) an e x t r e m e l y s m a l l f r a c t i o n t h a t w o u l d be e x p e c t e d t o be even f u r t h e r r e d u c e d by t h e components o f t h e o r t h o g o n a l m a t r i x (|U..| < 1, f o r a l l i , j ) , t h e o f f - d i a g o n a l e l e m e n t s o f w h i c h m i g h t be r e a s o n a b l y e x p e c t e d t o be q u i t e s m a l l . In t h e c a s e o f two f a m i l i e s , f o r example, t h e m a t r i x U c o u l d be w r i t t e n _ c o s e - s i n e (7.10) s i n e c o s e so t h e f a c t o r i n (7.9) becomes 2 2 2 £.U.,U m. = c o s e s i n e (m -m ) , (7.11) 1 l l l Z 1 2 1 i n agreement w i t h o t h e r work ( e . g . , Marshak, R i a z u d d i n and M o h a p a t r a , 1981). R e c e n t e s t i m a t e s do seem t o i n d i c a t e t h a t e i s s m a l l ( G e l m i n i , N u s s i n o v and R o n c a d e l l i , 1982; Dugan et al. , 1985). Thus t h i s d e c a y , f i r s t c a l c u l a t e d by Cheng and L i ( 1 9 8 0 b ) , i s h o p e l e s s l y b e y o n d any e x p e c t a t i o n o f e x p e r i m e n t a l v e r i f i c a t i o n , a c o n c l u s i o n i n d i f f e r e n t t o t h e s p e c i e s o f n e u t r i n o employed, whether o f t h e D i r a c t y p e employed by Cheng and L i o r t h e M a j o r a n a t y p e c o n s i d e r e d h e r e . L e t us now c o n s i d e r t h i s d e c a y u s i n g t h e r e s u l t o f t h e H i g g s p a r t i c l e B. I t s c o n t r i b u t i o n i s t h e h i g h e s t o r d e r t e r m from ( 5 . 4 . 1 3 ) , t h e one w i t h c o t e : r(u*B-»ey) « _ [Z.U..U (m./M ) 2 c o t 2 e ] 2 . (7.12) \ — r\r* i i l i " i " r ( n - » e w ) 96 IT From (7.4) we i m m e d i a t e l y s e e t h a t t h i s r a t i o i s o f t h e o r d e r 2 4 o f 10 t i m e s t h e p e s s i m i s t i c r e s u l t o f Cheng and L i , p r o v i d e d 120 t h a t M ~ M . From (7.7) one o b t a i n s B W r(u-»B-»eV) - 1 0 " 2 1 ( M / M j 4 (2.U.-.U ) 2 , (7.13) . . Tt D X X 1 X £ T( u-»ew) w h i c h i s c o n c e i v a b l y some 22 o r d e r s o f m a g n i t u d e g r e a t e r t h a n ( 7 . 9 ) , and n o t a l l t h a t much d i f f e r e n t f r o m (7.12) w h i c h was an e s t i m a t e b a s e d on t o t a l l y d i f f e r e n t a s s u m p t i o n s . In any c a s e b o t h a r e s t i l l a b out 10 t o 12 o r d e r s o f m a g n i t u d e below c u r r e n t d e t e c t a b i l i t y , w h i c h i s ( C o o p e r , 1978; K i n n i s o n et al., 1982) r(p.-*ey )/r(u->e Vv) < 1 0 _ 1 ° , (7.14) w i t h r(u-»ey*)/r(u->ei/ V) < 8 x l 0 9 (7.15) ( A z u e l o s , 1983). C u r r e n t hopes a r e t h a t t h e r a t i o i n (7.14) — 1 2 can be r e d u c e d t o ab o u t 10 i n t h e r e l a t i v e l y n e a r f u t u r e ( W e i n b e r g , 1984; Bowen, 1985). F o r t h e p. -» 3e d e c a y we have t h e t h r e e i l l u s t r a t i v e r a t e s c a l c u l a t e d i n Chap. VI. F o r t h e d e c a y v i a t h e W we have, from (6.3.28), l-(u-»W-»3e) _ 5 2 cx2 (S.U U m 2 ) 2 (7.16) r ( n - » e w ) 2 2 ( 2 w ) 2 m 2 M 2 1 1 - 3 x l 0 ~ 2 9 [£.U. U . _ ( m . / 1 0 e V ) 2 ] 2 , x x l x2 x where m and M a r e t h e e l e c t r o n and muon masses, r e s p e c t i v e l y . T h i s r a t e s h o u l d be compared w i t h t h e p. -» eV ( v i a W) r a t e o f (7.8): r(u-»W-»3e) _ 5 2 (2 2 )« MW - 10 1 5 . (7.17) r(M*W-»ey) " 3(2w) m 2M 2 Thus t h e W-mediated d e c a y r a t e f o r t h e d e c a y u -» 3e l o o k s v e r y much more p r o m i s i n g t h a n t h e r a d i a t i v e d e cay, b ut i s s t i l l v e r y 121 2 0 much (~10 ) beyond t h e l i m i t o f e x p e r i m e n t a l d e t e c t a b i l i t y w h i c h i s ( K o r e n c h e n k o et al., 1976; B o l t o n et al., 1984) r(u-»3e )/ r(u->e v v) < 1 0 ~ 1 0 , (7.18) w h i c h r a t e , t o o , i s e x p e c t e d t o improve by two more o r d e r s o f m a g n i t u d e i n t h e n e a r f u t u r e ( B e r t l et al., 1984; E i c h l e r , 1984). F o r t h e p. -» 3e d e c a y m e d i a t e d by t h e H i g g s b o s o n B we have, from (6.3.31), r(u->B-»3e) = oc2 MJL_ (2.U U m 2 c o t 2 e ) 2 (7.19) r ( n - » e w ) 3 2 ( 2 2 ) ( 2 i r ) 2 m 2 M 4 1 1 B u s i n g t h e a p p r o x i m a t i o n (7.7), o r ~ 10" 1 9 ( M / M N ) 4 [ Z . U . , U . o ( m . / 1 0 e V ) 2 ] 2 (7.20) W B I I 1 i 2 l u s i n g t h e a p p r o x i m a t i o n (7.4). Thus t h i s d e c a y mode seems t o be about 10 o r so o r d e r s o f m a g n i t u d e below c u r r e n t d e t e c t a b i l i t y . As a c o m p a r i s o n w i t h t h e B - m e d i a t e d d e c a y mode we have r(u-.B-»3e)/r(u-+B->e*) = (2«/3ir) ( M / m ) 2 * 60 , (7.21) n o t n e a r l y as d r a m a t i c a c o m p a r i s o n as (7.17). As m e n t i o n e d e a r l i e r , t h e s e r a t e s would be e x p e c t e d t o be even f u r t h e r r e d u c e d by t h e f a c t o r s s u c h as U 1 2 > Nor can one e x p e c t them t o be much r a i s e d by r e l a t i v e l y s m a l l v a l u e s f o r M : b e i n g e l e c t r i c a l l y c h a r g e d , s u c h b o s o n s s h o u l d be r e a d i l y B d e t e c t a b l e a t c u r r e n t a c c e l e r a t o r e n e r g i e s , and a l o w e r l i m i t o f 15 GeV o r so (Ade v a et al., 1982) has a l r e a d y been e s t a b 1 i s h e d . F o r t h e u-»3e d e c a y m e d i a t e d by t h e d o u b l y c h a r g e d H i g g s 122 boson H we have, from ( 6 . 3 . 3 5 ) , T(u-»H-»3e) _ (Bi 1 B 1 2 ) 2 . (7.22) r(u-»evi/) G 2 M* W i t h t h e a p p r o x i m a t i o n (7.7) we have r(p-»H-»3e) ~ 1 0 1 O e 4 ( M / M ) 4 (M / M ) 4 (7.23) T( p-»ei/v) ~ i o " 1 6 1 ' 2 w i t h t h e g e n e r o u s a s s u m p t i o n s M„ ~ (m M ) , 0, , ~ B, „ ~ B ~ H e p 1 1 1 2 e(M /M ), and M ~ M . F o r t h e o f f - d i a g o n a l B, 2 we m i g h t Jl W H W 1 ~2 presume t h a t B x 2 ~ 10 Bll o r s m a l l e r (Glashow and Manohar, — 2 0 1985), i n w h i c h c a s e (7.23) f a l l s t o 10 , t h e same r o u g h l e v e l o f t h e o t h e r p.-»3e t r a n s i t i o n r a t e s . Thus, o v e r a l l i t seems t h a t t h e p. •• ey, p -» 3e d e c a y modes a r e 10 o r more o r d e r s o f m a g n i t u d e b e l o w c u r r e n t d e t e c t a b i l i t y , a l t h o u g h t h i s i s a c o n s i d e r a b l e improvement o v e r t h e 30 o r so o r d e r s o f m a g n i t u d e t h a t f o l l o w s w i t h o u t t h e i s o - v e c t o r H i g g s f i e l d s . I t now r e m a i n s t o be s e e n w h e t h e r t h e model d e s c r i b e d i n t h i s t h e s i s has a n y t h i n g t o c o n t r i b u t e t o o t h e r p r o c e s s e s t h a t a r e w i t h i n t h e r e a c h o f c u r r e n t e x p e r i m e n t a l c a p a c i t y . 123 BIBLIOGRAPHY A b e r s , E.S. and B.W. Lee ( 1 9 7 3 ) . Phys. Rep. 9C, 1. Adeva, B. et al. ( 1 9 8 2 ) . Phys. L e t t . 115B, 345. A k h i e z e r , A . I . and V.B. B e r e s t e t s k i i ( 1 9 6 5 ) . Quantum Electrodynamics. ( I n t e r s c i e n c e , New Y o r k ) . A z u e l o s , G. et al. ( 1 9 8 3 ) . Phys. Rev. L e t t . 5_1, 164. A r n i s o n , G. et al. ( 1 9 8 3 a ) . Phys. L e t t . 122B, 103. A r n i s o n , G. et al. ( 1 9 8 3 b ) . Phys. L e t t . 126B, 398. A r n i s o n , G. et al. ( 1 9 8 3 c ) . Phys. L e t t . 129B, 273. A v i g n o n e , F.T. et al. ( 1 9 8 3 ) . Phys. Rev. L e t t . 50_, 721. B a g n a i a , P. et al. ( 1 9 8 3 ) . Phys. L e t t . 129B, 130. Banner, M. et al. ( 1 9 8 3 ) . P h y s . L e t t . 122B. 476. B a r g e r , V., P. L a n g a c k e r and J.P. L e v e i l l e ( 1 9 8 0 ) . Phys. Rev. L e t t . 45_, 692. B e r n s t e i n , J . ( 1 9 7 4 ) . Rev. Mod. Phys. 46, 7. B e r t l , W. et al. ( 1 9 8 4 ) . Phys. L e t t . 140B, 299. B i l e n k y , S.M. and B. P o n t e c o r v o ( 1 9 7 8 ) . Phys. Rep. 4_1, 225. B j o r k e n , J.D. and S.D. D r e l l ( 1 9 6 4 ) . Relativistic Quantum Mechanics. ( M c G r a w - H i l l , New Y o r k ) . B j o r k e n , J.D. and S.D. D r e l l ( 1 9 6 5 ) . Relativistic Quantum Fields. ( M c G r a w - H i l l , New Y o r k ) . B o l t o n , R.D. et al. ( 1 9 8 4 ) . Phys. Rev. L e t t . 53_, 1415. Bowen, T. ( 1 9 8 5 ) . Phys. Today, 38., 7, 23. Case, K.M. ( 1 9 5 7 ) . Phys. Rev. 107. 307. Cheng, T.P. and L.F. L i ( 1 9 8 0 a ) . Phys. Rev. D22., 2860. Cheng, T.P. and L.F. L i ( 1 9 8 0 b ) . Phys. Rev. L e t t . 45, 1908. Cheng, T.P. and L.F. L i ( 1 9 8 4 ) . Gauge Theory of Elementary Particle Physics. ( C l a r e n d o n , O x f o r d ) . 124 C o o p e r , M.D. (1978) i n Proceedings of Summer Insti tute on Particle Physics, p. 417, (M.C. Z i p f , E d . ) . (SLAC R e p o r t 2 1 5 ) . Dugan, M.J., G.B. G e l m i n i , H. G e o r g i and L . J . H a l l ( 1 9 8 5 ) . Phys. Rev. L e t t . 54, 2302. E i c h l e r , R.A. ( 1 9 8 4 ) . u n p u b l i s h e d c o m m u n i c a t i o n . E i c h t e n , E., I. H i n c h l i f f e , K. Lane and C. Q u i g g ( 1 9 8 4 ) . Rev. Mod. P h y s . 56, 579. Frampton, P.H. and P. V o g e l ( 1 9 8 2 ) . Phys. Rep. 82., 339. F e r m i , E. ( 1 9 3 4 ) . Z. P h y s i k 88. 161. Feynman, R.P. and M. G e l l - M a n n ( 1 9 5 8 ) . Phys. Rev. 109, 193. G e l m i n i , G.B. and M. R o n c a d e l l i ( 1 9 8 1 ) . Phys. L e t t . 99B, 411. G e l m i n i , G.B., S. N u s s i n o v and M. R o n c a d e l l i ( 1 9 8 2 ) . Nuc. Ph y s . B209. 157. G e o r g i , H.M., S.L. Glashow and S. N u s s i n o v ( 1 9 8 1 ) . Nuc. Phys. B193. 297. Glashow, S.L. and A. Manohar ( 1 9 8 5 ) . Phys. Rev. L e t t . 54, 2306. H a m i l t o n , J.D. ( 1 9 8 4 a ) . Am. J . Phys. 52., 56. H a m i l t o n , J.D. ( 1 9 8 4 b ) . J . Math. P h y s . 25, 1823. I t z y k s o n , C. and J.-B. Z u b e r ( 1 9 8 0 ) . Quantum Field Theory. ( M c G r a w - H i l l , New Y o r k ) . J a u c h , J.M. and F. R o h r l i c h ( 1 9 7 6 ) . The Theory of Photons and Electrons. (2nd ed.) ( S p r i n g e r - V e r l a g , New Y o r k ) . K a e m p f f e r , F.A. ( 1 9 6 5 ) . Concepts in Quantum Mechanics. ( A c a d e m i c , New Y o r k ) . K a y s e r , B. and A.S. G o l d h a b e r ( 1 9 8 3 ) . Phys. Rev. D2J8, 2341. K i n n i s o n , W.W. et al. ( 1 9 8 2 ) . Phys.. Rev. D25_, 2846. K i r s t e n , T. et al. ( 1 9 8 3 ) . Phys. Rev. L e t t . 50., 474. Ko r e n c h e n k o , S.M. et al. ( 1 9 7 6 ) . JETP 43, 1. L a n g a c k e r , P. ( 1 9 8 1 ) . Phys. Rep. 72., 185. 125 Lee, B.W. and R.E. S h r o c k ( 1 9 7 7 ) . Phys. Rev. D16., 1444. Lee, T.D. ( 1 9 8 1 ) . Particle Physics and Introduction to Field Theory. (Harwood, New Y o r k ) . Lee, T.D. and C.N. Yang ( 1 9 5 6 ) . Phys. Rev. 104, 254. L i , L.F. and F. W i l c z e k ( 1 9 8 2 ) . Phys. Rev. D25, 143. Lubimov, V.A. et al. ( 1 9 8 0 ) . Phys. L e t t . 94B, 266. M a r c i a n o , W.J. and A. S i r l i n ( 1 9 8 4 ) . Phys. Rev. D29, 945. Marshak, R.E. and E.C.G. S u d a r s h a n ( 1 9 5 8 ) . P h y s . Rev. 109, 1860. Marshak, R.E., R i a z u d d i n and R.N. M o h a p a t r a (1981) i n Proceedings of the TRIUMF Muon Physics/Facility Workshop, 1980 ( J . A . MacDonald, J.N. Ng and A. S t r a t h d e e , E d s . ) (TRIUMF, V a n c o u v e r ) . Nash, C. ( 1 9 7 8 ) . Relativistic Quantum Fields. ( A c a d e m i c , New Y o r k ) . N e i v e s , J . F . ( 1 9 8 2 ) . Phys. Rev. D26, 3152. P a l , P.B. and L. W o l f e n s t e i n ( 1 9 8 2 ) . Phys. Rev. D25., 766. P e r k i n s , D.H. ( 1 9 8 2 ) . Introduction to High Energy Physics. ( A d d i s o n - W e s l e y , R e a d i n g , M a s s a c h u s e t t s ) . R e u t e n s , P.G. et al. ( 1 9 8 5 ) . Phys. L e t t . 152B, 404. R o h l f , J . ( 1 9 8 5 ) . Phys. Today 3 8 ( 1 ) , S34. Salam, A. (1968) i n Elementary Particle Theory, p.367. (N. S v a r t h o l m , Ed.) ( A l m q v i s t and W i k s e l l , S t o c k h o l m ) , ( r e p r i n t e d i n New Particles: Selected Reprints, AAPT, S t o n y Brook, New York, 1981). Salam, A. ( 1 9 8 2 ) . I n t e r n a t i o n a l C e n t r e f o r T h e o r e t i c a l P h y s i c s ( P r e p r i n t IC/82/215) S c h e c h t e r , J . and J.W.F. V a l l e ( 1 9 8 0 ) . Phys. Rev. D22., 2227. S c i u l l i , F. (1980) i n Proceedings of Summer Institute on Particle Physics, p. 29, (A. Mosher, Ed.) (SLAC R e p o r t 2 3 9 ) . 126 S e n j a n o v i c , G. (1983) i n AIP Conference Proceedings No. 99 (19 8 3 ) , p.117, (V. B a r g e r and D. C l i n e , Eds.) (AIP, New Y o r k ) . Simpson, J . J . ( 1 9 8 2 ) . Phys. Rev. D24, 2971. Simpson, J . J . ( 1 9 8 4 ) . Phys. Rev. D30, 1110. T a y l o r , J.C. ( 1 9 7 6 ) . Gauge Theories of Weak Interactions. ( C a m b r i d g e U.P., C a m b r i d g e ) . t ' H o o f t , G. ( 1 9 7 1 ) . N u c l . Phys. B35, 167. Velt m a n , M.J. (1980) i n Proceedings of Summer Institute on Particle Physics, p. 1, (A. Mosher, E d . ) ( S L A C R e p o r t 2 3 9 ) . Weaver, D.L. ( 1 9 7 6 ) . J . Math. Phys. 17, 485. Wei n b e r g , S. ( 1 9 6 7 ) . Phys. Rev. L e t t . 19., 1264. Wei n b e r g , S. ( 1 9 8 4 ) . Phys. Rep. 104, 107. W o l f e n s t e i n , L. (1983) i n AIP Conference Proceedings No. 99 (1 9 8 3 ) , p. 53, (V. B a r g e r and D. C l i n e , Eds.) (AIP, New Y o r k ) . Yang, C.N. and R. M i l l s ( 1 9 5 4 ) . Phys. Rev. 96, 191. 127 APPENDIX A  NOTATION AND CONVENTIONS  F o u r - V e c t o r s and M e t r i c B a s i s 4 - v e c t o r s f o r a r e f e r e n c e frame a r e g e n e r a l l y d e n o t e d (V ) ( u = 0,1,2,3), where V i s a u n i t t i m e l i k e 4 - v e c t o r ( t h e u 0 4 - v e l o c i t y o f an o b s e r v e r a t t a c h e d t o t h e r e f e r e n c e frame) and where V. ( i = 1,2,3) a r e u n i t s p a c e l i k e 4 - v e c t o r s : l p. v uv 1, p=v=0 -1, u=v=l,2,3 ( A . l ) 0 , Ll*V A g e n e r a l 4 - v e c t o r s u c h as t h e 4-momentum p o f a p a r t i c l e o f mass m can t h e n be w r i t t e n P = P % = P V M (A.2) (where t h e u s u a l summation c o n v e n t i o n i s employed) w i t h V -V V a SV (A.3) and P 2 = p . p = p p M = ( p 0 ) 2 - ( p l ) 2 _ ( p 2 ) 2 _ ( p 3 ) 2 = m 2 . ( A . 4 ) The d i f f e r e n t i a l o p e r a t o r a w i t h components a = a/ax i s d e f i n e d by a = v^a = v a 1 1 • (A.5) The r e a d e r s h o u l d n o t e t h a t t h e (V ) a r e n o t m a t r i c e s . No m a t r i x r e p r e s e n t a t i o n o f t h e D i r a c a l g e b r a i s u s e d h e r e b e c a u s e t h e (V ) a r e g i v e n c e r t a i n a l g e b r a i c p r o p e r t i e s (App. B) t h a t makes u n n e s s a r y any use o f m a t r i c e s . 128 U n i t s U n i t s a r e employed h e r e s u c h t h a t = c = 1 , (A.6) and, as i s a l s o common i n f i e l d t h e o r y , t h e p r o t o n c h a r g e e>0 i s m easured i n r a t i o n a l i z e d u n i t s s u c h t h a t t h e f i n e s t r u c t u r e c o n s t a n t i s oc s e 2 / ( 4 i r * c ) = e 2 / ( 4 i r ) - 1/137 . (A.7) C o n j u g a t e s The n o t a t i o n (*) i s employed f o r t h e complex c o n j u g a t e s o f complex numbers and t h e h e r m i t i a n c o n j u g a t e s o f F o c k - s p a c e 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 . The n o t a t i o n ( + ) i s r e s e r v e d f o r a s p e c i a l d e f i n i t i o n o f a l g e b r a i c i m p o r t a n c e (App. B ) . 129 APPENDIX B THE GEOMETRIC ALGEBRA: CONVENTIONS AND APPLICATIONS M a t r i x r e p r e s e n t a t i o n s o f t h e D i r a c and P a u l i a l g e b r a s can be a v o i d e d t o t a l l y i f c e r t a i n a l g e b r a i c p r o p e r t i e s a r e p o s t u l a t e d f o r 4 - v e c t o r s i n M i n k o w s k i a n s p a c e - t i m e and 3-v e c t o r s i n o r d i n a r y E u c l i d e a n s p a c e , t h e i s o - s p a c e o f t h e SU(2) gauge t r a n s f o r m a t i o n s . T h i s a l g e b r a — a C l i f f o r d a l g e b r a — i s b a s e d on t h e d e f i n i t i o n , g i v e n two v e c t o r s A and B , A B = g-(AB + BA) + jCAB - BA) ( B . l ) = A «B + A A B = B «A - B A A , w h i c h g i v e s t h e t h r e e b a s i c p r o d u c t s (AB, A«B, A A B ) o f t h e a l g e b r a , AB and A A B b e i n g d e f i n e d as w e l l t o be a s s o c i a t i v e . The " D i r a c a l g e b r a " i s t h a t C l i f f o r d a l g e b r a b a s e d on 4-d i m e n s i o n a l s p a c e - t i m e , w h i l e t h a t u s i n g o r d i n a r y 3 - v e c t o r s i s c a l l e d t h e " P a u l i a l g e b r a " . A c c o u n t s o f t h e s e a l g e b r a s and t h e i r a s s o c i a t i o n w i t h s p i n o r s can be f o u n d i n H a m i l t o n (1984a,b) and r e f e r e n c e s c o n t a i n e d t h e r e i n . T h i s summary w i l l do no more t h a n l i s t t h e e s s e n t i a l s o f t h e s u b j e c t . The D i r a c and P a u l i A l g e b r a s The D i r a c a l g e b r a , a s s u m i n g t h a t r e a l numbers o n l y a r e b e i n g u s e d , i s a 1 6 - d i m e n s i o n a l v e c t o r s p a c e w i t h a g e n e r a l " m u l t i v e c t o r " M h a v i n g t h e form M = oc ( s c a l a r ) + V ( 4 - v e c t o r ) + B ( b i v e c t o r ) (B.2) + V 5 U ( t r i v e c t o r o r p s e u d o v e c t o r ) + tf5 B ( p s e u d o s c a l a r ) , 130 where oc, 8 a r e ( r e a l ) s c a l a r s , V, U a r e 4 - v e c t o r s , B = r B ^ V A V i s a . b i v e c t o r and where 2 u v v 5 * v 0 A v 1 A v 2 A v 3 = v 0 v , v 2 v 3 ( B . 3 ) w h i c h s a t i s f i e s V V = -V s V , V-V,. = -1 , (B.4) where t h e b a s i s 4 - v e c t o r s a r e d e n o t e d (V ; p.= 0 , l , 2 , 3 ) as e x p l a i n e d i n App. A. The P a u l i a l g e b r a , a g a i n a s s u m i n g r e a l c o e f f i c i e n t s and a b a s i s (<*^) f ° r 3 - d i m e n s i o n a l s p a c e , i s an 8 - d i m e n s i o n a l v e c t o r s p a c e w i t h a g e n e r a l m u l t i v e c t o r m h a v i n g t h e form m = oc ( s c a l a r ) + u ( 3 - v e c t o r ) + i v ( b i v e c t o r o r (B.5) p s e u d o v e c t o r ) + i B ( p s e u d o s c a l a r ) , where i = a, A a 2 A o3 = a 1 a 2 a 3 (B.6) s a t i s f i e s i a. = a. i , i 2 = -1 , (B.7) l l w h i c h i s why t h e symbol i i s u s e d h e r e . The P a u l i a l g e b r a i s a s u b - a l g e b r a o f t h e D i r a c a l g e b r a , w h i c h i s e s t a b l i s h e d by t h e i d e n t i f i c a t i o n c. s V.A Vn = V.Vfl . (B.8) In t h i s t h e s i s , however, e a c h a l g e b r a i s u s e d s e p a r a t e l y and i n d e p e n d e n t l y ( t h e D i r a c a l g e b r a f o r s p a c e - t i m e , t h e P a u l i a l g e b r a f o r t h e SU(2) 3 - v e c t o r s ) , so t h a t (B.8) i s r e a l l y o f no f u r t h e r i n t e r e s t h e r e . P a u l i S p i n o r s The P a u l i a l g e b r a a d m i t s b u t one s e t o f p r o j e c t i o n 131 o p e r a t o r s (as any o t h e r would not commute w i t h i t ) ^ ( 1 ± a 3 ) , CT3 b e i n g t h e c o n v e n t i o n a l u n i t 3 - v e c t o r t o use h e r e . A s u i t a b l e s p i n o r b a s i s w o u l d t h e n be x + , where 1(1 ± 0-3) x ± • x ± • (B.9) One d e f i n e s an o p e r a t i o n ( + ) by m + = m, b u t w i t h a l l v e c t o r p r o d u c t s r e v e r s e d , so t h a t a.* - a., (m, m„ ) + = ni * m *, whence 1 1 1 2 2 1 i + = - i and one d e f i n e s t h e d u a l s p i n o r s x + f by x + x = S . (B.10) r s r s W i t h t h e d e f i n i t i o n a 1 x + = x_ ( B . l l ) t o f i x an o t h e r w i s e u n d e t e r m i n e d p h a s e , one has o - 1 x = x , o - _ x = ± i x _ (B.12) j- — + + + and x + X +* = J d + c 3 ) , X + x_ + = ~(ox + i a 2 ) (B.13) x_ x _ + = \ ( l - a 3 ) , x_ x + + = j ( o - 1 - i a 2 ) . The b a s i c t heorem f o r s p i n o r a m p l i t u d e c a l c u l a t i o n s i s , g i v e n P a u l i s p i n o r s <|>, and a m u l t i v e c t o r m, <!>+ m >i> = 2 S [m ^ <t>+] , (B. 14) where S[»»»] means t h e complex s c a l a r p a r t ( t h e f i r s t and l a s t t e r m s o f ( B . 5 ) ) o f w h a t e v e r m u l t i v e c t o r i s e n c l o s e d i n t h e s q u a r e b r a c k e t s . From (B.14) f o l l o w s * *+ = Ut>* *) + |(0> f o-. *P) 0. , (B.15) 2 2 1 1 w h i c h d i s p l a y s t h e i m p o r t a n t s t r u c t u r a l c o n n e c t i o n between s p i n o r s and v e c t o r s . The Gauge F i x i n g Terms o f S e c t i o n I I . 6 The gauge f i x i n g terms o f Sec. I I . 6 , Eq. ( 2 . 6 . 1 ) , ( 2 . 6 . 2 ) , 132 were w r i t t e n i n terms o f t h e P a u l i a l g e b r a . Here t h e y w i l l be e v a l u a t e d i n more d e t a i l . G i v e n t h e H i g g s s p i n o r f i e l d * = *+ *+ + *o *_ ' (B.16a) w i t h <$> = (v/J"2) x_ (B.16b) $* = 4> x * + ** x * (B.16c) - + o <$•> = (v/J-2) x _ + , (B.16d) where (*) d e n o t e s t h e complex and h e r m i t i a n c o n j u g a t e s o f t h e f i e l d s ( s u c h as <J> = * + ) > we have S[<$> $+ - $ <$+>] (B.17) = (v/J"2) S[+ x x f + <l> x x + - <t> x x • - • x x + ] - - + o - - + + - o - -= ~ ( v / J - 2 ) (• - •*) , 2 o o where t h e s c a l a r p a r t s were o b t a i n e d from ( B . 1 3 ) . I f V[«««] s t a n d s f o r t h e ( r e a l p l u s pseudo-) v e c t o r p a r t o f (B.5) we a l s o have V[<$> - $ <*•>] = j(v/J " 2 ) (B.18) * [ ( • - <J> ) a, + <t> (a - i a - ) + * (f f . + i f f - ) ] . O O 3 — 1 2 + 1 2 a g a i n e m p l o y i n g ( B . 1 3 ) . B o t h o f t h e s e r e s u l t s were needed i n (2.6.8) and ( 2 . 6 . 9 ) . The H i g g s v e c t o r f i e l d H i s , from ( 2 . 6 . 4 a ) , H = H a = ( l / > T 2 ) H 0 ( f f 1 - i f f 2 ) + ( l / J - 2 ) H + + ( f f x + i f f 2 ) + H + < T 3 (B.19) w i t h <H> = \ v f l (ff x - i f f 2 ) (B.20) H + = ( l / J " 2 ) H * ( f f l + i f f 2 ) + ( l / J - 2 ) H _ _ ( f f 1 - i f f 2 ) + H _ < T 3 (B.21) 133 <3+> = \ v f l ( f f l + i a 2 ) . (B.22) I f we use t h e i d e n t i t i e s ( a x ± i a 2 ) 2 = 0 (B.23a) ( a 1 ± i a 2 ) ( < r 1 + i<x 2) = 2 (1 ± a 3 ) , (B.23b) we o b t a i n S[<H>3+ - H<H + >] = - ( l / J " 2 ) v T I ( H - H*) - (B.24) H o o and V[<H>H* - H<3+>] = \ v f l (B.25) x [J/2(H - H * ) a 3 + H ( a - i a 2 ) - H (o + i c )] . O O 3 — 1 2 + 1 2 T h e s e , t o o , were needed i n t h e e v a l u a t i o n o f (2.6.8) and ( 2 . 6 . 9 ) . D i r a c S p i n o r s A b a s i s o f t h e 4 - d i m e n s i o n a l s p a c e o f D i r a c s p i n o r s ( u + ( p ) , v + ( p ) ) i s d e f i n e d by ( H a m i l t o n , 1984b) \(1 + p/m) \ [ \ ± (-i)V 5Sp/m] u + = u + (B.26a) | ( 1 - p/m) | [ 1 ± (-i)V 5Sp/m] v_ = v_ , (B.26b) 2 2 o + + where p i s t h e 4-momentum o f a p a r t i c l e o f mass m and S 2 (S = -1, S«p = 0) i s a u n i t s p a c e l i k e 4 - v e c t o r t h a t p o i n t s i n the s p i n d i r e c t i o n i n t h e r e s t frame o f t h e p a r t i c l e . I f M o f (B.2) i s a g e n e r a l m u l t i v e c t o r o f t h e D i r a c a l g e b r a (M=S+V+B+T+P) one d e f i n e s M = M, w i t h a l l p r o d u c t s r e v e r s e d (B.27) = S + V - B - T + P . A s p i n o r e q u a t i o n <J> = ocMip f o r a r b i t r a r y s p i n o r s <|>, <V, a (com p l e x ) s c a l a r «, and a s s u m i n g M has r e a l c o e f f i c i e n t s -- t h e b a s i s v e c t o r s (V ) a r e d e f i n e d t o be r e a l — has as an 134 " a d j o i n t " * = aMil* <--• * = oc*4»M , (B.28) w i t h t h e b a s i s s p i n o r s s a t i s f y i n g u u = S = -v v (B.29a) r s r s r s u v = 0 = v u . (B.29b) r s r s One f u r t h e r d e f i n e s M = M, w i t h a l l v e c t o r s r e v e r s e d (B.30) = S - V + B - T + P , and t h e complex c o n j u g a t e o f <t> = ocMty i s t a k e n t o be * E „ M (B.31) ( a g a i n a s s u m i n g M t o have r e a l c o e f f i c i e n t s ) . ( I n (B.31) M i s u s e d r a t h e r t h a n M b e c a u s e i n t h e l a t t e r c a s e c o n t r a d i c t i o n s w ould ensue. An a l t e r n a t i v e would be t o use M, b u t w i t h V s - y : I s i m p l y p r e f e r ( B . 3 1 ) . ) E v a l u a t i o n o f S p i n o r A m p l i t u d e s The b a s i c theorem f o r t h e e v a l u a t i o n o f s p i n o r a m p l i t u d e s i s * M i|> = 4 S [ M <\> <J>] , (B.32) where <t>, a r e any two D i r a c s p i n o r s , M i s any m u l t i v e c t o r o f t h e D i r a c a l g e b r a and where S[«««] r e f e r s t o t h e s c a l a r p a r t , i n t h e s e n s e o f ( B . 2 ) , o f t h e g e n e r a l m u l t i v e c t o r i n s i d e t h e s q u a r e b r a c k e t s . In more c o n v e n t i o n a l m a t r i x t e r m i n o l o g y 4 S [ « » « ] i s e q u i v a l e n t t o T r a c e [ • • • ] . As an example o f t h e s e methods c o n s i d e r t h e a m p l i t u d e s q u a r e d t h a t must be e v a l u a t e d i n t h e p -» ei t r a n s i t i o n , t h e e x p r e s s i o n f r o m ( 5 . 4 . 5 ) : 135 I . |u ( p r ) Z qeu ( p ) | 2 (B.33) s p i n s e + p — — * = I . U S q€u U ( Z q€) U ] s p i n s e + p n + e = 1 [u Z,q€u U CqZ U ] s p i n s e + p p - e = 2 4 S [ Z qeu u eqZ u u ] . s p i n s + p p - e e From (B.26a) we have 2 . (uu) = Ul + p/m) , (B.34) s p i n s 2 and (B.33) becomes (2ar.-i„a €? ^ 4 s f z+<1 * jd+P/M >V qzkl+p'/m )] , (B.35) s p i n s (oc) (oc) + p 2 p v - z e a c a l c u l a t i o n t h a t w i l l be c o m p l e t e d i n App. E. F u r t h e r r e l a t i o n s n e e d ed a r e Z V = V Z_ (B.36a) ± p P + V P V V = V ( P « V V + P A V V ) (B.36b) = v P v + v - ( P A / ) + v A ( P A * V ) V V V = "2 P , where, i f M i s a m u l t i v e c t o r , V M = V ' M + V A M (B.37a) u P P and, i f A, B. a r e 4 - v e c t o r s , l A • (B A B A • • - A B ) = I. (-1) U + (A-B. ) (B. A • • ' A B ) (B.37b) 1 2 n l l 1 n where B. i s m i s s i n g f r o m t h e l a s t f a c t o r , l The f o r e g o i n g i s f a r from t h e l i s t o f v a l u a b l e r e l a t i o n s t h a t would be needed i n t h e g e n e r a l c a s e but i s a d e q u a t e f o r t h e needs o f t h i s t h e s i s . The b r i e f example above i l l u s t r a t e s how r e a d i l y t h e m a t r i x - f r e e D i r a c a l g e b r a can be e x p l o i t e d t o e v a l u a t e t h e u b i q u i t o u s " t r a c e s " o f r e l a t i v i s t i c quantum m e c h a n i c s . 136 A Theorem f o r S p i n o r A m p l i t u d e s An i d e n t i t y t h a t i s most u s e f u l i n e s t a b l i s h i n g t h e C and T i n v a r i a n c e p r o p e r t i e s o f L a g r a n g i a n s i s as f o l l o w s . L e t ip, <t> be quantum f e r m i o n f i e l d s and M any e l e m e n t o f t h e D i r a c a l g e b r a , n o t assumed h e r e t o have r e a l c o e f f i c i e n t s . Then 4>M4> = - <j>* v M V c ** (B.38) __ I • M * i f 1 k -<t> M <|> , i f M = S,T,P M = V,B where t h e n e g a t i v e s i g n a r i s e s f r o m t h e a n t i c o m m u t i v i t y o f t h e f e r m i o n f i e l d s ( t h e s i g n s a r e r e v e r s e d f o r non-quantum f i e l d s ) . The p r o o f o f (B.38) i s t e d i o u s b u t s t r a i g h t f o r w a r d : one need o n l y w r i t e , f o r example, ip=<x u + B v ( r = ± ) , (B. 39) r r r r where t h e « , P a r e a n t i c o m m u t i n g F o c k - s p a c e o p e r a t o r s and u , r r r v b a s i s s p i n o r s , and t h e n e s t a b l i s h (B.38) f o r e a c h o f t h e r f i v e d i f f e r e n t t y p e s o f m u l t i v e c t o r e l e m e n t s . 137 APPENDIX C MANIFESTLY COVARIANT C, P AND T TRANSFORMATIONS I t i s c o n v e n t i o n a l ( i f not u n i v e r s a l ) t o employ s p e c i f i c m a t r i x r e p r e s e n t a t i o n s o f t h e D i r a c a l g e b r a t o c o n s t r u c t t h e i m p o r t a n t C, P and T o p e r a t o r s o f quantum f i e l d t h e o r y . But t h i s i s n o t n e c e s s a r y : t h e D i r a c a l g e b r a d e s c r i b e d i n t h e p r e v i o u s a p p e n d i x c an be u s e d t o c o n s t r u c t a p p r o p r i a t e t r a n s f o r m a t i o n s t h a t a r e n o t o n l y f r e e o f any m a t r i x r e p r e s e n t a t i o n o f t h e a l g e b r a , b u t t h a t a r e m a n i f e s t l y c o v a r i a n t as w e l l . T h i s a p p e n d i x w i l l d i s p l a y s u c h o p e r a t o r s and, as an example', a p p l y them t o quantum e l e c t r o d y n a m i c s . S e c t i o n I I . 7 c o n t a i n s t h e i r a p p l i c a t i o n t o more g e n e r a l gauge t h e o r i e s . The p a r i t y o r P t r a n s f o r m a t i o n t o be u s e d h e r e i s q u i t e c o n v e n t i o n a l ( B j o r k e n and D r e l l , 1 9 6 5 , Sec. 1 5 . 1 1 ) . The a p p r o p r i a t e t r a n s f o r m a t i o n f o r a s p i n o r f i e l d i|> i s <l> P(t,?) = P i p ( t , ? ) P _ 1 (C. 1) = V 0 « K t , - r ) , where P i s a u n i t a r y F o c k - s p a c e o p e r a t o r made up o f t h e 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 o f t h e s p i n o r f i e l d ^. The e l e c t r o m a g n e t i c v e c t o r f i e l d A s a t i s f i e s A P ( t , r ) = P A ( t , r ) P _ 1 (C . 2 ) = V 0 A ( t , - ? ) V Q , and t h e e l e c t r o m a g n e t i c i n t e r a c t i o n i s s e e n t o be P - i n v a r i a n t i n t h e u s u a l way ( T a b l e I I ) . The c h a r g e c o n j u g a t i o n o r C t r a n s f o r m a t i o n u s e d t h r o u g h o u t 138 t h i s t h e s i s i s s i m p l e r t h a n t h e c o n v e n t i o n a l one ( B j o r k e n and D r e l l , 1965, S e c . 15.12 ) . I t i s i p C ( x ) s c *Kx) C _ 1 (C.3) s * (x) , * where C i s a l i n e a r , u n i t a r y F o c k - s p a c e o p e r a t o r and where 4> i s d e f i n e d i n ( B . 3 1 ) . The e l e c t r o m a g n e t i c v e c t o r f i e l d A s a t i s f i e s A C = C A C _ 1 = -A , (C.4) and t h e D i r a c e q u a t i o n i s t h e r e b y made C - i n v a r i a n t : C ( i a - m - e A ) C _ 1 C ^ C _ 1 = 0 (C.5) c c = ( i a - m - eA ) >P * = ( i a - m + eA)«l> = [ ( i a - m - eA) 4>] = ( i a - m - eA)4> , where c a r e must be t a k e n i n t h a t (B.31) r e v e r s e s t h e s i g n o f v e c t o r s u n d e r t h e (*) o p e r a t i o n . T a b l e I I l i s t s t h e i n v a r i a n c e p r o p e r t i e s o f t h e QED L a g r a n g i a n u n d e r t h e C t r a n s f o r m a t i o n . When a p p l i e d t o a s p i n o r f i e l d / .. „ , 3 , - i p * x ip«x * 4>(x) - Jd p (e * a u + e ^ b v ) (C.6) r r r r * w i t h b a s i s s p i n o r s u , v s a t i s f y i n g ( B . 2 6 ) , we have (u ) =v , r r \ r r from (B.26) and (B.31) ( w i t h an a r b i t r a r y p h a s e s e t e q u a l t o o n e ) , and so c, > */ v - ,3 , ip«x * -ip«x, . ,„ „. * ( x ) = + ( x ) ~ J \ l p ( e K a v + e * b u ) (C.7) r r r r w h i c h was needed i n Chap. VI. The CPT t r a n s f o r m a t i o n t o be a d o p t e d i s (e = c p t , 0 = TPC) 139 i|>° P t(x) H »iie(x) a © vj,(x) ® - 1 (C.8) a * (-X) (~X) , w h i c h r e q u i r e s t h e t i m e r e v e r s a l ( o r r e v e r s a l o f m o t i o n ) o r T t r a n s f o r m a t i o n t o be q^Ct.r) a T *(t,r) T " 1 (C.9) 5 V 0 * ( - t , r ) , i n marked c o n t r a s t t o t h e u s u a l T and CPT t r a n s f o r m a t i o n s ( B j o r k e n and D r e l l , 1965, S e c . 15.13). As u s u a l t h e o p e r a t o r s T and © a r e a n t i l i n e a r . T h a t t h i s p r o p e r t y i s c r u c i a l w i l l be s e e n when t h e f u l l i n v a r i a n c e o f t h e s p i n o r - v e c t o r e l e c t r o m a g n e t i c i n t e r a c t i o n i s d e m o n s t r a t e d ( T a b l e I I ) . The e l e c t r o m a g n e t i c v e c t o r p o t e n t i a l A t r a n s f o r m s as e -1 A (x) = © A ( x ) ® (C.10) = - A ( - x ) . T a b l e I I c o n t a i n s a d e t a i l e d l i s t o f t h e t r a n s f o r m a t i o n p r o p e r t i e s o f t h e v a r i o u s components o f t h e f e r m i o n - p h o t o n L a g r a n g i a n , e x c e p t f o r t h e k i n e t i c terms f o r t h e p h o t o n f i e l d w h i c h a r e r a t h e r t r i v i a l . One s h o u l d n o t e t h a t t h e i m p o r t a n t i d e n t i t y (B.38) has been u s e d w h e r e v e r ip a p p e a r s , and n o t e t h e i m p o r t a n c e o f t h e a n t i l i n e a r i t y o f T and ® i n t h e (iipaip) p a r t o f t h e f e r m i o n L a g r a n g i a n . O n l y r e a l c o e f f i c i e n t s have been i g n o r e d . Some o t h e r u s e f u l m a n i p u l a t i o n s t h a t a r e f r e e l y employed a r e x -• -x, t -» - t , r -» - r i n t h e a c t i o n i n t e g r a l J*d x C ( x ) , w h i c h a r e p e r m i t t e d by t h e a c t i o n i n t e g r a l ' s i n v a r i a n c e , and t h e p r o p e r t i e s o f t h e 4 - v e c t o r V Q and o f tf5 d i s c u s s e d i n App. 140 B. A l s o p e r m i t t e d i s C -» C + a (•••)". v The C, P and T p r o p e r t i e s o f t h e e l e c t r o m a g n e t i c i n t e r a c t i o n a r e c o n s i d e r e d i n d e t a i l b e c a u s e o f t h e a s s u m p t i o n made i n Sec. II.7 t h a t t h e v e c t o r gauge f i e l d s t r a n s f o r m e x a c t l y as does t h e e l e c t r o m a g n e t i c v e c t o r p o t e n t i a l . 141 TABLE I I ; C. P AND T INVARIANCE OF ELECTROMAGNETISM INVARIANCE LAGRANGIAN TERM .v — v . .7 =<I>V ip iipaip p Pj P PiipaipP 1 = ^ V 0 V V V 0 * ( - r ) = i ^ V 0 V V V o a v ^ ( - r ) = ( J ° . - / ) ( - ? > ( r -» - r ) -» iipaip c c j V 1 ciipaipc 1 — • V = <\> V ip , — • V = l H 3 * V = - i a iptf. V^y. ip 1/ 5 5 = - J = iipv va ip + a ( • • •) V V -> iipaip T Tj VT 1 Tiipa<pT 1 = g>*0 v v v 0ip(-t) . -t t = -up aip = (j°,-j1)(-t) = -i^P* 0* vV 0a v<P(-t) -» iipaip (as t -» -t) TPC .v -1 ©J ® ©iipaip® -1 — * v • . . ip * ip (-x) . -e , e = -up aip -iPV5* tfs*(-x) •up v a iP (-x) V - j"(-x) = i ( a iPVc V VciP) (-x) •i(+v a iP) (-x) + a ( • • •) -» iipaip (as x -» -x) 142 TABLE II (cont'd.) INVARIANCE LAGRANGIAN TERM ^ A t p P W P 1 P^AipP 1 = ^ V 0 V 04«(-?) = * * o V o A * o * o ^ -?) ( r -> - r ) ( r - r ) -» ipA^ 1 c ^ c - 1 cipA^C 1 -* * = <l> (-A) = -*v5v5* = ^ V 5 A V 5 * = ** = <M<P T T<M>T 1 TipA4«T 1 = *v0v0*(-t) = ^ V 0 V 0 A V 0 V 0 * ( - t ) (t - - t ) (t - - t ) -+ ipAip © i p ^ ® -* * = <i> <P (-x) = - ( * V s V s * ) ( - x ) (x •• -x) -» ©tpA^® -* * = «C (-A)* (-x) = (*V5 AV54») (-x) (x -» -x) -» <4>A^  143 APPENDIX D  ON INVARIANT SPACE-TIME INTEGRALS Most o f t h e i n t e g r a l s t h a t r e s u l t e d from t h e Feynman d i a g r a m s o f Chap. V, VI had t h e form f d 4 p f ( p 2 ) o r r/d 4 P p p f ( p 2 ) , ( D . l ) p v w h i c h a r e n o t a t a l l t r i v i a l t o e v a l u a t e . T h i r t y y e a r s o f e x p e r i e n c e w i t h quantum e l e c t r o d y n a m i c s has, however, r e s u l t e d i n s t a n d a r d methods o f c o m p u t i n g s u c h i n t e g r a l s when t h e y c o n v e r g e , and methods o f h a n d l i n g them, s u c h as 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 , when t h e y do n o t . T h i s a p p e n d i x w i l l l i s t t h e r e s u l t s t h a t a r e needed i n t h i s t h e s i s . F i r s t , i t i s n o t d i f f i c u l t t o s e e t h a t a l l s u c h i n t e g r a l s must have t h e form o f t h e f i r s t o f ( D . l ) b e c a u s e T d 4 p f ( p 2 ) p = 0 (D.2) p and ,fd 4p f ( p 2 ) p p = \ „ f/d4p p 2 f ( p 2 ) (D.3) p v * pv ( B j o r k e n and D r e l l , 1964, Chap.8; A k h i e z e r and B e r e s t e t s k i i , 1965, Sec. 4 7 . 1 ) . Next, one n o t e s t h a t a l l s u c h i n t e g r a l s t h a t a r e r e l e v a n t have p o l e s . One i n v o l v i n g a f e r m i o n p r o p a g a t o r w o u l d be f / d 4 P f ( P 2 ) = Jd*p f ( p 2 ) (p+m - ie ) (D.4) 2 2 (p-m+ie) p - (m - i e ) where t h e i € p r e s c r i p t i o n i s v e r y b r i e f l y e x p l a i n e d i n Sec. IV.3, w h i c h shows how t h e p o l e s a t p Q = ± ( m - i e ) a r e t o be a v o i d e d ( F i g . 2 7 ) . As t h e p o l e s can be a v o i d e d by a r o t a t i o n i n t h e complex p Q - p l a n e ( F i g . 27) t h e i n t e g r a l can be 144 IE F i g . 27: The complex p Q - p l a n e t r a n s f o r m e d i n t o one w i t h a E u c l i d e a n m e t r i c . From C a u c h y ' s theorem, s i n c e no p o l e s a r e e n c l o s e d w i t h i n t h e c o n t o u r c ( F i g . 27) , S f ( P 0 ) dp = 0 = (S°° + J""" 1) f ( p 0 ) d p Q , (D.5) C co col where t h e a s s u m p t i o n t h a t t h e i n t e g r a n d s v a n i s h s u f f i c i e n t l y r a p i d l y B S I P I oo has been made. W i t h a change o f v a r i a b l e , t h e n , P 0 s i P g , (D.6) we have N SZ F ( P 0 > d P o = 1 *Z f ( i P o ) dp^ . (D.7) 00 00 S i n c e P 2 = P 2 - I P I 2 = - ( p ; 2 + I P I 2 ) = - q 2 . ( ° - 8 ) 145 one can use t h e p a r a m e t r i z a t i o n p x = q sinot s i n e cos<|> (D.9) P 2 = q sinoc s i n e sin<|> P 3 = q sin<x c o s e P Q = q cosoc , (0<<J>£2w, Oie,<xiw) where (D.8) i s s a t i s f i e d , and 4 3 2 d p ' - dp dp dp 3dpg = q s i n oc s i n e dq doc de d<t> , (D.10) so on p e r f o r m i n g t h e a n g u l a r i n t e g r a t i o n s we have d V = 2 W 2 q 3 dq = TT 2 q 2 d ( q 2 ) . ( D . l l ) Thus we have, f i n a l l y , f d 4 p £ ( p 2 ) = i / d V f ( - q 2 ) (D.12) = i w 2 JC° q 2 d q 2 f (-q 2 ) o = i i r 2 J*" z f ( - z ) dz . o We a r e now i n a p o s i t i o n t o e v a l u a t e t h e t h r e e ( f i n i t e ) t y p e s o f i n t e g r a l s a p p e a r i n g i n Chap. V, VI and t h e n e x t two a p p e n d i c e s : d 4 p 1 = i TT2 (D.13) ( p 2 + A ) 3 2 A d 4 p 1 = i TT2 (D.14) ( p 2 + A ) 4 6 A 2 d 4 p p 2 = i TT2 (D.15) ( p 2 + A ) 4 3 A . w h i c h a g r e e w i t h t h e more g e n e r a l r e s u l t s o f Nash ( 1 9 7 8 ) . The method o f Feynman p a r a m e t e r s ( B j o r k e n and D r e l l , 1964, Chap. 8) i s u s e d t o r e d u c e i n t e g r a l s w i t h s e v e r a l p r o p a g a t o r s i n t o t h e form (D.12): 146 The s p e c i a l c a s e s t h a t a r e needed h e r e a r e _1 = f/JdocfJdB S ( « + B - l ) (D.17) (ax<x+a2B)2 a a 1 2 = 4 d « [ a x « + a 2 ( l - « ) ] 2 a a a 1 2 3 = 2rJdoc.rJde.rJdy S ( o c + B + V - l ) (D.18) ( a x <x+a2 B+a3 V ) 3 = 2rJdocjJ " K dB [a l 0 c+a 2 B+a 3 (l-oc-B) ] and a / 1 \ (D.19) a i a 2 a 3 ^ 1 \ S i a 2 a 3 = 6jJdocrJ _ , x d0 [ a l 0 c + a 2 B+a3 ( l - o c - B ) ] T h i s p a r a m e t r i z a t i o n t e c h n i q u e , a l o n g w i t h (D.12), i s a l l t h a t i s r e q u i r e d t o e x p l i c i t l y e v a l u a t e t h e i n t e g r a l s i n t h i s t h e s i s t h a t r e s u l t f r o m t h e Feynman d i a g r a m s o f Chap.V, VI. 147 APPENDIX E  DETAILS OF THE CALCULATION u -> e V T h i s a p p e n d i x c o n t a i n s t h e d e t a i l s o f t h e c a l c u l a t i o n s o f Sec. V.3, V.4 t h a t were o m i t t e d t h e r e b e c a u s e o f t h e i r l e n g t h and e s s e n t i a l l y t e c h n i c a l n a t u r e . The I n t e g r a l s (5.3.14-18) 6 —4 2 1 2 A p a r t f r o m a common f a c t o r o f [ i (2 * 0 g e U U m.] t h e * 1 1 1 1 £• 1 i n t e g r a l s t o be e v a l u a t e d a r e I(W)= -/d 4 k u £ tf (p - k ) V u < ^ P ( k-q ) * V a ( k ) € X r ( E . l ) e + p 4 v p Xo-p (P - k ) I ( S ) = Td 4 k u £ ( P ~ k ~ M p ) u 2k-€ (E.2) M 2 6 + ( P - k ) 2 » ( k 2-CM 2 ) [ ( k-q) 2-£M 2j W W W I(WS) = f d 4 k u E + V v u A y < J ( k ) € a (E.3) 6 ( p - k ) 2 » [ ( k-q) 2-£M 2] I(SW) = - f d 4 k u £ * / -1 + M p ( P " k ) | u A V g ( k-q) € a (E.4) e + V I . , . 2 , . 4 j P . 2 ,.2 . v (P - k ) (p - k ) ' (k -CM w) 1(B) = Td 4 k u E ( M P + ( p - k ) c o t 2 e \ u 2k«€ (E.5) M 2 6 + Wp - k ) 2 ( p - k ) 2 / * ( k 2-M 2 ) [ ( k-q) 2-M 2] where, from ( 5 . 2 . 1 ) , A(k) = A l ( k ) + A 2 ( k ) + A 3 ( k ) (E.6) ^ " ( k ) = - k M k V / [ M ; ( k 2 - M ; ) ] ^ v ( k ) = k V / [ M ; ( k 2 - c M ; ) ] and, f r o m ( 5 . 2 . 3 ) , r = (-q-k) -n + (2k-q) -n + (-k+2q) -n . (E.7) Xop p Xo X op o Xp 2 B e c a u s e q =0, q « e = 0 , one can show t h a t ( k-q ) P k C T € X r = 0 = ( k-q ) C T k P € X r , (E.8) Xcp Xop w h i c h means t h a t t h e o n l y n o n - v a n i s h i n g f a c t o r s i n ( E . l ) a r e 148 ^ P ( k - q ) ^ ° ( k ) , A ^ P ( k - q ) ^ C T 3 ( k ) , A M P 3 ( k - q ) ^ C T ( k ) . (E.9) F o r s i m p l i c i t y we d e f i n e ( i = 2 , 3 ) w i t h ^ V ( k ) = a . k * V / [M 2 ( k 2 - b . M f j ] ( E l O a ) 1 1 W l W a 2 = -1, a 3 = +1, b 2 = +1, b 3 = £, ( E . l O b ) so o n l y t h r e e i n t e g r a l s w i l l have t o be worked out t o e v a l u a t e ( E . l ) . The f i r s t p a r t o f ( E . l ) i s , from (D.19), I M 4 I , " ^ ,P/ [ _ e ( < l + k ) + 2k-€n + 6 (2q-k) ] ... - J d k u E V ( p - k ) V u o g gjz g a ( E . l l ) 6 + U ( p - k ) 4 ( k 2 - M 2 ) [ ( k - q ) 2 - M 2 3 = -J"d 4k 6J\xd«dB \ • • -1 { « ( p - k ) 2 + 0 [ ( k - q ) 2 - M 2 ] + ( l - o c - 8 ) ( k 2 - M 2 ) } 4 = - / d 4 k STocdocdB r « " 1 [ ( k - V ) 2 + B ] 4 where (D.19) has been u s e d , w i t h V a ap + Bq (E.12a) B = -V 2 + « p 2 - ( l - o c ) M 2 • - ( l - o t ) M 2 . (E.12b) w w We d e f i n e t h e new i n t e g r a t i o n v a r i a b l e K = k-V and o b t a i n - T d 4 K 6 .r<xd«dB u E y P ( p - K - V ) y C T u (.E.13) e + p. * [-€ (q+K+V) +n 2(K+V)-€+€ (2q-K-V) ] / ( K 2 + B ) 4 . a p ap p a 1 2 The n u m e r a t o r i n (E.13) has f a c t o r s K K -* n K ( f r o m ( D . 3 ) ) , P v 4 pv 2 K ( w h i c h v a n i s h , f r o m ( D . 2 ) ) , and a c o n s t a n t . E a c h o f t h e K P f a c t o r s g i v e s t h e s p i n o r a m p l i t u d e u E €u , w h i c h , f r o m e + p 2 ( 5 . 1 . 1 4 ) , has t h e wrong s t r u c t u r e and so t h e K terms a r e t h u s i g n o r e d . I n t e g r a l (E.13) t h e n becomes -/d 4K6J \xdocd0u E V P ( p - V ) V q u [-€ (q+V) + 2 V . € T I + € (2q-V) ] (E.14) 6 + ( K 2 + B ) 4 M ° P ° P 9 = - J T T 2 / « d « d B u E y p ( p - v ) y a u [-€ (q+V) +2V.€n +€ (2q-V) ] , 4 e + . 2 p a p ap P a \ (l-«) 149 from (D.14) and ( E . 1 2 b ) . The f i r s t t erm i s (scalar)«u Z €u e + p (wrong s t r u c t u r e , so i g n o r e ) , t h e s e c o n d becomes -1T T 2\ [-4p-€M ) u Z u J - ' cxd«J ,i~°td8 « 2 (1 -oc-B) (E.15) Ifl 1 e + p ° W 2 2 — = - lTT p •€ M u Z u , .,4 P e + n Mw and t h e t h i r d becomes - i T r 2 \ M u Z € P ' u r 'ocdocJ-^^dB r 2(l-«)-Bl (E.16) I**! P e + p 0 0 2 M w' (1-oc) = - 3 - i i T 2 u Z € p ' u M /M4 . 4 e + p p W The s e c o n d p a r t o f ( E . l ) u s e s t h e s e c o n d f a c t o r i n ( E . 9 ) : - . f d 4 k u Z V ( P-k) * u n M P a. k v k g (E.17) 6 + M ( P - k ) 4 v ^ ( k - q ) 2 - M ; ] 1 M a f ( k a - b i M i ) x [-€ (q+k) + 2 k . e - n + e (2q-k) ] <T p ap p CT = ^ 6 3 . r<xd«dB j ' d 4ku Z f-k •€ (g + k) + 2k »€k+e ( 2g-k-k 2 )1 ( p - k ) k u M 2 1 6 + { « ( p - k ) 2 + B [ ( k - q ) 2 - M 2 ] + ( l - o c - B ) ( k 2 - b . M 2 ) } 4 t l = ^6 . 8 . r«dtxdBj*d 4K u Z [-(K+V) •€ (q+K+V) + 2(K+V) •€(K+V ) M 2 1 E + W + € ( 2 q • ( K + V ) - ( K + V ) 2 ) ] ( p - K - V ) ( K + V ) u ( K 2 + B ) 4 M w i t h t h e s u b s t i t u t i o n s (E.12) and k -• K+V. In t h e n u m e r a t o r 4 — a l l K terms g i v e t h e wrong s p i n o r a m p l i t u d e u Z €u , so a r e e + p 3 d r o p p e d ; t h e K and K terms v a n i s h on i n t e g r a t i o n ; t h e c o n s t a n t g t e r m g i v e s an o v e r a l l l / ^ w f a c t o r w h i c h i s i g n o r e d b e c a u s e t h e K 2 f a c t o r i s t h e dominant f a c t o r ("1/M 4). Thus (E.17) becomes W -6a. rocdocdBJ*d4K u Z f 14 terms o f o r d e r K 2 1 u (E.18) W = ~6a./ i t r 2 \M u Z qeu f' ocdoc " " d B ( - o c 2 + oc/2 -ocB-5 B/4+1 /2) M 2 M -3M 2' M 6 + 1 1 [ b . ( l - o c ) + B d - b . ) ] W W i i where (D.15) was u s e d . T h i s i n t e g r a l a l o n e has t h e v a l u e s 150 1/48 , b 2 = 1 (E.19a) J L 1 f _ O n l ] + 1 i n L . b 3 = C . (E.19b) I 24 ( C - l ) 1 C - l / 6 C ~ l The t h i r d , and f i n a l , p a r t o f ( E . l ) u s e d t h e t h i r d f a c t o r i n ( E . 9 ) : - r d 4 k u E V (P-k) V u a, ( k - g ) U ( k - g ) P n v a (E.20) (p-k) M w t ( k - q ) - b i M w ] ( k - M ) x [-€ (q+k) + 2k'€n + € (2q-k) ] a p ap p CT = z 6 a . TocdocdBr d 4 K u Z [-(K+V-q) • ( q+K+V) (K+V-q) (p-K-V) e „ 2 i /.,2 „ v 4 e + M ( K + B ) W + 2€•(K+V)(K+V-q)(p-K-V)(K+V-q) + € - ( K + V - q ) ( K + V - q ) ( p - K - V ) ( 2 q - K - V ) ] u 2 where, a g a i n , o n l y t h e K f a c t o r s i n t h e n u m e r a t o r make t h e most i m p o r t a n t c o n t r i b u t i o n . We o b t a i n -6a./ i i r 2 l M u Z qeu <xd<x.r!;~ C CdB(-cx 2+ tx/2- txB+B/4) (E.21) M « X |- 3 Mw' " 6 + H [ d - c c ) + B ( b . - D ] W W 1 and t h e i n t e g r a l a l o n e has t h e v a l u e -5/48 , b 2 = 1 (E.22a) 12 C - l 24 C - l \ C - l ' b 3 = C . (E.22b) When ( E . 1 6 ) , (E.19) and (E.22) a r e added we o b t a i n t h e f i n a l r e s u l t f o r I(W) = ( - i T f 2 M |u Z qeu [1/4 - f ( C ) / 1 2 + g ( C ) / 2 ] (E.23) W where f ( C ) a ( l n C ) / ( C - l ) (E.24) g(C) = ( C l n C ) / ( C - D 2 - 1/(C-1) , as q u o t e d i n ( 5 . 3 . 2 1 a ) . The i n t e g r a l I ( S ) o f (E.2) i s , compared w i t h I(W) 151 i m m e d i a t e l y p r e c e d i n g , much s i m p l e r . We have, from (D.18), w „ s „ U Z ( p - k ~ M )u 2k«€ I ( S ) = 2/docderd k e + p u  { « ( p - k ) 2 + P [ ( k - q ) 2 - C M 2 J ] + ( l - o c - 0 ) ( k 2 - £ M 2 J ) } 3 W W = 4/dcxdp/d 4K u Z ( p - K - V - M )u (K+V) •€ (E.25) U M ( K 2 + B ) 3 where t h e s u b s t i t u t i o n k -» K+V i s a g a i n made, where V i s t h e 2 same as i n (E.12a) and B t h e same as i n ( E . 1 2 b ) , e x c e p t M -• W 2 2 £ M ^ . The K term i n t h e n u m e r a t o r g i v e s t h e wrong s p i n o r a m p l i t u d e u Z €u , and so i s i g n o r e d . Thus (E.25) becomes, e + p. u s i n g ( D . 13), I ( S ) = / 4 J T T 2 \ M u Z u p-€ r i d o c r i _ o c d p « ( - c x - B ) (E.26) U * M ) » 6 + » ( ! - « ) W and t h e i n t e g r a l a l o n e has t h e v a l u e (-5/12). T h i s gave t h e r e s u l t (5.3.21b) as q u o t e d . I n t e g r a l (E.3) i s (E.27) I(WS) = Td 4k V + V u € a ( p - k ) 2 [ ( k - q ) 2 - C M 2 ] W VO „ V O —31 . I. a, k_J< 2 2 + 1 1 2 2 2 k -Mtf M w ( k -b.M w), and we s e e a t once t h a t t h e f i r s t t e r m ( u s i n g •t\V<J) l e a d s t o t h e wrong s p i n o r s t r u c t u r e u Z €u and i s t o be i g n o r e d . The e + p. r e m a i n d e r i s I(WS) = I. ti. 2Jd«dBj'd4k U e Z + k U p k * € (E.28) 1 M 2 { < x ( p - k ) 2 + B [ ( k - q ) 2 - C M 2 ] + ( l - « x - B ) ( k 2 - b . M 2 ) } 3 W W 1 w = I . f l 2rd«dBJ"d4K u Z (K+V)u (K+V) «€  1 M 2 6 + » ( K 2 + B ) 3 w where, h e r e , B = - M 2 [ b ( l - « ) + B ( C - b ) ] . The K 2 term g i v e s t h e wrong s t r u c t u r e and we have, k e e p i n g o n l y t h e r e s t , I(WS) = I. S i / - i t r 2 \ 2p-e u Z u M J"'d<xf'_0CdB txU+B) (E.29) 1 < [ K' e + ^ ° b . ( l - c c ) + B ( C - b . ) W W i i and t h e i n t e g r a l a l o n e has t h e v a l u e s 152 1 [ 1 (2C-3) InC \ , b, = 1 (E.30) elc-i + (c-i)V 5_ 1 , b 3 = C , 12 C w h i c h , w i t h ( E . 2 4 ) , g i v e s t h e r e s u l t q u o t e d i n ( 5 . 3 . 2 1 c ) . I n t e g r a l (E.4) i s I(SW) = - J d 4 k u E V / -1 M u ( P ~ k M u € q (E.31) 6 + V 1(P-K) 2 + ( p - k ) 4 / " ( k 2 - C M 2 ) w * [ Jill S,a. (k-q) V ( k - q ) C T \ ( k - q ) 2 - M 2 M 2 [ ( k - q ) 2 - b . M 2 ] so, expanded, i s f o u r i n t e g r a l s . The p r o d u c t o f t h e f i r s t o f e a c h o f t h e two sums g i v e s t h e wrong s p i n o r s t r u c t u r e , and as pu =M u , a l l t h a t r e m a i n s o f t h e T\ term i s M J d 4 k u Z + € k u / 1 \/ 1 ) (E.32) 6 ( p - k ) 4 MU 2-CM;/l(k-q ,) 2-M; J = 6M T o c d « x d p f;d 4K u Z + € ( K + V ) u 6 ( K 2 + B ) 4 1 1 w i t h t h e u s u a l V and B = -M2 [ ? ( l - o c ) + B ( l - £ ) ] . The term l i n e a r W i n K v a n i s h e s and we have, d e l e t i n g terms w i t h (u epu ) (wrong e p s t r u c t u r e ) , 6M / i w M (-u E € P ' u ) J"* ceded"'""dp B (E.33) ^WJ e + » £ ( l - o < ) + B ( l - 0 w i t h t h e i n t e g r a l h a v i n g t h e v a l u e 1 _ InC ) . (E.34) 2 (c-l ( C - l ) 2 Back i n (E.31) t h e f i r s t t e r m i n t h e f i r s t sum, and t h e s e c o n d term i n t h e s e c o n d sum g i v e t h e wrong s t r u c t u r e , and what r e m a i n s i s 153 „ a.M _ J4, - Z ( k - q ) k , , „ £. i p Jd k u + u kj_€ ( E . 3 5 ) 1 M* 6 ( p - k ) 4 ( k 2 - C M 2 J ) ^ [ ( k - q ) 2 - b . M 2 ] W W l w a M 4 — = 6 7 , i u TocdocdB/d K u Z (K+V-q) (K+V) u (K+V)«€ 1 M 2 6 + ( K 2 + B ) 4 " w 2 2 where B h e r e i s -M ,[£ (1-oc) +B(b .-£ ) ] , w h i c h becomes, as o n l y K W I terms a r e needed, 6 7 . a i M u / i t r ^ ^ r ' t d c c T ^ ^ d B ( - l + 3oc) ( E . 3 6 ) M„ \-3M'/ £(l-oc) + B ( b . - £ ) W W 1 and t h e i n t e g r a l a l o n e i s 1 i n L . b 2 = 1 (E.37) 2 £-1 l/(2£) , b 3 = £ . The sum o f (E.34) and (E.37) g i v e s t h e v a l u e o f I(SW) q u o t e d e a r l i e r i n ( 5 . 3 . 2 1 d ) . The f i n a l i n t e g r a l , ( E . 5 ) , i s n ~ A A 0 r A * i " E _ J M + ( P ~ k ) c o t 2 e ] u 2k-e 1 ( B ) = 2 / d o t d B j d k e + p y (E.38) M 2 { « ( P - k ) 2 + p [ ( k - q ) 2 - M 2 ] + ( l - o c - B ) ( k 2 - M 2 ) } 3 = 2 J d o c d B/d 4 K u Z [ M u + ( p - K - V ) c o t % ] u 2 ( K + V ) . £ M ; ( K 2 + B ) 3 » 2 where B = -(l-oc)M , w h i c h i s v i r t u a l l y t h e same as ( E . 2 5 ) . I t s B f i n a l v a l u e i s K B ) = J T T 2 M (1 + j c o t 2 e ) i-u Z qeu (E.39) M 2 2 p. 6 2 e + p. W B as q u o t e d i n ( 5 . 3 . 2 1 e ) . The S p i n o r A m p l i t u d e (5.4.5) The e v a l u a t i o n o f t h e s p i n o r a m p l i t u d e I . |u Z qeu | 2 (E.40) s p i n s e + p i s n e i t h e r new n o r p a r t i c u l a r l y d i f f i c u l t , b u t w i l l be e v a l u a t e d u s i n g t h e n o v e l t e c h n i q u e s d e s c r i b e d i n App. B . We 154 use (5.4.4b) and (B.35) t o o b t a i n 2 . |u Z q€u | 2 = fV , 4 S [ Z q* _-(l+p/M )V qZ .-(l+p'/n )] s p i n s e + p oc (oc) (oc) + p.2 p v -z e = - n ^ S f Z q* PV qp' ] / M m = 2S [ \ ( 1 - i V 5 ) qpqp' ] / M m + p v p e p e = S [ q p q p r ] / M m = 2 q«p q-p' / M m ( E . 4 1 ) p e p e s i n c e q 2 = 0, and Z Z = 0 , Z Z = Z , and + - + + + qpqp' = qp(q«p' + q A p ' ) = q[pq*p' + p * ( q A p r ) + P A ( « A P ' ) -= (q-p + q A p ) q - p ' + q - [ p - ( q A p ' ) ] + q - ( p A q A p ' ) (E . 4 2 ) o f w h i c h o n l y t h e s c a l a r p a r t i s needed, o b t a i n e d f r o m ( B . 3 7 b ) . The I n t e g r a l (5.4.9) The phase s p a c e i n t e g r a l rd3q dV S 4 ( p - q - P f ) ( E . 4 3 ) w i l l be e v a l u a t e d u s i n g a t e c h n i q u e d e s c r i b e d i n K a e m p f f e r (1965, App. 6 ) . F i r s t i t i s r e w r i t t e n as J d 4 q S ( q 2 ) j [ l + € ( q ) ] dV S 4 ( p - q - p ' ) ( E . 4 4 ) where pi €(«> • {*}• q ° v. -1. q. >0 (E.45) i , q 0 < o , w h i c h becomes, b e c a u s e t h e q - i n t e g r a t i o n can be p e r f o r m e d , rdii J P : Now, s i n c e p ' $ r(p-P f ) 1 2 - [ l + €(p-p' )] . (E.46) Po ( P - P ' ) 2 = P 2 +P' 2 -2p-p' = M 2 + m2 - 2M p^ (E.47) p e p w h i c h v a n i s h e s a t p'Q = E', so t h a t as S [ ( P - P ' ) 2 ] J [ 1 + € ( P - P ' ) ] = / d ( p - p ' ) 2 x _ 1 dp: S ( P ; - E ' ) (E.48) = E R S ( p ' 0 - E ' ) 2M 1 5 5 and d V = 4ir|p' | 2 d | p ' | = 4 T T | P ' | p 0 r d P ; ( E . 4 9 ) we have 4 Wr|5 ' S ( P o " E , ) = (2 W/M ) ( E ' 2 - M 2 ) 1 / 2 ( E . 5 0 ) J p£ 2 M 1 1 . 6 U = (2ir/M ) [ ( M 2 - m 2 ) / ( 2 M ) ] • W P P e p. s i n c e M >>m . T h i s was t h e v a l u e c l a i m e d e a r l i e r , i n ( 5 . 4 . 9 ) . P e 156 APPENDIX F DETAILS OF THE CALCULATION p -> 3e T h i s a p p e n d i x c o n t a i n s t h e d e t a i l s o f t h e c a l c u l a t i o n s n e e d e d i n Chap. VI f o r t h e p. -» 3e Feynman d i a g r a m s , t r a n s i t i o n a m p l i t u d e and d e c a y r a t e . The I n t e g r a l (6.2.2) We now e v a l u a t e t h e i n t e g r a l T / n \ r . 4 , u Z ( p - k ) u u ( 2 k - q ) v 1(B) = fd k e + p e e_ ( F . l ) J ( p - k ) 2 [ ( k - q ) 2 - M 2 j ] ( k 2 - M 2 j ) j D r ^ n i " E ( p - k ) u u ( 2 k - q ) v = 2JaocdBJa k e + p e e { < x ( p - k ) 2 + B t ( k - q ) 2 - M 2 ] + ( l - o c - 0 ) ( k 2 - M 2 ) } 3 = 2 f/dcxdp.rdnK U e E + ( p " K " V ) U p u (2K+2V - q)v ( K 2 + B ) 3 where by 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 one imposes n<4 t o a l l o w t h e s h i f t i n t h e i n t e g r a t i o n v a r i a b l e , and where V = ap+0q, 2 2 B«-(l-ot)M . The K o r d i v e r g i n g p a r t as n-»4 i s , f r o m ( D . 3 ) , B - ( u E V u ) ( u V V v ) J d a d B ^ K K 2 ( F . 2 ) e + v p e e ( R 2 + B ) 3 as g i v e n i n ( 6 . 2 . 3 ) . The f i n i t e p a r t i s , f rom (D.13), ( - i 4 K \-2M / 1-oc - -docT„ dp u E (p-ocp-Bq)u u ocpv (F.3) ^ , ° e + u e e • 2M ; / (l-oc) m B b e c a u s e , as q = p 2+p', u ( P 2 ) q v (p')=u ( p 2 ) ( p . + p r ) v (p')=u ( p 2 ) ( m -m )v (p')=0. (F.4) e 2 e e 2 2 e e 2 e e e I n t e g r a l (F.3) becomes 4[irr 2/(-2M 2)]M (u E^u ) ( u pv ) fl doc/^""dBoct l - oc -B )/(1 - oc ) (F.5) B p e + p e e ° 0 and t h e i n t e g r a l a l o n e has t h e v a l u e 1/12, a l l o f w h i c h l e a d t o t h e r e s u l t q u o t e d i n (6.2.3). 157 The I n t e g r a l (6.2.4) W i t h t h e n e u t r i n o p r o p a g a t o r expanded as i n (5.3.1) t h e i n t e g r a l (6.2.4) becomes /. , 4 , u V pZ ( p - k ) u - x , _ A ( r e n o r m ) = _C fd k e x + P u V v (F.6) P q (P-k) (k - M ) n p j p j n i " V p £ ( p - k ) u - X = C Jdoc/d k e x + \x u V v M 2 q 2 [ c c ( p - k ) 2 + ( l - « x ) ( k 2 - M 2 I ) ] 2 6 6 = _ c _ /d«rdNK V X P ^ + ( P - K - C C P ) U u - v x y M V [ K 2 - ( I - O C ) M 2 ] 2 E E p B = _C M 2 (u Z V u ) ( u V X v ) rd«rdNK (1-cc) . , 2 2 p e + x p e e , „ 2 _ . 2 M q (K +B) as c l a i m e d i n ( 6 . 2 . 5 ) , s i n c e pu =M u . I f , i n F i g . 21, t h e v i r t u a l p h o t o n were t o emerge from t h e i n c o m i n g muon l i n e t h e d i a g r a m w o u l d have t h e a m p l i t u d e ( u s i n g t h e symbols o f F i g . 21) A' ( r e n o n ) = 7. f d4 k u (p. ) I i g U . .m. c o t e Z \( 1'z i \ ( F . 7 ) W 1 1 x / i g U , m .coteZ \( i \ ( i e * ) u (p) / i V-i-n \ \ 1 " " P i " * / X 1 1 ^ x u (p ) i e * v ( p r ) e 2 _ p e „ r j 4 , u Z (p - k ) ( p +M ) * u - x = C fd k e + 1 1 p x p u V v 2 /" 1 \ 2 / 1 2 M 2 W 2 M 2 \ 6 6 q (P,~k) (k - M ) ( p -M ) 1 B 1 p The d e n o m i n a t o r w i l l r e q u i r e k -» K+cxPx , b u t as t h e l i n e a r K term i n t h e n u m e r a t o r v a n i s h e s , and p, u = m u , t h e whole 1 e e e i n t e g r a l w i l l be o f o r d e r m /M << 1 r e l a t i v e t o (F.6) and i s e p t h e r e f o r e t o be i g n o r e d as was c l a i m e d i n t h e remarks j u s t p r e c e d i n g Eq. ( 6 . 2 . 4 ) . The I n t e g r a l (6.2.9) The f o l l o w i n g i n t e g r a l i s needed t o e s t a b l i s h t h e c o n t r i b u t i o n o f t h e W p a r t i c l e (and, b e c a u s e o f gauge 158 i n v a r i a n c e , t h e S H i g g s p a r t i c l e ) t o t h e p -» 3e t r a n s i t i o n a m p l i t u d e : K W ) = J"d 4 k u Z V P ( p - k ) V C T u F X q p ( F . 8 ) 6 + ^ ( p - k ) 4 [ ( k - q ) 2 - M 2 v ] ( k 2 - M 2 v ) where F i s g i v e n i n ( E . 7 ) . We have X < r p I(W) = 6 J ,cxd c<dB >rd 4k u Z [ - ( q + k ) ( p - k ) * +V ( p - k ) y C T ( 2 k - q ) + 6 + V x ( p - k ) ( 2 q - k ) l u u ( F 9 ) { < x ( p - k ) 2 + 0 [ ( k - q ) 2 - M 2 ] + ( l - o c - 0 ) ( k 2 - M 2 ) } 4 = 6fcxd<xdBr d 4 K u Z ( K 2 * -2* K* C TK +V K 2 ) u J ( K 2 + B ) 4 6 + X ° 2 where o n l y t h e h i g h e s t o r d e r terms - - t h e K terms - - have been 2 k e p t i n t h e n u m e r a t o r , and where B * -(l-oc)M . T h i s becomes, W from (D.15), I(W) = [ 6 i w 2 / ( - 3 M 2 ) ] 3u Z V u /'ocdocXo^dPt l/(l-«)] . (F.10) W e + x p u u The i n t e g r a l has t h e v a l u e 1/2 w h i c h g i v e s I ( W ) = ( - i u 2 / M 2 ) 3u Z V u , ( F . l l ) W e + x p as r e c o r d e d i n ( 6 . 2 . 1 3 ) . The I n t e g r a l (6.2.14) I n t e g r a l (6.2.14) becomes, when t h e n e u t r i n o p r o p a g a t o r i s r e d u c e d , A ( W S ) = C f d 4 k U e E + V x U p U e v X y e (F.12) 2 , , . 2 r , , . 2 „ 2 , . . 2 w 2 q ( P - k ) [ ( k - q ) - M w ] ( k - M w ) = C 2 TdocdB r _ _ A J L _ u Z V u u V v 2 / • T r 2 n ^ 3 e + x p e e .4„ - _ - x Iq"(K"+B) = C r / J T T 2 ^2 f d o c d B u Z V u u y X v \^7WJ hi-.) e + x * e e = ( - i r r 2 C ' / q 2 M 2 ) u Z V u u y \ W e + x p. e e as g i v e n i n ( 6 . 2 . 1 5 ) . 159 The I n t e g r a l ( 6 . 2 . 1 6 ) S i m i l a r l y , i n t e g r a l ( 6 . 2 . 1 6 ) r e d u c e s t o - C jVk u Z V f -1 M u ( P " k ) \ u V ^ e . ( F . 1 3 ) 6 + M(P-k)2 + (p - k ) 4 ; »q 2[(k- q ) 2-M;](k 2-M;) 2 2 The s e c o n d o f t h e s e t e r m s i s a f a c t o r M /M„ t i m e s t h e f i r s t , p W and s o i s n e g l i g i b l e , w h i l e t h e f i r s t i s i d e n t i c a l t o ( F . 1 2 ) . The S p i n o r A m p l i t u d e ( 6 . 3 . 1 ) The l e n g t h y s p i n o r a m p l i t u d e s q u a r e d ^ I . | [ u ( P , ) E V u (p) u ( p , ) V V v ( p ' ) ] / ( p - p , ) 2 - ( P l ^ P 2 ) | 2 s p i n s e 1 + v ii e 2 e 1 1 2 r e q u i r e d i n t h e p. -+ 3e t r a n s i t i o n r a t e c a n be e v a l u a t e d i n two p a r t s . The f i r s t r e q u i r e s t h e e v a l u a t i o n s p i n s u ( p , ) Z V u (p) u ( p 2 ) V V v ( p r ) e 1 + v p e 2 e x v ( p r ) * % (p ) u ( p ) V Z u (p ) e e 2 p p - e 1 = C-)4S[Z1 _-(l+p/M)V Z k l + p . / m ) ] ( F . 1 5 ) 2 + vi14 p — 2 1 x 4 S [ y V 2 - ( - l + p ' /m ) v l l 2-(l+p 2/m)] = (l/2mM) S [ Z V PV p.] S[-y ,'v' 1+vV v V / a 2 ] + v p 1 2 = ( l / 4 m 3 M ) (p p +p p —n p ' P . ) [p' V p 2 + P ' MP2--a M V(m 2+p' « P 2 ) ] V 1 p p 1 V pv ± 2 <2 <: = ( l / 2 m 3 M ) ( p ' P , P 1 « P 2 + P ' P 2 p ' * P 1 + m 2 P ' P 1 ) where m = m , M s M . e p By i n t e r c h a n g i n g p x and p 2 we ha v e two t e r m s o f t h e e x p a n s i o n . The c r o s s - t e r m f r o m ( F . 1 4 ) i s t w i c e t h e r e a l p a r t o f -1 . u (Pi>2,.* u (P) « ( P 2 ) V V v ( p F ) s p i n s e 1 + v p e 2 e x v ( p r ) V M u (p ) u (p)v Z u (p ) ( F . 1 6 ) e e 1 p p - e 2 2 2 d i v i d e d by ( p - p 1 ) ( p ~ P 2 ) . T h i s c a n be e v a l u a t e d b y a r e a r r a n g e m e n t : 160 -I . u ( P , ) E V u ( p ) u ( p ) V E u (p ) s p i n s e 1 + v p p p - e ^ x u ( p . ) V V v ( p ' ) v ( p ' ) V % (p ) ( F . 1 7 ) e 2 e e e 1 = (- 2-)4S[Z V 2 L ( 1 + P / M ) V E _ 2 - ( l + P 2 / m ) V V 2 - ( - l + p ' / m ) V l l 2 - ( l + P 1 / m ) ] = (-1/8M)S[E V PV ( l + P 2 / m ) V V ( - l + p ' /m) l+p_/m) ] + v p 2 1 = ( l / 4 m 3 M ) ( m 2 p - p x + m 2 p « p 2 - m 2 p « p r + 2 p « p ' p 1 « p 2 ) a f t e r a l e n g t h y b u t s t r a i g h t f o r w a r d m a n i p u l a t i o n . T w i c e t h i s 2 2 e x p r e s s i o n , d i v i d e d b y ( p - p ) ( p ~ P 2 ) , p l u s ( F . 1 5 ) ( i n c l u d i n g P x and p 2 i n t e r c h a n g e d ) g i v e t h e s p i n o r a m p l i t u d e s q u a r e d ( 6 . 3 . 5 ) . The S p i n o r A m p l i t u d e ( 6 . 3 . 3 ) The s p i n o r a m p l i t u d e s q u a r e d Z . l u ( P , ) Z u (p) u ( p 2 ) p v ( p ' ) / ( p - p . ) 2 - (P, —»P 2 ) I2 ( F . 1 8 ) s p i n s e 1 + p ' e 2 e 1 1 2 i s e v a l u a t e d s i m i l a r l y . The f i r s t t e r m " s q u a r e d " i s ( a g a i n , m sm, M SM) e p I . |u ( p , ) Z u (p) u ( P 2 ) p v ( p ' ) | 2 ( F . 1 9 ) s p i n s e 1 + p e 2 e = |Z . [u ( P , ) Z u ( p ) u ( p ) E u (p ) 2 s p i n s e 1 + p p - e 1 * u ( p 2 ) p v ( p r ) v (p' ) p u ( p 2 ) ] e 2 e e e 2 = \ 4 S [ E + 2 - ( l + p / M ) E _ 2 - ( l + p 1 / m ) ] 4 S \ v \ ( - 1 + p ' /m) p^ ( l + p 2 /m) ] = 2 " S C i " ( 1 ~ i V 5 ) p p i / m M ] S [ - P 2 + P P ' P P 2 / m 2 J = [ ( P ' P 1 ) / ( 4 m 3 M ) ] ( 2 p . p ' p . p 2 - m 2M 2 - M 2 p ' - p 2 ) . The s e c o n d t e r m " s q u a r e d " m e r e l y r e q u i r e s t h e i n t e r c h a n g e o f p x and p 2 . The c r o s s - t e r m i s t h e n e g a t i v e o f t w i c e t h e r e a l p a r t o f 161 s p i n s p e i 2 I . [u ( p , ) Z u (p) u ( p , ) p v ( P ' ) ] [ v ( p ' ) p u (p )u ( p ) E u (p )] s p i n s e 1 + p. e 2 e e e 1 p - e 2 = ( 2 - ) 4 S [ S + 2 - ( l + p / M ) S _ 2 - ( l + p 2 / m ) p 2 - ( - l + p ' / m ) p 2 - ( l + P 1 / m ) ] (F.20) = [ 4 p - p ^ « p 2 p - p ' - M 2 ( p « p 2 p ' •P 1 +P *P X p' • P 2 + P 1 , P 2 P * P r ) - m 2M 2 (p «p'-p » p x - p «p 2 ) ] / (16m 3M) , a f t e r a l e n g t h y , b u t s t r a i g h t f o r w a r d , r e d u c t i o n . The S p i n o r A m p l i t u d e (6.3.4) The s p i n o r a m p l i t u d e s q u a r e d I . |v ( p ) v (p') [u (p )v (p_) - u (p )v ( P l ) ] | 2 (F.21) s p i n s p e e x e 2 e 2 e 1 we e v a l u a t e i n p a r t s . The f i r s t f a c t o r i s r e a d i l y f o u n d t o be \l . |v ( p ) v ( p ' ) | 2 = 2 L 4 S [ 2 - ( - l + p ' / m ) 2 L ( - l + p / M ) ] (F.22) t(l+p.p'/mM) , w h i l e t h e s e c o n d f a c t o r I |u ( p . ) v (p_) - u (p )v ( P . ) | 2 , (F.23) r s r x s 2 s 2 r x where r , s = ±, seems u n l i k e o t h e r a m p l i t u d e s n o r m a l l y e n c o u n t e r e d i n quantum f i e l d t h e o r i e s b e c a u s e t h e c r o s s - t e r m i s not r e a d i l y r e d u c i b l e t o a " t r a c e " o v e r p r o j e c t i o n o p e r a t o r s , and must t h e r e f o r e be o b t a i n e d more i n d i r e c t l y . C o n s i d e r a s p i n o r b a s i s 2 - ( l + V 0 ) 2 - [ l ± ( - i ) V 5 V 3 V 0 ] < l > ± = * ± (F.24) 2 - d - V 0 ) 2 - [ l ± ( - i ) V 5 V 3 V 0 ] x _ = x_ w i t h V.* = oc x_ (F.25) 5 ± ± + and oe =1 t o d e f i n e a phase. Then we may t a k e |p | = 0 and + * u (p.) = <t> , v (p ) = x , (F.26) r 2 r r 2 r u (p, ) = L* , v (p ) = Lx , r 1 r r 1 r 162 where L i s t h e L o r e n t z b o o s t o p e r a t o r ( H a m i l t o n , 1984b) L = (1 + P^o/m) / [2(1 + e . / m ) ] 1 ' 2 , (F.27) and L 1 = (1 + V 0P x/m) / [2(1 + 6 , / m ) ] 1 ' 2 , (F.28) w i t h e = p•V 0• From (F.24) and (F.25) f o l l o w s V , x x = ± ( i / « _ ) * _ , (F.29) 3 ± + + and we a l s o have ( t o w i t h i n an i r r e l e v a n t p h a s e ) v x <r <b , v_x A «c + <b . (F.30) The a m p l i t u d e s q u a r e d (F.23) now becomes, w i t h p ^ p , L = L ( p ) , I |u (p )v ( P - ) - u ( p . ) v ( p , ) | 2 (F.31) r s r 1 s 2 s 2 r x = £ l«> L _ 1 x - i Lx | 2 r s r s s r = [ 2 m 2 ( l + € / m ) ] _ 1 S r 8 I V 0 P x s - * s P V 0 x r | 2 = [ 2 m 2 ( 1 + e / m ) ] _ 1 £ |i p V x + * P V X | 2 r s r i s s l r = [ 2 ( l + € / m ) ] - 1 [ 8 ( p X ) 2 + 8 ( P 2 ) 2 + 2 ( p 3 ) 2 |l-<x_| 2]/m 2 . Thus t o a c h i e v e a c o v a r i a n t r e s u l t we must make t h e s e l e c t i o n <x = -1 i n ( F . 2 5 ) . The a m p l i t u d e u s e d i n Sec. VI.3 i s t h u s I . |u(p ) v ( p ) - u ( p 2 ) v ( p ) | 2 (F.32) s p i n s 1 2 2 1 = 4 ( | p | 2 / m 2 ) / (l+€/m) = 4(€/m - 1) = 4 ( p x « p 2 / m 2 - 1) , on r e v e r t i n g b a c k t o t h e o r i g i n a l c o v a r i a n t n o t a t i o n . 163 The Phase Space I n t e g r a l (6.3.14) The t r a n s i t i o n r a t e o f ( 6 . 3 . 1 4 ) , 4 _4 , . s ,„ . d 3 p d 3p, 1 m ' (F.33) T = - J" | A | (2ir) S ( p - P j - P g - p ' ) m_ d 3 p ' m_ E' ( 2 r r ) 3 E x ( 2 T T ) 3 E 2 ( 2 T T ) 3 i s f i r s t i n t e g r a t e d o v e r t h e p o s i t r o n s t a t e s ( p r ) b e c a u s e t h e 2 a m p l i t u d e s q u a r e d |A| depends o n l y on E^ and E 2 . We use t h e same t e c h n i q u e as i n App. E and o b t a i n T = _ J L J" I A | 2 d 3 P i d 3 P 2 0 4 ( P - P i - p . - p ' ) S ( p ' 2 - m 2 ) ^ r i + € ( p f )1 d V ( 2 n ) 5 E x E 2 (F.34) J* IA | 2 S [ ( p - P l - p 2 ) 2 - m 2 ] 2 - [ l + € ( p - p i - p 2 ) ] d 3 P l d 3 l > 2 3 m ( 2 T T ) 5 E1 E 2 w h i c h i s ( 6 . 3 . 1 5 ) . We now i n t e g r a t e o v e r a l l a n g l e s . One r e p l a c e s d 3 p x w i t h 4 T T | P 1 | dlPj^ | = 4n|p 1 | E 1 d E 1 and d e f i n e s p x ' P 2 = c o s e 1 2 = c , (F.35) so t h a t f ( c ) = (P-P.-Pg ) 2-m 2 = M 2+m 2-2ME 1-2ME 2+2E 1E 2-2 |p x | |p 2 |c (F.36) and d e f i n e s f ( c Q ) a 0. (F.37) S i n c e I f ( c 0 ) | = 2 |p x | |p 2 | (F.38) we have T = 4TT m3 X | A | 2 |p\ | d E x S ( c ~ Q 2 r r | p 2 | d E 2 dc (F.39) ( 2 W ) S I*' ( O l = [m /(2T T) ] J*|A| d E x dE 2 = [ m 3 M 2 / 4 ( 2 T r ) 3 ] J | A ( x 1 , x 2 ) | 2 dx x d x 2 , where E. = ^Mx. f r o m (6.3.10) has been u s e d . Thus we have l 2 l d e r i v e d t h e e x a c t r e s u l t ( 6 . 3 . 1 6 ) . 164 APPENDIX G LIST OF VERTICES FOR FEYNMAN DIAGRAMS The v e r t i c e s computed i n S e c . V.4 t h a t were s u b s e q u e n t l y u s e d t o c o n s t r u c t t h e Feynman d i a g r a m s f o r muon d e c a y i n Chap. V, VI a r e l i s t e d h e r e f o r t h e c o n v e n i e n c e o f t h e r e a d e r . Q u e s t i o n s o f s i g n o r momentum d i r e c t i o n a r e t o be r e s o l v e d by r e f e r r i n g t o t h e o r i g i n a l d e r i v a t i o n . The n o t a t i o n u s e d h e r e i s Jl( o r H.) = l e p t o n ( i = l = e l e c t r o n , i = 2=muon, e t c . , o f mass M.); 1 1 x ( o r x.) = M a j o r a n a n e u t r i n o ( o f mass m.); A = p h o t o n ; Z = m a s s i v e , n e u t r a l e l e c t r o - w e a k gauge boson; W+ = m a s s i v e , c h a r g e d e l e c t r o - w e a k gauge boson; S + = u n p h y s i c a l , c h a r g e d H i g g s boson; B + = p h y s i c a l , c h a r g e d H i g g s b o s o n ; H = p h y s i c a l , d o u b l y - c h a r g e d H i g g s b o s o n . l + + Jl ietf i g y ( s i n e, [ E q . ( 4 . 4 . 9 ) ] W - [ E q . ( 4 . 4 . 1 1 ) ] H t± A-iL-H + + ivT20. . Z U ± [ E q . ( 4 . 4 . 2 2 ) ] A-x-W 165 igU..V Z [ E q . ( 4 . 4 . 1 3 ) ] A-x-S s - i ( g / M m ) U . . ( M . Z -m.Z ) [ E q . ( 4 . 4 . 1 5 ) ] W j i l + j --i(8/M )U..(M.Z_-m.Z +) [ E q . ( 4 . 4 . 1 6 ) ] A-x-B B ;B AT; i i A,Z K T ( o c ) W-W-A.Z ) ( P j A ^ w S-S-A B-B-A A P - v - . P* - 1 7 " s i ( g / M „ ) U . . ( M . t a n e Z +m.coteZ ) W j i l + j [ E q . ( 4 . 4 . 1 9 ) ] i ( g / M „ ) U . . ( M . t a n e Z +m.coteZ ) W l j j - I + [ E q . ( 4 . 4 . 2 0 ) ] + <-P.,-K.,>1 Q 1 [Eq.(4.4.28)] o » otp i e ( P ' + P ) [ E q . ( 4 . 4 . 3 3 , 3 4 ) ] S,8' S,B W-A-S A -ieM„-n [Eq. (4.4.36) ] W (it/ PUBLICATIONS J . Dwayne Hamilton: "The Classical Electrodynamics of Interacting Part ic les, " Am. J . Phys., 39, 1172-1177 (1971). J . Dwayne Hamilton: "The Conformal Invariance of the coupled Spinor-Vector F ie lds , " Can. J . Phys., 51^ , 316-321 (1973). J . Dwayne Hamilton: "The Uniformly Accelerated Reference Frame," Am. J . Phys., 46, 83-89 (1978). J . Dwayne Hamilton: "The Rotation and Precession of Relat iv is t ic Reference Frames," Can. J . Phys., 59, 213-224 (1981). J . Dwayne Hamilton: "Pauli Spinors and Hestenes1 Geometric Algebra," Am. J . Phys., 52, 56-60 (1984). J . Dwayne Hamilton: "The Dirac Equation and Hestenes1 Geometric Algebra," J . Math. Phys., 25, 1823-1832 (1984). 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0084906/manifest

Comment

Related Items