{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Science, Faculty of","@language":"en"},{"@value":"Physics and Astronomy, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Zakarauskas, Pierre","@language":"en"}],"DateAvailable":[{"@value":"2010-06-13T16:05:37Z","@language":"en"}],"DateIssued":[{"@value":"1984","@language":"en"}],"Degree":[{"@value":"Doctor of Philosophy - PhD","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"This thesis contains a study of the reaction proton+proton or proton-antiproton into a Higgs boson and a pair of heavy quarks, in the region of high energy and high momentum transfer. The Higgs boson mass is treated as a free parameter. Numerical results are obtained through a Monte Carlo integration. Several differential cross sections relevant to experiment are given.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/25622?expand=metadata","@language":"en"}],"FullText":[{"@value":"A QCD-PARTON CALCULATION OF ASSOCIATED HIGGS BOSON PRODUCTION IN HADRON-HADRON COLLISION by PIERRE ZAKARAUSKAS B . S c . , U n i v e r s i t e Du Quebec A C h i c o u t i m i , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department Of P h y s i c s We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June 1 9 8 4 \u00a9 P i e r r e Zakarauskas, 1 9 8 4 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying 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 not be allowed without my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date i i A b s t r a c t T h i s t h e s i s c o n t a i n s a study of the r e a c t i o n proton+proton or p r o t o n - a n t i p r o t o n i n t o a Higgs boson and a p a i r of heavy quarks, i n the region of high energy and h i g h momentum t r a n s f e r . The Higgs boson mass i s t r e a t e d as a f r e e parameter. Numerical r e s u l t s are o b t a i n e d through a Monte C a r l o i n t e g r a t i o n . S e v e r a l d i f f e r e n t i a l c r o s s s e c t i o n s r e l e v a n t to experiment are g i v e n . Table of Contents A b s t r a c t i i L i s t of Tables i v L i s t of F i g u r e s v Acknowledgement v i Chapter I INTRODUCTION 1 Chapter II LOCAL GAUGE TRANSFORMATIONS 8 Chapter III HIGGS MECHANISM 14 Chapter IV THE GLASHOW-WEINBERG-SALAM MODEL 20 Chapter V PARTON MODEL AND HADRON-HADRON COLLISION 33 Chapter VI HIGGS BOSON PHENOMENOLOGY 44 Chapter VII CALCULATION OF ASSOCIATED PRODUCTION OF HIGGS BOSON AND HEAVY FLAVOR IN PROTON-ANTIPROTON COLLIDERS 64 Chapter VIII RESULTS 77 Chapter IX DISCUSSION AND CONCLUSION 93 BIBLIOGRAPHY . ... 98 APPENDIX A - FEYNMAN DIAGRAMS AND QCD RULES 101 APPENDIX B - COLOR SUMMATION CALCULATION 107 APPENDIX C - THE MONTE-CARLO INTEGRATION ROUTINE 110 APPENDIX D - CALCULATION OF THE TRACE 122 APPENDIX E - PRINTOUT OF THE AMPLITUDE SQUARED OF THE PROCESS 127 APPENDIX F - QUARK AND GLUON DISTRIBUTION PARAMETRIZATIONS 1 44 APPENDIX G - HADRON-HADRON COLLIDERS 146 L i s t of Tables Number of charm quarks, n c , and number of charged l e p t o n s , n A , i n the f i n a l s t a t e p a r t i c l e s of r e a c t i o n (VII.1) a f t e r weak decays of the hadronsF, F and the Higgs boson 81 F i x e d t a r g e t c r o s s s e c t i o n f o r r e a c t i o n (VII.1) 82 Very high energy c r o s s s e c t i o n s 83 V L i s t o f F i g u r e s + - + -1. F e y n m a n d i a g r a m f o r W W \u2014>W W , w i t h o u t s c a l a r c o n t r i b u t i o n 6 2. S c a l a r c o n t r i b u t i o n t o t h e p r o c e s s W +W~\u2014>W + W 6 3. P o t e n t i a l I I I . 3 f o r t h e c a s e |> < 0 15 4. V e r t i c e s o f t h e s e l f - i n t e r a c t i n g s c a l a r m e s o n 18 5. V e c t o r - s c a l a r v e r t i c e s 19 6. F e y n m a n d i a g r a m f o r e l e c t r o n - m u o n s c a t t e r i n g 34 7. F e y n m a n d i a g r a m f o r e l e c t r o n - p r o t o n s c a t t e r i n g 35 8. V(
Y+ H d e c a y 54 1 1 . Z \u00b0 \u2014 > H \u00b0 + ICH d e c a y d i a g r a m 54 12. F e y n m a n d i a g r a m f o r e + e ~ \u2014 > H \u00b0 + Z \u00b0 55 o 1 3 . F e y n m a n d i a g r a m f o r g g \u2014 > H 57 14. C r o s s s e c t i o n s f o r p p \u2014 > H \u00b0 + X 59 o 1 5 . F e y n m a n d i a g r a m g +c \u2014 > c + H 60 o 1 6 . F e y n m a n d i a g r a m s f o r t h e p r o c e s s q q \u2014 > H q q 61 17 . C r o s s s e c t i o n s f o r c o m p t o n - l i k e p r o c e s s , f o r m H = 1 0 G e V \/ c 2 62 18. T o t a l c r o s s s e c t i o n f o r p r o c e s s e s 2 ) , 5) a n d 6) f o r mH = 4 1 0 G e V \/ c 1 \u2022 63 1 9 . F e y n m a n d i a g r a m s f o r t h e b a c k g r o u n d t o t h e p r o c e s s h a d r o n + h a d r o n \u2014 > H \u00b0 + a n y t h i n g 65 2 0 . F e y n m a n d i a g r a m s f o r q q \u2014 > F + F + H \u00b0 67 2 1 . F e y n m a n d i a g r a m s f o r g g \u2014 > F + F + H \u00b0 68 2 2 . T o t a l c r o s s s e c t i o n f o r t h e p r o c e s s ( V I I . 3 ) a s a f u n c t i o n o f Vs, f o r p p c o l l i s i o n , w i t h m 4 = 10 G e V \/ c * . 84 v i 23. T o t a l c r o s s s e c t i o n f o r the process (VII.4) as a f u n c t i o n of ^ s , with = 10 GeV\/c* 85 24. T o t a l c r o s s s e c t i o n i n pp from the sum of s u b r e a c t i o n ( V I I I . 3) and ( V I I I . 4 ) , as a f u n c t i o n of i\/s, with mH = 10 GeV\/c z 86 25. T o t a l c r o s s s e c t i o n f o r the process (VII.1) as a f u n c t i o n of the mass of the heavy quark produced with the Higgs boson 87 26. T o t a l c r o s s s e c t i o n f o r the process (VII.1) as a f u n c t i o n of the mass of the Higgs boson 88 27. D i f f e r e n t i a l c r o s s s e c t i o n dc\/ \/dEH f o r four d i f f e r e n t s e t s of the parameters m^ , m^ , and \\fs 89 28. D i f f e r e n t i a l c r o s s s e c t i o n do' \/dE\u00ab f o r four d i f f e r e n t set of the parameters m^ , mh , and Vs\" 90 29. D i f f e r e n t i a l c r o s s s e c t i o n da \/dh\u00b1 f o r four d i f f e r e n t s e t s of the parameters mh , mK and *Js^ 91 30. D i f f e r e n t i a l c r o s s s e c t i o n d c \/ d k i f o r four d i f f e r e n t s e t s of the parameters mH , mK, and ifs 92 31. Order of p a r t i c l e g e n e r a t i o n i n the Monte-Carlo method in p a r t i c l e p h y s i c s 111 v i i Acknowledgement I would l i k e to thank my r e s e a r c h s u p e r v i s o r , Dr. John Ng, f o r h i s abundant h e l p and guidance d u r i n g the course of t h i s work. His enthusiasm f o r r e s e a r c h has been a continuous source of m o t i v a t i o n f o r myself. I a l s o want to thank my wife, L o u i s e , f o r her continuous moral support. She always made sure I got enough of a l l the good t h i n g s l i f e has to o f f e r apart p h y s i c s . I g r a t e f u l y acknowledge f i n a n c i a l a s s i s t a n c e from the N a t u r a l Sciences and E n g i n e e r i n g Research C o u n c i l . 1 I . INTRODUCTION Within the l a s t decade the world has witnessed a t o t a l r e v o l u t i o n i n the understanding of p a r t i c l e p h y s i c s . U n t i l then, the weak i n t e r a c t i o n s (WI) were d e s c r i b e d by a phenomenological, non-renormalizable Fermi i n t e r a c t i o n of four f i e l d s at a p o i n t . The WI are the weakest, a f t e r g r a v i t a t i o n , of the b a s i c known f o r c e s of nature. They are r e s p o n s i b l e f o r the beta decay of the neutron, f o r example, and other r e l a t i v e l y slow processes i n n u c l e a r and p a r t i c l e p h y s i c s . On the other hand, the newly developed quark model of that time c o u l d account f o r the p r e v i o u s l y mind b o g g l i n g hundreds of \"elementary\" p a r t i c l e s produced i n str o n g i n t e r a c t i o n s ( h e r e a f t e r S I ) , or t h e i r decay p r o d u c t s . What was s t i l l badly needed however, was a theory of SI i t s e l f . The four known f o r c e s a c t i n g on matter i n the u n i v e r s e , - g r a v i t a t i o n , electromagnetism (EM), WI and SI, - d i d not seem to have much i n common. Then, at the end of the 60's, EM and WI were \" u n i f i e d \" w i t h i n the framework of a gauge model, the Glashow-Weinberg-Salam model (GWSM)1 . A few years l a t e r , i t was the t u r n of SI to be d e s c r i b e d by a gauge theory -quantum chromodynamics (QCD) 2 . Now, most models of p a r t i c l e i n t e r a c t i o n are based on the gauge i d e a . Among them, are the 1 For h i s t o r i c a l accounts and r e f e r e n c e s , see Nobel l e c t u r e s of (Glashow, 1980), (Weinberg S. 1980), (Salam 1980) 2 For a review of QCD, see (Reya, 1981) 2 g r a n d u n i f i e d t h e o r i e s ( G U T S ) 1 w h o s e g o a l i s t o u n i f y EM, WI a n d S I i n t o a s i n g l e i n t e r a c t i o n w i t h a n o n - a b e l i a n g a u g e g r o u p . I t s m a i n p r e d i c t i o n i s t h e i n s t a b i l i t y o f t h e p r o t o n , w h i c h i s b e i n g i n t e n s i v e l y t e s t e d i n many l a b o r a t o r i e s . O t h e r g a u g e m o d e l s a r e : t e c h n i c o l o r * , s u p e r g r a v i t y 3 , a n d s e v e r a l a l t e r n a t i v e s t o t h e GWSM . W h a t m a k e s t h e c o n c e p t o f g a u g e i n v a r i a n c e a t t r a c t i v e i s i t s i n h e r e n t e l e g a n c e . I t s m a i n f e a t u r e i s t h e f o l l o w i n g . Y o u s t a r t w i t h a s y m m e t r y y o u know t o b e v a l i d , ( o r h y p o t h e s i z e t o b e v a l i d ) , i n g e n e r a l i n a w o r l d w h e r e t h e m a t t e r f i e l d s a r e s p i n 0 b o s o n s o r s p i n 1\/2 f e r m i o n s . Y o u r e q u i r e t h i s s y m m e t r y t o b e c o n s e r v e d l o c a l l y , i . e . a t a n y p o i n t i n s p a c e - t i m e . T o d o s o , y o u m u s t i n t r o d u c e a new b o s o n f i e l d , w h i c h w i l l m e d i a t e some new i n t e r a c t i o n b e t w e e n t h e m a t t e r p a r t i c l e s , i n s u c h a way t h a t t h e s y m m e t r y r e m a i n s n o n - v i o l a t e d . H e n c e , y o u h a v e \" d e d u c e d \" a f o r c e f r o m t h e s y m m e t r y r e q u i r e m e n t . I t h a s b e e n known f o r q u i t e some t i m e t h a t EM c a n b e \" d e d u c e d \" t h i s way f r o m t h e p h a s e i n v a r i a n c e i n q u a n t u m m e c h a n i c s ( F o c k , 1 9 2 7 ) , ( W e y l , 1 9 2 9 ) . I t was s e e n a s m e r e l y a n e l e g a n t way o f l i n k i n g EM a n d QM. Y a n g a n d M i l l s ( 1 9 5 4 ) b r o a d e n e d t h e c l a s s o f s y m m e t r i e s t h a t c a n b e \" l o c a l i z e d \" t h i s way, t o i n c l u d e n o n - a b e l i a n s y m m e t r i e s . A n o n - a b e l i a n s y m m e t r y c a n b e c o m p a r e d t o a r o t a t i o n i n s p a c e - t h e o r d e r i n w h i c h y o u a p p l y t h e t r a n s f o r m a t i o n s i s i m p o r t a n t . I n t h i s a n a l o g y , a n a b e l i a n t h e o r y w o u l d b e a r o t a t i o n i n some F o r a r e v i e w o f GUTS a n d t h e i r p h e n o m e n o l o g y , s e e ( L a n g a c k e r , 1981 ) 2 s e e f o r e x a m p l e ( S u s s k i n d , 1 9 7 9 ) 3 F o r a r e v i e w o f s u p e r g r a v i t y , s e e ( v a n N i e w e n h u i z e n , 1 9 8 1 ) * F o r e x a m p l e ( G e o r g i a n d G l a s h o w , 1 9 7 4 ) , ( P a t i a n d S a l a m , 1 9 7 3 ) 3 plane. The f o r c e s generated by a non-abelian symmetry are much more com p l i c a t e d than those generated by an a b e l i a n one, mainly because the p a r t i c l e s or f i e l d s r e s p o n s i b l e f o r c a r r y i n g the i n t e r a c t i o n s are \"charged\" themselves. But the Y a n g - M i l l s theory d i d not a t t r a c t much a t t e n t i o n f o r a w h i l e , because the boson p a r t i c l e s you must i n t r o d u c e to c a r r y the i n t e r a c t i o n s must be massless, g i v i n g r i s e to long-range f o r c e s we do not observe. The f o r c e s generated by non-abelian symmetries d i d not seem to correspond to any of the known f o r c e s . One had to s o l v e the problem of g i v i n g a mass to the v e c t o r bosons i f one wants the theory to d e s c r i b e WI which are short range. The s o l u t i o n to t h i s problem had to wait t i l l 1964, when Higgs (1964) invented the spontaneous symmetry breaking (SSB) scheme. At the p r i c e of i n t r o d u c i n g a elementary s c a l a r f i e l d , the vacuum would be made to be n o n - t r i v i a l . Real p a r t i c l e s p ropagating through such a vacuum would i n t e r a c t with i t , g i v i n g them e f f e c t i v e l y a mass, i n much the same way as the apparent mass the of e l e c t r o n may be g r e a t l y a f f e c t e d when i t t r a v e l s through a l a t t i c e or a plasma. A few years l a t e r , Weinberg, Salam and Glashow came up independently with a model f o r the weak and 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 s , u s i n g a non-abelian symmetry based on i s o s p i n , r e p r e s e n t e d by an SU(2) group, and an a b e l i a n symmetry U(1). The SU(2) group has three g e n e r a t o r s , which i m p l i e s three bosons mediating the i n t e r a c t i o n s ; the U(1) group has one. The symmetry i s broken by i n t r o d u c i n g an i n t e r a c t i n g doublet of 4 c o m p l e x s c a l a r s , e n d o w e d w i t h a n e g a t i v e m a s s - s q u a r e d . O f t h e f o u r d e g r e e s o f f r e e d o m b r o u g h t i n b y t h e s c a l a r s , t h r e e a r e u s e d t o g i v e m a s s t o t h r e e o f t h e f o u r b o s o n s . T h e f o u r t h d e g r e e o f f r e e d o m a p p e a r s a s a p h y s i c a l e l e m e n t a r y f i e l d , w i t h o a r e a l m a s s . I t i s c a l l e d t h e H i g g s b o s o n , s y m b o l i c a l l y H . T h e GWS m o d e l a c c o u n t e d w e l l f o r w h a t was k n o w n a t t h e t i m e o f t h e WI, b u t i t p r e d i c t e d a new c o m p o n e n t t o t h e weak f o r c e ; a n e u t r a l o n e . A n e x a m p l e o f i t w o u l d b e t h e r e a c t i o n 1> q \u2014 > v q i n w h i c h a n e u t r i n o , a p a r t i c l e w h i c h i n t e r a c t s o n l y t h r o u g h WI, i n t e r a c t s w i t h a q u a r k a n d r e m a i n s a n e u t r i n o . T o s e e t h i s e x p e r i m e n t a l l y , o n e w o u l d s e n d a n e u t r i n o beam o n a t a r g e t , a n d w a i t t o d e t e c t a d e p o s i t i o n o f e n e r g y a n d momentum, w i t h n o l e p t o n p r o d u c e d . ( T h e c h a r g e d WI w o u l d p r o d u c e a c h a r g e d l e p t o n i n t h e f i n a l s t a t e ) . T h e s e n e u t r a l c u r r e n t i n t e r a c t i o n s h a v e e x t e n s i v e l y b e e n m e a s u r e d a n d s t u d i e d f r o m t h e i r d i s c o v e r y i n 1 973 t i l l now. S i n c e t h e n , t h e e x i s t e n c e o f t h e n e u t r a l b o s o n h a s b e e n c o m f i r m e d b y i t s s p e c t a c u l a r d i s c o v e r y i n p r o t o n - a n t i p r o t o n c o l l i s i o n s , a t t h e c o l l i s i o n b e a m f a c i l i t i e s a t C E R N ( C o n s e i l E u r o p e a n d e R e c h e r c h e N u c l e a i r e ) , i n t h e 1983 summer ( A r n i s o n e t a l . 1 9 8 3 ) . I t s d i s c o v e r y h a d t o w a i t s o l o n g b e c a u s e n o p a r t i c l e a c c e l e r a t o r i n t h e w o r l d c o u l d r e a c h t h e c e n t e r o f m a s s e n e r g y n e c e s s a r y t o i t s p r o d u c t i o n , s i n c e i t s m a s s was p r e d i c t e d t o b e 91 G e V \/ c . T h e n e x t v e r y i m p o r t a n t t a s k f a c i n g t h e e x p e r i m e n t a l i s t s i s t o l o o k f o r t h e H i g g s b o s o n . T h e d i s c o v e r y o f t h e H i g g s b o s o n w o u l d b e a b a d l y n e e d e d c o n f i r m a t i o n t h a t t h e m e c h a n i s m w h i c h e n d o w s t h e g a u g e b o s o n s w i t h m a s s e s i s t h e s p o n t a n e o u s s y m m e t r y 5 b r e a k i n g m e c h a n i s m . T h i s i s a c o r n e r s t o n e o f t h e GWS m o d e l , a n d i n d e e d , o f n e a r l y a l l u n i f i c a t i o n t h e o r i e s b a s e d o n t h e g a u g e p r i n c i p l e . T h e m a i n o b s t a c l e t o i t s d i s c o v e r y , i f i t e x i s t s , i s t h a t u n l i k e t h e i n t e r m e d i a t e v e c t o r b o s o n W +, W a n d Z \u00b0 , i t s m a s s a n d d e c a y p r o d u c t s a r e f r e e p a r a m e t e r s o f t h e t h e o r y . T h e s e f a c t o r s m ake i t s p r o d u c t i o n , a n d e s p e c i a l l y i t s i d e n t i f i c a t i o n , v e r y d i f f i c u l t . S e v e r a l p r o d u c t i o n m e c h a n i s m s h a v e a l r e a d y b e e n s u g g e s t e d . T h o s e p e r t i n e n t t o h a d r o n - h a d r o n c o l l i s i o n s g e n e r a l l y l a c k a c l e a r s i g n a t u r e . H o w e v e r , i f t h e H \u00b0 i s t o o m a s s i v e , i t s p r o d u c t i o n w i l l n o t y e t b e p o s s i b l e i n t h e c l e a n e r e l e c t r o n - p o s i t r o n c o l l i d e r r i n g s . F o r t h e e + e ~ c o l l i d e r s t h a t a r e p l a n n e d now t h e h i g h e s t e n e r g y o f 2 0 0 G e V w o u l d b e r e a c h e d b y L E P I I a t C E R N . On t h e o t h e r h a n d , a h a d r o n c o l l i d e r o f c m . e n e r g y 5 t o 40 T e V (1 T e V = 1000 G e V ) i s b e i n g p l a n n e d . O n e m o r e a r g u m e n t may b e g i v e n i n f a v o r o f t h e e x i s t e n c e o f a n e l e m e n t a r y s c a l a r , i n d e p e n d e n t l y o f t h e s p o n t a n e o u s s y m m e t r y b r e a k i n g s c h e m e . I t c o n c e r n s t h e h i g h - e n e r g y b e h a v i o r o f t h e t h e o r y ( H a l z e n a n d M a r t i n , 1 9 8 4 ) . T h e p r e d i c t e d c r o s s - s e c t i o n f o r a n y p r o c e s s m u s t n o t d i v e r g e , i . e . t h e p r o b a b i l i t y o f o c c u r e n c e o f t h i s p r o c e s s m u s t r e m a i n l e s s t h a n o n e . I f o n e c a l c u l a t e s t h e c r o s s - s e c t i o n f o r t h e e l a s t i c s c a t t e r i n g o f a p a i r o f c h a r g e d W , f r o m t h e t h r e e d i a g r a m s o f F i g u r e 1. 6 -V - + -F i g u r e 1 - Feynman diagram f o r W W \u2014>W W , without s c a l a r c o n t r i b u t i o n one f i n d s that t h e i r sum d i v e r g e s as s\/M^ as s \u2014 > \u00a9 \u00b0 , (where the square of the t o t a l energy i s denoted by s ) . A simple s o l u t i o n i s to i n t r o d u c e a s c a l a r p a r t i c l e to c a n c e l t h i s d i v e r g e n c e , through the diagram of F i g u r e 2. F i g u r e 2 - S c a l a r c o n t r i b u t i o n to the process W W \u2014>W W The c o u p l i n g of the h p a r t i c l e must be p r o p o r t i o n a l to the W mass to c a n c e l the divergences of the other diagrams. T h e r e f o r e , i f we had not i n t r o d u c e d the Higgs boson to g i v e mass to the gauge bosons, a l a SSB mechanism, we would have 7 been f o r c e d to invent i t to c a n c e l out d i v e r g e n c e s i n other p r o c e s s e s ! T h i s t h e s i s i s d i v i d e d i n t o two p a r t s . The f i r s t one covers the background m a t e r i a l p e r t i n e n t to Higgs mechanism and phenomenology, and i n c l u d e s the f i r s t s i x c h a p t e r s . Chapter II g i v e s a g e n e r a l treatment of gauge t h e o r i e s . The t h i r d chapter i n t r o d u c e s the phenomena of spontaneous symmetry breaking and the important Higgs mechanism. The Glashow-Weinberg-Salam model i s developed i n chapter IV. Chapter V b r i n g s i n the hadron c o n t r i b u t i o n . There i s presented the extremely u s e f u l , yet simple parton model. Using i t , one may use p e r t u r b a t i v e QCD and d e r i v e u s e f u l p r e d i c t i o n s f o r experiments. We get to the core of the s u b j e c t i n chapter VI with the known phenomenology of the \"standard\" Higgs boson. T h i s i s where i s rooted any a n a l y s i s of Higgs boson p r o d u c t i o n . The whole work r e l i e s h e a v i l y on i t . The second p a r t of the t h e s i s i n c l u d e s c h a p t e r s seven through n i n e . The s t a r t i n g p o i n t of the c a l c u l a t i o n s i s d e s c r i b e d i n chapter V I I , and the r e s u l t s are to be found i n chapter V I I I . The d e t a i l s of the c a l c u l a t i o n s , i n p a r t i c u l a r the matrix element squared, and the Monte-Carlo i n t e g r a t i o n r o u t i n e developed, have been c o n f i n e d to appendices. I summarize the work and suggest p o s s i b l e routes of e x t e n s i o n s i n chapter IX. 8 I I . LOCAL GAUGE TRANSFORMATIONS Because l o c a l gauge i n v a r i a n c e i s at the heart of today's attempts to u n i f y and\/or e x p l a i n fundamental i n t e r a c t i o n s i n p h y s i c s , we w i l l s t a r t with a b r i e f account of t h i s important s u b j e c t . GENERAL CASE; FERMIONS: We s t a r t with the Lagrangian f o r f r e e fermions. We demand that Jf. be l o c a l l y i n v a r i a n t under t r a n s f o r m a t i o n s of itee a simple L i e group G, and Y transforms as a c e r t a i n r e p r e s e n t a t i o n of G. The generators of G have r e p r e s e n t a t i o n m a t r i c e s T a which s a t i s f y [ T*,TJ - i Cube Tc ( I I . 2 ) where the C a b c a r e t n e t o t a l l y antisymmetric s t r u c t u r e c o n s t a n t s . I f the fermion f i e l d s , under i n f i n i t e s i m a l t r a n s f o r m a t i o n s , t r a n s f o r m as ( I I . 3 ) i t i s easy to check that the f r e e Lagrangian d ^ r e e i s not 9 i n v a r i a n t u n d e r t h i s t r a n s f o r m a t i o n . T h e d e r i v a t i v e i n t r o d u c e s w h i c h s p o i l s t h e i n v a r i a n c e o f $ i r t e . T h e l o c a l p r o p e r t y o f t h e s y m m e t r y i s e x p r e s s e d b y t h e x - d e p e n d e n c e i n 9. T o make i n v a r i a n t , o n e i n t r o d u c e s t h e c o v a r i a n t d e r i v a t i v e D h ; w h e r e a s e t o f new 4 - v e c t o r \" g a u g e \" f i e l d s h a v e b e e n i n t r o d u c e d . Now, i f o n e d e m a n d s t h a t t h e c o v a r i a n t d e r i v a t i v e h a s t h e same t r a n s f o r m a t i o n p r o p e r t y a s ^ i t s e l f , i . e . ( I I . 5 ) t h e n o n e m u s t i n t r o d u c e v e c t o r g a u g e f i e l d s w h i c h t r a n s f o r m u n d e r i n f i n i t e s i m a l t r a n s f o r m a t i o n s a s ; I n t h i s e x p r e s s i o n , t h e s e c o n d t e r m i s t h e t r a n s f o r m a t i o n l a w f o r t h e a d j o i n t m u l t i p l e t u n d e r G. T h i s i m p l i e s t h a t t h e g a u g e f i e l d s A ^ c a r r y t h e n o n - a b e l i a n q u a n t u m n u m b e r s , i . e . t h e y a r e 10 \" c h a r g e d \" . We n e e d now t o i n t r o d u c e i n t h e l a g r a n g i a n a k i n e t i c t e r m f o r t h e v e c t o r g a u g e f i e l d s . I n a n a l o g y w i t h t h e a b e l i a n c a s e ( Q E D ) , a p o s s i b l e a n t i s y m m e t r i c s e c o n d r a n k t e n s o r f o r t h e f e r m i o n f i e l d i s ; ( I I . 7 ) w h i c h l e a d s u s t o d e f i n e h fi c ( I I . 8 ) a U n d e r i n f i n i t e s i m a l t r a n s f o r m a t i o n , t r a n s f o r m s a s a m u l t i p l e t u n d e r G; >a _ r a i r c ( I I . 9 ) P u ^ ~ f ~ + Cahc 9 F T h e c o m b i n a t i o n F F^,^ i s t h e n i n v a r i a n t u n d e r G. N o t i c e t h a t ; 1: A m a s s t e r m f o r t h e g a u g e f i e l d w o u l d n o t b e i n v a r i a n t ( u n l e s s t h e g a u g e f i e l d was i n v a r i a n t u n d e r G ) . 2: T h e k i n e t i c e n e r g y t e r m f o r t h e g a u g e f i e l d i m p l i e s t r i p l e a n d q u a d r u p l e v e r t i c e s , s i n c e b A C (11.10) 11 The G - i n v a r i a n t Lagragian i s f i n a l l y ; ABELIAN CASE; U(1) SYMMETRY: The U ( l ) case i s simply QED. There i s only one generator; t h e r e f o r e the s t r u c t u r e constant i s 0 and the gauge f i e l d tensor i s ; F M J , = ()hA\u00bb - XAh) (II-12) The complete Lagrangian i s then j u s t the usual QED Lagrangian, with the f i e l d A^ being r e a d i l y r e l a t e d to the vect o r p o t e n t i a l of electromagnetism. SU(2) CASE; T h i s i s the Yang M i l l s case where SU(2) i s u s u a l l y taken to be an i s o s p i n symmetry, r e l a t i n g to \" i n t e r n a l \" i s o s p i n quantum numbers. Equations (II.2) to (11.11 ) h o l d with the i d e n t i f i c a t i o n Ca be ~ ^ a- b c and the two-dimensional r e p r e s e n t a t i o n m a t r i c e s can be chosen to be the u s u a l P a u l i m a t r i c e s , i = l,2,3. SU(3) CASE; The l o c a l SU(3) symmetry has found an a p p l i c a t i o n 1 2 i n t h e a t t e m p t t o d e v e l o p a f u n d a m e n t a l t h e o r y o f s t r o n g i n t e r a c t i o n s ( R e y a , 1 9 8 1 ) . T h e f e r m i o n s a r e q u a r k s , t h e e i g h t g a u g e p a r t i c l e s a r e c a l l e d g l u o n s , a n d t h e i n t e r n a l q u a n t u m n u m b e r o n w h i c h t h e s y m m e t r y i s b a s e d i s c a l l e d c o l o r . QCD i s a n e x a c t l y l o c a l l y i n v a r i a n t t h e o r y , i . e . t h e L a g r a n g i a n ( 1 1 . 1 1 ) a p p l i e s w i t h o u t a n y m o d i f i c a t i o n . T h e r e a r e e i g h t g e n e r a t o r s o f t h e S U ( 3 ) g r o u p , u s u a l l y l a b e l l e d \"X,-, i = 1 , . . . , 8 . T h e m o s t i m p o r t a n t p r o p e r t i e s o f QCD a r e a s y m p t o t i c f r e e d o m a n d c o n f i n e m e n t : I t s e f f e c t i v e c o u p l i n g c o n s t a n t , a t a g i v e n momentum t r a n s f e r s q u a r e d Q , i s g i v e n b y : _ L - _ L | ( I I . M ) i n t h e l e a d i n g a p p r o x i m a t i o n , a n d w h e r e sn^ i s t h e n u m b e r o f q u a r k f l a v o r s . A s l o n g a s 16, o^(Q) g r o w s s m a l l e r a t l a r g e 2 Q . T h i s i s c a l l e d a s y m p t o t i c f r e e d o m , a n d i s a m o s t u s e f u l f e a t u r e o f QCD, a s i t p e r m i t s p e r t u r b a t i v e t r e a t m e n t o f many \" h a r d \" s c a t t e r i n g p r o c e s s e s . I n f a c t , QCD i s t h e o n l y c a n d i d a t e t h e o r y w h i c h e x p l a i n s t h i s b e h a v i o r o f t h e S I c o u p l i n g c o n s t a n t , c o r r e s p o n d i n g t o t h e p h e n o m e n o n o f \" s c a l i n g \" i n e x p e r i m e n t s ( s e e c h a p t e r V ) . I f t h e c o u p l i n g c o n s t a n t g r o w s s m a l l e r a t l a r g e Q a n d c o r r e s p o n d i n g l y s h o r t d i s t a n c e s , t h e o p p o s i t e i s a l s o t r u e . L o w e r e n e r g y t r a n s f e r i n t e r a c t i o n s c o r r e s p o n d t o l a r g e r d i s t a n c e s a n d l a r g e c o u p l i n g s , w h i c h l e a d s t o t h e n o t i o n o f q u a r k c o n f i n e m e n t . Q u a r k c o n f i n e m e n t m e a n s t h a t q u a r k s a r e 1 3 f o r e v e r c o n f i n e d w i t h i n hadrons and cannot appear i s o l a t e d . Confinement has not been d e r i v e d from QCD y e t , but the behavior of the QCD c o u p l i n g constant makes i t q u a l i t a t i v e l y p l a u s i b l e . Asymptotic freedom and confinement are the most important reason QCD i s now c o n s i d e r e d the complete theory of strong i n t e r a c t i o n s . 14 I I I . H I G G S MECHANISM T h e H i g g s m e c h a n i s m c a n c a u s e t h e s p o n t a n e o u s s y m m e t r y b r e a k i n g o f some l o c a l l y i n v a r i a n t L a g r a n g i a n s ( H i g g s , 1 9 6 4 ) . B u t b e f o r e t o s t u d y i n g t h i s c a s e , o n e h a s t o s e e t h e e f f e c t o f t h e s p o n t a n e o u s b r e a k i n g o f a g l o b a l l y i n v a r i a n t l a g r a n g i a n . S P O N T A N E O U S L Y BROKEN SYMMETRY: L e t u s c o n s i d e r t h e c a s e o f t w o r e a l s c a l a r f i e l d s a n d ; T h e e f f e c t i v e p o t e n t i a l i s c h o s e n f o r i l l u s t r a t i o n t o b e ; ( i n . 1 > w h i c h i s i n v a r i a n t u n d e r r o t a t i o n U; ( I I I . 2 ) ( I I I . 3 ) a n d o n e c a n d i s t i n g u i s h t w o c a s e s : - c a s e 1: p. > 0. T h e m i n i m u m o f V o c c u r s a t ^ = ^ = 0 a n d t h i s w i l l g i v e s i m p l y a d e g e n e r a t e d o u b l e t o f m a s s - c a s e 2: p. < 0. T h e m i n i m u m o c c u r s a t .1 ( I I I . 4 ) 15 and there i s a continuum of degenerate s t a t e s at the minimum. The p o t e n t i a l f o r t h i s case i s represented i n F i g . 3 below. F i g u r e 3 - P o t e n t i a l I I I . 3 f o r the case ^ < 0 One can always d e f i n e c o o r d i n a t e s so that the p h y s i c a l vacuum i s at i n the c l a s s i c a l f i e l d t heory, that i s , i n the quantum f i e l d theory; <\u00b0| 0, while the ^ p a r t i c l e i s massless. T h i s i s an example of the Goldstone theorem, which s t a t e s t h a t i f a theory has an exact continuous symmetry of the Lagrangian which i s not shared by the vacuum, a massless p a r t i c l e must occur. HIGGS MECHANISM: In the case of a l o c a l l y i n v a r i a n t gauge theory, there i s no massless Goldstone boson when the symmetry i s spontaneously broken. The would-be Goldstone boson combines with the massless gauge boson to g i v e a massive v e c t o r boson. T h i s i s the Higgs mechanism. To i l l u s t r a t e t h a t p o i n t , l e t us c o n s i d e r the simple case of the Abelian. gauge theory with Lagrangian ( I I I . 7 ) ( I I I . 8 ) where The Lagrangian ( I I I . 7 ) d e s c r i b e s a charged s c a l a r i n t e r a c t i n g with i t s e l f , and with a gauge f i e l d A^. I f j i \/ < 0, i t 17 d e s c r i b e s s c a l a r QED. T h e L a g r a n g i a n i s i n v a r i a n t u n d e r t h e l o c a l t r a n s f o r m a t i o n s ( I I I . 9 ) 2 When > 0, | o > ^ ; v' i ( i n . i o > L e t u s u s e p o l a r v a r i a b l e s t o p a r a m e t r i z e cp , a n d e x p a n d a b o u t a s p e c i f i c v a c u u m p o i n t . T h i s i s d o n e t o show m o r e c l e a r l y t h e p h y s i c a l c o n t e n t o f t h e t h e o r y . T h e new s e t o f c o o r d i n a t e s i s 0. 0179 AK3(4) - -H3<4) 0180 C 0181 C TRANSFORM THE 4-MOMENTA BACK INTO FRAME 2 1 0182 C 0183 CALL MULT
0839 M1SA - M1B\/