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UBC Theses and Dissertations

A QCD-parton calculation of associated Higgs boson production in hadron-hadron collision 1984

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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 © 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 —>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~—>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(<P) f o r d i f f e r e n t v a l u e s o f fx 48 9. B r a n c h i n g r a t i o o f t h e H ° i n f u n c t i o n o f mH 51 10. F e y n m a n d i a g r a m f o r Z ° — > Y+ H d e c a y 54 1 1 . Z ° — > H ° + 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 ~ — > H ° + Z ° 55 o 1 3 . F e y n m a n d i a g r a m f o r g g — > H 57 14. C r o s s s e c t i o n s f o r p p — > H ° + X 59 o 1 5 . F e y n m a n d i a g r a m g +c — > 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 — > 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 • 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 — > H ° + 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 — > F + F + H ° 67 2 1 . F e y n m a n d i a g r a m s f o r g g — > F + F + H ° 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 m̂  , m̂  , 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« f o r four d i f f e r e n t set of the parameters m̂ , 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± 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 — > 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 ° , 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 ° 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 —>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 — > © ° , (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 —>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» - 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; <°|<M°> ~~^r Ol<f>a/0> = ° ( I I I . 5 ) To do p e r t u r b a t i o n theory around the c l a s s i c a l minimum, one has to expand i n powers of cf> = (h - V i n s t e a d of Cp . <p of course, i s s t i l l expanded around the value z e r o . ( I l l . 6 ) 16 The important f e a t u r e here r e s i d e s i n the mass terms. The f i e l d 11 has a c q u i r e d a (mass) - -2 jA > 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, <p d e v e l o p s a g a i n a v a c u u m e x p e c t a t i o n v a l u e . I t i s < o | ' < t > | 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 <f>00 = ^ iv + 7 f x H e ( i n . I D C o n s i d e r now t h e g a u g e t r a n s f o r m a t i o n ( I I I . 9 ) w i t h x ) / V . T h e t r a n s f o r m e d f i e l d s a r e : a n d A * A ; = A*- - ^ M 18 ( I I I . 13) T h i s i s r e f e r e d to as the U-gauge i n the l i t e r a t u r e (Abers and Lee, 1973). If one s u b s t i t u t e s these new e x p r e s s i o n s f o r the f i e l d s i n t o the Lagrangian ( I I I . 7 ) , and expands, one g e t s : Kinetic t-- i & i m ) - w f ; \ } 3 a 5 terins •TniSS in tetacti'oti tet-ms Self - inteticton of *l ( I I I . 14) The s c a l a r meson has a c q u i r e d a mass 3 AV - ̂  = 2 ju. and s e l f - i n t e r a c t i o n s represented by the v e r t i c e s of F i g . 4. \ )• 7 F i g u r e 4 - V e r t i c e s of the s e l f - i n t e r a c t i n g s c a l a r meson 19 P The i n t e r a c t i o n s between the v e c t o r gauge boson and the s c a l a r meson w i l l g ive r i s e to the v e r t i c e s of f i g u r e 5. F i g u r e 5 - V e c t o r - s c a l a r v e r t i c e s The gauge boson has a l s o a c q u i r e d a mass, which i s the aim of t h i s mechanism. We can h e n c e f o r t h b u i l d gauge t h e o r i e s g i v i n g r i s e to short-range i n t e r a c t i o n s . Moreover, the theory, a l t o u g h having i t s symmetry e x p l i c i t e l y broken, i s s t i l l r e n o r m a l i z a b l e . T h i s was demonstrated by t ' H o f f t (1971). The Higgs mechanism f i n d s i t s best a p p l i c a t i o n s i n the GWS model, which w i l l be d e s c r i b e d i n the next chapter. 20 IV. THE GLASHOW-WEINBERG-SALAM MODEL The GWS model i s u s u a l l y i n t r o d u c e d f i r s t with one doublet and one s i n g l e t of fermions o n l y . The other known fermions can e a s i l y be int r o d u c e d t h e r e a f t e r . T h i s i s the path I w i l l f o l l o w . BASIC LAGRANGIAN: One wants to i d e n t i f y the massive v e c t o r bosons a r i s i n g i n the Higgs mechanism with the in t e r m e d i a t e v e c t o r bosons (IVB) c a r r y i n g the WI. In the phenomenologically s u c c e s s f u l IVB theory, the l a g r a n g i a n f o r weak i n t e r a c t i o n s i s given by: r jUvA k.c.) where J , = 1/ )/A) -L )e i s the l e p t o n i c charged c u r r e n t i n i t s A e ,\ a s o - c a l l e d V-A form, and h.c. stands f o r "hermitian conjugate". On the other hand, the l a g r a n g i a n f o r the el 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 i s given by < L c . = C l L A-X ( I V ' 2 ) where J p / = e / e i s the e l e c t r o m a g n e t i c c u r r e n t . Then, t o u n i f y EM with WI, one needs at l e a s t 3 gauge — o t •*. bosons, W , W , Z , to couple with the c u r r e n t s , and J e / e c . The s i m p l e s t group with three such generators i s S U ( 2 ) . However, i f and C are to form a doublet under S U ( 2 ) , as 21 s u g g e s t e d b y t h e f o r m o f t h e c u r r e n t J / p r , Q.elec c a n n o t b e a g e n e r a t o r o f t h e g r o u p , b e c a u s e t h e e l e c t r i c c h a r g e s o f t h e d o u b l e t d o n o t a d d u p t o z e r o , w h e r e a s a l l 2 X 2 S U ( 2 ) r e p r e s e n t a t i o n s m a t r i c e s a r e t r a c e l e s s . O n e i s t h e n l e d t o i n t r o d u c e a f o u r t h g a u g e b o s o n Z . T h e s m a l l e s t g r o u p i s now S U ( 2 ) ® U ( 1 ) . A s s u m i n g t h a t o n l y V - A ± i n t e r a c t i o n s o c c u r f o r W , o n e t a k e s t h e g e n e r a t o r s o f t h e a. S U ( 2 ) g r o u p s t o b e t h e i s o s p i n o p e r a t o r s T , w h o s e 2-D r e p r e s e n t a t i o n may b e t a k e n t o be t h e P a u l i m a t r i c e s d i v i d e d b y t w o . ( I V . 3 ) L = (^e/^ ^ s t h e n a d o u b l e t u n d e r S U ( 2 ) a n d R = e R i s a s i n g l e t , a s w a n t e d - T h e s u b s c r i p t R o r L m e a n s t h a t o n l y t h e r i g h t - h a n d e d o r l e f t - h a n d e d c o m p o n e n t o f t h e l e p t o n w a v e f u n c t i o n i s s e l e c t e d . One d o e s i t b y m u l t i p l y i n g t h e s p i n o r b y t h e p r o j e c t i o n o p e r a t o r (1 - ^ s ) t o o b t a i n t h e l e f t - h a n d e d c o m p o n e n t , o r (1 + V s ) f o r t h e r i g h t - h a n d e d c o m p o n e n t , i . e . 1.. •= ( W ( I V . 4 ) T h e g e n e r a t o r o f U ( 1 ) i s c h o s e n s u c h t h a t t h e e l e c t r i c c h a r g e i s a l i n e a r c o m b i n a t i o n o f t h e U ( 1 ) g e n e r a t o r a n d t h e g e n e r a t o r T 3 o f S U ( 2 ) . One c a n c h o o s e 22 ( I V . 5 ) as generator of the CJ(1) group. The b a s i c Lagrangian L of the GWS model may be s p l i t i n t o 4 p a r t s as f o l l o w ; J = J * J + J J ( I V . 6 ) The gauge p a r t of the Lagrangian i s ; with oc fc •a m) ( I V - 8 > GAUGE Fi'fiO TfyJafi The l e p t o n i c p a r t of the Lagrangian i s ; titrmon ~ L L I % 8 ~ \ }Z Aw) L t R a K r ^ + ^ ' 6 j K , I V - 9 ) 23 w h e r e g = c o u p l i n g c o n s t a n t a s s o c i a t e d w i t h S U ( 2 ) g ' = c o u p l i n g c o n s t a n t a s s o c i a t e d w i t h U ( 1 ) N o t i c e o n e c a n n o t h a v e a b a r e m a s s t e r m o f t h e f o r m e _ e . , w h i c h i s f o r b i d d e n b y S U ( 2 ) i n v a r i a n c e . A l s o , t h e t e r m s e ^ e R a n d e i eL v a n i s h , b e c a u s e t h e y c o n t a i n t h e p r o d u c t s o f o r t h o g o n a l o p e r a t o r s (1 - Y$ ) a n d (1 + Ys ). a. T o g i v e a m a s s t o t h e g a u g e b o s o n s A ^ a n d t h e e l e c t r o n , l e t u s i n t r o d u c e a d o u b l e t o f c o m p l e x H i g g s s c a l a r s ( I V . 1 0 ) T h e y h a v e Y = 1 t o s a t i s f y ( I V . 5 ) , a n d t r a n s f o r m a s a d o u b l e t u n d e r S U ( 2 ) . T h e s c a l a r p a r t o f t h e l a g r a n g i a n i s t h e n ( I V . 1 1 ) T h e m o s t g e n e r a l r e n o r m a l i z a b l e H i g g s p o t e n t i a l V(<£) i s ( F l o r e s a n d S h e r , 1 9 8 2 ) ; V ( 9 ) = M^V + Ktfvf <IV-12> 24 A l s o , one i s f r e e to add a c o u p l i n g between the s c a l a r doublet and the l e p t o n s , of the form; I t has been i n t r o d u c e d by Yukawa (1935) to e x p l a i n the nuc l e a r b i n d i n g f o r c e between nucleons, through the exchange of bosons. There cannot be terms of the form L t L because, being the product of three SU(2) d o u b l e t s , they cannot form an SU(2) i n v a r i a n t . One needs to i n c l u d e both a s i n g l e t and a doublet, hence the form of (IV.13). One now must spontaneously break the SU(2) ® U ( 1 ) symmetry. SPONTANEOUS SYMMETRY BREAKING (SSB): Assume once more that p. < 0. The two minima are at|<?/ = v/ 2 with v = One now r e q u i r e s the n e u t r a l s c a l a r f i e l d t o develop a vacuum e x p e c t a t i o n value (VEV). One must l e t the VEV of the charged s c a l a r v a n i s h , i n order not to have a charged vacuum. T h i s leads t o 1 YL/KAWA (IV.13) The c o u p l i n g of the form i s known as the Yukawa c o u p l i n g . (IV.14) and again one expresses the s c a l a r f i e l d s i n p o l a r c o o r d i n a t e s 25 to b r i n g out the p h y s i c a l meaning of the theory. Using the u n i t a r y t r a n s f o r m a t i o n ; l u $ ) = e ( I V - 1 5 ) the s c a l a r f i e l d s read i n the new c o o r d i n a t e s ; (IV.16) The four r e a l components of $ are now d i s t r i b u t e d i n 3 components f o r and one f o r the s c a l a r °l. The symmetry breaking scheme that Weinberg and Salam adopted breaks both SU(2) and U(1)y, but pr e s e r v e s U ( l ) e / f c . One can check t h i s u s ing the c o n d i t i o n f o r a generator 21 t o leave the vacuum i n v a r i a n t ; where ($) i s the 2 X 2 matrix r e p r e s e n t a t i o n of the operator o - 0 (IV.17) For the generators of SU(2) X U(1), we f i n d . (IV.18) 26 But CX̂X = i ( z 3 H ) < 4 \ ~-o <iv . , 9 ) The photon, and only the photon w i l l then remain massless. Transforming now to the U-gauge (IV.15): 4 > - 4 ' - - Utf)<|> = U)/2 L - * 1' = U i \ e R : ^ R , ' 8^ r Bfj. (iv. 20) r A ; - T°A';= ̂ n[T/\;-iu"(f)̂ w«)]u'(f) and A^ s t i l l transforms a c c o r d i n g to ( I I . 6 ) . 'In terms of the new f i e l d s , the Lagrangians become; •̂P J . r ' a r- I o l G A U G E I pi> The mass of the e l e c t r o n a r i s e s from the Yukawa term; 2 7 ( I V . 2 3 ) a n d i s s e e n t o b e m e = vGe/*{2. T h e c o u p l i n g o f t h e r e m a i n i n g H i g g s b o s o n t t o t h e e l e c t r o n i s i/a ?y~ (iv.24) w h i c h w i l l b e o f p r i m a r y i m p o r t a n c e t o p r o d u c e a n d d e t e c t i t . T h e s c a l a r f i e l d w i l l g i v e r i s e t o g a u g e b o s o n m a s s e s v i a t h e t e r m w i t h L e t u s i s o l a t e t h e v e c t o r m a s s t e r m s , w h i c h a r e t h o s e t e r m s q u a d r a t i c i n v e c t o r f i e l d s , i n t o a L a g r a n g i a n J1 , s u b s e t o f 28 I f » e d e f i n e ^ r ^ ^ + M ^ , 3 w i t h a , o >a\ at ( I V . 2 8 ) T h e L a g r a n g i a n ̂  b e c o m e s X* -" All w I m * * t M i ? * ? " < I V - 2 9 ) ( I V . 3 0 ) T h e m a s s l e s s v e c t o r g a u g e b o s o n c a n t h e n b e i d e n t i f i e d w i t h t h e p h o t o n . F o r c o n v e n i e n c e , o n e u s u a l l y i n t r o d u c e s a n a n g l e dw ( W e i n b e r g a n g l e ) , w h i c h r e l a t e s t h e c o u p l i n g g o f t h e S U ( 2 ) ^ t o t h e c o u p l i n g g ' o f t h e U ( 1 ) V g r o u p . E x p l i c i t l y 29 J hn 6w ( i v .3D So t h a t 4 f * V 1 /COS 6. <!»•"> T h e i n t e r a c t i o n b e t w e e n t h e l e p t o n s a n d t h e g a u g e f i e l d s c a n now b e r e a d o f f f r o m t h e l e p t o n l a g r a n g i a n i n ( I V . 2 2 ) . R e - e x p r e s s i n g $\Kftoh i n f u n c t i o n o f t h e new f i e l d s W~, a n d A r , o n e g e t s , u s i n g ( I V . 3 2 ) ; L u , - JL* i ^ ( \ ( i ^ ^ / e r i u \ t j \ ) 3 COS 01* a? > n ' ^ ^ y - * ( I V . 3 3 ) E q u a t i n g t h e c o u p l i n g b e t w e e n t h e p h o t o n A ^ a n d t h e e l e c t r o n t o t h e e l e c t r o m a g n e t i c c o u p l i n g e g i v e s t h e r e l a t i o n : S sin Qw = €. ( I V . 3 4 ) T h e I V B c o u p l i n g i s c o n s i s t e n t w i t h l o w e n e r g y p h e n o m e n o l o g y p r o v i d e d we i d e n t i f y 30 3 - _ ! - ( I V . 3 5 ) I/a ml ^ 3 T h e o n l y m i s s i n g p i e c e i s t h e m a s s o f t h e H i g g s b o s o n w h i c h c a n b e w o r k e d o u t f r o m <$M\t* • T h i s i s e q u a l t o : z ( I V . 3 6 ) t h a t i s , c o m p l e t e l y u n d e t e r m i n e d i n t h i s m o d e l . H o w e v e r , u p p e r a n d l o w e r l i m i t s h a v e b e e n d e r i v e d , w h i c h we w i l l c o n s i d e r i n c h a p t e r V I . A D D I T I O N OF QUARKS: L e t u s i n t r o d u c e t h e f i r s t t w o f a m i l i e s o f q u a r k s . T h e i n c o r p o r a t i o n o f t h e t h i r d f a m i l y f o l l o w s t h e same l i n e o f a r g u m e n t . A c c o r d i n g t o C a b b i b o ' s p i c t u r e o f WI ( C a b i b b o , 1 9 6 3 ) , t h e h a d r o n i c c h a r g e d c u r r e n t s a r e r e p r e s e n t e d b y t h e w e a k - i s o s p i n d o u b l e t s L ( J ' l > U - (1\ ' S'"/ W S ( I V . 3 7 ) > . ci - J cos 6c + S  st'n 6<l w h e r e 5' -'J Sin Bz + S cos 0c a n d & c i s r e f e r e d t o a s t h e ( C a b b i b o ) m i x i n g a n g l e . T h e L a g r a n g i a n t e r m s c o r r e s p o n d i n g t o ( I V . 2 2 ) a n d ( I V . 2 3 ) a r e 31 (IV.38) q = u, d, s, c and (IV.39) r e s p e c t i v e l y . The Lagrangian p i e c e (IV.38) g i v e s r i s e to e x a c t l y the same kind of r e s u l t s as f o r the l e p t o n case. I f we now perform SSB by r e p l a c i n g by i t s e x p e c t a t i o n value (IV.14) we o b t a i n a s e r i e of mass terms e q u i v a l e n t t o (IV.23). (IV.39) We must now chose the Yukawa c o u p l i n g s Gj ,...,G6 so that u,d,s and c are mass e i g e n s t a t e s ; Gi -" ^ 1/2 G 3  1 %s 1/3 AT (IV.40) Gt, - ^ c ifa /at 6s- : " G2 t*f> Be Gc - f G2 cot^n Bc 32 The g e n e r a l i z a t i o n to three g e n e r a t i o n s i n t r o d u c e s two more quark mixing angles, but the r e s t of the procedure s t a y s e s s e n t i a l l y the same. 33 V . PARTON MODEL AND HADRON-HADRON C O L L I S I O N I n o r d e r t o c a l c u l a t e t h e p r o d u c t i o n r a t e s i n h a d r o n - h a d r o n a n d l e p t o n - h a d r o n c o l l i s i o n s , some s i m p l i f y i n g h y p o t h e s e s a r e n e e d e d a b o u t t h e s t r u c t u r e o f h a d r o n s . S u c h a s e t o f h y p o t h e s e s , w e l l s u p p o r t e d b y e x p e r i e n c e , f o r m s t h e p a r t o n m o d e l 1 . L e t u s d e s c r i b e i t s s o u r c e s , l i n k s w i t h QCD a n d a p p l i c a t i o n s t o h a d r o n - h a d r o n c o l l i s i o n . When h i g h e n e r g y e l e c t r o n s o r n e u t r i n o s a r e s c a t t e r e d f r o m n u c l e o n s t h e i r a n g u l a r d i s t r i b u t i o n s l o o k a s i f t h e y w e r e s c a t t e r i n g f r o m h a r d , p o i n t l i k e c o n s t i t u e n t s i n s i d e t h e n u c l e o n s . I t i s a r e p e t i t i o n , a t h i g h e r e n e r g i e s , o f R u t h e r f o r d ' s e x p e r i m e n t . T h e s e p o i n t l i k e c o n s t i t u e n t s o f n u c l e o n s h a v e b e e n g i v e n t h e name " p a r t o n s " , a n d t h e p a r t o n s i n t e r a c t i n g e l e c t r o m a g n e t i c a l l y o r w e a k l y w i t h t h e l e p t o n s h a v e b e e n i d e n t i f i e d ( t h e o r e t i c a l l y ) a s q u a r k s . A n o t h e r c l a s s o f n u c l e o n c o n s t i t u e n t s , m e d i a t i n g t h e QCD f o r c e b e t w e e n t h e q u a r k s , a r e " g l u o n s " . T h e g l u o n s d o n o t i n t e r a c t t h r o u g h WI, a n d a r e e l e c t r i c a l l y n e u t r a l . T h e y a r e s p i n - 1 b o s o n s , a n d c a r r y t h e " c o l o r " c h a r g e w h i c h g i v e s r i s e t o s t r o n g i n t e r a c t i o n s . T h e r e f o r e , g l u o n s i n t e r a c t w i t h g l u o n s , m a k i n g q u a n t i t a t i v e p r e d i c t i o n s o f QCD v e r y d i f f i c u l t . S t i l l , t h e m e t h o d o f l a t t i c e g a u g e t h e o r y 2 m a n a g e d t o p r o d u c e a n a c c e p t a b l e h a d r o n s p e c t r u m . T h e l a t t i c e g a u g e t h e o r y i s a n o n - p e r t u r b a t i v e way o f g e t t i n g p r e d i c t i o n s f r o m a t h e o r y w i t h a l a r g e c o u p l i n g c o n s t a n t , l i k e QCD a t l o w momerttum t r a n s f e r . 1 s e e f o r e x a m p l e ( C l o s e , 1 9 7 9 ) 2 f o r a r e v i e w s e e ( D r o u f f e a n d I t z y k s o n , 1 9 7 8 ) 34 A l s o , when experiments reach very high energy, the phenomenon of s c a l i n g occurs, and one may use p e r t u r b a t i v e QCD to c a l c u l a t e p r o d u c t i o n r a t e s . SCALING: The parton model p i c t u r e stems from a property of le p t o n - p r o t o n s c a t t e r i n g , c a l l e d s c a l i n g . Here are the foundations of i t . When one c a l c u l a t e s the amplitude f o r l e p t o n - l e p t o n s c a t t e r i n g , say — > e , one gets from the amplitude c o r r e s p o n d i n g to the diagram of F i g . 6 F i g u r e 6 - Feynman diagram f o r electron-muon s c a t t e r i n g the c r o s s - s e c t i o n : Q C M 6 x U SVn'fi a w i 1 (v.D where E : energy of the incoming e l e c t r o n i n the l a b frame E': energy of the s c a t t e r e d e l e c t r o n i n the l a b frame V : energy t r a n s f e r t o the muon i n the l a b frame z Q : minus one times the momentum t r a n s f e r squared 35 0 : s c a t t e r i n g angle of the e l e c t r o n i n the l a b frame M : r e s t mass of the muon On the other hand, i f one wants to c a l c u l a t e i n c l u s i v e e l e c t r o n - p r o t o n s c a t t e r i n g , one i s f o r c e d to in t r o d u c e s t r u c t u r e f u n c t i o n s i n the hadronic r a t e t e n s o r s W . The hadronic tensor W i s the pi e c e which must be i n t r o d u c e d i n the spin-summed amplitude squared at the l o c a t i o n of the photon-proton v e r t e x i n e v a l u a t i n g the r a t e corresponding to the Feynman diagram of F i g . 7 ) F i g u r e 7 - Feynman diagram 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 . (See appendix B f o r i n t r o d u c t i o n to Feynman diagrams). I t serves to parametrize our ignorance of the form of the c u r r e n t at the proton end of the photon propagator. The most g e n e r a l form f o r the proton s t r u c t u r e f u n c t i o n i s 36 p being the proton 4-momentum. The W are i n gene r a l 2 pis f u n c t i o n s of V and Q . Gauge i n v a r i a n c e q W = 0 g i v e s us some r e l a t i o n s between the c o e f f i c i e n t s , and one i s l e f t with only two independent s t r u c t u r e f u n c t i o n s ; (V.3) assuming p a r i t y - c o n s e r v i n g i n t e r a c t i o n s . C o n t r a c t i n g with the l e p t o n i c r a t e tensor L and e x p r e s s i n g the c r o s s - s e c t i o n i n the l a b o r a t o r y frame where the i n i t i a l proton i s at r e s t y i e l d s ; J V = faXei J o / I w.(v.q')+3 w s v , ( (v.4) One can now compare e x p r e s s i o n s (V.1) and (V.4) and deduce that i f the v i r t u a l photon s c a t t e r s o f f a p o i n t l i k e D i r a c p a r t i c l e , the s t r u c t u r e f u n c t i o n s reduce t o : the equations being w r i t t e n t h i s way to form dimensionless 2 r a t i o CO= 2MV/Q o n l y . 37 For s c a t t e r i n g from the proton, i n g e n e r a l one expects a 2 dependence of these f u n c t i o n s on 2/ and Q s e p a r a t e l y . But, at high momentum t r a n s f e r the phenomena of s c a l i n g o c c u r s , i . e . i s observed to h o l d e m p i r i c a l l y (Bjorken, 1969). The energy and momentum-transfer dependence of the process behaves e x a c t l y as i f the e l e c t r o n s were s c a t t e r i n g o f f hard, p o i n t l i k e c o n s t i t u e n t s i n s i d e the protons, i . e . l i k e (V.5). T h i s i s why i t was s a i d e a r l i e r t h a t i t i s a r e p e t i t i o n , at higher e n e r g i e s , of the R u t h e r f o r d s c a t t e r i n g experiment. Impulse approximation: The parton model c o n t a i n s i m p l i c i t l y the e q u i v a l e n t of the impulse approximation i n n u c l e a r p h y s i c s . The b a s i c assumptions are the f o l l o w i n g : 1. During the time of i n t e r a c t i o n one can n e g l e c t i n t e r a c t i o n s between the p a r t o n s . 2. F i n a l s t a t e i n t e r a c t i o n s can be i g n o r e d . That i s to say the parton i s q u a s i - f r e e i n the p r o t o n , and can be c o n s i d e r e d f r e e at very h i g h energy. The e f f e c t of confinement a c t s much l a t e r , when the s c a t t e r e d parton has moved a d i s t a n c e of the same order as the s i z e of the proton. In terms of Feynman diagrams, t h i s means t h a t we c o n s i d e r 3 f o r f i x e d Co and Q >, 1 GeV; (V.6) 38 s u b p r o c e s s e s w i t h d i f f e r e n t i n i t i a l o r f i n a l s t a t e s a s b e i n g n o n - i n t e r f e r i n g . O n e d o e s n o t h a v e t o w o r r y a b o u t t h e o t h e r " s p e c t a t o r " p a r t o n s t o t h e l o w e s t o r d e r . T h e s e a s s u m p t i o n s a r e e x t r e m e l y u s e f u l f o r c a l c u l a t i o n s o f p r o c e s s e s i n h a d r o n - h a d r o n s c a t t e r i n g . A n o t h e r e l e m e n t t h a t o n e n e e d s i s t h e p a r t o n m omentum d i s t r i b u t i o n , w h i c h w i l l b e d i s c u s s e d b e l o w . E l e c t r o m a g n e t i c s p i n 1/2 s t r u c t u r e f u n c t i o n : Now we w i l l e x p r e s s ( V . 1 ) a n d ( V . 4 ) u s i n g t h e M a n d e l s t a m i n v a r i a n t s s , t T h i s s e t o f v a r i a b l e s m a k e s e x p l i c i t t h e L o r e n t z i n v a r i a n c e o f a n y q u a n t i t y e x p r e s s e d i n t e r m s o f i t . T h e r e l a t i o n ( V . 1 ) b e c o m e s a n d u : t - Cp-p' 7/ - ( P-9 ) (V.1) ( V . 8 ) I t w i l l b e u s e f u l l a t e r t o know t h a t 5 r t c ( V . 9 ) I f o n e w a n t s t o c o m p a r e ( V . 8 ) t o i n e l a s t i c e l e c t r o n - p r o t o n s c a t t e r i n g c r o s s - s e c t i o n ( V . 4 ) , o n e u s e s t h e p a r t o n m o d e l , 39 w h e r e i t i s h y p o t h e s i s e d t h a t i n e l a s t i c , e l e c t r o n - p r o t o n s c a t t e r i n g c o m e s f r o m t h e sum o f i n c o h e r e n t e l a s t i c s c a t t e r i n g o f e l e c t r o n s o n t h e p a r t o n s i n t h e t a r g e t . I f t h e s e p a r t o n s h a v e s p i n 1/2 a n d c o u p l e t o t h e p h o t o n t h e same way t h e ju~ c o u p l e s t o t h e p h o t o n , t h e n o n e c a n e a s i l y o b t a i n a n e x p r e s s i o n f o r t h e c r o s s - s e c t i o n . G o i n g i n t o a r e f e r e n c e f r a m e w h e r e t h e p r o t o n h a s i n f i n i t e m omentum, o n e e f f e c t i v e l y " f r e e z e s " t h e s l o w i n t e r a c t i o n s . O n e d e f i n e s (V.10) T h e r e l a t i o n (V.8) c a n b e w r i t t e n i n t e r m s o f t h e momentum o f a p a r t o n w i t h t h e s u b s t i t u t i o n s s — > x s u — > x u t — > t dV̂J fa* I / S J + U 4 N ! X 8(f + X ( 5 * U ) ) (V . 1 1 ) J t du I t 2 2 \ S* T h e p r o t o n b e i n g s u p p o s e d l y made o f s e v e r a l p a r t o n s , d e n o t e d b y t h e i n d e x i , o n e h a s t o sum o v e r t h e s e , a n d i n t e g r a t e o v e r t h e p r o b a b i l i t y f u n c t i o n f ( x ) f o r a p a r t o n i t o h a v e a momentum f r a c t i o n b e t w e e n x a n d x + d x , t o g e t t h e c r o s s - s e c t i o n 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 : i t i u / e r * t 2 2 £z J L s+u  ( v - 1 2 ) 40 where the r e l a t i o n (V.9) has been used i n r e w r i t i n g the d e l t a f u n c t i o n . We are now ready to compare with the general e x p r e s s i o n f o r e-p s c a t t e r i n g , eq. (V.4) which we r e c a s t i n t o the form (V.13) Comparing the c o e f f i c i e n t s i n (V.12) and (V.13), one gets (V.14) T h i s i s the C a l l a n - G r o s s formula ( C a l l a n and Gross, 1969) f o r s p i n 1/2 parton model. I d e n t i f y i n g the s p i n 1/2 partons with the quarks, and denoting f^ by the symbol q., one puts f o r the proton and the neutron (V.15) i f 7 ( * ) ^ i U v + J * ) - * H ^ k ^ ^ + s * ) * - Sum r u l e s and momentum parton d i s t r i b u t i o n s : The fundamental r e l a t i o n s the quarks d i s t r i b u t i o n s u(x) and d(x) must obey come 41 f r o m t h e i s o s p i n p r o p e r t i e s a n d z e r o n e t s t r a n g e n e s s o f t h e p r o t o n a n d n e u t r o n : 0 r f.'dx LSOO - SrioJ 1 0 ( V . 1 6 a ) O r 5 - I J x l u u ) - u a ) J J = J o J* E «kx) - Jc*). ( V . 1 6 b ) A s i m i l a r r e l a t i o n f o r e l e c t r i c a l l y n e u t r a l p a r t o n s ( g l u o n s ) c o m e s f r o m momentum c o n s e r v a t i o n i ( V . 1 7 ) w h e r e £ i s t h e f r a c t i o n o f momentum c a r r i e d b y t h e g l u o n s . T h e n t h e g l u o n momentum d i s t r i b u t i o n G ( x ) m u s t o b e y L ix x Gcx) - £ (V.18) T h e v a l u e o f £ t u r n e d o u t i n e x p e r i m e n t s t o b e a b o u t 42 £ - 0 . 5 ( S m i t h , 1 9 7 4 ) . T h a t m e a n s t h a t h a l f o f t h e p r o t o n momentum i s c a r r i e d b y t h e g l u o n s . T h e q u a r k momentum d i s t r i b u t i o n s c a n a l s o b e d e d u c e d f r o m e x p e r i m e n t s a f t e r i n v e r t i n g e q u a t i o n s ( V . 1 . 5 ) e/v  & P 1 ( V . 1 9 ) Jcx) - i l (x) - F 2 (x)J A IS A f e w p a r t i c u l a r p a r a m e t r i z a t i o n s a r e g i v e n i n a p p e n d i x F . T h e m o s t u s e f u l c l u e s c o m e f r o m t h e l>- h a d r o n a n d e - h a d r o n d e e p i n e l a s t i c s c a t t e r i n g . H a d r o n - h a d r o n s c a t t e r i n g ; T h e c a l c u l a t i o n o f a p r o c e s s i n QCD f o r h a d r o n - h a d r o n s c a t t e r i n g p r o c e e d s a c c o r d i n g t o t h e f o l l o w i n g s c h e m e . a ) C a l c u l a t e t h e s u b p r o c e s s i n a p e r t u r b a t i v e way, u s i n g QCD r u l e s , a n d o t h e r m o d e l s , s u c h a s 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 g i v e s a s u b - c r o s s - s e c t i o n C , ( x • , u , s , t ) w h e r e x . i s t h e J jab ' ' f r a c t i o n o f t h e m o m e n t a u , s a n d t c a r r i e d b y t h e i n c o m i n g p a r t o n i . b ) C o n v o l u t e & i u 6 w i t h t h e p a r t o n d i s t r i b u t i o n s f ( x ) o f t h e i n c o m i n g h a d r o n s , w h i c h r e a d s ; 43 Let us now come back to the sub j e c t of Higgs boson, to examine i t s p r o p e r t i e s i n more d e t a i l . T h i s w i l l enable us to c a l c u l a t e the s u b - c r o s s - s e c t i o n s r e l a t e d to Higgs p r o d u c t i o n i n hadron c o l l i s i o n s . 44 VI . HIGGS BOSON PHENOMENOLOGY Here, we are coming t o the heart of the s u b j e c t -the Higgs boson p r o p e r t i e s . The knowledge of these i s e s s e n t i a l t o the development of a s t r a t e g y i n the Higgs boson "hunt". I t s mass and c o u p l i n g s to matter are needed to p r e d i c t the p r o d u c t i o n mechanisms and d e t e c t i o n modes. Mass of the Higgs boson: There e x i s t s e v e r a l arguments g i v i n g r i s e to upper and lower mass bounds on the Higgs boson. Only one, a lower bound, i s d e r i v e d from s o l i d experimental f a c t s . A l l other ones depend on t h e o r e t i c a l e x p e c t a t i o n s . One upper bound on the mass of the Higgs boson comes from the u n i t a r y r e s t r i c t i o n i n the e l a t i c s c a t t e r i n g W+W —>W^W (Lee et a l . , 1977). The s u b s c r i p t L denotes a l o n g i t u d i n a l l y p o l a r i z e d p a r t i c l e . As we have seen i n chapter I, the process v i o l a t e s u n i t a r i t y , and t h i s i s removed by the s c a l a r c o n t r i b u t i o n . The s c a t t e r i n g amplitude f o r t h i s process i s , a f t e r c a n c e l l a t i o n of the divergences T (VI.1) The amplitude T has the usual p a r t i a l - w a v e expansion T ~ l& i r Ltytl) t Please) (VI.2) P a r t i a l wave u n i t a r i t y r e q u i r e s jt.| < 1. Here, j = 0, and 45 fal < Hidi ~ l.S TeV/c* <VI-3) Of A m o r e r e f i n e d c a l c u l a t i o n y i e l d s m M < 1 T e V / c ( L e e , Q u i g g a n d T h a c k e r , 1 9 7 7 ) . I f mH l i e s a b o v e t h i s l i m i t , p e r t u r b a t i o n e x p a n s i o n b r e a k s down a n d h i g h e r o r d e r t e r m s a r e a s i m p o r t a n t a s t h e l o w e s t o r d e r o n e . I t c a n b e sho w n ( V e l t m a n , 1 9 7 7 ) t h a t a s mH i n c r e a s e s , t h e p a r a m e t e r o f t h e s c a l a r p o t e n t i a l ( I V . 1 1 ) i n c r e a s e s , a n d when /)>>1, p e r t u r b a t i v e t h e o r y i s m e a n i n g l e s s . T h i s h a p p e n s a t a r o u n d m H= 1 T e V / c ' . T h e r e i s n o t h i n g w r o n g i n i t s e l f f o r t h e p e r t u r b a t i v e e x p a n s i o n t e c h n i q u e t o b r e a k d o w n . A p e r t u r b a t i v e t h e o r y i s m e r e l y a d e s i r a b l e c o n d i t i o n . I t h a s b e e n s h o w n t h a t a n o n p e r t u r b a t i v e H i g g s s e c t o r w o u l d show v e r y l i t t l e e f f e c t i n c u r r e n t p h e n o m e n o l o g y ( A p p e l q u i s t a n d B e r n a r d , 1 9 8 0 ) . T h e H i g g s s e c t o r i t s e l f c o u l d g i v e r i s e t o some new p h y s i c s , f o r e x a m p l e w i t h b o u n d s t a t e s o f e l e m e n t a r y H i g g s b o s o n s . A n o t h e r u p p e r b o u n d o n t h e H i g g s c o m e s f r o m a s t u d y o f t h e h t r i v i a l i t y o f t h e s c a l a r ?\^ i n t e r a c t i o n ( C a l l a w a y , 1 9 8 3 ) . T h e r e i s e v i d e n c e t h a t t h e t h e o r y i s a t r i v i a l t h e o r y , i . e . t h e i n t e r a c t i o n s c r e e n s i t s e l f a n d i s e q u i v a l e n t t o a f r e e f i e l d t h e o r y . T h e ̂  i n t e r a c t i o n c o u p l e d w i t h f e r m i o n s a n d / o r v e c t o r b o s o n s m i g h t n o t b e t r i v i a l h o w e v e r . T h i s c o u l d h a p p e n o n l y w i t h i n c e r t a i n l i m i t s , o n e o f w h i c h b e i n g a b o u n d o n t h e o H m a s s . F o r t h e s t a n d a r d m o d e l : 46 d^A <: \2J fm.^^O &V/c* ( v i . 4 ) L o w e r b o u n d s o n t h e H i g g s m a s s h a v e b e e n p r o p o s e d f r o m s e v e r a l s o u r c e s . T h e d e c a y K — > 7 T 1 1 g i v e s a l o w e r b o u n d o f a b o u t 3 2 5 M e V / c j f o r t h e m a s s o f t h e H i g g s b o s o n ( W i l l e y a n d Y u , 1 9 8 2 ) . I f t h e H i g g s b o s o n was a n y l i g h t e r , i t w o u l d a p p e a r a s a r e s o n a n c e p e a k i n t h e i n v a r i a n c e m a s s o f t h e l e p t o n p a i r . O t h e r l o w e r b o u n d s c o me f r o m s t u d i e s o f t h e r a d i a t i v e l y c o r r e c t e d H i g g s p o t e n t i a l . H o w e v e r , a l l c o n c l u s i o n s d e r i v e d f r o m t h e s t u d i e s o f t h e H i g g s p o t e n t i a l a r e s u s p e c t 1 . T h e t e c h n i q u e o f t h e s c a l a r p o t e n t i a l i s d e v e l o p e d i n t h e f o l l o w i n g w a y . We i n t r o d u c e d i n ( I V . 1 1 ) t h e c l a s s i c a l s c a l a r p o t e n t i a l V(<£)= fJ. I$>j t ^ / ^ / w h i c h i s t h e m o s t g e n e r a l r e n o r m a l i z a b l e s c a l a r f i e l d e x p r e s s i o n . T h e " e f f e c t i v e " p o t e n t i a l f o r t h e q u a n t u m f i e l d c a n b e w r i t t e n i n t e r m s o f t h e c l a s s i c a l f i e l d . ^ c ( J o n a - L a s i n i o , 1 9 6 4 ) , ( C o l e m a n a n d W e i n b e r g , 1 9 7 3 ) . A c c o r d i n g t o q u a n t u m f i e l d t h e o r y , o n e c a n c a l c u l a t e e f f e c t i v e p o t e n t i a l s a t t h e n o r d e r l e v e l , c o r r e s p o n d i n g t o n - l o o p g r a p h s . F o r e x a m p l e , t h e e f f e c t i v e p o t e n t i a l c o r r e s p o n d i n g t o t h e z e r o t h o r d e r c o r r e c t i o n ( t r e e g r a p h ) i s t h e c l a s s i c a l p o t e n t i a l ( I V . 1 0 ) . T h e f i r s t o r d e r q u a n t u m c o r r e c t i o n i n c l u d e s t h e sum o v e r a l l o n e - l o o p g r a p h s o f t h e t h e o r y , e t c . A t t h e f i r s t o r d e r , i n c l u d i n g s c a l a r + v e c t o r l o o p s , t h e e f f e c t i v e p o t e n t i a l i s ( J a c k i w , 1 9 7 4 ) 1 N g , p r i v a t e c o m m u n i c a t i o n 47 y i i i w h e r e B = 3 [ 2 g + ( g + g ' ) 3/1 024 77" a n d i s t h e m i n i m u m o f t h e p o t e n t i a l . W i t h t h i s p o t e n t i a l , o n e f i n d s u. -- 4H ol<J>c = f X ^ 3 B J c r ' < V I - 6 ) T h e e f f e c t i v e p o t e n t i a l f o r d i f f e r e n t v a l u e s o f t h e p a r a m e t e r s i s p l o t t e d i n f i g u r e 8, e x t r a c t e d f r o m ( F l o r e s a n d S h e r , 1 9 8 2 ) . 3 N o t i c e t h a t o n e c a n h a v e ^ n e g a t i v e , ( n o n - t a c h y o n i c s c a l a r m a s s ) a n d s t i l l a c h i e v 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 . T h e a e l e g a n t h y p o t h e s i s jA - 0, d u e t o C o l e m a n a n d W e i n b e r g ( 1 9 7 3 ) , g i v e s a c a l c u l a b l e v a l u e m c^ = 10.4 G e V / c . I n t h i s h y p o t h e s i s , n o m a s s s c a l e i s i n t r o d u c e d a t t h e l e v e l o f t h e b a r e L a g r a n g i a n . A l s o if ja < 4 B c r , t h e s p o n t a n e o u s l y b r o k e n v a c u u m i s n o t s t a b l e . I t c o u l d " t u n n e l t h r o u g h " a l o w e r l y i n g v a c u u m a t <£= 0. T h e p r o b a b i l i t y o f t u n n e l l i n g i n c r e a s e s a s m^ d e c r e a s e s , a n d t h i s b r i n g s v a r i o u s l i m i t s o n m , d e p e n d i n g o n w h a t o n e a s s u m e s a b o u t t h e c o n d i t i o n s p r e v a i l i n g a t t h e b e g i n n n i n g o f t h e u n i v e r s e . F i g u r e 8 - V(<£) f o r d i f f e r e n t v a l u e s of 49 If one assumes that the u n i v e r s e has somehow been brought i n the asymmetric-state vacuum a f t e r i t s b i r t h , one r e q u i r e s the l i f e t i m e of t h i s s t a t e to be more than the age of the u n i v e r s e . T h i s l e a d s to a lower bound of mH > 260 MeV/c*. If one i n s t e a d takes the p o s i t i o n that the u n i v e r s e was i n the symmetric s t a t e j u s t a f t e r i t s b i r t h , and underwent a phase t r a n s i t i o n to the spontaneously symmetry-broken vacuum, then t h i s l a t e r must l i e below the ^ = 0 p o i n t . The lower l i m i t on mH becomes 7 GeV/c Z (Weinberg, 1976). I t was a l s o p o i n t e d out by Linde (1976) that the l i f e t i m e of t h i s t r a n s i t i o n must be s u b s t a n t i a l l y s m a l l e r than the age of the u n i v e r s e , and he got a l i m i t m > 0.99 m„ . The i n c l u s i o n of fermion loops makes drop by 6 Mev(m f/15 GeV/c*) . T h e r e f o r e , f o r m̂ < 30 GeV/c 2, the fermion c o n t r i b u t i o n i s n e g l i g i b l e . I f the top quark mass (or any other heavy quark mass) i s l a r g e r than 30 GeV/c*, m,̂  w i l l drop s i g n i f i c a n t l y . I f m̂ > 100 GeV/c 2, mCK, becomes nega t i v e , and more care i n the Coleman-Weinberg mechanism i s needed to d e r i v e meaningful r e s u l t s . Couplings of the Higgs boson: The c o u p l i n g (IV.24) was d e r i v e d i n the model with one fermion doublet and s i n g l e t . We saw i n (IV.39) that many Yukawa c o u p l i n g terms , each with i t s constant G- must be i n t r o d u c e d . A d j u s t i n g the G4- to reproduce the fermion mass spectrum, one gets f o r the c o u p l i n g of the Higgs boson to fermions: (VI.7) 50 which means that the p r o b a b i l i t y of a r e a c t i o n producing a Higgs boson w i l l be p r o p o r t i o n a l to the square of the mass of the fermion i t i s coupled t o . I t a l s o means that the Higgs w i l l decay almost e x c l u s i v e l y i n t o the h e a v i e s t p a r t i c l e k i n e m a t i c a l l y allowed. The c o u p l i n g of the Higgs to photons and gluons i s made only through loop diagrams. The c o u p l i n g to + o W" and Z bosons, however, i s ; - 2 c Nil (Gf 4z) <VI-e) and w i l l be dominant when the a v a i l a b l e energy a l l o w s i t . Decay of the Higgs; The decay r a t e s f o r the Higgs boson i n t o l e p t o n s are given by (Sudaresan and Watson, 1972); fVz r r \ 1 mH I For quarks, simply m u l t i p l y by t h r e e , because of the c o l o r degree of freedom. A p l o t of the branching r a t i o s of the H° i s given i n f i g . 9 e x t r a c t e d from ( E l l i s , G a i l l a r d and Nanopoulos, 1976). F i g u r e 9 - B r a n c h i n g r a t i o ' o f t h e H i n f u n c t i o n o f f r o m ( E l l i s , G a i l l a r d a n d N a n o p o u l o s , 1 9 7 6 ) 52 T h e d e c a y r a t e o f t h e H i g g s b o s o n i n t o v e c t o r b o s o n s i s fOA V V 0 = Gfkl /m„ C b x i a ( 3 ^ - V x + V) (vi.10) 2 2 w h e r e x = 4 m v/m H. T h i s r a t e b e c o m e s r a p i d l y v e r y l a r g e a s m H i n c r e a s e s . T h e c o n s e q u e n c e i s t h a t , b e c a u s e o f t h e i r w i d t h , H i g g s o f m a s s g r e a t e r t h a n 7 0 0 - 8 0 0 G e V / c may n e v e r b e o b s e r v e d ( A l i , 1 9 8 1 ) . H i g g s s i g n a t u r e : O n c e o n e h a s p r o d u c e d t h e H i g g s b o s o n , how d o e s o n e know a b o u t i t ? O ne c h a r a c t e r i s t i c o f t h e H i g g s b o s o n i s i t s s t r o n g t e n d e n c y t o d e c a y i n t o t h e h e a v i e s t p a r t i c l e k i n e m a t i c a l l y a l l o w e d . S o , e v e n b e f o r e a n a l y s i n g i n d e t a i l t h e s p i n a n d a n g u l a r d i s t r i b u t i o n s o f i t s r e a c t i o n p r o d u c t s , o n e m i g h t s u s p e c t H i g g s b o s o n s h a d b e e n p r o d u c e d i f t h e f i n a l s t a t e c o n t a i n e d a n a n o m a l o u s l y l a r g e f r a c t i o n o f h e a v y p a r t i c l e s . I f t h e H i g g s m a s s l i e s a b o v e t h e b - q u a r k t h r e s h o l d , b u t b e l o w t h e t o p q u a r k o n e , t h e n i t w o u l d d e c a y p r e d o m i n a n t l y i n t o b - q u a r k p a i r s . I f i t s m a s s i s b e l o w t w i c e t h e W m a s s , b u t a b o v e t - q u a r k t h r e s h o l d , i t w o u l d d e c a y m o s t l y i n t o t - q u a r k p a i r s , w h i c h w o u l d t h e n d e c a y i n t o b - q u a r k s . T h u s , t h e o b s e r v a t i o n o f e v e n t s w i t h o n e o r t w o j e t s o f i n v a r i a n t m a s s m^, c o n t a i n i n g a t l e a s t t w o b o t t o m q u a r k s w o u l d b e t h e s i g n a t u r e f o r a H i g g s o f m a s s 2m f c< m^ < 2m^. M o r e o v e r , e a c h p r o d u c t i o n m e c h a n i s m w i l l p r o d u c e s o m e t h i n g d i f f e r e n t a l o n g w i t h t h e H i g g s , a n d c a n b e u s e d t o d i s c r i m i n a t e i t f r o m t h e b a c k g r o u n d . 53 Production of the Higgs boson; Here w i l l be presented the p r i n c i p a l mechanisms that have been proposed up to now, i n the search f o r the Higgs boson. The s t r a t e g y i s to produce p a r t i c l e s t h a t have very l a r g e c o u p l i n g s to the H , and look o f o r the s i g n a l of a H which c o u l d be produced with i t , r a d i a t e d from i t , or decay from i t , dependent on the p r o c e s s . The f i r s t three processes are more p e r t i n e n t to e*~e machines, and the l a s t two are a p p r o p r i a t e f o r hadron-hadron c o l l i s i o n s . 1) Decay of the Z°: a) The Z° can decay i n t o a Higgs and photon, through a fermion loop or a W" loop (Cahn et a l . , 1978) represented by the diagram of f i g u r e 10. 54 %/ 0 F i g u r e 10 - F e y n m a n d i a g r a m f o r Z — > j + H d e c a y . w i t h a r a t i o ml (VI.11) 0 - + s. H o w e v e r , t h e b a c k g r o u n d f o r t h i s p r o c e s s , Z -> 1 1 / i s s o l a r g e t h a t t h e p r o c e s s 1a) w o u l d b e b u r i e d i n i t ( B a r b i e l l i n i e t a l . , 1 9 7 9 ) b) Z d e c a y a l o n g t h e c h a n n e l Z — > H + 1 1 ( B j o r k e n , 1 9 7 6 ) r e p r e s e n t e d b y t h e d i a g r a m o f f i g u r e 11, F i g u r e 11 - Z — > H + XX d e c a y d i a g r a m . 55 where Z denotes a v i r t u a l Z . The branching r a t i o f o r t h i s decay channel i s : which i s observable f o r a h i g h - l u m i n o s i t y Z f a c t o r y . A Z f a c t o r y i s an e*e~ c o l l i d e r where the center-of-mass c o l l i s i o n energy can be tuned to the Z mass, a l l o w i n g a very l a r g e Z p r o d u c t i o n r a t e . The process peaks at l a r g e dimuon mass. The angular d i s t r i b u t i o n and d i l e p t o n mass d i s t r i b u t i o n may be used to d i s t i n g u i s h Higgs bosons and other s c a l a r p a r t i c l e s , elementary or not (Kalyniak et a l . , 1984). The main drawback to the process 1b) i s t h a t i t works only i f the mass of the Higgs i s l e s s than about 60 GeV/c z. To circumvent the problem one needs to produce a Higgs together 0 . o with a Z , from a v i r t u a l Z . O 4. _ O o 2) Bremsstrahlung from a v i r t u a l Z : e e —>H + Z i l l u s t r a t e d by the diagram of f i g u r e 12. 4. _ , O o F i g u r e 12 - Feynman diagram f o r e e — > H + Z . 56 The p r o d u c t i o n r a t e peaks atjs =m? + 2mH (Glashow et a l . , 1978) and the t o t a l r a t e i s encouraging f o r e + e ~ machines, p r e d i c t i n g -35 2 2 a c r o s s - s e c t i o n of 4 X 10 cm f o r a Higgs of 10 GeV/c f o r a center-of-mass energy of 104 GeV. The problem i s that one has to wait f o r the LEP II p r o j e c t to be completed. For the pp c o l l i d e r s , the p r o d u c t i o n r a t e s c o r r e s p o n d i n g to t h i s process are below the minimum accept a b l e ( E l l i s et a l . , 1976). 3) Decay of quarkonia: The form of the Higgs c o u p l i n g to fermions makes i t worthwhile to i n v e s t i g a t e heavy quarkonia decay. For the u p s i l o n p a r t i c l e r a d i a t i v e decay has been c a l c u l a t e d u sing a n o n - r e l a t i v i s t i c quark model (Wilczek, 1977). However, i f the Coleman-Weinberg estimate i s c o r r e c t , t h i s decay i s not a c c e s s i b l e . One of the best ways to produce a r e l a t i v e l y l i g h t Higgs i s through the decay of the ( s t i l l unobserved) toponium s t a t e : -r 0 (VI.13) an upper l i m i t branching r a t i o (VI.14) which has a huge branching r a t i o f o r mT < m : 57 r(JT~" n y) y (vi .15) P C JT — HA* In the case m̂  > m̂ , the J f w i l l decay mostly through weak decay, thus d e p l e t i n g the branching r a t i o f o r Ĵ . — > H + Y . A l s o , a l l of the above processes share the same c h a r a c t e r i s t i c that t h e i r r a t e s are i n s i g n i f i c a n t i n a hadron-hadron c o l l i d e r , because of the i m p o s s i b i l i t y of " s i t t i n g on" a resonance, as with e*e~ c o l l i d e r s . The f o l l o w i n g processes may l e a d t o more s i z e a b l e c r o s s - s e c t i o n at hadron-hadron c o l l i d e r s . 4) Gluon-gluon f u s i o n : The f u s i o n of two gluons i n t o a s i n g l e Higgs boson through a quark loop as i n the Feynman diagram of f i g u r e 13 allows the use of the important gluon component of .the hadrons (Georgi et a l . , 1978). F i g u r e 13 - Feynman diagram f o r gg — > H . I t a l s o e x i b i t s the i n t e r e s t i n g f e a t u r e of counting a l l p o s s i b l e quark l o o p s , even f o r quarks that are so heavy they 58 would not be produced i n the l a b o r a t o r y . T h i s comes from the p a r t i c u l a r form of the Higgs c o u p l i n g to fermions, which being p r o p o r t i o n a l to the fermion mass, c a n c e l out a fermion mass term i n the denominator of the phase space i n t e g r a t i o n to y i e l d a c r o s s - s e c t i o n which i s not s e n s i t i v e t o the quark mass, but p r o p o r t i o n a l to the square of the number of heavy quarks. T h i s process has been d e s c r i b e d as a "heavy quark counter" because of t h i s f e a t u r e . For Higgs bosons of masses l e s s than 2MW, the background i s s e v e r a l orders of magnitude l a r g e r than the s i g n a l (Keung, 1981), and there i s no hope to d i s c r i m i n a t e them. The background process i s the c r e a t i o n of a fermion p a i r through quark-antiquark a n n i h i l a t i o n . The c r o s s - s e c t i o n f o r _ o o p p — > H + X through g g —> H and the estimated D r e l l - Y a n background are given i n F i g . 14, from (Keung, 1981) 59 z y/s ( G e V ) F i g u r e 14 - C r o s s s e c t i o n s f o r p p — > H + X t h r o u g h t h e p r o c e s s g g — > H ° ( s o l i d c u r v e ) a n d b a c k g r o u n d 2 ( d o t t e d c u r v e s ) f o r m M = 10 G e V / c f r o m ( K e u n g , 1 9 8 1 ) 60 H o w e v e r , i f t h e m a s s o f t h e H i g g s i s s u c h t h a t i t c a n d e c a y i n t o a p a i r o f v e c t o r b o s o n s , t h e s i g n a l may b e c o m e m o r e i m p o r t a n t t h a n t h e b a c k g r o u n d ( C a h n a n d D a w s o n , 1 9 8 4 ) . M o r e c a l c u l a t i o n s a r e n e e d e d . 5) C o m p t o n - l i k e p r o c e s s e s : T h i s i s t h e c o m p t o n s c a t t e r i n g o f g l u o n s f r o m h e a v y s e a q u a r k s , i l l u s t r a t e d i n f i g . 15 T h e s i g n a l a n d b a c k g r o u n d a r e e s t i m a t e d i n ( B a r g e r a n d a l . , 1 9 8 2 ) a n d r e p r o d u c e d i n f i g . 1 7 . T h e a u t h o r s c l a i m t h e f i n a l s t a t e w o u l d p r o d u c e a d r a m a t i c s i g n a t u r e . T h e f i n a l s t a t e w o u l d b e t h e same a s t h e o n e d e s c r i b e d a n d c a l c u l a t e d l a t e r i n t h i s t h e s i s . H o w e v e r , f o r t h e r e a c t i o n 5) t h e r a t e i s v e r y l o w , o f o r d e r 1 p i c o b a r n o r l e s s , b e c a u s e o f t h e v e r y s m a l l c - q u a r k c o n t e n t o f t h e p r o t o n a n d a n t i p r o t o n . 6) V e c t o r - b o s o n f u s i o n : C a h n a n d D a w s o n ( 1 9 8 4 ) h a v e p r o p o s e d a n o t h e r m e c h a n i s m a s p a r t o f a s t u d y o f v e r y m a s s i v e H i g g s b o s o n p r o d u c t i o n . I t m a k e s u s e o f t h e l a r g e v e c t o r b o s o n c o u p l i n g t o t h e H i g g s b o s o n , a c c o r d i n g t o t h e p r o c e s s i l l u s t r a t e d i n F i g . 16 61 F i g u r e 16 - Feynman diagrams f o r the process qq — > H qq The t o t a l c r o s s s e c t i o n f o r m^ = 5 Myy f o r t h i s process, together with processes 2) and 5) at SSC e n e r g i e s , are shown i n F i g . 18, e x t r a c t e d from (Cahn and Dawson, 1984). A f t e r having examined the p r i n c i p a l channels suggested up to now f o r Higgs p r o d u c t i o n , we are now i n p o s i t i o n to i n t r o d u c e the new mechanism on which t h i s work i s based. 62 100 pp or pp — HX L (from gc — He) T 1 1—I I I I 1 1 1—i—i r i T J Q 10b- 0.1 10 d i f f r p c t i v e c h a r m F i g u r e 17 - C r o s s s e c t i o n s fo'r c o m p t o n - l i k e p r o c e s s , f o r m H • 10 G e V / c z . T h e s o l i d c u r v e r e p r e s e n t s t h e s i g n a l , t h e d o t t e d c u r v e i s t h e b a c k g r o u n d f o r t h e H i g g s d e c a y i n g i n t o a t a u p a i r . F i g u r e 18 - T o t a l c r o s s s e c t i o n f o r processes 2 ) , 5) and 6) f o r mu = 410 GeV/c \ 64 V I I . C A L C U L A T I O N OF A S S O C I A T E D PRODUCTION OF H I G G S BOSON AND HEAVY F L A V O R I N PROTON-ANTIPROTON C O L L I D E R S I n t h e l a s t c h a p t e r , we s u r v e y e d s e v e r a l m e c h a n i s m s t h r o u g h w h i c h t h e H i g g s b o s o n c o u l d b e p r o d u c e d i n p r e s e n t o r p l a n n e d a c c e l e r a t o r s . I t was s u g g e s t e d t h a t H ° b r e m s s t r a h l u n g f r o m Z ° b o s o n s i n e * e ~ a n n i h i l a t i o n a t t h e Z° r e s o n a n c e w i l l o p r o v i d e t h e c l e a n e s t s i g n a l , i f t h e H m a s s , m^, i s l e s s t h a n o t h e Z m a s s . I n f a c t t h e l u m i n o s i t i e s o f c u r r e n t l y p l a n n e d m a c h i n e s s u c h a s L E P a n d S L C w i l l r e s t r i c t t h e d e t e c t a b i l i t y t o z m^ < 50 G e V / c . I t b e c o m e s i m p o r t a n t t o know w h a t a r e t h e p o s s i b i l i t i e s o f p r o d u c i n g a n d d e t e c t i n g t h e H i n p r o t o n - a n t i p r o t o n c o l l i d e r s , s u c h a s t h e CERN S P S c o l l i d e r , t h e F N A L T e v a t r o n o r e v e n t h e S S C ( s e e a p p e n d i x G f o r p r o p e r t i e s o f t h e s e c o l l i d e r s ) . T h e s e m a c h i n e s w i l l b e c a p a b l e o f t a k i n g t h e o z s e a r c h f o r t h e H u p t o t h e m a s s r a n g e o f m «- 1 T e V / c , f a r b e y o n d t h e r a n g e r e a c h e d b y e 4 e ~ c o l l i d e r s a v a i l a b l e i n t h e f o r e s e e a b l e f u t u r e . T h i s i s why e s t i m a t e s o f t h e p r o d u c t i o n c r o s s - s e c t i o n s o f t h e H • i n h a d r o n - h a d r o n c o l l i d e r s a r e now v e r y i m p o r t a n t f o r t h e p l a n n i n g a n d t h e d e s i g n i n g o f c o l l i d e r s e x p e r i m e n t s . A l l o f t h e f o l l o w i n g w i l l t h u s b e c o n c e r n e d w i t h p r o t o n - a n t i p r o t o n o r p r o t o n - p r o t o n c o l l i s i o n o n l y . We saw i n t h e l a s t c h a p t e r t h a t t h e H p r o d u c t i o n m e c h a n i s m w i t h t h e h i g h e s t c r o s s - s e c t i o n was t h e g l u o n - g l u o n f u s i o n . I f mH< 2 m w o n e e x p e c t s t o o b s e r v e t w o b a c k - t o - b a c k j e t s w h i c h a r e i s o t r o p i c w i t h r e s p e c t t o t h e beam d i r e c t i o n , e a c h c o n t a i n i n g a t l e a s t o n e h e a v y f l a v o r e d p a r t i c l e . U n f o r t u n a t e l y , t h e p r o c e s s i l l u s t r a t e d i n F i g . 19 c a n a l s o l e a d 65 to two heavy quarks and i t i s estimated to be an overwhelming background, Q F i g u r e 19 - Feynman diagrams f o r the background to the process hadron+hadron —> H ° + a n y t h i n g . One can attempt to suppress the background by c o n s i d e r i n g H p r o d u c t i o n i n c o n j u n c t i o n with heavy quarks. I f one has enough incoming energy a p o s s i b i l i t y i s f? irons (VII.1) where F(F) denotes a hadron which c o n t a i n s a heavy quark such as the b- or t-quark. One expects t h a t the remaining hadrons (VII.1) do not c o n t a i n heavy quarks, as i n d i c a t e d i n SPS c o l l i d e r d a t a . S e q u e n t i a l weak decays w i l l then l e a d to up to twenty c-quarks or 4 b-quarks and 4 c-quarks, or 4 c-quarks p l u s 8 charged l e p t o n s i n the f i n a l s t a t e f o r the case of t-quarks. Table I g i v e s the number of c-quarks and charged l e p t o n s o b t a i n a b l e a f t e r F(F) and H° decays. Other i n t e r m e d i a t e combinations of c-quarks and charged l e p t o n s are 66 a l l p o s s i b l e . O n e e n d s t h e c h a i n o f s e q u e n t i a l d e c a y s a t t h e c - q u a r k s i n a n t i c i p a t i o n t h a t t h e t a g g i n g o f c h a r m o r b e a u t y h a d r o n s may b e c o m e a p o s s i b i l i t y w i t h r a p i d l y d e v e l o p i n g v e r t e x d e t e c t o r s ( S t o n e , 1 9 8 3 ) W i t h i n t h e f r a m e w o r k o f QCD p a r t o n m o d e l ( s e e a p p e n d i x A f o r QCD r u l e s ) t h e p r o d u c t i o n o f M ° t h a t w i l l r e s u l t i n a c c o m p a n y i n g h e a v y q u a r k f i n a l s t a t e s c a n p r o c e e d v i a a t l e a s t t h r e e m e c h a n i s m s : 1) g l u o n - h e a v y q u a r k s c a t t e r i n g ( B a r g e r e t a l . , 1 9 8 2 ) h+ f.-lf.-) - fj(fj) + H" W I . 2 ) 2) H i g g s b r e m s s t r a h l u n g f r o m h e a v y q u a r k s i n l i g h t q u a r k s (q (-) a n n i h i l a t i o n ?« f; + £ + H° ( V I I . 3 ) 3) H i g g s b r e m s s t r a h l u n g f r o m h e a v y q u a r k s i n g l u o n - g l u o n f u s i o n a a L I U° ( V I 1.4) w h e r e t h e s u b s c r i p t s d e n o t e c o l o u r i n d i c e s o f t h e g l u o n s a n d q u a r k s a n d f ( f ) i s a h e a v y q u a r k ( a n t i q u a r k ) s u c h a s t h e b - o r t - q u a r k . T h e F e y n m a n d i a g r a m s d e p i c t i n g p r o c e s s e s 2) a n d 3) 67 are given i n F i g s . 20 and 21. Mechanism 1) has been i n t r o d u c e d i n the l a s t chapter. F i g u r e 20 - Feynman diagrams f o r qq —> f + f + H . F i g u r e 21 - Feynman diagrams f o r gg — > / + ? +H 69 I t makes e s s e n t i a l use of the heavy quark (antiquark) content i n the sea component of the hadron wave f u n c t i o n . The a c t u a l s i z e of t h i s component i s not very w e l l known but can be estimated from hadronic charm p r o d u c t i o n data ( F i e l d and Feynman, 1977). In g e n e r a l the p r o b a b i l i t y of f i n d i n g a heavy quark i n the proton ( a n t i p r o t o n ) i s expected to be very s m a l l : l e s s than a per cent or so. However, i t i s a two-body f i n a l s t a t e and hence l e s s suppressed by phase space. Reaction (VII.2) g i v e s a r a t e which i s of order Ol^y where ® s i s the c o l o u r f i n e - s t r u c t u r e constant and y denotes the Yukawa c o u p l i n g of the Higgs boson to the quarks and i s given by y = 2 m fG F . On the other hand r e a c t i o n s (VII.3) and (VII.4) are both ,z 2 , of the order CXS y ; hence are down by a f a c t o r ws compared to the p r e v i o u s mechanism. They a l s o have three-body f i n a l s t a t e phase space s u p p r e s s i o n . These are compensated by the l a r g e p r o b a b i l i t y of f i n d i n g l i g h t quarks or a n t i q u a r k s and gluons i n both the p and p. Hence, one would expect the mechanisms (VII.2) to (VII.4) to g i v e comparable r a t e s of H° p r o d u c t i o n i n pp a n n i h i l a t i o n s . In t h i s t h e s i s the r e s u l t s of a complete c a l c u l a t i o n of the processes (VII.1) using both r e a c t i o n s (VII.3) and (VII.4) as the fundamental subprocesses are g i v e n . The parton p i c t u r e has been assumed to convolute the i n i t i a l quark and gluon d i s t r i b u t i o n s over the fundamental subprocesses. The c a l c u l a t i o n s f o r both mechanisms are presented i n t h i s c h a p t e r . In chapter VIII the r e s u l t s of the numerical c a l c u l a t i o n s are 7 0 g i v e n . D i s t r i b u t i o n s i n energy of the Higgs boson and the behaviour of the p r o d u c t i o n r a t e as a f u n c t i o n of the center-of-mass energy, and other e s s e n t i a l k i n e m a t i c a l v a r i a b l e s are a l s o g i v e n . Chapter IX c o n t a i n s d i s c u s s i o n s of the r e s u l t s and t h e i r experimental i m p l i c a t i o n s are g i v e n . C a l c u l a t i o n of the s u b r e a c t i o n amplitude: As the s u b r e a c t i o n s (VII.3) and (VII.5) r e s u l t i n the same f i n a l s t a t e , they are i n d i s t i n g u i s h a b l e at the macroscopic l e v e l . Since they have d i f f e r e n t i n i t i a l s t a t e s at the parton l e v e l they add i n c o h e r e n t l y . Together they g i v e us a f i r s t - o r d e r QCD estimate f o r the s e m i - i n c l u s i v e process (VII.1). In the f i n a l s t a t e , the meson F and i t s charge conjugate c o n t a i n at l e a s t a heavy quark of f l a v o u r c,b or t . We w i l l c o n c e n t r a t e on the s i x - q u a r k s model. The case where the f f forms a resonance such as toponium (T) r e s u l t s i n p t p -* H°4 J t / , a j r o h S ( V I I . 5 ) which has been e s t i m a t e d to be s m a l l 1 . T h i s i s due to the smallness of the wave f u n c t i o n s at the o r i g i n f o r t h i s p r o c e s s . We w i l l now d i s c u s s the two mechanisms (VII.3) and (VII.4) s e p a r a t e l y . 1 Ng and Zakarauskas, unpublished 71 Quark-antiquark a n n i h i l a t i o n mechanism:'" The u- and d-type quarks are mainly r e s p o n s i b l e f o r t h i s p r o c e s s , s i n c e they are the dominant quark components of the proton wave f u n c t i o n . o A l s o due to the small c o u p l i n g t h a t the H has with u- and d-quarks, H° bremsstrahlungs o f f the i n i t i a l quarks can be n e g l e c t e d . To lowest order i n 0(i one needs only to c a l c u l a t e the diagrams d e p i c t e d i n F i g . 20. W i t h i n the framework of p e r t u r b a t i v e QCD model, the c r o s s s e c t i o n f o r the r e a c t i o n (VII.1) i s given by f i r s t c a l c u l a t i n g the elementary subprocesses ( V I I . 3 ) , then c o n v o l u t i n g with the quark and antiquark d i s t r i b u t i o n s i n the proton and a n t i p r o t o n . The amplitude f o r (VII.4) i s (see f i g . 20 f o r kinematics) + per imutsi ion s C ic «* k ") where i , j , k , l = 1-3 (quark c o l o r i n d i c e s ) a,b = 1-8 (gluon c o l o r i n d i c e s ) l\,l> = 1-4 (Lorentz i n d i c e s ) T* = ^ < L a r e the SU(3) m a t r i c e s , i n t r o d u c e d by Gell-Mann (see appendix I ) . The c r o s s s e c t i o n C^f-» j f ° r t h i s elementary process i s given by 72 ( V I I . 1 ) 36 V? ir w h e r e s = Q ? ( Q J ^ + Q L ) • F o r t h e v a l u e o f c(i we u s e d t h e r u n n i n g c o u p l i n g c o n s t a n t ( 1 1 . 1 4 ) w i t h n t h e n u m b e r o f q u a r k s f l a v o r s e q u a l t o 6 . T h e v a l u e A - 0 . 2 G e V h a s b e e n c h o s e n f o r t h i s Q C D p a r a m e t e r . T h e m a t r i x e l e m e n t e l e m e n t s q u a r e d i s g i v e n b y 31 2 A ~ * 1 + ^ 2 , 2 ^ Q ^ ^ K ) AT 2 ( V I I . 8 ) T h e s p i n a n d c o l o r d e g r e e s o f f r e e d o m h a v e b e e n summed o v e r . A l s o t h e F e y n m a n g a u g e i s u s e d f o r t h e g l u o n p r o p a g a t o r . T h e c o n t r i b u t i o n t o t h e c r o s s s e c t i o n o f ( V I I . 1 ) s t e m m i n g f r o m V I I . 3 i s t h e n : ( V I I . 9 ) w i t h s = x ( x 2 s a n d x ( a n d b e i n g , r e s p e c t i v e l y , t h e f r a c t i o n s 73 of momenta c a r r i e d by the quark and a n t i q u a r k i n t h e i r parent hadrons. For numerical c a l c u l a t i o n s we have used two d i f f e r e n t p a r a m e t r i z a t i o n s of the quark d i s t r i b u t i o n f u n c t i o n s , to get an estimate of the u n c e r t a i n t y i n t r o d u c e d by the quark d i s t r i b u t i o n s . They are w r i t t e n e x p l i c i t l y i n equations (F.1a) to (F.2b). The d i f f e r e n c e s i n the r e s u l t s i n using one or the other p a r a m e t r i z a t i o n were no more than a few per c e n t . The c o n t r i b u t i o n s from the sea quarks in the proton or a n t i p r o t o n have a l s o been omitted, s i n c e t h e i r importance i s of the few percent l e v e l . The i n t e g r a t i o n s are performed u s i n g a Monte-Carlo method d e s c r i b e d i n Appendix C. Gluon-gluon f u s i o n mechanism: T h i s mechanism takes advantage of the l a r g e gluon component i n both proton and a n t i p r o t o n wave f u n c t i o n s as w e l l as the l a r g e c o u p l i n g of Higgs boson to heavy quarks i n order to compensate f o r phase space and &s suppression d i s c u s s e d b e f o r e . As a r e s u l t i t a l s o c a r r i e s with i t the not so well-measured gluon d i s t r i b u t i o n f u n c t i o n s , thus l e a d i n g to u n c e r t a i n t i e s i n the estimates of the p r o d u c t i o n c r o s s s e c t i o n s . We w i l l f u r t h e r d i s c u s s these p o i n t s l a t e r and a l s o e x h i b i t q u a n t i t a t i v e l y these u n c e r t a i n t i e s . The c a l c u l a t i o n proceeds by e v a l u a t i n g the Feynman diagrams shown i n f i g . 21. The amplitudes are given by ( k * - * k ) (VII.10) 74 f o r t h e d i a g r a m o f f i g . 2 1 a , M ^ - A T * T « ; U X K ) / , v ; . ( * ) ( V I I . 1 1 ) f o r t h e d i a g r a m o f f i g . 2 1 b , a n d 2 '/2 '/«< f o r t h e d i a g r a m o f f i g . 2 1 c , w i t h A = g^m^G^ 2 H e r e , c, a n d a r e t h e p o l a r i z a t i o n 4 - v e c t o r s o f t h e i n c o m i n g g l u o n s . T h e S U ( 3 ) s t r u c t u r e c o n s t a n t s a r e g i v e n b y t h e f 4 ( , c . T o e v a l u a t e t h e s q u a r e o f t h e a m p l i t u d e g i v e n b y 2 ( V I I . 1 3 ) t h e t r a c e s a r e o b t a i n e d b y u s i n g t h e s y m b o l m a n i p u l a t i o n p r o g r a m R E D U C E . T h e RED U C E p r o g r a m w r i t t e n f o r t h i s i s g i v e n i n a p p e n d i x D. T h e g a u g e i n v a r i a n c e o f t h e r e s u l t h a s b e e n 75 checked by making the s u b s t i t u t i o n <f,—> g ( or c\—> g t . The i n i t i a l gluon p o l a r i z a t i o n and c o l o u r s t a t e s are then averaged, and the f i n a l s t a t e s p i n s and c o l o u r f a c t o r s are summed over. The r e s u l t i n g output f o r the amplitude squared i s given at l e n g t h i n appendix E. The t o t a l c r o s s s e c t i o n (/(s,mH ,ny ) f o r the subprocess i s o o b t a i n e d by i n t e g r a t i n g over the phase space f o r the H , f and f . Using the parton model assumptions one c o n v o l u t e s over the gluon d i s t r i b u t i o n s v i a (VII.14) to o b t a i n the t o t a l p r o d u c t i o n r a t e . The lower l i m i t s of the x ( and x. z i n t e g r a l s are given by the k i n e m a t i c a l requirements of z x x s > (mt + 2m,) . (VII.15) The c o n d i t i o n of eq. (VII.15) r e q u i r e s that the events generated i n the Monte-Carlo c a l c u l a t i o n s s a t i s f y the kinematics f o r heavy p a r t i c l e p r o d u c t i o n . The p r o d u c t i o n c r o s s s e c t i o n depends on the gluon d i s t r i b u t i o n s . From g e n e r a l CPT arguments i t i s expected that G (x) has the same form as G ( x ) ; thus any u n c e r t a i n t y i n the gluon d i s t r i b u t i o n s w i l l be doubled i n the c r o s s s e c t i o n O'(s). We w i l l study t h i s below. 76 To t h i s end, two s p e c i f i c p a r a m e t r i z a t i o n s r e p r e s e n t i n g extreme cases (F.3 and F.4) are chosen. The d i f f e r e n c e s i n the c r o s s s e c t i o n s coming from the use of one or the other of the gluon momentum d i s t r i b u t i o n s g i v e s an estimate of the approximate s i z e of the u n c e r t a i n t y i n the r e s u l t s due to an incomplete knowledge of t h i s d i s t r i b u t i o n . In a d d i t i o n t o r e s t r i c t i n g the generated events to be p h y s i c a l ones, one has to take i n t o account that QCD p e r t u r b a t i v e c a l c u l a t i o n s have s t r i c t v a l i d i t y only i n the high energy deep i n e l a s t i c r e g i o n . One should t h e r e f o r e a v o i d the region of phase space where the gluons or quarks become s o f t and thereby i n v a l i d a t e the use of the parton model. Hence the i n t e g r a t i o n s have been r e s t r i c t e d to take p l a c e in the re g i o n of h i g h momentum t r a n s f e r . The event g e n e r a t i n g r o u t i n e requests t h a t a l l s c a l a r products between the 4-momenta of the incoming and outgoing p a r t i c l e s be l a r g e r than 3 GeV . More s t r i n g e n t c u t s may be imposed to reproduce experimental c o n f i g u r a t i o n s . 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 have a l s o been generated by the Monte-Carlo i n t e g r a t i o n r o u t i n e . These may be extremely u s e f u l i n s e l e c t i n g experimental c u t o f f s and hence reducing the background. The 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 c a l c u l a t e d are those r e l a t i v e t o the Higgs boson and heavy quark's k i n e t i c e n e r g i e s and t r a n s v e r s e momenta. 77 V I I I . RESULTS In t h i s chapter are presented the r e s u l t s of the Monte-Carlo c a l c u l a t i o n of the two processes d e s c r i b e d l a s t c h a p t e r . The f r e e parameters of the theory are m̂ , the mass of the top quark, and m^. The v a r i a b l e s on which the t o t a l c r o s s - s e c t i o n depends are the center-of-mass energy of the p r o t o n - a n t i p r o t o n p a i r , and the lower c u t o f f on r e l a t i v e t r a n s v e r s e momenta of the produced p a r t i c l e s . F i g . 22 to 24 d i s p l a y the p r o d u c t i o n c r o s s s e c t i o n versus the c m . energy of the pp f o r = 10 GeV/c f o r two values of mR, corresponding to the b-quark with m̂, = 4.5 GeV/c* and a 35 GeV/c* t-quark. The c r o s s s e c t i o n s from quark-antiquark a n n i h i l a t i o n and gluon-gluon f u s i o n are shown s e p a r a t e l y i n f i g . 22 and 23, and they are added i n f i g . 24. The quark-antiquark channel reaches a peak i n the p i c o b a r n range around Vs = 2 TeV. However, i t dominates over the gluon f u s i o n mechanism at lower e n e r g i e s i n the range < 60 GeV. At these r e l a t i v e l y low e n e r g i e s one i s r e q u i r e d to use partons with l a r g e x i n order to produce the f i n a l s t a t e p a r t i c l e s . The gluon momentum d i s t r i b u t i o n i s s t e e p l y peaked toward small x as opposed to the quark d i s t r i b u t i o n s . T h i s can be u n i d e r s t o o d by n o t i c i n g t h a t the gluons are r e d i a t e d from the quarks, and t h e r e f o r e must show a r a d i a t i v e spectrum. As a r e s u l t t here are fewer gluons at l a r g e x. On the other hand, the gluon f u s i o n i s t o t a l l y dominating at high e n e r g i e s , where small x s t i l l makes the incoming parton very e n e r g e t i c . There are two types of curves i n a l l of the f i g s . 22 to 78 3 0 . T h e d o t t e d l i n e s r e p r e s e n t t h e r e s u l t s o f t h e c a l c u l a t i o n s d o n e u s i n g t h e s c a l e - v i o l a t i n g g l u o n momentum d i s t r i b u t i o n g i v e n b y ( F . 3 ) . T h e c o n t i n u o u s l i n e s a r e r e s u l t s u s i n g t h e s c a l i n g d i s t r i b u t i o n ( F . 4 ) . T h e d i f f e r e n c e b e t w e e n t h e t w o i s i n d i c a t i v e o f t h e e f f e c t s o f s c a l i n g v e r s u s s c a l e v i o l a t i n g g l u o n d i s t r i b u t i o n s . T h e i n t e r s e c t i o n p o i n t s a r e r e f l e c t i o n s o f t h e p a r t i c u l a r v a l u e s o f x w h e r e t h e t w o p a r a m e t r i z a t i o n s o f G ( x ) c r o s s e a c h o t h e r . E x p l i c i t l y , f o r t h e c a s e o f p p c o l l i d e r a t t h e T e v a t r o n , 2 t h e p r o d u c t i o n o f 10 G e V / c H i g g s b o s o n i n c o n j u n c t i o n w i t h a t - q u a r k p a i r o f m a s s 35 G e v / c * i s w e l l o v e r 100 p b . I n t e r e s t i n g l y , t h e p r o d u c t i o n i n c o n j u n c t i o n w i t h t w o b - q u a r k s h a s t h e same c r o s s s e c t i o n , i n s p i t e o f t h e f a c t t h a t t h e Y u k a w a c o u p l i n g i s p r o p o r t i o n a l t o t h e q u a r k m a s s . T h i s s u p p r e s s i o n i s h e r e o v e r c o m e b y k i n e m a t i c s a n d q u a r k d y n a m i c s . T h e k i n e m a t i c r e a s o n i s t h a t t h e s u b e n e r g i e s o f t h e t w o g l u o n s m u s t b e s u c h t h a t > 80 G e V f o r t h e t - q u a r k c a s e a n d a n d t h i s i s h i n d e r e d b y t h e r a p i d l y f a l l i n g g l u o n d i s t r i b u t i o n f u n c t i o n s . T h e n t h e d y n a m i c a l e n h a n c e m e n t o c c u r s f o r t h e c r o s s s e c t i o n v i a t h e p r o p a g a t o r e f f e c t w h i c h f a v o u r s s m a l l e r q u a r k m a s s e s . T h i s r e s u l t s i n t h e c r o s s i n g o v e r o f t h e p r o d u c t i o n c r o s s s e c t i o n a t Vs = 2 T e V . A d e t a i l e d e x a m i n a t i o n o f 0̂ - a s a f u n c t i o n o f m^ i s g i v e n 2 i n F i g . 25 f o r m^ = 10 G e V / c . T h e u p p e r c u r v e s a n d p o i n t s c o r r e s p o n d t o t h e e x p e c t e d p r o d u c t i o n r a t e a t F N A L , t h e l o w e r o n e s t o S P S c o l l i d e r . H e r e , t h e r e i s a r i s e i n t h e c r o s s s e c t i o n w h i c h r e a c h e s a p e a k a t m f = f o r V s = 540 G e V a n d 79 irij =20 G e V / c " f o r Vs = 2 T e V . I n t a b l e I I a r e g i v e n t h e t o t a l c r o s s s e c t i o n s r e l e v a n t f o r l o w e r e n e r g i e s ( V s = 45 G e V ) f o r d i f f e r e n t v a l u e s o f m^ a n d m K. T h i s w i l l b e o f r e l e v a n c e f o r a 1 T e V p s c a t t e r i n g o n f i x e d t a r g e t w h e r e o n e c a n p r o b e m u c h s m a l l e r c r o s s s e c t i o n s t h a n p o s s i b l e w i t h c o l l i d e r s , d u e t o t h e h i g h e r l u m i n o s i t y . T h e p r o d u c t i o n c r o s s s e c t i o n i s a l s o a v e r y s e n s i t i v e f u n c t i o n o f mh a n d t h i s i s d e p i c t e d i n f i g . 2 6 . H e r e we h a v e c h o s e n t h e r e f e r e n c e v a l u e o f m = 4.5 G e V / c . T h e b e h a v i o u r s e e n a s m^ v a r i e s i s m a i n l y d u e t o t h e p r o p a g a t o r e f f e c t o f t h e h e a v y q u a r k . F r o m e q u a t i o n s ( V I I . 7 ) a n d ( V I I . 1 0 ) t o ( V I I . 1 2 ) we s e e t h a t f o u r o f s i x d e n o m i n a t o r s i n t h e a m p l i t u d e h a v e t h e i r m i n i m u m v a l u e s n e a r m H when e i t h e r h - k o r h-ic i s s m a l l . o T h i s c o r r e s p o n d s t o c o l l i n e a r H b r e m s s t r a h l u n g f r o m t h e h e a v y q u a r k ( o r a n t i q u a r k ) . We a l s o c a l c u l a t e d t h e c r o s s - s e c t i o n f o r c m . e n e r g i e s o f 10, 20 a n d 40 T e V , f o r a w i d e r a n g e o f m^, u p t o 1 T e V / c 2 . T h e s e e n e r g i e s a r e r e l e v a n t t o t h e p l a n n e d S S C ( S u p e r S u p r a c o n d u c t o r C o l l i d e r ) , a p p o r p p c o l l i d e r w h i c h w o u l d b e b u i l t i n U . S . b e f o r e 1 9 9 5 . T h e r e s u l t s a r e r e p r o d u c e d i n t a b l e I I I , f o r t w o v a l u e s o f t h e c u t o f f o n 4 - momenta s c a l a r 2 Z p r o d u c t s , 3 G e V a n d 100 G e V . T h e f o r m e r v a l u e i s t h e QCD c u t o f f , i n t r o d u c e d l a s t c h a p t e r , g u a r a n t e e i n g a p p l i c a b i l i t y o f p e r t u r b a t i v e QCD. T h e 100 G e V 2 v a l u e may b e m o r e r e l e v a n t t o e x p e r i m e n t a l c u t o f f s , e s p e c i a l l y a t t h e S S C . C o m i n g b a c k t o p r e s e n t d a y e n e r g i e s , i n f i g s . 27 t o 30 a r e p l o t t e d 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 , f o u r d i f f e r e n t 80 v a l u e s of" m̂  , mK and / s . The d i s t r i b u t i o n s i n e n e r g i e s of the H° and the heavy quark are compared i n f i g s . 27 and 28. In g e n e r a l , the H° has an average energy higher than the heavy o quarks, the mean value being 40-50 GeV f o r the H , and 10-15 GeV f o r the fermions, i n the case of a 10 Gev/c* Higgs produced with a p a i r of b-quarks at the F e r m i l a b c o l l i d e r . The t r a n s v e r s e momentum of H° i s shown i n f i g . 29. I t i s seen that these d i s t r i b u t i o n s are peaked at hx = 15 GeV at Fer m i l a b and h i = 5 GeV at CERN, and the peak i n c r e a s e s f o r hea v i e r H ° . S i m i l a r l y , the t r a n s v e r s e momenta of the heavy quarks produced are given i n f i g . 30. They have the same f e a t u r e s as h i with the peak l o c a t e d at kj. = 5 GeV f o r CERN and kj. = 1 5 GeV fo r F e r m i l a b , which i s s t i l l a high v a l u e . The numerical c a l c u l a t i o n s needed to produce these curves have been performed on a VAX-780, and n e c e s s i t a t e d a p p r o x i m a t i v e l y 60 hours of CPU time. 81 Table 1 - Number of charm quarks, n c , and number of charged l e p t o n s , n^, 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. The f i r s t column denotes the heavy quark f l a v o u r c o n t a i n e d i n F. The second, t h i r d and f o u r t h column e n t r i e s g i v e the valu e s of n c i f the heavy quark decays n o n - l e p t o n i c a l l y and the valu e s of (n c ,njj ) i f they decay s e m i l e p t o n i c a l l y . The headings of columns g i v e mass ranges of H °. F 2mc < m H < 2mi 2m b < m H < 2m̂ . m H > 2m̂ c 4 6 12 (4,2) (4,4) b 6 8 14 (4,2) (4,4) (4,6) t 12 1 4 20 (4,4) (4,6) (4,8) 82 Table 2 - 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 ( V I I . 1 ) . Cross s e c t i o n , i n p i c o b a r n s , f o r the p r o d u c t i o n of a Higgs boson of mass m̂  and a p a i r of charm quarks (m c = 1.5 GeV/c 2), or bottom quarks (m k = 5 GeV/c*), 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 , with centre-of-mass energy (/s~ = 45 GeV, corr e s p o n d i n g to a f i x e d t a r g e t experiment using a 1 TeV a n t i p r o t o n beam. m \ m H (GeV/c7) 10 5 2 1 0.5 1 .5 1 .3 X -H 10 9.0 X -3 10 0.4 3.5 20.0 5 2.0 X -3 10 1.0 X -2 10 0.18 0.65 1 .5 83 Table 3 - Very high energy c r o s s s e c t i o n s T o t a l c r o s s s e c t i o n , i n Picobarns, f o r the process pp or p p — > H° + F + F, with mt = 35 GeV/c Z, f o r d i f f e r e n t v a l u e s of / s and mH, and two values of the s c a l a r product c u t o f f . The gluon d i s t r i b u t i o n used i s the s c a l e v i o l a t i n g one (F.4). c u t o f f (GeV ) s (TeV) m (GeV/c ) / / ° 50 100 250 500 1000 2 159 1 . 1 7.2X10 Z ~3 1.3X10 -s 2X1 0 3 10 s 10 3 8X1 0 800 13 0.18 -3 3X10 20 s 4X1 0 V 4X1 0 3 10 1 000 8.5 0.2 40 5X1 0 6 7 10* 5 2X1 0 2X10 V 410 5.4 2 18 0.6 -2. 7X1 0 -3 10 -5 2X1 0 100 10 1000 380 100 8.4 0.17 - 3 3X1 0 20 3000 1000 600 1 40 4.5 0.18 40 5500 3700 2500 1000 350 1 .7 84 I 0 2 10 s I0 4 Vs GeV F i g u r e 22 - 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 . 85 8 6 Ss (GeV) F i g u r e 24 - T o t a l c r o s s s e c t i o n i n p p f r o m t h e sum o f s u b r e a c t i o n ( V I I I . 3 ) a n d ( V I I I . 4 ) , a s a f u n c t i o n o f v's, w i t h m H - 10 G e V / c * . 87 I t J 10 100 c b Mx (GeV/c2) F i g u r e 25 - 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 . 1 ) a s a f u n c t i o n o f t h e m a s s o f t h e h e a v y q u a r k p r o d u c e d w i t h t h e H i g g s b o s o n , f o r m = 10 G e v / c . D i s c r e t e p o i n t s r e f e r t o t h e v a l u e o f t h e c r o s s s e c t i o n a t t h e m a s s e s o f t h e c - a n d b - q u a r k s . T h e c o n t i n u u m p o r t i o n , s t a r t i n g a t m K = 20 GeV/c* c o r r e s p o n d s t o t h e t - q u a r k c o n t r i b u t i o n . T h e d o t t e d l i n e h a s b e e n a d d e d t o g u i d e t h e e y e . 88 F i g u r e 2 6 - 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 . 1 ) a s a f u n c t i o n o f t h e m a s s o f t h e H i g g s b o s o n . F i g u r e 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 d ^ / d E n ° f o r f o u r d i f f e r e n t s e t s o f t h e p a r a m e t e r s m H, m^, a n d 90 F i g u r e 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 d c r / d E * f o r f o u r d i f f e r e n t s e t o f t h e p a r a m e t e r s mH, m M a n d V i . F i g u r e 2 9 - 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 / d h i f o r f o u r d i f f e r e n t s e t s o f t h e p a r a m e t e r s m H, m K a n d Vs. F i g u r e 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 do- / d k j f o r f o u r d i f f e r e n t s e t s o f t h e p a r a m e t e r s m H, m K, a n d ys. 93 IX. DISCUSSION AND CONCLUSION T h i s t h e s i s p r e s e n t s the QCD-parton model c a l c u l a t i o n of the p r o d u c t i o n c r o s s s e c t i o n of a Higgs boson p l u s a heavy quark p a i r i n p r o t o n - a n t i p r o t o n or proton-proton c o l l i s i o n s . We employed QCD and the Weinberg-Salam model of electroweak i n t e r a c t i o n s , where the SU(2) ® U ( 1 ) group i s broken by a doublet of s c a l a r f i e l d s . A f t e r the spontaneous symmetry breaking o c c u r s , one i s l e f t with one r e a l s c a l a r p a r t i c l e , c a l l e d the Higgs boson. Now that the W~ and Z bosons have been found at CERN, the Higgs boson i s the only p a r t i c l e p r e d i c t e d by the Weinberg-Salam model yet to be d i s c o v e r e d . Thus, i t i s the whole concept of spontaneous symmetry breaking which would be confirmed i n the event of a p o s i t i v e i d e n t i f i c a t i o n of a Higgs p a r t i c l e . T h i s mechanism g i v e s masses to p a r t i c l e s i n the popular gauge t h e o r i e s , hence the great importance of g e t t i n g some experimental evidence s u p p o r t i n g or i n v a l i d a t i n g i t . The t o t a l p r o d u c t i o n r a t e f o r the r e a c t i o n (VII.1) has been c a l c u l a t e d f o r center-of-mass e n e r g i e s ranging from 45 GeV to 40 TeV, as w e l l as f o r a wide range of the Higgs mass, from 1 GeV/c to 1 TeV/c . The dependence of the c r o s s s e c t i o n on m̂ , h± , k J f E/, and E^ has been c a l c u l a t e d , and should be u s e f u l to p l a c e experimental c u t s or d i s c r i m i n a t e a g a i n s t the background. At t h i s p o i n t we compare the r e s u l t s of our c a l c u l a t i o n s of r e a c t i o n ( V I I . l ) with t h a t of the estimate u s i n g the bremsstrahlung technique of E l l i s et a l (1976). There the 94 p r o d u c t i o n c r o s s s e c t i o n i s given by (IX.1) i n the r e s t system of the heavy quark. The d i f f e r e n t i a l c r o s s s e c t i o n dG\j i s that of p a i r p r o d u c t i o n of heavy quarks without the Higgs boson, i . e . , _ _ (IX.2) p + p _^ F + F + y If we take t h i s c r o s s s e c t i o n to s c a l e l i k e mj* , then we see that C T H w i l l be governed by the charm-quark p a i r p r o d u c t i o n . Using CT ~ 10 cm , one o b t a i n s c r ~ 10 cm f o r m, =10 GeV/c c " hj and Vs = 540 GeV.This i s about two orders of magnitude l a r g e r than our c a l c u l a t i o n . The phenomenological estimate given above i n c l u d e s a l l p o s s i b l e mechanisms f o r the p h y s i c a l p r o c e s s to o c c u r , i n c l u d i n g ( V I I . 2 ) , (VII.3) as w e l l as ( V I I . 4 ) . Our c a l c u l a t i o n s are only good to f i r s t order i n QCD. Furthermore, eq. (IX.1) g i v e s an overestimate s i n c e i t does not take i n t o account the t r a n s v e r s e momentum of the heavy quark and other k i n e m a t i c - s u p p r e s s i o n f a c t o r s . One can expect the r e a l value of the c r o o s s e c t i o n to l i e somewhere i n between the two c a l c u l a t i o n s . The r a t e s f o r h i g h c m . energy and high mH, even with the 2 100 GeV c u t o f f , remain q u i t e impressive compared to the three other p r o d u c t i o n r a t e s , c a l c u l a t e d by Cahn and Dawson(1984), 95 and reproduced i n F i g . 17. A comparison i s d i f f i c u l t , because i t i s not c l e a r what c u t o f f ( s ) , i f any, has been used by Cahn and Dawson. A a n a l y s i s of these p r o c e s s e s and t h e i r background i s under way 1 . Any p r o d u c t i o n r a t e l a r g e r than 1 pb. i s l a r g e enough f o r the corresponding process to be observable i n present or planed c o l l i d e r r i n g s (see appendix G). Then one must deduce from i t a l l the c u t s needed to d e t e c t the s i g n a l and d i s c r i m i n a t e i t a g a i n s t the background; c e r t a i n c u t s may depend on the d e t e c t o r s used, l i k e minimum t r a n s v e r s e momenta or e n e r g i e s . The background r a t e s must a l s o be c a l c u l a t e d and compared to the s i g n a l r a t e . I f the former i s l a r g e r than the t o t a l s i g n a l r a t e , there i s s t i l l a chance that the s i g n a l and background have markedly d i f f e r e n t angular d i s t r i b u t i o n s , energy spectrum or some other kinematic v a r i a b l e dependence. A c a r e f u l a n a l y s i s of the s i g n a l and i t s background i n the d i f f e r e n t decay channels of the Higgs boson i s needed. T h i s work c o u l d a l s o be extended i n assuming a d i f f e r e n t Higgs s e c t o r i n the symmetry breaking mechanism, l e a d i n g to s e v e r a l Higgs bosons, of both charged and n e u t r a l t ypes. The c o u p l i n g c o n s t a n t s i n these a l t e r n a t e models are very n e a r l y f r e e however, i n c o n s t r a s t to the minimal s c a l a r f i e l d case of the Weinberg-Salam model. T h i s would i n t r o d u c e one or more new f r e e parameters to the c a l c u l a t i o n s . The main sources of u n c e r t a i n t y on the c a l c u l a t i o n s presented i n t h i s t h e s i s are brought by the gluon d i s t r i b u t i o n 1 Ng, Bates and Zakarauskas 9 6 and the behavior of the amplitude squared i n the low momentum t r a n s f e r r e g i o n . The u n c e r t a i n t i e s r e l a t i v e to the gluon d i s t r i b u t i o n have been e x p l i c i t e l y c a l c u l a t e d i n most cases. Only the SSC region c a l c u l a t i o n s (Table I I I ) have been covered u s i n g only one gluon d i s t r i b u t i o n ( F . 4 ) , because the s c a l i n g one, ( F . 3 ) , i s no longer a p p r o p r i a t e at these e n e r g i e s . The low momentum t r a n s f e r r e g i o n has been completely avoided by using the 3 GeV 3 c u t o f f on s c a l a r p r o d u c t s . A l l events w i t h i n t h i s r e g i o n , which correspond o f t e n to l a r g e r s u b - c r o s s - s e c t i o n s , have been d i s c a r d e d by the Monte-Carlo phase space generator. But because these events have g e n e r a l l y low p^ or a small opening angle between two of the f i n a l s t a t e components, they would a l s o be d i s c a r d e d i n r e a l experiments. What has been done i n t h i s t h e s i s i s the complete f i r s t - o r d e r c a l c u l a t i o n of the process ( V I I . 1 ) . I t p o i n t s out the importance of t h i s process i n the search f o r the Higgs boson. Second order QCD c o r r e c t i o n s are expected to be of order ds times the c a l c u l a t e d r a t e , or l e s s . The stro n g c o u p l i n g constant ols i s a p p r o x i m a t i v e l y 0 . 2 i n the kinematic r e g i o n c o n s i d e r e d . Of course, a more s p e c i f i c c a l c u l a t i o n of the s i g n a l and bakcground r a t e s , i n c l u d i n g a l l kinematic c u t s , geometry of d e t e c t o r s , decay and h a d r o n i z a t i o n of r e a c t i o n p r o d u c t s , would have to be done before an experiment l o o k i n g f o r the H° proceeds on a p a r t i c u l a r set of a c c e l e r a t o r and d e t e c t o r . I t must be p o i n t e d out that other models of electroweak i n t e r a c t i o n s and grand u n i f i e d t h e o r i e s a l l i n c l u d e at l e a s t 97 one s c a l a r boson which corresponds to the Higgs boson i n " the Weinberg-Salam model. Thus, the c a l c u l a t i o n exposed i n t h i s t h e s i s i s r e l e v a n t to a l l these t h e o r i e s . 9 8 BIBLIOGRAPHY Abers,E.S., Lee,B.W. (1973). Phys. Reports 9 ,1. A l i , A . (1981). Report No. BNL-51443, pl94 A p p e l q u i s t , T . and Bernard,C. (1980). Phys. Rev. D22, 200 Arnison,G. et a l . (1983). Phys. 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P o l o n i c a B8, 475 Weinberg,S. ( 1 976). Phys. Rev. L e t t . 3_6, 294 Weinberg,S. (1980). Rev. Mod. Phys. 52, 515 Weyl,H. (1929). Z.f.Physik 56, 330 Wilczek,F. (1977). Phys. Rev. L e t t . 39, 1304 Will e y , R . S . and Yu,H.L. (1982). Phys. Rev. D26, 3287 Yang,C.N. and M i l l s , R . L . (1954). Phys. Rev. 96, 191 101 APPENDIX A - FEYNMAN DIAGRAMS AND QCD RULES In t h i s appendix are c o l l e c t e d the set of r u l e s and conventions used throughout t h i s work, as w e l l as an i n t r o d u c t i o n to Feynman diagrams and c r o s s - s e c t i o n c a l c u l a t i o n s . Appendix C. g i v e s more d e t a i l s about c r o s s - s e c t i o n i n t e g r a t i o n . The conventions used i n the c a l c u l a t i o n s r e l a t i v e to D i r a c m a t r i c e s and c r o s s - s e c t i o n c a l c u l a t i o n s have been adopted from Bjorken and D r e l l (1964). The s c a l a r product of two 4-vectors, p and q, i s d e f i n e d as: The gamma ma t r i c e s must be i n t r o d u c e d when s o l v i n g the problem of d e s c r i b i n g the motion of s p i n 1/2 p a r t i c l e s ( f e r m i o n s ) . The equation one must s o l v e i s the D i r a c equation ( f o r the n o n - i n t e r a c t i n g case) (A.1 ) where g i s the metric (A.2) 0 (A.3) The f are s p i n o r s , which are u s u a l l y represented as 4-component 1 02 v e c t o r s . The / are the s o - c a l l e d gamma m a t r i c e s , obeying the f o l l o w i n g anticommutation r e l a t i o n s { f . f } - - vYyV- i f i I being the 4 X 4 u n i t m a t r i x . A c c o r d i n g to quantum f i e l d theory, i n t e r a c t i o n s between p a r t i c l e s d e s c r i b e d by a Lagrangian g i v e r i s e to s c a t t e r i n g amplitudes which may be represented by an expansion i n terms of Feynman diagrams, examples of which are given in f i g s . (VII.1) to ( V I I . 5 ) . To each Lagrangian corresponds a unique set of Feynman diagrams at each order of the expansion parameters g 's, the g 's being c o n s t a n t s measuring the s t r e n g t h of an i n t e r a c t i o n . I f the i n t e r a c t i o n i s very weak, the f i r s t order expansion w i l l y i e l d a good approximation. I f not, one has to c a l c u l a t e the next order, e t c . The Feynman diagrams are c o n s t r u c t e d i n the f o l l o w i n g way. From each Lagrangian may be d e r i v e d a set of v e r t i c e s and p r o p a g a t o r s . The v e r t i c e s are nodes i n a diagram; the propagators are l i n e s i n between the nodes. To each v e r t e x i s a s s o c i a t e d a s t r e n g t h gt-. An expansion at the n~" order i n ĝ . i s the set of a l l p o s s i b l e diagrams i n which the product of the g(- i s g{- or l e s s . One looks f o r a l l the ways to combine a v a i l a b l e propagators and v e r t i c e s from a given Lagrangian, with the same i n i t i a l s t a t e A and f i n a l s t a t e B. The sum of these c o n s t i t u t e s the A -—> B process n order expansion. 103 The Feynman diagrams r e a l l y are elegant mnemonic d e v i c e s , a l l o w i n g one to represent v i s u a l l y any given process, and f i n d a l l the a l l o w a b l e ways to combine v e r t i c e s and propagators i n t o a c a l c u l a b l e e x p r e s s i o n . To each vertex or propagator corresponds e i t h e r a s c a l a r , a g or a gamma matrix times a s c a l a r . To e x t e r n a l l i n e s are as s i g n e d e i t h e r s p i n o r s , f o r fermions, or p o l a r i z a t i o n 4 - v e c t o r s f o r spin-1 bosons. A l l the terms are then combined to form the t r a n s i t i o n amplitude of the p r o c e s s . The matrix element squared expresses the p r o b a b i l i t y f o r the t r a n s i t i o n to happen. One then sums over allowed c o n f i g u r a t i o n s i n phase space, m u l t i p l i e s by the a p p r o p r i a t e kinematic f a c t o r s t o get the t o t a l c r o s s - s e c t i o n c/, or some d i f f e r e n t i a l c r o s s s e c t i o n r e l a t i v e to any d e s i r e d v a r i a b l e s x ( , dO'/dxJ dx 2.. .dx,, . The i n t e g r a t i o n over phase space using the Monte-Carlo i n t e g r a t i o n method i s e x p l a i n e d i n appendix C. The term "on-mass s h e l l " r e f e r s to r e a l p a r t i c l e s , as opposed to v i r t u a l ones; f o r the former, t o t a l energy, momentum and mass obey the r e l a t i v i s t i c r e l a t i o n (A.5) 1 04 The r u l e s a s s o c i a t e d with v e r t i c e s and propagators i n QCD are ( P o l i t z e r , 1974); Propagators gluon K2 J (A.6) fermion i v e r t i c e s (A.8) 9 (A.9) T(-- as d e f i n e d below i , j , k , l = 1-3 (quark c o l o r ) a,b = 1-8 (gluon c o l o r ) P o l a r i z a t i o n sum (A.10) 105 Color sum The qqV vertex i n v o l v e s the f a c t o r T a= Âtt where the are the SU(3) m a t r i c e s . The T<x' s obey the commutation r u l e s [ j a , T J ~ 1 iaU Tc (A. 1 1) { L , L l r H.ilm^lu L (A-12) where f a b c are antisymmetric and the are symmmetric under interchange of any two i n d i c e s . I ( 3 j i s the 3 X 3 u n i t matrix. Some i d e n t i t i e s that w i l l be used in.appendix B i n v o l v i n g the m a t r i c e s T a and symbols f a t c a r e : Tr (TaTb) = i&i ( A* 1 3 ) Tr (TaTt l a Tc ) ~ " /2 C§ U (A.15) 0 ' (A.16) 106 f a c e / fhe i ~ 5 5d (A.17) 107 APPENDIX B - COLOR SUMMATION CALCULATION Having i n t r o d u c e d the QCD r u l e s i n appendix A, we are now i n a p o s i t i o n t o c a l c u l a t e the c o l o r f a c t o r s f o r d i f f e r e n t terms of the amplitudes. For the amplitude (VI.6), u s i n g the r e l a t i o n s (A.13) to (A.17), the c o l o r f a c t o r of the matrix element squared i s Because we are averaging over i n i t i a l c o l o r , we must a l s o d i v i d e by a f a c t o r 9 f o r quarks. T h i s i s the t o t a l number of d i f f e r e n t c o l o r s t a t e s f o r two incoming quarks. Hence, the f a c t o r 2/9 i s obtained a f t e r averaging over quark c o l o r s . In the process ( V I I . 4 ) , there are two gluons i n the i n i t i a l s t a t e . T h e r e f o r e , to average over c o l o r , we must d i v i d e by an o v e r a l l f a c t o r of 64, which i s the number of d i f f e r e n t c o l o r combinations f o r two incoming gluons. For the c a l c u l a t i o n of the c o l o r f a c t o r s , the d i f f e r e n t terms a r i s i n g from the squ a r i n g of the amplitude (VII.10) to (VII.12) are d i v i d e d i n t o f i v e c l a s s e s . For the squares of M, to Mfc, and the c r o s s terms i n between M, to M 3, or M v to M 6, the c o l o r f a c t o r i s : (B.1) (B.Z) 108 use eq. (A.12) = Tr(TVrrVif .icUrr'r) use eqs. (A.14) and(A.15) r " ̂ 3 Cut ~ i fabc i ( SAkc * i i a i c ) use eq. (A.16) we get with eq. (A.17) >k/3 For the c r o s s terms between one of M , M 2 or M 5 on one hand, and M 5, Mg or M ? on the ot h e r , the c o l o r f a c t o r i s : Tr(TaTVT4)=-ii<$.i -~Ah For Mo and M* squared: filc ru r« • = f f urn (B.4) For c r o s s terms between one of My, M 5, or M 6 on one hand, and M^ or M^ on the other, the c o l o r f a c t o r i s : (B.5) ^ ate And f i n a l y , f o r c r o s s terms i n between My, M^ or M^ on one hand, and M^ or Mg on the o t h e r : 109 , ( B . 6 ) These c o l o r f a c t o r s are s u b s t i t u t e d when summing the d i f f e r e n t terms of the amplitude squared. They are e s s e n t i a l to get the c o r r e c t gauge i n v a r i a n c e . V (Jake "* i Ld) - " L 7 ^ & * : - 6 1 10 APPENDIX C - THE MONTE-CARLO INTEGRATION ROUTINE In t h i s appendix i s given a b r i e f o u t l i n e of the Monte-Carlo method, followed by a l i s t i n g of the program used in t h i s work to i n t e g r a t e over phase space and parton momenta. For a complete d e s c r i p t i o n of the Monte-Carlo method i n p a r t i c l e p h y s i c s , see ( B y c k l i n g and K a j a n t i e , 1973). What we want to do i s to i n t e g r a t e the amplitude squared over phase space to get the sub c r o s s s e c t i o n & , and then over x 2 and x ? to get the t o t a l c r o s s s e c t i o n . The Monte-Carlo method c o n s i s t s of g e n e r a t i n g ( s i m u l a t i n g ) events, and then c a l c u l a t e the p r o b a b i l i t y of i t to happen, through the amplitude squared. The average of these f o r a l a r g e number of events converges toward the t o t a l c r o s s s e c t i o n f a s t e r than standard i n t e g r a t i o n method when the d i m e n s i o n a l i t y D of the i n t e g r a l i s l a r g e . In our case D = 7. To get 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 i s as easy. Suppose you want to c a l c u l a t e the d e r i v a t i v e of the c r o s s s e c t i o n r e l a t i v e to some angle 6. You c r e a t e a v e c t o r V of dimension 100 or so, d e f i n e 9 from the 4-vectors generated by the s i m u l a t i o n , and c a l c u l a t e f o r each event i n which p o s i t i o n i n V the event f a l l s . That i s , i f f o r one event the a n g l e d i s in between 0 and 1.8 , you add the p r o b a b i l i t y corresponding to i t i n t o the f i r s t element of V. The p r o b a b i l i t y a c t s here as a weight to the event. The c r o s s s e c t i o n of a process Q +Q --> k + k + h where k, k and h are the heavy quark, antiquark and Higgs boson 4-momenta r e s p e c t i v e l y i s : 111 2 JaK J3I< A M ^ K (inlEt W E , ( C D where F = 2 s i s t h e f l u x a n d |M| i s t h e a m p l i t u d e s q u a r e d . The d e l t a f u n c t i o n i n s u r e s t h a t t h e c o n s e r v a t i o n o f e n e r g y momentum i s r e s p e c t e d . What t h e M o n t e - C a r l o p r o c e s s d o e s i s g e n e r a t e 4-momenta of r e a l p a r t i c l e s i n t h e c e n t e r o f mass o f d e c a y i n g o r v i r t u a l p a r t i c l e s , a n d t h e n b o o s t them ba c k i n t o t h e l a b o r a t o r y c.m. The s i m u l a t i o n h a p p e n s a s i f t h e r e a c t i o n was t a k i n g p l a c e a s i n t h e d i a g r a m o f f i g u r e 3 1 . K A F i g u r e 31 - O r d e r o f p a r t i c l e g e n e r a t i o n i n t h e M o n t e - C a r l o method i n p a r t i c l e p h y s i c s . T h e r e f o r e , i n o u r c a s e , i t i s a s i f t h e i n c o m i n g q u a r k and a n t i q u a r k were p r o d u c i n g a f i c t i t i o u s p a r t i c l e G, w h i c h w o u l d d e c a y i n t o a r e a l h e a v y q u a r k k and a f i c t i t i o u s p a r t i c l e X. The 4-momentum o f k i n t h e G c e n t e r o f mass i s s t o r e d , a s w e l l a s a b o o s t m a t r i x t o p a s s f r o m t h e G c.m t o t h e l a b o r a t o r y c.m. We t h e n go i n t h e X c.m. and l e t i t d e c a y i n t o a h e a v y a n t i q u a r k a n d a r e a l H i g g s b o s o n . T h e i r 4-momenta i n t h e X c.m. a r e c a l c u l a t e d , t h e n b o o s t e d back t o t h e G c.m. A l l 1 1 2 t h r e e p a r t i c l e s ' 4-momenta a r e f i n a l l y b o o s t e d b a c k t o t h e l a b o r a t o r y c m . where t h e y a r e u s e d t o c a l c u l a t e t h e c r o s s s e c t i o n . To t h i s e n d , a l l p o s s i b l e s c a l a r p r o d u c t s b e t w e e n t h e 4-momenta a r e f o r m e d a nd s u b s t i t u t e d i n t o t h e a m p l i t u d e s q u a r e d . The r e s u l t i s t h e n s t o r e d a n d t h e w h o l e p r o c e s s i s r e p e a t e d N t i m e s . To c a s t t h e i n t e g r a l ( C . l ) i n t o a f o r m c o m p a t i b l e w i t h t h e M o n t e - C a r l o t e c h n i q u e , one i n t e g r a t e s i t o v e r t h e & f u n c t i o n 2 and i n t r o d u c e s a f i c t i t i o u s mass M x = (k + h) , c o r r e s p o n d i n g t o a v i r t u a l p a r t i c l e w i t h 4-momentum X = k +h. One must a l s o make a n o t h e r c h a n g e o f v a r i a b l e s t o s p h e r i c a l c o o r d i n a t e s o f t h e k i n t h e G c m . and t h e h i n t h e X c m . What i s l e f t i s an i n t e g r a l o v e r t h i s M x v a r i a b l e a n d t h e a n g u l a r c o o r d i n a t e s o f t h e two r e a l p a r t i c l e s , a s i n & r ± JM]_2 J_ JM* JUxo. d/l*„ i l l i l l (C.2) F 2SF (irrf I 2 •± _ -» where |k| i s t h e k momentum i n t h e X c e n t e r o f mass, and |k| i s t h e k momentum i n t h e G c.m. To p r o d u c e o n l y p h y s i c a l e v e n t s , t h e X p a r t i c l e i s r e s t r i c t e d t o have a mass o f a t l e a s t m H + m K, b u t a t most f s - m . The M o n t e - C a r l o p r o g r a m g e n e r a t e s X's m a s s e s a n d r e a l p a r t i c l e s a n g u l a r d i s t r i b u t i o n s r a n d o m l y , u s i n g a random number g e n e r a t o r c a l l e d GGUBFS o u t of t h e IMSL l i b r a r y . GGUBFS a c t s on an a r g u m e n t SEED, w h i c h i s c h a n g e d e v e r y t i m e i t i s u s e d . The J a c o b i a n f o r t h e v a r i a b l e c h a n g e f r o m M x a n d t h e XL's t o t h e random numbers g e n e r a t e d 1 1 3 b e t w e e n 0 and 1 i s (4TT) [ E - mu - 2m.,]. The M o n t e - C a r l o r o u t i n e h a s been t e s t e d on t h e p h a s e s p a c e (|M|Z 5 1 ) , where any m i s t a k e i n t h e p r o g r a m w o u l d show up a s a s u s p e c t momentum d i s t r i b u t i o n , o r an a s y m m e t r y b e t w e e n p a r t i c l e s . The t o t a l p h a s e s p a c e i n t e g r a t i o n h a s been c o m p a r e d t o a n a l y t i c a l c a l c u l a t i o n a n d f o u n d t o a g r e e w i t h i n 1 %. 1 14 0001 REAL'S P. XI. VI. THETA1. PHI1,SEED.V2,XI. X2, CI, 02. C1P 0002 REAL*B G1M. C2P. C2M, CUTOFF, CUT2, VOLUME, ST 0003 REAL*8 DCADRE, F, A, B, AERR. RERR. ERROR. SC 0004 REAL'S BK4.4), E2, MH, MK, MH2. MK2, MX, MX2, S, PI 0005 REAL'S K(4). AK(4), EX2, PX2.B2<4,4).XI2, THETA2. PHI2.H(4> 0006 REAL'B K2<4). AK2<4),H2(4>,AK3<4>.H3(4).SUPX 0007 REAL'S C0STHETA1,C0STHETA2,WI,FLUX, C.CONSTANTE OOOB REAL'B DI, D2. D3, D4, D5, D6, BUMW 0009 REAL'8 HSP1. HSP2. C1SH, 02SH,C1SP1.01SP2.C2SP1, C2SP2. P1SP2 0010 REAL'B HC, LAMBDA1, LAMBDA2. W, 8UMW2, INTECRALE 0011 REAL'B EH(100).EK(100).EAK(100>.ETH(100>.ETK<100>.ETAK(100) 0012 REAL'B XFE(100), RAP(100), Y, XF 0013 REAL'B ENERGY,TRANSH, TRANSK, TRANSAK, BINR 0014 REAL'S Rl( 4 ) , R2(4), RT1,RT2, UPS1,UPS2,PHIT1,PHIT2, PQ 0015 REAL'8 DISTRIBUTION, DENSITY, STOT 0016 REAL'S MAI.MA2.MA3.MA4.MAS. 0017 C Ml1A, M12A, M13A, M14A,M15A,M16A.M17A,M1BA.M22A, M23A. M24A, M25A, 0018 C M26A. M27A.M2SA.M33A.M34A.M35A.M36A.M37A.M38A. M44A. M45A, M46A. 0019 C M47A,M48A.M55A,M56A,M57A,M5BA,M66A.M67A,M68A, M77A, M78A, M88A 0020 INTEOER START,DATA, IER 0021 EXTERNAL F 0022 COMMON C,STOT,ST 0023 COMMON EH, EK, EAK, ETH, ETK, ETAK 0024 C 0025 C RECEIVE PARAMETERS 0026 C — 0027 C GIVE THE CHOICE DF THE FORM OF ENTRY 0028 C 0029 16 WRITE(6,19) 0030 19 FORMAT< ' DO YOU WANT TO START A NEW CALCULATION (TYPE 0) OR 0031 C CONTINUE A PREVIOUS ONE (TYPE 1)') 0032 READ(5.18) DATA 0033 18 FORMAT (II) 0034 IF (DATA . NE. O .AND. DATA . NE. 1) WRITE<6,17), STOP 0035 17 FORMAT(' YOU MUST ENTER 0 OR 1') 0036 C 0037 C READ ENTRIES THROUGH TERMINAL 0038 C 0039 WRITE(6,20) 0040 20 FORMAT(' ENTER BEPERATELY CUTOFF,P. MH AND MK') 0041 READ(5.10) CUTOFF,P, MH, MK 0042 WRITE(6,29) 0043 29 FORMATC ENTER STEEPNESS OF OLUON DISTRIBUTION') 0044 READ< 5. 10)8T 0045 WRITE (6,30) 0046 30 FORMAT(' ENTER NUMBER OF EVENTS DESIRED') 0047 READ (5,12)N 0048 IF (DATA . EQ. 0) THEN 0049 START - 1 0050 BUMW «= O. 0051 6EED - 12345. O 0052 END IF 0053 C 0054 C READ ENTRIES THROUGH FILES 20 AND 21 0055 C 0056 IF (DATA . EQ. 1) THEN 0057 WRITE (6.11) 1 1 5 0058 11 FORMAT<' ENTER IJ, SEED AND SUMW'> 0059 READ(5.12) START 0060 12 FORMAT (17) 0061 READ(5.9)SEED. SUMW 0062 9 F0RMAT(D1B. 10) 0063 10 FORMAT(Fl5. 8) 0064 DO 15 1*1. 100 0065 15 READ<27.96)EH(I).EK(I).ETH(I),ETK(I),RAP(I), XFE(I) 0066 END IF 0067 C 0068 C INITIALIZE VARIABLES 1 0069 C 0070 WRITE(20. 98) 0071 98 FORMAT ( ' P MK MH CUTOFF ST ') 0072 URITE(20.99) P.MK.MH,CUTOFF. ST 0073 99 FORMATUD12. 5, D8. 3) 0074 WRITE(20. 97) 0075 97 FORMAT(' I J SEED 0076 C BUMW X-SECTI ON ') 0077 MH2 = MH • MH 0078 MK2 - MK * MK 0079 C - MH «• 2. * MK 0080 8T0T - 4. * P#P 0081 RT1-0. 0082 RT2=0. 0083 C • 0084 C ESTABLISH 4-VECT0R OF QUARKS IN LAB SYSTEM 1 0085 C 0086 100 DO 1000 IJ-START,N 0087 150 XI - OOUBFS(SEED) 0088 C1M=1. 0089 CIP-EXP(-ST) 0090 XI * -LOC(Q1M+XHMC1P-01M> )/ST 0091 G l - 8T» EXP(-ST«X1> 0092 X2 - OCUBFS(8EED) 0093 CUT2 - C*C/(ST0T*X1) 0094 C2M=EXP(-ST»CUT2) 0095 C2P - C1P 0096 X2«= -L0C(C2M+X2*(C2P-C2M) >/ST 0097 C2 « ST*EXP(-ST*X2) 0098 UPS1 • XI - (RT2*«2.>/(X2 # 2 «P> 0099 UPS2 - X2 - (RT1**2. )/(Xl * 2. *P) 0100 IF(UP81 LE. 0.) 00 TO 150 0101 IF(UPS2 LE. O. ) 00 TO 150 0102 C 0103 C PICK-UP ANOLE FOR TRANS. MOMENTUM OF PARTONS 1 0104 C 0105 PI • 3. 141592654 0106 PHIT1 - 2. • PI * OCUBFS(SEED) 0107 PHIT2 - 2. • PI * OOUBFS(SEED) 0108 C 0109 C CALCULATE PARTONS 4-MOMENTA IN LAB SYSTEM 1 0110 C 0111 R l ( l > - UPS1*P • RT1*«2. /<4. »UPS1»P) 0112 R K 2 ) - RT1 * COS(PHITl) 0113 R K 3 ) - RT1 » SIN(PHITl) 0114 Rl(4) - UPS1*P - RT1*»2. /(4. »UPB1*P) 0115 R2(l> - UPS2*P • RT2««2. /(4. «UPS2»PJ 0116 R2(2> - RT2 * C0S(PHIT2) 0117 R2(3> - RT2 * 8IN(PHIT2> 0118 R2(4) - -UPS2*P + RT2**2. /(4.*UPS2*P> 0119 C 0120 C CALCULATE VELOCITY VI AND RAPIDITY XI OF CM IN LAB FRAME 1 0121 C ; 0122 PG - SQRT((Rl(2>+R2(2>)**2. •MR2(3)+R1(3))**2. +(R2(4>+Rl(4))*«2. ) 0123 VI - PQ / ( R l ( l ) + R 2 ( l ) ) 0124 XI - L0C((1. + V l ) / ( 1 . - V l ) ) / 2 . 0125 C 0126 C E2 IS ENERGY IN CM 1 0127 C ; 0128 S - X1«X2*ST0T 0129 E2 - SQRT(S) 0130 C 0131 C CHECK IF ENOUGH ENERGY IS AVAILABLE FOR REACTION TO OCCUR 1 0132 C 0133 IF(S . LE. (MH + 2»MK)»*2. )00 TO 150 0134 C 0135 C PICK-UP ANGLES OF ROTATION 1 0136 C 0137 PI * 3. 141592654 0138 C0STHETA1 • 2.« GGUBFS(SEED > -1. 0139 THETA1 - ACOS<COSTHETA1) 0140 PHU - 2. * PI • GGUBFS(SEED> 0141 C 0142 C COMPOSE ROTATION MATRIX 1 0143 C 0144 CALL BOOST(BI. XI. THETA1. PHI 1> 0145 C 0146 C PICK UP INVARIANT MASS OF THE DFF MASS-SHELL QUARK 1 0147 C 0148 SUPX- E2 - HK 0149 MX - SUPX - OGUBFB(SEED>« (SUPX - (MH+HK)) 0150 MX2»MX»MX 0151 C 0152 C CALCULATE 4-MOMENTA OF X AND K QUARKS IN QUARKS CM 1 0153 C 0154 K2(l) - (8 • MK2 - MX2)/ (2. *E2) 0155 K2<2> - O. 0156 K2(3) - 0. 0157 K2(4) - 8QRT(K2(1> *K2(1> - MK2) 0158 EX2 - E2 - K2(l> 0159 PX2 » -K2(4) 0160 C —* 0161 C COMPOSE BOOST MATRIX BETWEEN QUARKS CM AND X CM 1 0162 C 0163 V2 • PX2/EX2 0164 XI2« L0G((1. • V2>/(1. - V2))/2. 0165 C0STHETA2 - 2. * GGUBFS(SEED)-1. 0166 THETA2 - ACOS(CDSTHETA2) 0167 PHI2 - 2 • PI* OGUBFS(SEED) 0168 CALL BOOST(B2, XI2, THETA2. PHI2) 0169 C 0170 C CALCULATE 4-MOMENTA OF HIGGS AND ANTI-QUARK IN X CM 1 0171 C 1 1 7 0172 H3<1) - (MX2+ MH2 - HK2)/(2. »MX) 0173 H3<2) -0. 0174 H3<3) - 0. 0175 H3<4> - SORT<H3(1) * H3(l) - MH2) 0176 AK3U) • MX - H3(l) 0177 AK3(2) - 0. 0178 AK3(3) > 0. 0179 AK3(4) - -H3<4) 0180 C 0181 C TRANSFORM THE 4-MOMENTA BACK INTO FRAME 2 1 0182 C 0183 CALL MULT<B2.H3» H2> 0184 CALL MULT(B2, AK3, AK2) 0185 C 0186 C TRANSFORM THE 4-MOMENTA BACK INTO LAB FRAME 1 0187 C 0188 CALL MULT(B1. H2, H> 0189 CALL MULT(B1,AK2.AK) 0190 CALL MULT(B1. K2. K) 0191 C 0192 C CALCULATE THE SCALAR PRODUCTS 1 0193 C 0194 CALL SCALP(AK, Rl, C2SP2) 0195 CALL SCALP(AK, R2, 01SP2) 0196 CALL SCALP(K. R l . C2SP1) 0197 CALL SCALP(K. R2. ©1SP1) 0198 CALL SCALP(Rl, H. 02SH) 0199 CALL SCALP(R2. H,C1SH) 0200 CALL SCALP(K. AK. P1SP2) 0201 CALL SCALP(K, H. HSP1> 0 2 0 2 CALL 8CALP(AK, H. H8P2) 0203 C 0204 C PUT CONDITIONS ON VALIDITY OF CALCULATION 1 0205 C IN PERTURBATIVE QCD 1 0206 C 0207 IF (01SP1 .LT. CUTOFF) 00 TO 150 0208 IF (G1SP2 . LE. CUTOFF) 00 TO 150 0209 IF (02SP1 .LE. CUTOFF) 00 TO ISO 0210 IF (C2SP2 . LE. CUTOFF) 00 TO 150 0211 IF (P18P2 . LE. CUTOFF) 00 TO 150 0212 IF (C1SH LE. CUTOFF) 00 TO 150 0213 IF (02SH LE CUTOFF) 00 TO 150 0214 IF (HSP2 . LE. CUTOFF) 00 TO 150 0215 IF (HSPl . LE. CUTOFF) 00 TO 150 0216 C 0217 C CALCULATE THE DENOMINATORS OF AMPLITUDE 1 0218 C 0219 DI - MH2 • 2. *HSP2 0220 D2 - MH2 + 2. *HSP1 0221 D3 - -2 •018P1 0222 D4 - -2.•02SP2 0223 DS - -2. •01SP2 0224 D6 - -2. •02SP1 0225 CALL. AMPL(S. DI. 02. D3. D4. D5. D6. 0226 C MK2.MH2.HSPl.HSP2. 01BH. 02SH.018P1, 0227 C 01SP2. 02SP1. 02SP2. PISP2. HC.MAI.HA2. MA3. MA4. MAS. M11A. M12A. M13A. 0228 C M14A.M15A.M16A.M17A.niBA.M22A.M23A.M24A.M25A.M26A.M27A.M28A, 1 18 0229 C M33A. H34A. M35A, M36A.M37A,MSBA,M44A. M4SA. M46A. M47A, M48A. M55A. 0230 C M56A. M57A. H58A. M66A, M67A.M6BA.M77A,H7BA.M88A) 0231 C 0232 C CALCULATE THE SCALE VIOLATING GLUON DISTRIBUTION 1 0233 C 0234 SC - LOG(25. *S> 0235 BC - SC / LOO(125.0) 0236 SC * LOG (SO 0237 CALL DIST<XI.X2<SC.DISTRIBUTION) 0238 C . — 0239 C CALCULATE PHASE SPACE DENSITY AND ELEMENT OF INTEORALE 1 0240 C 0241 LAMBDA1 - SORT((8 - MK2 - HX2>**2-4. *MX2»MK2) 0242 LAMBDA2 - 8QRT((MX2 - MH2 - MK2>»*2 - 4.*MK2*MH2) 0243 VOLUME « (G1P-C1M)»(G2P-G2M> 0244 DENSITY « VOLUME/(Gl*02) 0245 CONSTANTE - 4. 488D+04 • MK2/(LOO(25. *S))*«2 0246 FLUX - 2. *S 0247 Ul - ((4. •PI)*«2«(E2 - 2«MK - MH)/(32. »S»MX)) • LAMBDA1 *LAMBDA2 0248 W - WI *HG * DISTRIBUTION • CONSTANTE • DENSITY /<(2.«PI>#»5»FLUX) 0249 SUMW - SUMW + W 0250 88 FORMAT( ' I J = M 1 0 ) 0251 C 0252 C WRITE OUT designed EVENTS 1 0253 C 0254 IF ( I J . LE. 10) THEN 0255 WRITE(17. 89) 0256 89 FORMAT( ' I J H(I) AMI) K ( D ' ) 0257 DO 90 I • 1.4 0258 WRITE(17,91> IJ. H(I). AMI), M I ) 0259 91 FORMAT<19. 3D20. 8) 0260 90 CONTINUE 0261 WRITE(17.92) 0262 92 FORMAT(' ') 0263 WRITE(17. 93) 0264 93 FORMAT(' I J HG WI 0265 C E2 DISTRIBUTION W XI 0266 C X2'> 0267 WRITE(17,94) IJ, HO,WI, E2. DISTRIBUTION, W. XI, X2 0268 94 FORMAT(17, 7D15. 5) 0269 WRITE(17,92) 0270 END IF 0271 0272 IF (XI .LE. -.01) THEN 0273 WRITE(17,89) 0274 DO 85 I - 1,4 0275 WRITE(17,91> IJ. H(I), AMI), M I ) 0276 85 CONTINUE 0277 WRITE(17. 92) 0278 WRITEU7.93) 0279 WRITE(17.94) IJ, HC.W1. E2. DISTRIBUTION, W. XI, X2 0280 WRITE(17.92) 0281 END IF 0282 0283 IF (W . LE. 0. > THEN 0284 WRITEU7.93) 0285 WRITEU7. 94) IJ, HG, WI, E2. DISTRIBUTION. W. XI. X2 1 19 0386 END IF 0287 0288 IF <W . LE. 0) THEN 0289 WRITE(6,8) 0290 WRITE(17. 7)IU. HC 0291 WRITE(17. 9)MA1,MA2. MAS. MA4. MAS, Ml 1A. M12A. M13A. M14A, M15A, M16A, M17A, 0292 C M1BA.M22A.M23A,M24A,M2SA,M26A,M27A.M28A,M33A. M34A. M3SA. M36A. M37A, 0293 C M38A. M44A. M45A, M46A.M47A. M4BA,MSSA. M56A.MS7A. M58A, M66A, M67A. M68A, 0294 C M77A,M7BA. M8BA 0295 B FORMAT<' THERE 16 A NEGATIVE CROSS-SECTION') 0296 7 FORMAT ( UO. D20. 8) 0297 END IF 0298 C 0299 C BIN THE ENEROY AND TRANSVERSE ENERGY OF H.K.AK 1 0300 C • 0301 ENERGY «( 2*P - MH - 2»MK>/3. 0302 TRANSH • SORT(H(2)*H(2) • H(3)»H(3)> 0303 TRANSK - SORT(K(2)*K(2) + K(3)»K(3)> 0304 CALL BIN(H(1), EH. 0.. ENERGY. U> 0305 CALL BIN(Kd), EK. 0. , ENERGY. W> 0306 CALL BIN(TRANSH. ETH. 0. , ENERGY, U) 0307 CALL BIN(TRANSK, ETK, 0. . ENERGY, W> 0308 C 0309 C BIN THE RAPIDITY Y AND FEYNMAN SCALING VARIABLE XF OF HIGGS 1 0310 C 0311 Y - ABS(0.5*L00((H(1)+H(4))/(H(1)-H(4)))) 0312 XF - ABS(H(4)/P) 0313 CALL BIN(Y,RAP,0. , 4. . W) 0314 CALL BIN(XF.XFE,O. , 1. .W) 0315 C 0316 C WRITE DOWN ANSWER EVERY 10O0 EVENTS. IN CASE 8YSTEM 1 0317 C BREAKS DOWN 1 0318 C 0319 RAT - MOD(IJ,1000) 0320 IF (RAT . EQ. 0) THEN 0321 WRITE(20,95)IJ,SEED.BUMW,SUMW/IJ 0322 OPEN (UNIT « 27) XFORT-I-DEFSTAUNK, Default STATUS- 'UNKNOWN' used i n OPEN »tate»ent CPEN (UNIT - 27)3 i n module OLUONtMAIN at l i n e 322 0323 DO 112 1-1. 100 0324 WRITE(27,96)EH(I).EK(I).ETH(I).ETK(I).RAP(I>, XFE(I> 0325 95 FORMAT (17, D25. 10, D20. 10, D15. 6) 0326 96 FORMAT(6D15. 8) 0327 112 CONTINUE 032B CLOSE (UNIT - 27) 0329 END IF 0330 1000 CONTINUE 0331 0332 C CALCULATE THE INTEORALE 0333 0334 INTEGRALE- SUMW/N 0335 WRITE(17.43)N 0336 43 FORMAT( ' N «• ', 17) 0337 WRITE(17,42)CUT0FF 033B 42 FORMAT(' CT0FF='D15. 6) 0339 WRITE(17»44)P 120 0340 44 FORMAT( ' p - ', D15. 6> 0341 WRITE(17, 45)MH 0342 45 FORMAT< ' MH - ',015.6) 0343 WR1TE<17,46)MK 0344 46 FORMAT( ' MK • '.D15. 6) 0345 WRITE(17,55)1NTEGRALE 0346 55 FORMAT( ' X—SECTION - '.D15.6, ' PICOBARN') 0347 BINR » 0. 0348 DO 111 I- 1,100 0349 WRITE<18,56) BINR*ENERCY/100. . EH(I)/N, EK(I)/N 0350 WRITE(19,56) BINR*ENER0Y/100., ETH(I)/N,ETK<I)/N 0351 WRITE(22,57) BINR»4./100. ,RAP(I)/N 0352 WRITE(23,57) BINR/100. ,XFE(I)/N 0353 57 FORMAT(2D15. 4) 0354 BINR BINR + 1 0355 56 FORMAT (3D 15. 4) 0356 U l CONTINUE 0357 END 0001 0002 SUBROUTINE DIST(XI.X2,SC.DISTRIBUTION) 0003 C CO04 C CALCULATES THE SCALE VIOLATING GLUON DISTRIBUTION FROM 1 0005 C PARAMETERS XI, X2 AND THE SCALE PARAMETER SC 1 0006 C 0007 REAL*8 XI.X2.SC.DISTRIBUTION, E l , E2 0008 E l - -0. 93*SC + 0. 36*SC»*2 CDC? E2 - 2. 9 + l.B3»SC 0010 DISTRIBUTION « X1«»E1 * X2**E1 0011 DISTRIBUTION - DISTRIBUTION • (1-X1)«*E2 • d-X2)*»E2 0012 DISTRIBUTION - DISTRIBUTION * (2.01 - 2. 73*SC • 1. 29*SC«»2)*«2 0013 DISTRIBUTION - DISTRIBUTION / (X1«X2> 0014 RETURN 0015 END 0001 0002 SUBROUTINE B0OST(B.XI,THETA,PHI) 0003 REAL «B B(4,4). XI. THETA, PHI 0004 B d . 1)- COSH(XI) 0005 B(1.2)= -BINH(XI) • BIN(THETA) 0006 B(1.3> = O 0007 B d . 4)» BINH(XI )»COS(THETA) OOOB B(2. 1)* 0. 0009 B(2.2>« COS(PHI)•COS(THETA) 0010 B(2.3)« -SIN(PHI) 0011 B(2. 4>- SIN(THETA>*C0S(PHI) 0012 B(3.1)- 0. 0013 B(3. 2)« BIN(PHI)*COS(THETA) 0014 B(3.3>« COS(PHI) 0015 B(3. 4 ) - BIN(THETA)*SIN(PHI) 0016 B(4. 1>- SINH(XI) 0017 B(4,2)- -COSH(XI)*SIN(THETA> 0018 B(4. 3 ) - O. 0019 B(4.4)- COSH(XI)*COS(THETA) 0020 RETURN 0021 END 1 2 1 0001 0002 SUBROUTINE BIN(F.AR.INF.SUP. W) 0003 C 0004 C CLASSES F INTO ONE OF 100 BINS BETWEEN INF AND SUP AND PUT 1 0005 C IT INTO ARRAY AR 1 0006 C 0007 REAL*8 F. AR<100), INF. SUP,POS 0008 COMMON EH.EK.EAK,ETH. ETK, ETAK 0009 POS - INT<100. # (F — INF)/<SUP - INF)) • 1. 0010 IF (POS . OT. 100. ) POS - 100. 0011 AR(POS) - AR(POS) + W 0012 RETURN 0013 END 0001 0002 SUBROUTINE SCALP(VI,V2, B) 0003 C 0004 C TAKE THE SCALAR PRODUCT OF THE TWO 4-VECTORS VI 1 0005 C AND V2 AND PUT THE RESULT INTO 8 1 0006 C 0007 REAL*B VI(4), V2(4).B 0008 8 - V l ( l ) * V 2 ( l ) - V1(2)*V2(2> - V1(3)»V2(3> - V1(4)»V2(4> 0009 RETURN 0010 END OOOl 0002 SUBROUTINE MULT(B, VI,V2> 0003 C 0004 C CALCULATES THE PRODUCT BETWEEN THE MATRIX B AND 1 0005 C VECTOR Ml AND PUTS RESULT INTO V2 1 0006 C 0007 REAL«8 B(4.4), Vl(4>, V2(4). PH 0008 DO 300 1-1,4 0009 PH-O. 0010 DO 301 J»l. 4 0011 301 PH - B U . J ) • V K J ) • PH 0012 V2(I) - PH 0013 300 CONTINUE 0014 RETURN 0015 END 1 22 APPENDIX D - CALCULATION OF THE TRACE Trace c a l c u l a t i o n s come i n e v a l u a t i n g Feynman diagrams i n v o l v i n g fermions. Standard methods f o r c a l c u l a t i n g the t r a c e s are given i n Bjorken and D r e l l (1964). When the number or le n g t h of t r a c e s to e v a l u a t e become too l a r g e to manage, one may now use one of a few number of programs designed to t h i s end. One of them i s REDUCE (see UBC REDUCE), which was used i n a program f o r e v a l u a t i n g the amplitude squared of the process ( V I I . 4 ) . In t h i s appendix i s given a l i s t i n g of t h i s REDUCE program. The input i s i n c l u d e d in the program, and c o n s i s t s of the numerators of the amplitude ( V I 1 . 1 0 ) to (VII.12). The output i s a FORTRAN code, in term of the s c a l a r products of the outgoing p a r t i c l e s 4-momenta. 123 1 »SICN0N OUCH TOM PAGES=80 PROUTE=PHYS 2 •**» 3 »SQURCE »REDUCE 4 OFF ECHO; 5 X 6 1 1 7 1 THIS PROGRAM CALCULATES THE AMPLITUDE 1 6 1 SQUARED FOR THE PROCESS 1 9 1 CLUON+CLUON — > QUARK •*• ANT I-QUARK • HIGGS 1 10 1 1 11 1 1 12 1 1 13 1 DEFINE VECTORS AND MASSES OF COMPONENTS 1 14 1 ; 15 MASS 01=0. 02=0. P1»MK. P2=MK, H-MH; 16 MSHELL Gl,G2,PI.P2, Hi 17 VECTOR E1.E2; IB LET G1.E1=0. G2. E2=0; 19 LET Gl. G2=S/2i 20 OPERATOR V2» U2> GM> GMH, 21 X ' 22 1 GIVE THE RULES FOR SUMMATION 1 23 1 OVER POLARIZATION OF GLUONS 1 24 1 ; 25 LET E l . E l • -2; 26 FOR ALL P LET El.P * El.P = -P.P + 2 *( P.C1*Q C2+P. C2»Q. Gl)/S; 27 FOR ALL R, Q LET El.R • El.Q = -R.Q • 2 * (R. C1*Q. C2 + R.G2+Q.Gl>/S; 2B LET E2. E2 = -2; 29 FOR ALL P LET E2. P * E2. P = -P. P • 2 *( P.G1*Q G2+P. 02*0 Gl)/S; 30 FDR ALL P. Q LET E2. P * E2 Q = -P. Q + 2 » (P.G1*Q. 02 + P C2+Q. Cl)/S, 31 XOFF MCD; 32 FACTOR S, P1.G1, P1.C2. P2. Gl. P2. 02, PI. P2, H. Gl, H C2. H. PI. H P2; 33 ^ 34 1 DEFINE NEW OPERATORS TO SIMPLIFY THE 1 35 1 TYPING OF THE AMPLITUDE 1 36 1 ; 37 FOR ALL T. U LET 0M(T-MJ> = C(L. T)*C(L, U) • MK; 38 FOR ALL H LET OMH(H) = C(L. H) 2«MK; 39 LET V2 = C(L,PI) - MK; 40 LET U2 = C<L,P2) + MK; 41 LET P1C1 • 0M<G1 - P l ) i 42 LET P2C1 - 0M<P2 - 01)i 43 LET P2G2 = 0M(P2 - G2>; 44 LET P1G2 = CM<C2 - Pl)» 45 LET 0E2 = G(L,E2)i 46 LET 0E1 - G(L,E1)< 47 LET VERTEX - 2*C1. E2*0E1 + El.E2«(G(L.G2)-G(L. Gl)) 48 -2*02. E1»GE2; 49 LET MH*«2 - MH2; 50 LET MK*«2 » MK2; 51 OFF NAT; 66 X 67 1 WRITE THE AMPLITUDE COMPONENTS 1 68 I AND THE SUM OVER U AND V SPINORS 1 69 1- . 70 LET Ml • U2 * CMH(H) * GE2 • P1C1 * 0E1 • V2i 71 LET M2 • U2 * 0E2 * P202 • P1G1 • ©El • V2i 72 LET M3 « U2 * CE2 * P2C2 * CE1 • CMH(-H) * V2; 124 73 LET M4 - U2 » 0E1 * P2G1 » GE2 • ©MH(-H) » V2; 74 LET M5 = U2 • GE1 » P2G1 » P1G2 * GE2 * V2; 75 LET M6 « U2 » OMH(H) * OE1 * P1G2 • GE2 «V2; 76 LET M7 - U2 * VERTEX « GMH(-H) • V2; 77 LET MS - U2 * GMH(H) * VERTEX * V2; 78 X 79 1 WRITE THE COMPLEX CONJUGATE OF THE AMPLITUDE 1 80 1 ; 81 LET MIR * GE1 • P1G1 * GE2 • GMH(H); 82 LET M2R - GE1 • P1G1 * P2C2 • GE2; 83 LET M3R - OMH(-H) * 0E1 • P2C2 • CE2; 84 LET M4R « CMH(-H) • GE2 * P2G1 • CE1; 85 LET M5R • CE2 » P102 * P2G1 * 0E1; 86 LET M6R - GE2 • P1G2 • 0E1 « GMH(H); B7 LET M7R - CMH(-H) • VERTEX; 88 LET M8R - VERTEX * OMH(H); 89 X 90 1 WRITE THE SQUARE OF THE AMPLITUDE 1 91 1 (WITHOUT THE DENOMINATORS) 1 92 1 ; 93 OFF NAT; 94 OFF ECHO; 166 XWRITE "M23H - ", M2*M3R; 167 XWRITE "M33H = ", M3*M3R; 168 XWRITE "M34H = ". M3*M4R; 169 XWRITE "M35H «= ". M3*M5R; 170 XWRITE "M36H = ". M3*M6R; 171 XWRITE "M37H = ". M3»M7R; 172 XWRITE "M3BH - M3*MBR; 173 XWRITE "M44H - ", M4#M4R; 174 XWRITE "M45H • M. M4*M5R; 175 XWRITE "M46H - ". M4»M6R; 176 XWRITE "M47H • ". M4»M7R; 177 XWRITE "M48H • ", M4»MBR; 17B XWRITE -M55H - ", M5»M5R; 179 XWRITE "M56H - M5*M6R; 180 XWRITE "M57H - '*. M5»M7R; 181 XWRITE "M58H - ". M5*M8R; 182 XWRITE "M66H • ". M6»M6R; 183 XWRITE "M67H « ". M6»M7R; 184 XWRITE "M6BH - "» M6*MBR; 185 XWRITE "M77H = ". M7«M7R; 186 XWRITE "M8BH • ". MB»MBR; 187 XSHUT ZGLU0NRES23; 188 XON NAT; 189 X 190 1 REWRITE THE AMPLITUDE IN FILE CLU0NRES2 1 191 I IN A FORM READABLE BY FORTRAN 1 192 1 193 XON FORT, 194 XOUT GLU0NRES2; 218 XWRITE "LET M46H »,,,M46H; 219 XWRITE "LET M47H «",M47H; 220 XWRITE "LET M48H «".M48H; 221 XWRITE "LET M55H «"»M55H; 222 XWRITE "LET M56H -M,M56H; 223 XWRITE "LET M57H -".M57H; 224 XWRITE "LET M58H M58H; 125 225 XWRITE "LET M66H -". M66H; 226 "/.WRITE "LET M67H »",M67H; 227 XWRITE "LET M68H -".M68H; 228 XWRITE "LET M77H "",M77H; 229 XWRITE "LET M78H *".M78H; 230 XWRITE "LET MB8H M88H; 231 XSHUT CLU0NRES2; 232 XON ECHO; 2 3 3 x 234 1 DEFINE DENOMINATORS 1 235 1 AND DENOMINATORS SQUARED 1 236 1 ; 237 LET DI - S - 2*P1. (G1+G2); 238 LET D2 = S - 2*P2. (G1+C2); 239 LET D3 • -2*01. PI; 240 LET D4 - -2*P2. 02; 241 LET D5 = -2*P2. CI; 242 LET D6 = -2*P1. C2; 243 LET D12 = D l * D l i 244 LET D22 = D2*D2; 245 LET D32 • D3»D3; 246 LET D42 - D4*D4; 247 LET D52 » D5»D5; 248 LET D62 = D6*D6; 249 7.IN GLU0NRESULT2; 250 LET H «• Gl + G2 - PI - P2, 251 LET P1.P2 = (Gl + 02). (PI + P2) - (S + 2*MK*»2 - MH»*2)/2; 252 LET MH#*2 - MH2; 253 LET MK**2 = MK2; 254 %0UT GLU0NRESULT2; 255 -/.WRITE " M i l « "; 256 V.M11H; 327 7. 328 1 PUT ALL COMPONENTS OF AMPLITUDE SQUARED 1 329 1 OVER SAME DENOMINATOR 1 330 1 ; 331 LET MUA = M i l * D22 * D42 • D52 * D62 » S2; 332 LET M12A « M12 * DI * D22 * D4 * DS2 * 062 • S2; 333 LET M13A = M13 * DI * D2 * D3 * D4 * D52 • D62 • S2; 334 LET M14A = M14 * DI * D2 * D3 * D42 » D5 * D62 • S2; 335 LET M15A = Ml5 * DI * D22 * D3 * D42 * D5 * D6 * S2; 336 LET M16A = M16 • D22 » D3 * D42 • DS2 * D6 * S2; 337 LET M17A - M17 * DI * D2 * D3 * D42 » D52 * D62 * S; 338 LET M18A = M18 • D22 * 03 * D42 * D52 • D62 • S; 339 LET M22A = M22 • D12 • D22 » D52 * D62 • S2i 340 LET M23A = M23 • D12 • D2 • D3 » DS2 • D62 • S2; 341 LET M24A = M24 • D12 * D2 * D3 * D4 • D5 * D62 * S2; 342 LET M25A - M25 * D12 * D22 » D3 * D4 * D5 * D6 * S2; 343 LET M26A » M26 * DI * D22 » D3 * D4 * DS2 • D6 • S2; 344 LET M27A = M27 * D12 « D2 * D3 * D4 * D52 * D62 *S; 345 LET M28A = M28 * DI • D22 » D3 * D4 • D52 • D62 *S; 346 LET M33A = M33 » D12 * D32 * D52 * D62 * S2; 347 LET M34A = M34 » D12 • D32 » D4 * D5 » D62 • S2; 348 LET M35A - M35 * D12 » D2 * D32 * D4 * D5 * D6 * S2; 349 LET M36A - M36 * DI * D2 * D32 * D4 • D52 * D6 * S2; 350 LET M37A = M37 » D12 • D32 * D4 * D52 * D62 »S; 351 LET M3BA » M38 * DI * D2 * D32 * D4 » D52 * 062 «S; 352 LET M44A = M44 » D12 * D32 » D42 » D62 * S2; 126 353 LET M45A - M45 • D12 * D2 » D32 • D42 » D6 « S2; 354 LET M46A - M46 • 01 * 02 • D32 • D42 * D5 » D6 » S2; 355 LET M47A - M47 * D12 »D32 • 042 • D5 • D62 *S; 356 LET M48A = M48 * DI » 02 • D32 • D42 * D5 • D62 * S i 357 LET M55A - M55 • D12 * D22 » D32 • D42 • S2; 358 LET M56A - M56 * DI • D22 • D32 • D42 • D5 • S2; 359 LET M57A - M57 « D12 # D2 » D32 • D42 * D5 » D6 »S; 360 LET M5BA = M58 * DI * D22 * D32 • D42 * D5 * D6 *S; 361 LET M66A - M66 * D22 • D32 • D42 • D52 • S2; 362 LET M67A - 1167 » DI • D2 « D32 * D42 • D52 • D6 *S; 363 LET M6BA - M68 * D22 • D32 • D42 * D52 • D6 *S; 364 LET M77A - M77 • D12 * D32 » D42 » D52 » D62; 365 LET M78A - M78 • DI * D2 * D32 • D42 • D52 • 062. 366 LET M88A = M88 * D22 * D32 * D42 • D52 • D62. 367 LET MB8A • M88 • D22 « D32 • D42 • DS2 * D62; 368 X 369 1 REGROUP THE TERMS ACCORDING TO 1 370 1 COLOR FACTOR 1 371 1 ; 372 LET MAI - MilA + M22A + M33A + M44A + M55A + M66A + 373 2»(M12A + M13A • M23A •»• M45A • M46A * M56A), 374 LET MA2 - 2*(M14A + M15A + M16A + M24A • M25A + M26A 375 + M34A + M35A + M36A); 376 LET MA3 « M77A •»• M88A + 2*M78A; 377 LET MA4 - 2*(M17A + M18A + M27A + M28A • M37A + M38A); 378 LET MA5 - 2»(M47A • M48A + M57A + M58A •»• M67A + M68A); 379 XOUT CLU0NRESULT3J 380 XWRITE "MAI MAI; 381 XWRITE "MA2 «".MA2; 382 XWRITE "MA3 •".MA3; 383 XWRITE "MA4 •". MA4; 384 XWRITE "MA5 «",MA5; 385 XWRITE "M77A »".M77A; 386 XWRITE "Mil -".Mils 387 XWRITE "D12 D12; 388 X16*MAl/3 -2*MA2/3 • 12*MA3 • 6«MA4 - 6«MA5; 389 XSHUT CLU0NRESULT3; 390 MTS; 391 SIG 127 APPENDIX E - PRINTOUT OF THE AMPLITUDE SQUARED OF THE PROCESS Here i s given the amplitude squared of the process (VII.4), c a l l e d by the subroutine AMPL of the r o u t i n e GLUON, whose l i s t i n g appears in appendix D. The denominators i n s e r t e d i n l i n e s 852 to 887 come from the propagators i n the amplitude (V .10) to (VII.12). The c o l o r f a c t o r s i n l i n e 896 have been c a l c u l a t e d i n appendix C. The v a r i a b l e s i n the numerator are d e f i n e d as f o l l o w : S = s MK2 = m^ MH2 = G1SP1 = j ' f l HSP1 H- p i e t c . 128 0001 SUBROUTINE AMPL<S, DI. D2. D3. D4. 05. 06. 0002 C MH2. NH2. HSPl. HSP2. 01SH. 02SH. 01SP1. 0003 C 01SP2,C2SP1.02SP2. P1SP2. HG> 0004 C 0005 C CALCULATES THE AMPLITUDE OF THE PROCESS 1 0006 C 0007 0008 REAL«8 S, MK2, MH2. HSPl. HSP2.01SH.G2SH. C1SP1. 01SP2, C2SP1. C2SP2. P1SP2 0009 REAL*8 Ml 1. M12, M13.M14. M15.M16.M17.M18.M22. M23. M24. M25. M26, M27, M28 0010 REAL*8 M33.M34,M35.M36.M37,M3B.M44.M45.M46. M47. M48. MSS. M56. M57, M58 0011 REAL*8 M66.M67,M68.M77.M7B.MBS.DI.D2. D3. D4. D5. D6. HG 0012 REAL*8 Ml1A.M12A.M13A,M14A.Ml5A,M16A, M17A. M18A 0013 REAL»8 M22A,M23A,M24A.M2SA.M26A,M27A. M28A 0014 REAL»B M33A,M34A. M35A,M36A.M37A,M38A 0015 REAL*8 M44A, M45A. M46A, M47A, M4BA 0016 REAL*8 M55A, M56A, M57A. M5BA 0017 REAL*8 M66A,M67A.M6BA 0018 REAL*8 M77A.M7BA.MBBA 0019 REAL*B MAI.MA2, MA3. MA4. MA5 0020 0021 M i l «<-32. *S**2*MK2**3 0022 C -8. *S**2*MK2**2*MH2+32. *S*»2*MK2**2*(-HSP2+C1SH+ 0023 C GlSP2)-8. *S**2*MK2*MH2*C1SP2+16.*S**2*MK2*(HSP2*01SH+2. *G1SH* 0024 C G1SP1+2. *GlSPl*01SP2>-8. *S**2*MH2*01SP1*C1SP2+16. *S**2*HSP2*Q1SH 0025 C *01SPl+64. *S*MK2**2*<-ClSH*C2SPl-©2SH*ClSPl+2. *C1SP1*02SP1-61SP1 0026 C *02SP2-C1SP2»C2SP1> + 16.*S*MK2*MH2*C2. »©1SP1*Q2SP1+01SP1*02SP2+ 0027 C 61SP2»C2SP1)+32. *S*MK2*(-HSP2*01SH*C2SPl-HSP2*02SH»ClSPl+4. *HSP2 0028 C *QlSPl*02SPl-4. *GlSH*01SPl*Q2SPl-4. *01SP1*01SP2*02SP1)+32. *S*MH2 0029 C »GlSPl*GlSP2*G2SPl-64. *S*HSP2*C1SH*01SP1*02SP1+256. *MK2*C1SP1* 0030 C 02SP1»<G1SH#G2SP1+Q2SH»G1SPH-G1SP1*C2SP2+G1SP2*02SP1)-<64. *MH2* 0031 C G1SP1«G2SP1)*(01SP1*C2SP2+G1SP2*C2SP1> + 128. *HSP2*01SP1*C2SP1*< 0032 C C1SH*C2SP1+C2SH*G1SP1>)/S**2 0033 M12 »8. *S**3*MK2**2+4.*S**3*HK2*(HSP2+2. »G1SP1) 0034 C +4. *S**3*HSP2*C1SP1+ 0035 C 16. *S**2*P1SP2*MK2**2+16. *S**2*P1SP2*MK2*(HSP2+01SP1> + 16. *S**2* 0036 C P1SP2*HSP2*G1SP1-16.»S**2*MK2**3+B. *S**2*MK2**2»<-HSP2-HSP1+Q1SH 0037 C -C2SH-2. *C2SPl-2. *Q2SP2)+B. «S**2*MK2*(-HSP2*01SP2-HSP2*e2SPl- 0038 C HSPl*01SPl+01SH*ClSPl+01SH*Q2SP2-02SH*©lSP2-4. *01SP1*C2SP1>+B. *S 0039 C **2*01SPl*(-HSP2*01SP2-2. *HSP2*©2SP1+»1SH*02SP2-02SH*©1SP2> + 16. * 0040 C S*PlSP2*MK2»<-GlSH»G2SP2-G2SH«QlSP2-4. •018P1»02SP1> + 16. »S*P1SP2* 0041 C ClSPl»<-4. *HBP2*62SP1-Q1SH*Q2SP2-G2SH*01SP2)+16. *S*MK2*«2*<C1SH* 0042 C Q2SP2+Q2SH*QlSP2+4.*01SPl*G2SPl-2. *01SPl»02SP2-2. *C1SP2*G2SP1+4. 0043 C *C1SP2*G2SP2> 0044 M12=M12+16. *S*MK2*<2. *HSP2*01SP1*02BP1-H8P2*G1SP1*02SP2- 0045 C HSP2*GlSP2*02SPl+2.»HSPl*01SPl»02SPl+2. •HSP1»01SP2»G2SP2-G1SH* 0046 C GlSPl*G2SPl-01SH*01SP2*02SP2-C18H*G2SPl*G2SP2+02SH*ClSPl**2+2. • 0047 C 02SH*C1SP1*©2SP1+02SH*C1SP2**2+C2SH*Q1SP2*02SP1-2. *01SP1**2* 0048 C ©2SP2-2. *©lSPl*C19P2*C2SPl+4. »01SPl*G2SPl**2+4. *©1SP1»02SP1* 0049 C Q2SP2>+16.»S*GlSPl*(-2.*HSP2*01SPl*62SP2+2. *HSP2*G2SP1**2+2. * 0050 C HSPl*ClSP2*G2SP2-01SH*01SP2*G2SP2-2. «G1SH*C2SP1*C2SP2+G2SH*C1SP2 0051 C **2+2. *02SH*01SP2*Q2SP1)+64.•P1SP2*Q1BP1*Q2SP1*<01SH*©2SP2+G2SH* 0052 C 019P2>+64. *MK2*01SP1*02SP1*<-01SH*C2SP2-02SH»GlSP2+2. *eiSPl* 0053 C 02SP2+2. *01SP2*02SPl-4. *01SP2*02SP2>+32. »01SP1*(2. *H8P2*01SP1* 0054 C ©2SPl*Q2SP2+2. »HSP2*GlBP2*G2SPl**2-4. *HSP1*01SP2*02SP1*02SP2+ 0055 C 01SH*GlSPl*G2SP2**2+GlSH*GlSP2*G2SPl*02SP2+2. »G1SH*G2SP1**2* 0056 C 02SP2-G2SH*01SPl»018P2*G2SP2-G2SH*01SP2**2*02SPl-2. *02SH*C1SP2* 0057 C ©2SP1**2) 1 29 0038 M12«t112/S»»2 0039 W13 — 2 . *S**3*P1SP2*MH2 0060 C +8. «S**3»MK2»»2*4. «S»*3*MK2*(HSP2+HSPn+4. *S*# 0061 C 3»HSP2»HSP1 + 16. *S**2*PlSP2«MK2*«2+4. •S«»2*P1SP2*HK2*MH2+16.*S»«2 0062 C *PlSP2*MK2*<HBP2+HSPl>+4.»S**2*PlSP2*l1H2*tClSP2+02SPl >+8. »S**2* 0063 C P1SP2*<2. *HSP2»HSP1+01SH*C2SH>-16. •S»«2*MK2»«3+4. *S * *2»MK2**2» 0064 C MH2- (16. #B»*2*HK2»*2) * (HSP2+HSP 1+01 SP I+G2SP2 > +4. *6«*2*HK2* (-2. • 0065 C HSP2»«2-2. •HSP2*HSPl-HSP2*01SH+HSP2*02SH-2. •HSP2»01SP1-2. «HSP2* 0066 C C1SP2-2. *HSPl*«2+HSPl*GlSH-HSPl*02SH-2. *HSPl»G2SPl-2. *HSP1*C2SP2 0067 C -2. *01SH#02SPl+2. *QlSH*02SP2+2. •Q2SH»ClSPl-2. »02SH*018P2>+4. •S** 0068 C 2»MH2*<ClSPl*G2SP2*018P2*02SPl>-<8. •S**2)*(HSP2*HBP1*01SP2>HSP2* 0069 C HSPl*G2SPl+HSP2*GlSH«02SPl+HSPl*02SH*QlSP2>+32. •S*P1SP2*MK2*< 0070 C C1SP1«C2SP2+C1SP2*C2SP1>-<8. *S*P1SP2*HH2)*(C1SP1*G2SP2+01SP2* 0071 C G2SP1) 0072 tl 13=M13-(16. *S*P1SP2)»< HSP2«01SH*G2SP1+HSP2*C2SH*G1SP1•HSP1»G1SH» 0073 C C2SP2+HBP1»C2SH*C1SP2+Q1SH#C2SH#C1SP2+G1SH#C2SH*02BP1>*32. »S*MK2 0074 C **2*<C1SH»02SP1+Q1SH*02SP2+G2SH*01SP1+©28H#01SP2*2. •G1SP1*02SP1- 0075 C GlSPl*C2SP2-QlSP2*02SPl+2. *C1SP2*G2SP2>+8. *S»MK2*MH2*<-2. *C1SP1* 0076 C 02SPl-01SPl*02SP2-01SP2*02SPl-2. *GlSP2*02SP2>+8. *S*HK2*<2. »HSP2* 0077 C 01SH*Q2SPl+2. »HSP2*GlSH#G2SP2+2.•HSP2*G2SH«01SPl+2. *HSP2»G2SH* 0078 C 01SP2+B. •HSP2*01SPl«G2SPl-4.*HSP2»ClSPl#G2SP2-4. »HBP2*G1SP2« 0079 C 02SP1+2. •HSPl»01BH*G2SPH-2. »HSP1*C1SH*02SP2*2. »HSP1«Q2SH*G1SP1 + 0080 C 2. *HSPl*Q2SH*GlSP2-4. *HSPl*GlSPl»G2SP2-4. »HSP1*01SP2*C2SP1+B. • 0081 C HBP 1*C1SP2«G2SP2-G1SH**2*G2SP1+01SH**2*02SP2+01SH*Q2SH*Q1SP1 + 0082 C GlSH*C2SH*QlSP2+GlSH*02SH»Q2SPl+QlSH«C2SH*Q2_P2+2. •Q1BH*018P1« 0083 C G2SP2+2. *01SH*ClSP2*C2SP2+2. »G1SH«Q2SP1**2+2. *C18H*02SP1*C2SP2+ 0084 C C2SH»*2»GlSPl-C2SH**2*GlSP2+2.*G2SH»QlSPl*ClSP2+2. *Q2SH*G1SP1* 0085 C 02SP1+2. •02SH*GlSPl*02SP2+2. *02SH*G1SP2**2+B. •01SP1»C1SP2»G2SP2+ 0086 C 8. *G1SP1*Q2SP1*G2SP2> 0087 M13-M13-(B. *S*MH2)«(G1SP1*01SP2*G2SP2+C1SP1«G2SP1* 0088 C 02SP2+C1SP2*«2*G2SP1+01SP2*G2SP1**2>+16. »S*<2. *HSP2**2*01SP1* 0089 C G2SP1-HSP2*HSP1*G1SP1*G2SP2-HSP2*HSP1*G1SP2»G2SP1+HSP2*G1SH* 0090 C ClSP2*C2SPl+HSP2*ClSH*C2SPl**2+2.*HSP1**2*C1SP2*C2SP2+HSP1*G2SH* 0091 C 01SP2«*2+HSPl*02SH*01SP2*G2SPl)+32.*P1SP2*<C1SH**2*G2SP1*02SP2+ 0092 CGISH*C2SH*01SP1*02SP2+01SH*G2SH*C1SP2*G2SP1*C2SH**2*C1SP1»C1SP2) 0093 C +32. »MK2*(-G1SH**2*G2SP1*C2SP2-01SH*C2SH*C1SP1*G2SP2-G1SH*C2SH* 0094 C ClSP2*G2SPl-4. *QlSH»ClSPl*C2SPl*C2SP2-4. *C1SH*C1SP2*Q2SP1*C2SP2- 0095 C G2SH**2»01SPl*ClSP2-4. *C2SH*GlSPl*GlSP2*G2SPl-4. *C2SH*G1SP1* 0096 C GlSP2«02SP2-2. *C19P1»»2#C2SP2**2-12. #019Pl*ClSP2*02SPl*C2SP2-2. • 0097 C G1SP2»*2*C2SP1»»2>+16. *MH2»<GlSPl**2*G2SP2**2+6. *G1SP1*01SP2* 009B C C2SPl*C2SP2+GlSP2*«2*C2SPl*«2>-64.*<HSP2«C1SH»C1SP1*C2SP1*C2SP2+ 0099 C HSP2*G2SH*G1SP1*C1SP2*02SP1+HSP1*01SH*G1SP2*C2SP1*02SP2+HSP1• 0100 C G2SH*G1SP1*01SP2*02SP2> 0101 M13«M13/S**2 0102 M14 -8. *S**2*P1BP2**2*MH2 0103 C +32. *S**2*P1SP2*MK2**2+16. *S**2*P1SP2*MK2«< 0104 C HSP2+HSP1-2.»GlSH-01SPl-01SP2>+8.*S**2»PlSP2*01SH**2+4. *S**2*MK2 0105 C *MH2*(01SPl+ClSP2)+8. *B**2*MK2*(-HBP2*G1SH+HSP2*C1SP1-HSP2*G1SP2 0106 C -HSP1*Q1SH-HSP1*01SP1+HSP1*G1SP2)+B. *S**2*MH2*0lBPl*QlSP2-<8. *S 0107 C **2*GlSH)*<HSP2#GlSPl+HSPl*GlSP2>-32.»S*PlSP2»*2*ClSH*02SH+32. »S 0108 C *P1SP2»MK2»(01SH*02SH+01SH*02SP1+01SH*02SP2+02SH*01SP1+02SH» 0109 C C1SP2+2. #01SPl»Q2SP2+2. •01SP2*02SPl>-<32. *S*P1SP2*MH2)*(01SP1* 0110 C G2SP2+01SP2*G2SP1)+16. *8*P1SP2*(HSP2*G19H*G2SP1+HSP2*C2SH*01SP1 + 0111 C HSP1*01SH*C2SP2+HSP1*C2SH*01SP2-01SH**2*02SP1-G1SH**2»02SP2)-( 0112 C 64. *S*MK2**2)*(01SP1*02SP2+G1SP2*02SP1) 0113 M14-M14+16. *S*MK2*(-2. *HSP2*G1SP1 0114 C *02SP2-2. *HSP2*GlSP2*02SPl-2.•HSPl*01SPl»C2SP2-2. •HSP1»C1SP2* 1 30 0115 C 02SPl+QlSH**2*02SPl+01SH»«2»02SP2+GlSH«©lSPl»©2SPl+3.*01SH#01SP1 0116 C •C2SP2+3. *G1SH*G1SP2*G2SP1+C1SH*C1SP2»G2SP2+C2SH*01SP1»»2-2. • 0117 C G2SH»ClSPl»GlSP2+02SH*GlSP2*»2+2. *QlSPl»»2*02SP2+2. »Q1SP1»G1SP2» OUS C C2SP1+2. *QlSPl*01SP2*02BP2+2. *G1SP2*«2*02SP 1)-(16. *9*MH2*01SPi» 0119 C 01SP2)*<G2SP1+02SP2)+16. *S*Q1SH»<HSP2»01SP1»G2SP1+HSP2«01SP1» 0120 C 02SP2+HSPl«01SP2»02SPl+HSPl«GlSP2*G2SP2)+64. *P1SP2*01SH*Q2SH*< 0121 C Q1SP1*C2SP2+G1SP2»Q2SP1>+64. *MK2»<-G1SH»G2SH*G1SP1*02SP2-G1SH» 0122 C G2SH»C1SP2*G2SP1-G1SH*G1SP1*G2SP1»02SP2-01SH«01SP1*02SP2**2-01SH 0123 C *01SP2*02SP1*#2-C1SH«Q1SP 2«02SP1*G2SP2-G2SH*01SP1*«2«G2SP2-G2SH* 0124 C ClSPl*GlSP2*02SPl-02SH«GlSPl*ClSP2*G2SP2-C2SH«G18P2««2«G2SPl-2. « 0125 C 01SPl**2»G2SP2*«2-4. •GlSPl»GlSP2»G2SPl*02SP2-2. *01SP2»»2»02SP1** 0126 C 2)+32 *MH2*<ClSPl««2»C2SP2«*2+2. •C1SP1«C1SP2*C2SPI*C2SP2+G1SP2** 0127 C 2*G2SPl««2)-32.*(HSP2*G1SH«01SP1*G26P1*02€P2+HSP2«01SH«G1SP2* 0128 C G2SP1##2+HSP2*02SH#01SP1»»2»G2SP2+HSP2»G2SH»GISP1*01SP2*02SP1+ 0129 C HSP1»01SH»01SP1*C2SP2»»2+HSP1*01SH*01SP2*G2SP1*G2SP2+HSP1»C2SH» 0130 C G1SP1#01SP2»C2SP2+HSP1*02SH»C1SP2»«2*G2SP1> 0131 M14«M14/B*»2 0132 M15 — 8 . »S»*3#P1SP2*MK2 0133 C +4. »S*«3»PlSP2*QlSH-4. *S»«3«MK2*HSPl-4. »S»*3* 0134 C HSP1*C1SP2-16. •S»»2«PlSP2«*2«MK2+8. •S**2*P1SP2**2*G1SH+16 *S**2* 0135 C PlSP2*MK2**2+8. •S**2»P1SP2*MK2»(HSP2-HSP1-G1SH+G2SH+01SP1-G1SP2+ 0136 C 2. •02SP1+2. *02SP2)+8. *8»»2*P1SP2*<HBP2*02SP1-HSP1»Q1SP2-01SH# 0137 C 01SPl-ClSH»02SPl-01SH*02SP2>-<8.•S**2*MK2**2>*(G18H+G1SP1+G1SP2> 0138 C +4. •S*»2*nK2#(-HSP2*01SPl+HSP2»ClSP2+HSPl*GlSPl+HSPl*GlSP2+2. » 0139 C GlSH»C2SPl+2. *02SH#C18Pl+4.«GlSPl»02SP2+4. •01SP2*02SP1)+8. »S**2* 0140 C (HSP1*C1SP1*C1SP2+HSP1*C1SP2*G2SP1+HSP1*01SP2*C2SP2-01SH»C1SP1* 0141 C C2SP2+G2SH*01SPl*ClSP2>-<16.«S«P1SP2*«2)«<G1SH«02SP1+G2SH*01SP1> 0142 C +16. •S«PlSP2*MK2»<GlSH*C2SPl+G2SH*QlSPl-4. #G1SP1*C2SP1+4. •01SP1* 0143 C G2SP2+4. »G1SP2»C2SP1> + 16. »S»PlSP2»C-2. •HSP2*G1SP1*G2SP1+HSP1* 0144 C G1SP1»G2SP2+HSP1»C1SP2«C2SP1+G1SH*C1SP1«02SP1-01SH«01SP1«G2SP2- 0145 C GlSH*G1SP2»G2SP1+G1SH*G2SP1*G2SP2+G2SH«0ISP 1«G1SP2-02SH«G1SP2* 0146 C G2SP1) 0147 M15-M15-<32. *S«MK2««2)«<G1SP1»G2SP2+G1SP2*G28P1)+B. »S»MK2«<-2. • 0148 C HSP2»GlSPl*G2SP2-2. *HSP2«GlSP2»G2SPl+2. •HSP1*G1SPl»02SP2+2. *HSP1 0149 C #GlSP2*G2SPl+3. *GlSH*GlSPl*G2SPl+3. •G1SH*G1SP1»G28P2+G1SH«G1SP2* 0150 C G2SPl+01SH*GlSP2*G2SP2-G2SH«GlSPl»«2-2 *G2SH*GlSPl«GlSP2-2. *G2SH 0151 C •01SPl»G2SP2-G2SH*01SP2*«2-2. *G2SH*GlSP2*G2SPl+4. *G1SP1*«2*G2SP1 0152 C -2. *GlSPl**2*G2SP2+2. •01SPl»01SP2*Q2SPl+2. •GlSPl*GlSP2*G2SP2-4. • 0153 C GlSPl*G2SPl»G2SP2-4. »GlSPl*G2SP2**2+2. »G1SP2*«2»G2SP1-4. «G1SP2* 0154 C 02SPl»*2-4. •01SP2«02SP1*02SP2> + 16. •8«<HSP2*01SP1»*2»02SP1-HSP2« 0155 C GISP1*C1SP2»C2SP1-HSP2*C1SP1*C2SP1»C2SP2-HSP2*01SP2*G2SP1*»2- 0156 C HSP1»G1SP1»01SP2*C2SP1+HSP1»G1SP2»»2*02SP1+01SH»G1SP1•»2*02SP2+ 0157 CGISH*G1SP1»G2SP1»G2SP2+01SH*C1SP1•G2SP2**2-G2SH»G1SP1**2«G1SP2- 0158 C G2SH*GlSPl»01SP2*G2SPl-02SH*ClSPl*01SP2*G2SP2>+32. *P1SP2«(G1SH* 0159 C C1SP1»G2SP1*02SP2+G1SH»01SP2*02SP1**2+Q2SH*C1SPH»«2*©2SP2+Q2SH* 0160 C 01SP1*G18P2*02SP1> 0161 Ml5-M15+32.*MK2*(-G1SH*01SP1*G2SP1*02SP2-01SH*G1SP2* 0162 C 02SPl*»2-02SH«ClSPl»«2*G2SP2-G2SH*GlSPl»01SP2*02SPl+4. *01SP1«»2» 0163 C 02SPl»C2SP2-2.«01SPl*»2*02SP2«»2+4. •01SPl*018P2*028Pl**2-4. • 0164 C ClSPl«ClSP2»C2SPl*C2SP2-2. •C1SP2»»2»G2SP1»«2)*32. »<2 •HSP2»C1SP1 0165 C ••2»G2SPl*02SP2+2. »HSP2»01SP1«01SP2*02SP1»«2-HSP1»G1SP1*»2#02SP2 0166 C »«2-2. •HSPl«OlSPl*01SP2»028Pl*02SP2-HSPl*G18P2»»2*02SPl»»2-2. * 0167 C 01SH»01SP1»*2*02SP1•02SP2-01SH»C1SP1»02SP1#02SP2«»2-G1SH«G1SP2« 0168 C 02SPl*«2»G2SP2+2. •G2SH*G1SP1»*2*G1SP2»02SP1+G2SH*G1SP1«01SP2* 0169 C G2SP1»G2SP2+G2SH*G18P2»«2*02SP1*»2) 0170 M15-M15/S»»2 0171 M16 — 8 . «S»«3*PISP2»MK2 131 0172 C +2. *S»*3»P1SP2*MH2-B. •S#»3»t1K2»HSPl-4. *S»#3» 0173 C HSP2«HSP1-M6. »S**2«PlSP2«MK2*«2-4. *S»«2»P1SP2*MK2»MH2+16. «S»»2« 0174 C P1SP2«MK2»<C1SP1+02SP1>-<4.«S«»2*P1SP2»MH2)»<01SP1«-02SP1 >-16. «S 0175 C •*2*MK2**3-4. •S»»2«MK2«*2«MH2+16. •S*«2»MK2«-»2« <-H8P2+HSP1)-M3. *S 0176 C »*2*MK2*(HSP2*H8Pl+2. *HSPl»01SPl+2. »HSPl*02SPl+2. •018H*C2SPl+2. • 0177 C C2SH*ClSPl+2. •C1BPl»C2SP2+2. *G1SP2»G2SP1)-(4. •S»*2*HH2)»<01SP1* 0178 C 02SP2+01SP2*02SP1)+8. •S««2»HSP2»<HSP1«C1SP1+HSP1*C2SP1+01SH* 0179 C G2SPl+G2SH*ClSPl)-64. *S*P1SP2#MK2*G1SP1*C2SP1+16. •S»P1SP2«MH2« 0180 C GlSPl*G2SPl+64. •S*MK2*«2*(-Q1SH»G2SP1-Q2SH*G1SP1+01SP1»G2SP1- 0181 C 01SP1•C2SP2-01SP2*G2SP1) +16. *5*MK2*MH2»(G1SP1*G2SP1•©1SP1«C2SP2+ 0182 C C1SP2*G2SP1) 0183 M16-M16+32. *S*MK2«(-HSP2*ClSH*G2SPl-HSP2*G2SH*01SPl+2. »HSP2 0184 C »C1SP1«G2SP1-2. *HSP1*G1SP1«G2SP1-C1SH*C1SP1*G2SP1-01SH*C2SP1*»2- 0185 C G2SH*C1SP1••2-G2SH*G1SP1*G2SP1-01SP1••2*G2SP2-G1SP1»01SP2»G2SP1- 0186 C 01SP1*C2SP1*C2SP2-G1SP2*C2SP1*»2)+B. «S#MH2«(C1SP1»«2»C2SP2+G1SP1 0187 C *G1SP2»G2SP1+Q1SP1«Q2SP1*Q2SP2+G1SP2»G2SP1*»2>+16. *S*HSP2»<-2. « 0188 C HSP1*G1SP1*G2SP1-01SH*G1SP1*G2SP1-01SH»C2SP1**2-G2SH*G1SP1»»2- 0189 C G2SH«G1SP1*G2SP1>+256.•HK2»C1SP1«G2SPHMG1SH«G2SP1+G2SH»C1SP1+ 0190 C 01SPl*C2SP2+01SP2»G2SPl)-(64. *HH2«G1SP1»62SP1)»(G1SP1»02SP2+ 0191 C Q1SP2»G2SP1)+12B. *HSP2»01SP1«02SP1«<G1SH»C2SP1+02SH«C1SP1) 0192 M16»M16/S«»2 0193 M17 -4. #S**2»P1SP2»MK2 0194 C -S««2»PlSP2*f1H2+4. *S»«2*MK2#*2+S*«2*MK2«MH2+4. • 0195 C S»*2»MK2»<HSP2+HSP1>+2.•S»»2*HSP2*HSP1+8. *S*P1SP2»MK2*(G1SH-C2SH 0196 C +C1SP1-02SP1>+2. »S«P1SP2*MH2»<-G1SP1+02SP1>+4. *S*P1SP2»Q1SH»<- 0197 C GlSH+G2SH)+8. »S«MK2»«2«(01SP2-G2SP2)+2. *S»MK2«MH2*(-G1SP2+C2SP2) 0198 C +4. «S«MK2»(HSP1»G1SH-HSP1*02SH+2.«HSP1»G1SP1-2. «HSP1»02SP1+01SH 0199 C ••2-ClSH*G2SH+2. »ClSH#01SPl+2.•ClSH*01SP2-2. »01SH»02SPl-2. «G2SH* 0200 C G1SP2+4. *0ISP 1*01SP2-2. *01SP1*G2SP2-2.»G1SP2*G2SP1)+2. •S*MH2«(- 0201 C 2. »G1SP1»01SP2+G1SP1«G2SP2+Q1SP2*G2SP1)+4. »S«(HSP2*HSP1*C1SP1- 0202 C HSP2»HSP1"028?1+HSP2»G1SH*01SP1-HSP2»C1SH»G2SP1+HSP1*01SH«01SP2- 0203 C HSP1»G2SH*C1SP2>+B. »P1SP2«(G1SH**2*G2SP1+G1SH*G2SH*G1SP1-G1SH* 0204 C G2SH*G2SP1-02SH*«2*G1SP1> 0205 M17-M17+B. »MR2#<-C1SH»«2»G2SP1-01SH*02SH*C1SP1 0206 C +01SH«G2SH»C2SPl-2. •GlSH»GlSPl»G2SPl-2. •GlSH*GlSPl*G2SP2-2. «G1SH 0207 C •GlSP2*G2SPl+2. *G1SH*G2SP1**2+G2SH**2*G1SP1-2. »C2SH*GlSPl««2+2. • 0208 C G2SH*01SPl«C2SPl+2. »G2SH*GlSPl*C2SP2+2. *C2SH*ClSP2«G2SPl-2. » 0209 C 01SPl»*2#02SP2-6. •GlSPl*01SP2*C2SPl+6. «ClSPl»C2SPl*02SP2+2. * 0210 C QlSP2*02SPl»«2>+4. »MH2*(G1SPl«*2*G2SP2+3. •GlSPl*GlSP2*G2SPl-3. • 0211 C 01SP1»02SP1*02SP2-01SP2»G2SP1**2)+B. *(-HSP2»01SH*01SPl»02SPl + 0212 C HSP2»C1SH«G2SP1*«2-HSP2»G2SH»G1SP1»*2+HSP2*G2SH*01SP1«02SP1-HSP1 0213 C »01SH»G1SP1»02SP2-HSP1»C1SH»G1SP2*G2SP1+HSP1•C2SH»G1SP1«C2SP 2+ 0214 C HSP1*G2SH*G1SP2»G2SP1) 0215 M17-M17/S 0216 M1B -(4.»S**2*P1SP2»MK2 0217 C -S#«2«PlSP2«MH2+4. #S«*2«MK2»«2+S»»2«MK2«MH2*4. • 021B C S»«2*MK2»(HSP2+H8P1)+2. •S*»2*HSP2»HSP1+B. •S«P1SP2*«K2*(01BP1- 0219 C G2SP1)+2. •S»PlSP2*MH2»(-GlSPl+G2SPl>+8.•S»MK2*»2»CC1SH-C2SH+ 0220 C 01SP2-G2SP2)+2. *S*MK2*MH2#<-01SP2+02SP2)+4. *S»MK2«<HSP2»C1SH- 0221 C HSP2»G2SH+2. *HSPl«GlSPl-2. •HSPl*G2SPl+4. •GlSH»01SPl-2. «G1SH» 0222 C 02SP1-2. «02SH*01SPl+4. •01SPl«01SP2-2. •01SPl*C2SP2-2. •01BP2*G2SP1 0223 C >+2. »S»MH2»(-2. »G1SP1*C1SP2+G1SP1»G29P2+01SP2»02SP1)+4. •S*HSP2*< 0224 C HSPl»ClSPl-HSPl*C2SPl+2. •01SH»01BP1-01SH*Q2SP1-02SH#C1SP1>+16. # 0225 C MK2»(-3. #01SH«01SPl*G2SPl+01SH*G2SPl**2-02SH»GlSPl«»2+3. *G2SH» 0226 C ClSPl«G2SPl-GlSPl»*2»02SP2-3.»01SPl#01SP2»02SPl«-3. •01SP1402SP1* 0227 C 02SP2+GlSP2»C2SPl»«2)+4. •MH2»(01SP1»»2»G2SP2*3. »01SP1«G1SP2» 0228 C G2SP1-3. «019P1«G2SP1«G2SP2-01SP2*G28P1**2)-H3. •HSP2»(-3. •C1SH* 1 3 2 0229 C G1SP1»C2SP1+C1SH*G2SP1**2-C2SH*G1BP 1**2+3. *G2BH*Q1BP1*02SP1>>/S 0230 M22 «<B. *S**3»HK2**2 0231 C +8. *S*«3*MK2*<ClSPl+Q2SP2>+8. *S**3*Q1SP1*C2SP2+16. 0232 C «S»*2*P1SP2*MK2**2+16.*S**2*P1SP2*MK2* < 01SP1+02SP2) +16. *S**2* 0233 C P1SP2*01SP1*02SP2-16.*S**2*I1K2**3-<16. *S**2*MK2**2)*<01SP2+02SP1 0234 C > + 16. *S**2*MK2*(-QlSPl*01SP2-2.*01SPl»C2SPl-2. *C1BP2*C2SP2-C2SP1 0235 C *02SP2)-<32. *S**2*G1SP1*02SP2)*<G1SP2+028P1)-<64. *S*P1SP2*MK2>*< 0236 C 01SPl*Q2SPl+01SP2*C2SP2)-<64. *S*P1SP2*C1SP1*C2SP2)«<C1SP2+G2SP1) 0237 C +64. *S*MK2**2*<01SPl*02SPl+QlSP2*G2SP2)+32. *S*MK2*<-0ISP 1**2* 0238 C G2SP2+GlSPl*GlSP2*G2SPl+2. *GlSPl*G2SPl**2-01SPl*02SP2**2+2. * 0239 C 01SP2**2»G2SP2+01SP2*02SPl*G2SP2>+32. *S*CISP1*02SP2*<-Q1SP1» 0240 C 02SP2+2. *ClSP2**2+3. »C1SP2*C2SP1+2. »C2SP1**2)+256. *P1SP2*C1SP1* 0241 C GlSP2*Q2SPl»Q2SP2-256. *HK2»C1SP1»01SP2»C2SP1»G2SP2+128. *G1SP1* 0242 C G2SP2*< 01SP1*G1SP2*02SP2+01SP1»02SP1*02SP2-01SP2**2*C2SP1-01SP2* 0243 C G2SP1*«2))/8**2 0244 M23 -8.»S**3*MK2**2 0245 C +4. »S**3*f1K2*(HSPl+2. *02SP2)+4. *S**3*HSP1*Q2SP2+ 0246 C 16. «S**2*P1SP2*MK2««2+16.*S**2*P1SP2*MK2*< HSP1+02SP2> +16. *S**2» 0247 C P1SP2*HSP1*Q2SP2-16.*S**2*MK2**3+B. *S**2»MK2**2*<-HSP2-HSP1-01SH 0248 C +02SH-2. »GISP1-2. *01SP2)+B. *S**2*MK2*<-HSP2*02SP2-HSP1*C1SP2- 0249 C HSPl*Q2SPl-01SH*C2SPl+02SH«QlSPl+Q2SH*C2SP2-4. *©1SP2*Q2SP2)+B. »S 0250 C »»2*G2SP2*(-2. *HSP1»01SP2-HSP1*C2SP1-G1SH»02SP1+02SH»G1SP1> + 16. * 0251 C S*PlSP2*MK2*<-01SH*Q2SPl-02SH»018Pl-4. »01SP2»G2SP2)+ 16. »S»P1SP2» 0252 C 02SP2*(-4. »HSP1»01SP2-G1SH»02SP1-02SH*G18P1) + 16. *S*MK2**2*<01SH* 0253 C 02SPl+02SH»01SPl+4. *016Pl*02SPl-2. •01SPl*G2SP2-2. »©lSP2*C2SPl+4. 0254 C *©1SP2*C2SP2)+16. *S»MK2*<2. *HSP2*01SP1*02SP1+2. *HSP2*C1SP2*©2SP2 0255 C -HSPl»QlSPl»Q2SP2-HSPl*01SP2*©2SPl+2. »HSP1»G1SP2»G2SP2+G1SH* 0256 C 01SP2*02SPl+2. »C1SH*C1SP2*C2SP2+C1SH*C2SP1**2+C1BH*C2SP2»*2-G2SH 0257 C *01SPl*GlSP2-G2SH*GlSPl*G2SPl-G2SH*ClSP2*G2SP2+4. *01SP1*C1SP2* 0258 C G2SP2-2. *QlSPl*C2SP2**2+4. *ClSP2*»2*C2SP2-2. *C1SP2*C2SP1»G2SP2)+ 0259 C 16.»S*G2SP2»<2.»HSP2»C1SP1»C2SP1-2.*HSPl*GlSPl*G2SP2+2. *HSP1* 0260 C ClSP2**2+2. *G1SH*C1SP2*C2SP1+G1SH*C2SP1**2-2. *G2SH*G1SP1*G1SP2- 0261 C G2SH*G1SP1*G2SP1) 0262 M23-M23+64. *P1SP2*01SP2*02SP2*(01SH*G2SP1+02SH*C1BP1> + 0263 C 64. *HK2*01SP2*G28P2*(-GlSH*G2SPl-G2SH*018Pl-4. *G18Pl»G28Pl+2. * 0264 C GlSPl*G2SP2+2. *G1SP2»G2SP1)+32.*G2SP2*(-4. *HSP2*G1SP1*G1SP2* 0265 C 02SP1+2. *HSPl»G18Pl*01SP2*G2SP2+2. *HSP1*01SP2**2*G2SP1-G1SH* 0266 C GlSPl*02SPl*G2SP2-2. *01SH*G1SP2**2*G2SP1-G1SH*G1SP2*G2SP1**2+ 0267 C G2SH*GlSPl**2*02SP2+2. *02SH*G1SP1*G1SP2**2+028H*01SP1*G1SP2* 0268 C G2SP1) 0269 M23»M23/B**2 0270 M24 —B.*S*«3*P1SP2*HK2 0271 C +4. *S**3*PlSP2*01SH-4. *S**3*MK2*HSP2-4. *S**3* 0272 C H8P2«G1SP1-16. *S**2«P1SP2**2*MK2+B. *S**2*P1SP2**2*018H+16. *S**2« 0273 C PlSP2»MK2**2+8. *S**2*P1SP2*MK2*(-HSP2+HSP1-G1SH+C2SH-01SP1+01SP2 0274 C +2. *02SPl+2.*G2SP2)+8. *S**2*P1SP2*(-HSP2*C1SP1+HSP1*G2SP2-G1SH* 0275 C 01SP2-01SH*02SP1-01SH*02SP2)-(B.»S**2*MK2»*2>*(01SH+G1SP1+C1SP2> 0276 C +4. *S**2*MK2*(HSP2*GlSPl+HSP2*ClSP2+HSPl*01SPl-HSPl*GlSP2+2. * 0277 C GlSH»G2SP2+2. *G2SH»01SP2+4. *01SPl*02SP2+4. *G1SP2*02SP1)+8. *S**2* 0278 C (HSP2*G1SP1*G1SP2+HSP2*G1SP1*02SP1+HSP2*01SP1*G2SP2-G1SH*G1SP2* 0279 C G2SP1+02SH*01SP1*01SP2>-(16.*S*P19P2**2)*(018H»G2SP2+02SH*G1SP2> 0280 C +16. »S*P18P2*HK2*(018H*02SP2+G2SH*GlSP2+4. *01SPl»02SP2+4. *G1SP2* 0281 C G2SP1-4. *G1SP2*02SP2) + 16. *S*P1SP2*(HBP2*018P1*02SP2+HSP2*G1SP2* 0282 C 02SP1-2. *HSP1*G1SP2*G2SP2-G1SH*01SP1*G2SP2-G1SH*018P2»G2SP1+G1SH 0283 C «G1SP2*02SP2+01SH*02SP1*G2SP2+02SH*G1SP1*01SP2-02SH*01SP1*G2SP2 > 0284 C -<32. *S*HK2**2)*(G1SP1*02SP2+01SP2*02SP1> 0285 M24-M24+B. *S*t1K2*(2. *HSP2* 1 33 0286 C QlSPl»02SP2+2. *HSP2#01SP2*02SPl-2. •HSPl»01SPl»02SP2-2. *HSP1» 0287 C 01SP2*02SP1+015H*01SP1*02SP1+01SH»C1SP1*G2SP2+3. *C1SH*C1SP2* 0288 C 02SP1+3. •01SH*01SP2«02SP2-C2SH»GlSPl»»2-2. •C2SH»01SPl»ClSP2-2. • 0289 C 02SH»QlSPl»C2SP2-02SH»ClSP2»«2-2.»02SH»Q1SP2*C2SP1+2. *01SP1**2* 0290 C 02SP2+2. *0tSPl»0lSP2»02SPH-2.•0lSPl»QlSP2*C2SP2-4. *018P1«G2SP1* 0291 C 02SP2-4. »01SPl*C2SP2»»2-2. •GlSP2**2»G2SPl+4. •GlSP2*»2*C2SP2-4. * 0292 C QlSP2»C2SPl»*2-4. »01SP2»G2SP1»02SP2) + 16. •S*(HSP2*G1SP1»*2*02SP2- 0293 C HSP2»C1SP1*C1SP2*C2SP2-HSP1*0ISP1*01SP2»C2SP2-HSP1*C1SP1*02SP2*» 0294 C 2+HSP1#01SP2»*2*C2SP2-HSP1*C1SP2«C2SP1•G2SP2+C1SH*C1SP2*«2*C2SP1 0295 C +01SH*C1SP2*C2SP1••2+01SH»G1SP2»02SP1•02SP2-02SH*01SP1»Q1SP2««2- 0296 C ©2SH«01SPl»ClSP2*G2SPl-G2SH»01SPl»QlSP2»02SP2>+32. •P18P2«(G1SH« 0297 C 01SP1«G2SP2««2+G1SH«G1SP2*C2SP1•02SP2+C2SH«G1SP1*01SP2*G2SP2+ 0298 C G2SH»CIBP2**2»C2SP1> 0299 M24-M24+32. «MK2«C-01SH*01SP1»02SP2*»2-01SH*016P2* 0300 C 02SPl*G2SP2~G2SH*GlSPl*GlSP2*G2SP2-G2SH*GlSP2*»2*G2SPl-2. •01SP1 0301 C ••2«G2SP2»«2-4. •GlSPl»01SP2»02SPl»02SP2+4. •01SP1»01SP2»02SP2»»2- 0302 C 2. #01SP2*»2«C2SPl»«2+4. *ClSP2»*2*G2SPl»G2SP2>+32. •(-HSP2 0303 C *Q1SP1**2 0304 C «02SP2»«2-2. *HSP2«Q1SP1*01SP2*02SP1*G2SP2-HSP2*01SP2**2*C2SP1**2 0305 C +2. *HSP1*01SP1*G1SP2«C2SP2*«2+2. *HSP1»G1SP2*»2»02SP1•C2SP2-01SH» 0306 C GlSPl*G2SPl»G2SP2»*2-2. •C1SH*G1SP2»*2*G2SP1*G2SP2-01SH*G1SP2* 0307 C G2SPl**2«02SP2+02SH*G18Pl«*2*G2SP2»*2+2. •02SH*01SP1*01SP2*«2* 0308 C 02SP2+02SH*G1SP1*G1SP2*02SP1«G2SP2) 0309 t124=M24/S*»2 0310 M25 -2. *S»«4*P18P2+8. »S»*3*P1SP2»»2 0311 C -B. *S*»3»PlSP2»MK2-(4. *S*»3*P1SP2) 0312 C •<01SP1+G1SP2+G2SP1+02SP2>-B.»S»«3*MK2«»2-<4. •S»*3)»(G1SP1*G2SP2 0313 C +01SP2*02SP1)+16. »S»»2«P1SP2»»3-16. •S**2*P1SP2*«2«MK2-(B. »S*»2» 0314 C P1SP2*«2)*<01SP1+G1SP2+G2SP1+G2SP2)+B. •S«»2«P1SP2«MK2»(G1SP1+ 0315 C C1SP2+C2SP1+G2SP2)+B.•S»«2*P1SP2*(2.*GlSPl*GlSP2+GlSPl*G2SPl-3. • 0316 C GlSPl*G2SP2-3.•GlSP2*G2SPl+01SP2*02SP2+2. »C2SPl»02SP2>+8. »S*»2» 0317 C MK2«(3. »ClSPl*C2SPl+2. •GlSPl»02SP2+2. •GlSP2*G2SPl+3. »G1SP2»C2SP2 0318 C )+8. »S««2»<C1SP1«*2»02SP2+C1SP1«01SP2»C2SP1+C1SP1*G1SP2»02SP2+ 0319 C G1SP1*G2SP1•G2SP2+C1SP1»C2SP2»«2+C1SP2«»2*02SP1+C1SP2*G2SP1**2+ 0320 C 01SP2»02SPl*02SP2>-<64. *S*P1SP2*»2>*(C1SP1»G2SP2+C1SP2*G2SP1)• 0321 C 64. •S»PlSP2*MK2»<ClSPl»C2SP2+ClSP2*C2SPl>+32. #S»P1SP2*(GISP 1**2* 0322 C G2SP2+G1SP1*02SP2**2+G1SP2«*2*G2SP1+G1SP2«G2SP1**2>-<16. •S»MK2>* 0323 C < 01SP1##2«02SP2+G1SP1*C1SP2«G2SP1+G1SP1»C1SP2*C2SP2+C1SP1*02SP1• 0324 C G2SP2+G1SP1«G2SP2**2+G1SP2»»2»C2SP1+G1SP2»G2SP1••2+01SP2*C2SP1• 0325 C G2SP2) 0326 M23-M23+16. »S» <-2. #G1SP1*»2»C1SP2*G2SP2-G1SP1•*2»C2SP1*C2SP2+ 0327 C QlSPl»»2«C2SP2«»2-2.•G1SP1*G1SP2**2*G2SP1-G1SP1*G1SP2»G2SP1**2- 0328 C 2. •01SPl*GlSP2*G2SPl*G2SP2-GlSPl*G18P2»G2SP2««2-2. •01SP1»C2SP1» 0329 C 02SP2»«2+GlSP2»»2»G2SPl»»2-GlSP2»*2»G2SPl»G2SP2-2. •C1SP2»G2SP1»» 0330 C 2*C2SP2>+64. #PlSP2»<01SPl«*2«02SP2»*2+2. «C1SP1»C1SP2*C2SP1»C2SP2 0331 C +01SP2»»2»C2SPl*»2>+64.•MK2#<-01SPl«*2«©2SP2«»2-2. »©1SPI«01SP2* 0332 C C2SPl«02SP2-ClSP2»»2«Q2SPl»«2>+32. »<-GlSPl«»3«C2SP2««2+GlSPl««2« 0333 CGISP2»C2SP2**2+G1SP1»*2»G2SP1•C2SP2»»2-C1SP1•»2»G2SP2»«3+G1SP1• 0334 CGISP2««2*G2SP1••2-G18P2*«3*G2SP1••2-01SP2**2*G2SP1••3+01SP2»«2» 0335 C C2SP1»»2»02SP2> 0336 M23-M25/B«»2 0337 M26 — 8 . *S**3*P1SP2«MR2 0338 C +4. »S*»3»PlSP2»02SH-4. •S«»3»MK2»HSPl-4. *S*«3« 0339 C HSP1«C2SP2-16. *S»*2«P1SP2«*2»MK2+B. •S»»2»P1SP2»»2»C2SH+16. »S»»2* 0340 C P1SP2»MK2*«2+B. *S»»2»P18P2»t1K2»(HSP2-HSPl+01SH-G2SH+2. •01SP1+2. • 0341 C G1SP2+C2SP1-02SP2)+B. »6»»2»P1SP2«<HSP2*G1SP1-HSP1»02SP2-©2SH« 0342 C G1SP1-G2SH*G1SP2-G2SH*G2SP1)-(B.*S»«2*MK2»»2)»(02SH+C2SP1+G2SP2) 1 34 0343 C +4. »S*#2«MK2#<-HSP2#02SPl+HSP2#02SP2+HSPl#02SPl+HSPl#G2SP2+2. » 0344 C 01SH#02SPl+2. #02SH#01SP1+4. #GlSPl#C2SP2+4. •01SP2*02SP1>+8. #S#*2* 0345 C <HSP1#01SP1#02SP2+HSP1#Q1SP2#G2SP2+HSP1#02SP1#02SP2+C1SH»C2SP1# 0346 C 02SP2-02SH#C1BP2*G2SP1)-(16.»S»P15P2*#2)•(01SH»02SP1+02SH*01SP1) 0347 C +16. #S#PlSP2#MK2#<01SH#G2SPl+02SH#01SPl-4. #01SPl#C2SPl+4. #01SP1* 0348 C 02SP2+4. *01SP2#C2SP1) + 16.*S#PlSP2#<-2.#HSP2#01SP1*02SP1+HSP1» 0349 C 01SP1#Q2SP2+HSP1#QISP2#C2SP1-01SH*G1SP1*Q2SP2+01SH*©2SP1»02SP2+ 0350 C 02SH#C1SP1*C1SP2+C2SH«01SP1«02SP1-02SH#01SP1*G2SP2-02SH*01BP2* 0351 C G2SP1) 0352 M26-M26-(32. •S#MK2**2>#(G1SP1#02SP2+G1SP2#G2SP1>+a •S»MK2«(-2. » 0353 C HSP2#ClSPl«02SP2-2. #HSP2«ClSP2#C2SPl+2. #HSPl*01SPl#Q2SP2+2. #HSP1 0354 C *01SP2*02SPl-2. #GlSH#01SPl#G2SP2-2.#G1SH#C1SP2#G2SP1-01SH#G2SP1 0355 C «*2-2. #ClSH#C2SPl»02SP2-ClSH»02SP2##2+3. #Q2SH*C1SP1»C2SP1+C2SH# 0356 C 01SPl#C2SP2+3. #G2SH*01SP2»02SPl+C2SH#ClSP2*C2SP2-4. #C1SP1#»2* 0357 C G2SP2-4. #GlSPl#ClSP2#G2SPl-4.#ClSPl#GlSP2*G2SP2+4. #01SP1#02SP1*» 0358 C 2+2 •GlSPl#02SPl»G2SP2+2.#01SPl#02SP2##2-4. #01SP2*#2#G2SPl-2. « 0359 C ClSP2»G2SPl##2+2. •C1SP2»G2SP1#C2SP2)+16. #S#<-HSP2*G1SP1##2«G2SP2 0360 C -HSP2«G1SP1#01SP2«G2SP1+HSP2*01SP1«02SP1##2-HSP2»G1SP1»G2SP1» 0361 C G2SP2-HSP1#01SP1#C2SP1*G2SP2+HSP1#01SP1»C2SP2#»2-G1SH»01SP1• 0362 C 02SP1#02SP2-01SH*©1SP2»02SP1#G2SP2-G1SH#G2SP1##2*C2SP2+Q2SH» 0363 C ©lSPl*QlSP2#©2SPl+C2SH*01SP2##2»02SPl+C2SH#ClSP2«02SPl##2>+32. * 0364 C P1SP2# < 01SH»01BP 1#02SP1#02SP2+01SH#01SP2*G28P1**2+02SH*G1SP1##2» 0365 C G2SP 2+G2SH*G1SP1*01SP 2»02SP1) 0366 M26-M26+32. #MK2# <-01SH*G1SP1#©2SP1#G2SP2- 0367 C 01SH»01SP2»02SP1#«2-02SH#C1SP1•#2#G2SP2-C2SH«C1SP1»01SP2*C2SP1+ 036B C 4. *GlSPl«#2#G2SPl#G2SP2-2. #ClSPl##2#G2SP2#*2+4. #01SP1#01SP2 0369 C *G2SP1 0370 C ##2-4. •01SPl#ClSP2#02SPl*02SP2-2. #GlSP2##2*02SPl#«2>+32. «(2. • 0371 C HSP2#01SPl##2#C2SPl#G2SP2+2.»HSP2#C1SP1#G1SP2#G2SP1**2-HSP1# 0372 C GlSPl##2#G2SP2»#2-2.#HSP1#01SP1*G1SP2#G2SP1*©2SP2-HSPI#01SP2##2* 0373 C e2SPl##2+eiSH»GlSPl»*2#G2SP2#*2+GlSH#GlSPl#GlSP2*G2SPl#G2SP2+2. » 0374 C 01SH«G1SP1*G2SP1••2#Q2SP2-02SH#C1SP1##2#01SP2#©2SP2-C2SH*C1SP1• 0375 C ClSP2*#2#G2SPl-2. »G2SH#G1SP1#01SP2*C2SP1*#2) 0376 M26=M26/S##2 0377 M27 =4. *S*#2*P1SP2*MK2 0378 C -2. »S*#2#PlSP2#ClSH+4. #S#*2#MK2**2+2. #S##2#MK2# 0379 C <HSP2+HSP1+01SH+2. »01SPl+2. #G2SP2)+2. #S##2#(HSP2*G1SPI+HSP1# 0380 C G2SP2>+4. #S#PlSP2##2#<-01SH+02SH>+4. #S#P15P2«MK2»<ClSH-02SH+2. * 0381 C G1SP1-2. *01SP2-2. #G2SPl+2. *G2SP2)+4. #S#P1SP2#<HSP2*01SP1-HSP2* 0382 C Q2SP1-HSP1#G1SP2+HSP1#02SP2+01SH*01SP2+©1SH*G2SP1>+4. #S*MK2#(- 0383 C 01SH*C2SPl-GlSH#C2SP2-02SH#ClSP2+©2SH#©2SP2-2. »©lSPl#C2SPl-2. * 0384 C GlSPl#G2SP2-2. #GlSP2»C2SPl-2.#C1SP2#G2SP2)+4.»S*(-HSP2»G1SP1* 0385 C GlSP2-HSP2 #01SP1*02SP1-HSP1#01SP2*G2SP2-HSP1#©2SP1•C2SP2+C1SH* 0386 C G1SP2#G2SP1-01SH#G2SP1#C2SP2-C2SH#G1SP1*G1SP2+G2SH#G1SP1#G2SP2)+ 0387 C 8. •PlSP2*<01SH#ClSPl#©2SP2+eiSH#eiSP2#©2SPl-02SH#ClSPl#02SP2 0388 C -C2SH 0389 C •01SP2*02SP1) 0390 M27-M27+8. #MK2#<-©1SH#©18P1#02SP2-01SH#©1SP2*©2SP1+©2SH# 0391 C G1SP1#C2SP2+C2SH#G1SP2#C2SP1-2.»G1SP1##2#©2SP2-2. •©1SP1•©1SP2* 0392 C G2SP1+2. #©1SPl#GlSP2#G2SP2+2.#G1SP1#Q2SPl#C2SP2-2. #018P1#©2SP2*» 0393 C 2+2. #ClSP2#«2#G2SPl+2. #01SP2#©2SPl##2-2. •01SP2*G2SP1#©2SP2)+B. •( 0394 C -HSP2*01SP1••2#©2SP2-HSP2*01SP1•©!SP2*02SP1+HSP2»G1SP1•02SP1• 0395 C ©2SP2+HSP2*01SP2#02SP1*»2+HSP1*©1SP1#©1SP2«©2SP2-HSPI«01SP1« 0396 C G2SP2*«2+HSP1#G1SP2»»2»G2SP1-HSP1*01SP2»G2SP1•G2SP2-01SH*0ISP2»* 0397 C 2*02SP1-C1SH#01SP2*02SPI##2+01SH#G1SP2#02SP1#02SP2+C1SH#G2SP1#*2 0398 C «02SP2+G2SH#01SP1#01SP2##2+02SH*01SP1*G1SP2#G2SP1-02SH#G1SP1« 0399 C G1SP2#G2SP2-02SH#G1SP1#G2SP1*G2SP2) 1 35 0400 M27-M27/S 0401 M2B «4. #S*«2*P1SP2«HK2 0402 C -2. »S*«2»PlSP2«C2SH+4. •S««2»MK2«2+2. *S«*2«MK2* 0403 C (HSP2+HSP1+02SH+2. *01SPl+2. «C2SP2)+2. •S**2*(HSP2*01SP1+HSP1* 0404 C G2SP2>+4. »S»PlSP2«»2#<01SH-G2SH)+4. »S*P1SP2*MK2»(-01SH+G2SH+2. • 0403 C Q1BP1-2. *ClSP2-2. »02SPl+2. »C2SP2)+4. #S»P1SP2»(HSP2*C1SP1-HSP2» 0406 C 02SP1-HSP1*C1SP2+HSP1•G2SP2+G2SH«01SP2+G2SH«G2SP1)+4. •S«MK2»( 0407 C 01SH»01SP1-C1SH*C2SP1-02SH*61SP1-C2SH#C1SP2-2. *01SP1*02SP1-2. * 0408 C 01SPl»02SP2-2. •01SP2»G2SPl-2. »Q1SP2»Q2SP2>+4. *S»<-HSP2»G1SP1» 0409 C 01SP2-HSP2*C18P1*C2SP1-HSP1*G1SP2*C2SP2-HSP1*C2SP1*C2SP2+C1SH* 0410 C01SP1*C2SP2-01SH»G2SP1#02SP2-02SH«01SP1•©1SP2+G2SH*G1SP 2«02SP1> + 0411 C B. *PlSP2*<-ClSH*01SPl»G2SP2-ClSH*019P2»02SPl+G2SH»01SPHfG2SP2+ 0412 C G2SH*C1SP2»Q2SP1> 0413 M28-M28+8. *MK2#(G1SH*01SP1*Q2SP2+01SH*01SP2»02SP1-C2SH 0414 C *ClSPl»C2SP2-C2SH»GlSP2»02SPl-2. »ClSPl**2»G2SP2-2. *C1SP1»C1SP2» 0415 C C2SP1+2. »01SPl«ClSP2«G2SP2+2. •GlSPl*G2SPl*G2SP2-2. »01SP1*G2SP2*« 0416 C 2+2. •GlSP2**2»G2SPl+2. *01SP2»G2SPl*»2-2. •G1SP2*02SP1»G2SP2)+8. •< 0417 C -HSP2»GISP1*»2»G2SP2-HSP2*01SP1»G1SP2«G2SP1+HSP2*C1SP1»G2SP1* 0418 C C2SP2+HSP2«Q1SP2*02SP1*»2+HSP1*01SP1»G1SP2*C2SP2-HSP1»C1SP1• 0419 C G2SP2««2+HSP1•©1SP2*»2»C2SP1-HSP1•©1SP2*G2SP1«G2SP2-C1SH*G1SP1* 0420 C 01SP2*02SP2-01SH*01SP1*02SP1#C2SP2+01SH#G1SP2«G2SP1*C2SP2+G1SH* 0421 C 028P1•*2*02SP2+02SH»Q1SP1»01SP2»»2+02SH*01SP1*G1SP2»02SP1-G2SH* 0422 C 01SP2**2*02SP1-G2SH*©1SP2*028P1**2> 0423 M28-M2B/S 0424 M33 -(-32. «S»*2*MK2««3 0425 C -8. «S*»2*MK2**2«MH2+32. *S»*2*MK2*«2*(-HSP1+©2SH+ 0426 C ©2SP1)-8. *S»«2*MK2«MH2*C2SP1 + 16. *S«*2*MK2«(HSP1*©2SH+2. *C2SH* 0427 C C2SP2+2. •G2SPl«GSSP2)-8. •S**2«MH2*G2SP1*C2SP2+16. »S*»2*HSP1*G2SH 0428 C *G2SP2+64.»S*MK2**2»(-C1SH»C2SP2-C2SH«C1SP2-C1SP1»©2SP2-C1SP2* 0429 C G2SP1+2. *G1SP2*G2SP2> + 16. *S«MK2*MH2*(G1SP1«G2SP2+G1SP2*G2SP1+2. • 0430 C GlSP2«C2SP2)+32. *8«MK2*<-HSPl«GlSH*C2SP2-HSPl*C2SH»01BP2+4. «HSP1 0431 C »GlSP2»G2SP2-4. •G2SH»GlSP2*G2SP2-4.•01SP2«G2SPl*G2SP2>+32. »S«MH2 0432 C •GlSP2»G2SPl«G2SP2-64. •S*H5Pl*G2SH*GlSP2*C2SP2+236. *MK2»C1SP2« 0433 C 02SP2»(01SH*G2SP2+02SH«G1SP2+01SP1«G2SP2+G1SP2#028P1)-(64. •MH2* 0434 C G1SP2*G2SP2)*<01SP1*G2SP2+G1SP2*Q2SP1>+128. *HSP1*01SP2«02SP2»< 0433 C C1SH*G2SP2+G2SH*G1SP2))/S»»2 0436 M34 — 8 . «9*«3*P1SP2*MK2 0437 C +2. •S*«3#PlSP2*HH2-8.•S#»3»MK2*HSP2-4. «S»«3» 0438 C HSP2«HSP1 + 16. •S«*2«PlSP2*HK2*«2-4. #S*»2*P1SP2«MK2#MH2+16. «S*«2* 0439 C P18P2*MK2»<ClSP2+C2SP2>-(4. *S««2*P1SP2*MH2)»<G1SP2+C2SP2>-16. «S 0440 C »«2»MK2*«3-4. «S«»2«MK2«*2*MH2+16. *B»#2*MK2«»2*<HSP2-HSP1)+8. *S»* 0441 C 2*MK2»CHSP2»HSPl+2.»HSP2»01SP2+2.»HSP2»G2SP2+2. *01SH#G2SP2+2.• 0442 C C2SH*ClSP2+2. *019Pl»C2SP2+2. *G1SP2*C2SP1> 0443 M34-M34-(4. «S»»2*MH2)•(G1SP1• 0444 C Q2SP2+G1SP2»C2SP1>*8. *S#*2*HSP1*(HSP2*G1SP2+HSP2*02SP2+©1SH« 0443 C C2SP2+02SH*0lSP2>-64. •8»P1SP2*MK2*C1SP2*G2SP2+16. •S«P1SP2»MH2* 0446 C 01SP2«G2SP2+64. •S*HK2#»2»<-01SH»Q2SP2-G2SH*C1SP2-G1SP1«C2SP2- 0447 C 01SP2*Q2SP1+01SP2»G2SP2>+16. »S»MK2*MH2»<G1SP1»C29P2+G1SP2*G2SP1+ 0448 C G18P2«02SP2>+32. «S*HK2*(-2. *HSP2*01SP2*028P2-H8P1»G1SH«02SP2- 0449 C HSPl»C2SH«ClSP2+2. *HSP1*C1SP2»G2SP2-C1SH»C1SP2*C2SP2-C1SH«C2SP2 0450 C •*2-G2SH*01SP2**2-Q2SH«G1SP 2*C2SP2-Q1SP1*G1SP2*G2SP2-G1SP1*G2SP2 0451 C •«2-01SP2«*2*02SPl-01SP2*02SPl»02SP2)+8. »S»MH2»<01SP1*01SP2» 0452 C 02SP2+C1SP1*©2SP2«*2+01SP2**2«G2SP1+C1SP2*Q2SP1*G2SP2> + 16. *S» 0453 C HSPHM-2. »HSP2#C1SP2*02SP2-C1SH*01SP2»02SP2-C1SH*C2SP2**2-G2SH* 0454 C GlSP2»*2-C2SH»GlSP2»02SP2)+256. »MK2*C1SP2»02SP2»(01SH»G2SP2+02SH 0455 C •C1SP2+G1SP1*G2SP2+G1SP2*C2SP1)-(64. •MH2»C1SP2*02SP2)»(C1SP1« 0456 C 02SP2+01SP2«02SP1)+128.•HSP1*G1SP2*02SP2*(01SH*G2SP2+02SH*01SP2) 136 0457 M34-M34 /S«*2 0458 M35 — 8 . »S*»3»P1SP2*MK2 0459 C +4. *S*»3*PlSP2«C2SH-4. »S*#3*MK2»HSP2-4. *B*«3» 0460 C HSP2*02SP1-16. *S»*2»P1SP2»*2*MK2+B. »S»*2»P1SP2»*2*026H+16. #S*»2» 0461 C P1SP2*MK2«*2+B. •S»»2*PlSP2*MK2«<-HSP2+HSPl+GlSH-G2SH+2. »ClSPl+2. 0462 C *01SP2-C2SP1+C2SP2>+S. •S»«2«P1SP2*<-HSP2*G2SP1+HSP1«G1SP2-G2SH* 0463 C 01SPl-C2SH»GlSP2-G2SH»02SP2>-<8. *S*#2*MK2»*2>*<©2SH+02SP1+C2SP2) 0464 C +4. *S»#2*MK2«(HSP2»02SPl+HSP2«Q2SP2+HSPl»Q2SPl-HSPl*G2SP2+2. * 0465 C ©lSH*Q2SP2+2. *C2SH*01SP2+4. •01SPl*©2SP2+4. *©1SP2»©2SP1)+B. *S**2* 0466 C (HSP2*01SP1*02SP1+HSP2»Q1SP2«©2SP1+HSP2«C2SP1•C2SP2+01SH#02SP1* 0467 C ©2SP2-C2SH*©1SP1*C2SP2)-<16.•S»P1SP2**2)*(C1SH*C2SP2+C2SH#C1SP2) 0468 C +16. *S*PlSP2»MK2*C01SH*©2SP2+©2SH*ClSP2+4. »©lSPl*©2SP2+4. »01SP2* 0469 C G2SP1-4. »01SP2#C2SP2) + 16. *S#P1SP2»<HSP2»©1SP1»02SP2+HSP2»©1SP2* 0470 C G2SP1-2. *HSP1*G1SP2*G2SP2-01SH»G1SP2#02SP1+G1SH*C2SP1*©2SP2+C2SH 0471 C »C1SP1»01SP2-G2SH*G1SP1•C2SP2-G2SH»01SP2*G2SP1+02SH»©1SP2«G2SP2) 0472 C -(32.•S»MK2»»2)»(GISP1»G28P2+G1SP2»G2SP1) 0473 M35-M35+B. •S»MK2*<2.*HSP2» 0474 C 01SPl*02SP2+2. *HSP2»01SP2«G2SPl-2. #HSPl»GlSPl*02SP2-2. »HSP1* 0475 C GlSP2»02SPl-2. »01SH*01SPl»02SP2-2.»C1SH*C1SP2»02SP1-01SH«C2SP1«* 0476 C 2-2. »01SH»02SPl*02SP2-01SH»02SP2«»2+02SH*01SPl»02SPl+3. «©2SH» 0477 C ©lSPl«©2SP2+©2SH*©lSP2*©2SPl+3.*©2SH»©lSP2*©2SP2-4. »G1SP1»«2« 0478 C 02SP2-4. •01SPl*ClSP2»62SPl-4.•01SPl«©lSP2»©2SP2+2. *G1SP1*G2SP1* 0479 C G2SP2-2. »C1SPl*C2SP2»*2-4. *01SP2*«2»©2SP1+2. *©lSP2*C2SPl««2+2. * 0480 C GlSP2*Q2SPl»Q2SP2+4.*G1SP2»G2SP2»»2>+16.»S»(HSP2»C1SP2*G2SP1*»2- 04B1 C HSP2*01SP2*02SP1*02SP2-HSP1*6ISP 1*61SP2*C2SP2-HSP1»01SP2»*2» 0482 C C2SP1-HSP1•©1SP2*C2SP1*02SP2+HSP1*G1SP2*C2SP2**2-C1SH*C1SP1• 0483 C G2SP1»G2SP2-G1SH*C1SP2*G2SP1•C2SP2-C1SH*G2SP1•C2SP2»*2+G2SH* 0484 C GlSPl»»2*C2SP2+G2SH#GlSPl»ClSP2»C2SP2+C2SH#GlSPl»C2SP2*»2>+32. • 04B5 C P1SP2« < C1SH»©1SP1*©2SP2**2+©1SH*C1SP2»©2SP1»©2SP2+C2SH»G1SP1• 0486 C G1SP2*G2SP2+G2SH*G1SP2*«2*C2SP1) 0487 M35=M35+32. »MK2« (-C1SH«G1SP1#G2SP2»»2- 0488 C C1SH«C1SP2»C2SP1*02SP2-C2SH«C1SP1#C1SP2*02SP2-©2SH*C1SP2*«2* 0489 C C2SP1-2. »01SPl*«2»©2SP2**2-4.•01SPl*01SP2»©2SPl»C2SP2+4. »01SP1* 0490 C GlSP2*C2SP2**2-2. •GlSP2«*2«G2SPl*«2+4. *01SP2**2*G2SP1*02SP2)+32. 0491 C #(-HSP2*01SPl«*2*02SP2**2-2.•HSP2*01SP1»01SP2*02SP1*02SP2-HSP2« 0492 C ©lSP2*«2*02SPl»«2+2. *HSPl*ClSPl*GlSP2*C2SP2»*2+2. •HSP1«C1SP2**2* 0493 C G2SP1•C2SP2+01SH*C1SP1*01SP2*G2SP1*02SP2+G1SH*G1SP2**2*G2SP1**2+ 0494 C 2. *G1SH«G1SP2*02SP1*G2SP2*«2-C2SH«G1SP1**2*C1SP2*02SP2-02SH 0495 C *©1SP1 0496 C »01SP2*«2»02SPl-2. •02SH*G1SP1*G1SP2»02SP2*«2) 0497 M35-M35/S**2 0498 M36 -8. »S*»2*P1SP2*«2»MH2 0499 C +32. »S**2»P1SP2«MK2*«2+16.»S«*2»P1SP2»HK2*( 0500 C HSP2+HSP1-2. »G2SH-02SP1-02SP2)+8. »S*»2»PlSP2»02SH»»2+4. »S»*2»MK2 0501 C •MH2*<G2SP1+G2SP2)+B. »S*»2»MK2«<-HSP2»C2SH+HSP2*C2SPl-HSP2»C2SP2 0502 C -HSP1*C2SH-HSP1*02SP1+HSP1«©2SP2)+B. •S*«2*MH2«G2SPl*G2SP2-<8. «S 0503 C **2#G2SH)•(HSP2*G2SP1+HSP1»02SP2)-32. »S*P1SP2»*2»01SH*02SH+32. *S 0504 C »P1SP2*MK2»(G1SH*G2SH+G1SH*G2SP1+G1SH*G2SP2+G2SH»G1SP1+G2SH* 0505 C 01SP2+2. *01SPl*02SP2+2. *C1SP2*02SP1)-(32. #S*P1SP2»MH2)«(G18P1* 0506 C G2SP2+01SP2*G2SP1) + 16. •S*P1SP2*(HSP2*G1SH*G2SP1+HSP2*G2SH*G1SP1 + 0507 C HSP1»C1SH«C2SP2+HSP1«C2SH*C1SP2-G2SH«*2*G1SP1-C2SH»«2»01SP2 >-( 0508 C 64. •S*HK2*«2)»(01SP1»02SP2+G1SP2«02SP1> + 16. »S*MK2»(-2. *HSP2 0509 C «G1SP1 0510 C «C2SP2-2. *HSP2»GlSP2*02SPl-2.#HSPl»ClSPl*G2SP2-2. *HSP1*G1SP2« 0511 C G2SPl+01SH*02SPl**2-2. »01SH*02SP1»02SP2+018H*02SP2«*2+02SH««2« 0512 C GlSPl+02SH*«2*01SP2+G2SH*01SPl*02SPl+3. »02SH»GlSPl»02SP2+3. »02SH 0513 C •GlSP2*G2SPl+G2SH*GlSP2»G2SP2+2.*GlSPl*02SPl»02SP2+2. »01SP1* 1 37 0314 C C2SP2«#2+2. *Q1SP2»G2SPl*«2+2.•C1SP2»C2SP1*G2SP2> 0313 M36-M36-(16. *S*MH2« 0316 C Q2SP1«02SP2)*(01SP1+01SP2)+16.•S*G2SH*(HSP2*G1SP1*02SP1+HSP2* 0517 C QlSP2*C2SPl+HSPl«ClSPl*02SP2+H8Pl#GlSP2»02SP2)+64. •P1SP2«01SH* 0518 C 02SH*(01SP1»Q2SP2+01SP2«02SP1)+64. »MK2*(-01SH*02SH*01SP1»02SP2- 0519 C 01SH*C2SH*C1SP2»C2SP1-C1SH*01SP1»C2SP1•C2SP2-C1SH*01SP1»C2SP2**2 0520 C -01SH*C1SP2*G2SP1*»2-G1SH»C1SP2*C2SP1«G2SP2-G2SH*C1SP1**2*G2SP2- 0521 C 02SH«01SP1»G1SP2*02SP1-C2SH«01SPH>C1SP2*C2SP2-02SH#C19P2««2* 0522 C 02SP1-2. »01SPl*»2»02SP2**2-4.*01SPl»ClSP2»C2SPl*C2SP2-2. *C1SP2»» 0523 C 2»02SPl»»2)+32. »MH2*(01SPl*»2»02SP2*#2+2. *G1SP1»G1SP2»G2SP1* 0524 C 02SP2+G1SP2«*2*G2SP1*«2)-32.•(HSP2*01SH*G1SP1*02SP1«02SP2+HSP2» 0525 CGISH*G1SP2*G2SP1*»2+HSP2*02SH»GlSPl*«2«C2SP2+HSP2*G2SH*Q1SP1• 0526 C 01SP2»C2SP1+HSP1*01SH*G1SP1»C2SP2*»2+HSP1*C1SH»G1SP2»G2SP1*C2SP2 0 527 C +HSP1»C2SH«C1SP1»C1SP2*C2SP2+HSP1*C2SH#01SP2*»2»02SP1) 0528 M36-H36/S**2 0529 M37 «(4. *S**2*P1SP2«MK2-S»«2»P1SP2«MH2 0530 C +4. »S*«2*MK2**2+S#*2«MK2*MH2+4. * 0531 C S*»2»MK2»(HSP2+HSP1>+2.•S*»2»HSP2»HSP1+8. *S»P1SP2»MK2»(-01SP2+ 0532 C 02SP2)+2. *S*PlSP2*MH2*(018P2-02SP2>+8. •S*MK2*«2*(-01SH+02SH- 0533 C 01SP1+G2SP1)+2. *S«MK2»MH2«(01SP1-G2SP1)+4. •S*HK2*(-2. *HSP2*G1SP2 0534 C +2. •HSP2*02SP2-HSPl*018H+HSPl»02SH-2. •01SH*02SP2-2. •02SH*61SP2+ 0535 C 4. *02SH*02SP2-2.*0lBPl»028P2-2.«©lSP2»Q2SPl+4. »02SP1*02SP2) 0536 C +2. *S* 0537 C MH2»(QlBPl«02SP2+01SP2»02SPl-2.#02SPl»G2SP2)+4. »S*HSP1»(-HSP2* 0538 C 01SP2+HSP2«G2SP2-GlSH»02SP2-02SH»GlSP2+2. •02SH*G2SP2) + 16. »MK2»( 0539 C 3. •GlSH*01SP2»G2SP2-01SH«G2SP2*«2+G2SH*01SP2««2-3. «02SH»01SP2« 0540 C G2SP2+3. •01SPl»ClSP2*Q2SP2-GlSPl*02SP2»*2+01SP2»»2»©2SPl-3. • 0541 C GlSP2*02SPl*G2SP2)+4. *MH2*(-3.*0ISP 1*01SP2«02SP2+0ISP1*02SP2**2- 0542 C 01SP2»*2«02SPl+3. •01SP2*02SPl»02SP2)+8.•H8P1»(3. »G1SH#01SP2« 0543 C G2SP2-QlSH«02SP2**2+G2SH*GlSF2«*2-3.*02SH«01SP2»02SP2)>/S 0544 1138 -4.•S**2»P1SP2*MK2-S*«2*P1SP2*MH2 0545 C +4. #S**2«HK2**2+S*«2*MK2»l1H2+4. # 0546 C S««2«MK2«(HSP2+HSPl>+2. «S*»2«HSP2«HSPl+8. »S»P1SP2«MK2#(-01SH+ 0547 C G2SH-GlSP2+G2SP2)+2.•S*PlSP2»MH2»(01SP2-C2SP2)+4. •S*P1SP2*G2SH*( 0548 C 01SH-02SH)+8. »S»MK2»*2»(-01SP1+02SP1>+2. *S»MK2*MH2*(01SP1-02SP1) 0549 C +4. •S*MK2«(-HSP2*ClSH+HSP2*G2SH-2.»HSP2»C18P2+2. •HSP2*G2SP2-G1SH 0550 C »G2SH-2. #01SH*C2SPl+02SH«*2-2.»G2SH«GlSP2+2 •02SH»C2SP1+2. »02SH« 0551 C 02SP2-2. #01SPl»02SP2-2. •GlSP2*G2SPl+4. *G2SP1»C2SP2)+2. •S»MH2»( 0552 C ©lSPl*02SP2+ClSP2»C2SPl-2. *G2SPl»02SP2>+4. •S*(-HSP2*HSP1»G1SP2+' 0553 C HSP2«HSP1»02SP2-HSP2»018H*G2SP1+HSP2«02SH*G2SP1-HSP1»G2SH«01SP2+ 0554 C HSP1*C2SH»G2SP2>+B.•P1SP2«(-G1SH**2*G2SP2-G1SH*G2SH*G1SP2+01SH« 0555 C ©2SH»G2SP2+©2SH»*2*©1SP2> 0556 M3B-M38+B. «MK2*(01SH«*2*02SP2+Q1SH*C2SH*C18P2- 0557 C 01SH*G2SH*02SP2+2. •01SH»01SPl*02SP2+2.•01SH*01SP2»02SPl+2. *01SH* 0558 C 01SP2*02SP2-2. •01SH*02SP2**2-©2SH«*2»©lSP2-2. *©2SH»©1SP1#©2SP2+ 0559 C 2. •G2SH*GlSP2*«2-2. •02SH*01SP2*02SPl-2. •02S++»01BP2«02SP2 0560 C +6. •G1SP1 0561 C »01SP2»C2SP2-2. *ClSPl»©2SP2««2+2.•01SP2»«2*C2SPl-6. *©1SP2*02SP1# 0562 C 02SP2>+4. »MH2*<-3. »01SP1*©1SP2»02SP2+©1SP1*G2SP2»*2-©1SP2»*2« 0563 C 02SP1 +3. •© 1SP2»C2SP1*G2SP2 > +8.•(HSP2*C1SH*C1SP1»©2SP2+HSP2»©1SH* 0564 C 01 SP2«02SP 1 -HSP2*02SH*01 SP 1 •02SP2-H8P2*02SH«C 1 SP2»02SP 1+HSP1 • 0565 C C1SH»C1SP2»G2SP2-HSP1«Q1SH»02SP2**2+HSP1»G2SH«G1SP2»»2-HSP1«G2SH 0566 C «01SP2«G2SP2) 0567 M3B-M3B/S 0568 M44 -(-32. »S*»2»MK2*»3 0569 C -8. •S*«2#MK2*«2*MH2+32. »S**2»MK2**2»(-HSP1+01SH+ 0570 C 01SPD-8. *S»«2«MK2»MH2*0ISP 1 + 16. *S*»2*MK2»(HSPl«01BH+2. *©1SH« 138 0571 C Q1BP2+2. •01SP1»01SP2)-B. *S»*2»MH2»Q1SP1*C1SP2+16. »S««2«HSP1»C1SH 0572 C »01SP2+64. •S«MK2««2*(-Q1SH*G2SP2-Q2SH»C1SP2-01SP1»G2SP2-G1SP2» 0573 C 02SP1+2. •G1SP2»02SP2)+16. •S*riK2*MH2*<GlSPl*02SP2+01SP2*G2SPl+2. • 0574 C Q18P2*02SP2>+32. •S»MK2«(-HSPl»01SH*G2SP2-HSPl*02SH*QlSP2+4. »HSP1 0575 C *01SP2*02SP2-4. •GlSH»GlBP2»C2SP2-4. »ClSPl*01BP2*G2SP2>+32. •S#MH2 0576 C •01SPl*01SP2«02SP2-64. •S»HSPl»G18H»01SP2*Q2SP2+256. *HK2*01SP2« 0577 C 02SP2«<GlSH*C2SP2+G2SH»GlBP2+01SPl»02SP2+QlSP2*Q2SPl>-<64. *MH2* 0578 C 01SP2»Q2SP2>*<Q1SP1»Q2SP2+01SP2*02SPI>+128. •HSP1*G1SP2»G2SP2»< 0579 C 01SH*C2SP2+G2SH#01SP2>)/S»*2 0580 M4S -8. *S«*3*MK2»«2 0581 C >4. »S«*3*HK2«(H8P1+2. #018P2)+4. •B*«3«HSP1»C1SP2+ 0582 C 16. •S»*2«P1SP2»MK2»*2+16. *S*«2»P1SP2«MK2*<HSP1+Q1SP2>+16. *S*»2* 0583 C P1SP2*HSP1»01SP2-16.•S«»2*MK2»*3+B. •S**2*MK2*«2«(-HSP2-HSP1+01SH 0584 C -02SH-2. •C2SP1-2. #G2SP2)+B. *S#*2*MK2»<-HSP2«C1SP2-HSP1»G1SP1- 0585 C HSPl»C2SP2+GlSH*ClSP2+01SH*G2SPl-02SH»GlBPl-4. »ClSP2*02SP2>+8. #8 0586 C *»2«01SP2«<-HSPl*01SPl-2. •HSP1»02SP2+G1SH«C2SP1-Q2SH«G1SP1>+16. * 0587 C S»PlSP2«MK2»<-01BH«G2SPl-C2SH*ClSPl-4. *C1SP2*C2SP2)+16. •S*P1SP2* 0588 C 01SP2*(-4. *HSP1»G2SP2-01SH*Q2SP1-Q2SH*01SP1>+16. »S»MK2**2»(01SH* 0589 C C2SPl+02SH»01SPl+4. «G18Pl#02SPl-2. »01SPl*02SP2-2. *C1BP2»Q2SP 1+4. 0590 C »C1SP2*C2SP2) 0591 M45-M45+16. *S«MK2*<2. *HSP2*01SP1*02SP1+2. *HSP2»C1SP2*C2SP2 0592 C —HSP1*G1SP1»02SP2-HSP1*01SP 2*02SP1+2. *HSP1»G1SP2«02SP2-01SH* 0593 C G1SP1»02SP1-GlSH»01SP2»C2SP2-C1SH*C2SP1»C2SP2+G2SH»G1SP1••2+02SH 0594 C •01SPl«G2SP2+G2SH*GlSP2**2+2.•02SH*ClSP2»02SP2-2. »018P1*C1SP2» 0595 C 02SP2-2. •GlSP2«*2«G2SPl+4.*GlSP2*G2SPl*G2SP2+4. •G1SP2*G2SP2**2> + 0596 C 16. •S*01SP2*(2. »HSP2*01SPl*C2SPl-2. •HSPl*GlSP2«G2SPl+2. *HSP1» 0597 C 02SP2»#2-01SH*01SPl»02SPl-2.»01SH»G2SPl»02SP2+02SH»GlSPl»»2+2. • 0598 C C2SH*01BPl»C2SP2)+64. »P18P2*G1SP2*G2SP2*(01BH*G2SP1+02SH*G1SP1) + 0599 C 64. *MK2«GlSP2»C2SP2»<-GlSH»02SPl-02SH»C16Pl-4. »GlSPl»02SPl+2. • 0600 C ClSPl«G2SP2+2. •C1SP2«C2SP1)+32.»C18P2»<-4. #HSP2«G1SP1«G2SP1» 0601 C G2SP2+2. »HSPl»GlSPl»02SP2**2+2.»HSP1»C1SP2*02SP1»C2SP2+01SH» 0602 C 01SPl«G2SPl»G2SP2+01SH*GlSP2»G2SPl**2+2. *G1SH«G2SP1*G2SP2*«2- 0603 C 02SH»01SPl**2«G2SP2-G2SH*GlSPl*GlSP2«G2SPl-2. •G2SH*01SP1*G2SP2«» 0604 C 2 > 0605 M45«M45/S««2 0606 M46 — 2 . •S«*3»P1SP2#MH2 0607 C +8. *S»*3*MK2»«2+4. *S*«3»MK2*(HSP2+HSP1>+4. »S»* 060B C 3*HSP2*HSP1 + 16. *S««2«P18P2*MK2**2+4. *S«»2»P1SP2»MK2*MH2+16. «S*»2 0609 C *PlSP2»MK2»<HSP2+HSPl)+4.*S»»2*PlSP2*MH2»(GlSPl+G2SP2)+8. »S»»2« 0610 C P1SP2#(2. •HSP2»HSP1+01SH*G2SH)-16. »S*»2«MK2**3+4. *S*«2*MK2»«2» 0611 C MH2-C16. •S«»2*MK2**2>*(HSP2+HSP1+G1SP2+02SP1>+4. *S««2*MK2»(-2. • 0612 C HSP2**2-2. »HSP2»HSPl+HSP2*ClSH-HSP2»C2SH-2. *HSP2»C2SP1-2. »HSP2* 0613 C 02SP2-2. •HSPl«*2-HSPl*GlSH+HSPl«02SH-2. •HSPKG1SP1-2. •HSP 1*0ISP2 0614 C +2. •GlSH*G2SPl-2. •GlSH»G2SP2-2. *G2SH*GlSPl+2. «02SH*GlSP2>+4. *S*« 0615 C 2*MH2«<QlSPl»G2SP2+01SP2»C2SPl>-<8. »S»»2)»<HSP2»HSP1»G1SP1+HSP2» 0616 C HSPl»02BP2+HSP2*02SH*ClSPl+HSPl*018H*02SP2)+32. •S»P1BP2»MK2»< 0617 C GlSPl*02SP2+ClSP2*C2SPl)-<8.•S»P1SP2»MH2)«(G1SP1»C2SP2+01SP2« 0618 C C2SP1)-(16. *S*P1SP2>*<HSP2*G1SH*C2SP1+HSP2*C2SH*G1SP1+HSP1»C1SH* 0619 C 02SP2+HSPl*G2SH*G18P2+01BH»02SH*GlSPl+01SH*G2SH*028P2>+32. »S»MK2 0620 C **2*< G1SH*02SP1+01SH»02SP2+02SH»01SP1+02SH*G1SP2+2. *01SP1»02SP1- 0621 C QlSPl»G2SP2-QlSP2«02SPl+2. *01SP2*02SP2) 0622 M46-M46+8.*S*MK2»MH2*<-2. *01SP1• 0623 C G2SPl-01SPl*G2SP2-01SP2*G2SPl-2. *01SP2«02SP2)+B. •S»HK2*<2. *HSP2* 0624 C GlSH*C2SPl+2.*HSP2*G18H»G2SP2+2.*HSP2»02SH*ClSPl+2. *HSP2»02SH« 0625 C 01SP2+8. *HSP2*GlSPl*02SPl-4.*HSP2*G1SPl*02SP2-4. «HSP2*G1SP2« 0626 C G2SP1+2. »HSPl»01SH»C2SPl+2.•HSPl*GlSH*G2SP2+2. •HSP1*02SH*01SP1+ 0627 C 2. «HSPl»C2SH*01SP2-4. «H8Pl*GlSPl*G2SP2-4. *HSPl*ClSP2»C2SPl+8. * 139 0628 C HSP1*C1SP2*C2SP2+01SH**2*Q2SP1-C1SH**2*C2SP2+C1SH*C2SH*01 SP 1 + 0629 C 01SH*C2SH*ClSP2+ClSH»Q2SH»02SPl+ClSH*C2SH*C2SP2+2. *01SH*C1SP1* 0630 C 02SP1+2. *QlSH*QlSP2*02SPl+2.*01SH*G2SPl*C2SP2+2. *G1SH*C2SP2**2- 0631 C 02SH**2*ClSPl+Q2SH**2«ClSP2+2.»C2SH*C1SP1**2+2. *02SH*C1SP1*G1SP2 0632 C +2. *C2SH*01SP2*G2SP1+2. »C2SH*C1SP2*C2SP2+8. *C1SP1*G1SP2*G2SP1+8. 0633 C *01SP2*02SPl»C2SP2)-(8. *S*MH2)*(01SP1**2*C2SP2+01SP1*01BP2*C2SP1 0634 C +018P1*02SP2**2+01SP2*02SP1*G2SP2>+16. *S*(2. *HSP2**2*G1SP1*G2SP1 0635 C -HSP2*HSP1*01SP1*02SP2-HSP2*HSP1*01SP2*02SP1+HSP2*C2SH*G1SP1**2+ 0636 C HSP2*02SH*QlSPl*Q2SP2+2. *HSP1**2*G1SP2*02SP2+HSP1*01SH*01SP1* 0637 C C2SP2+HSP1*C1SH*C2SP2**2> 0638 H46-M46+32. *P1SP2*<G1SH**2*Q2SP1*02SP2+C18H* 0639 C 02SH*01SPl*C2SP2+01BH*G2SH*ClSP2*C2SPl+C2SH**2*ClSPl*01SP2>+32. * 0640 C MK2*(-01SH**2*02SP1*02SP2-01SH*02SH*G1SP1«G2SP2-01SH*G2SH*G1SP2* 0641 C 029P1-4. *G1SH*01SP1*02SP1*02BP2-4. *C1SH*G1SP2*02SP1*G2SP2-02SH*« 0642 C 2*01SPl*01SP2-4. *02SH*01SP1*G1SP2»G2SP1-4. *02SH*G1SP1*G1SP2* 0643 C 02SP2-2. »01SP1**2*02SP2**2-12.*01SP1*01SP2*02SPl*G2SP2-2. *G1SP2 0644 C **2*02SP1»*2>+16. *HH2*(QlSPl**2*C2SP2**2+6. *01SP1*01SP2*G2SP1* 0645 C 02SP2+GlSP2**2*02SPl**2)-64.*(HSP2*01SH*01SP1*02SP1*©2SP2+HSP2* 0646 C 02SH»C1SP1*01SP2*02SP1+HSP1*01SH»01SP2*02SP1*G2SP2+HSP1*G2SH* 0647 C 01SP1*01SP2*G2SP2) 0648 M46-M46/S**2 0649 M47 -<-4. *S»«2*P1SP2»HK2+S**2*P1SP2*MH2 0650 C -4.*S**2»MK2**2-S**2*MK2*MH2-( 0651 C 4. *S**2*MK2)*(HSP2+HSP1)-2.*S**2#HSP2*HSP1+8. *S*P1SP2*MK2*( 0652 C -01SP2 0653 C +02SP2)+2. »S*PlSP2»MH2*<01SP2-02SP2)+8. *S*«K2**2»<-G1SH+02SH- 0654 C 01SP1+02SPD+2. *S*MK2*MH2*(G1SP1-02SP1 >+4. *S*MK2»(-2. *HSP2*G1SP2 0655 C +2. *HSP2*G2SP2-HSPl*01SH+HSPl*G2SH-4. *GlSH*GlSP2+2. *G1SH*G2SP2+ 0656 C 2. *02SH*01SP2-4. *01SPl*GlSP2+2.*G1SPl*G2SP2+2. *01SP2*02SP1) 0657 C +2. *S* 0658 C MH2*<2. *01SP1*01SP2-01SP1*02SP2-01SP2*G28P1)+4. *S*HSP1*(-HSP2* 0659 C 01SP2+HSP2*G2SP2-2. *G1SH*G1SP2+G1SH*G2SP2+G2SH*G1SP2> + 16. *MK2*( 0660 C 3. *GlSH*GlSP2*G2SP2-GlSH*G2SP2**2+G2SH*GlSP2**2-3. *C2SH*G1SP2* 0661 C G2SP2+3. «01SPl*01SP2*02SP2-GlSPl*G2SP2**2+01SP2**2*G2SPl-3. * 0662 CGISP2*02SP1*02SP2)+4. *HH2*(-3.*01SP1*C1SP2*02SP2+01SP1*02SP2**2- 0663 C GlSP2**2*02SPl+3. *01SP2*02SPl»G2SP2)+8. *HSP1*(3. *G1SH*G1SP2* 0664 C G2SP2-01SH*G2SP2»*2+C2SH*01SP2**2-3. *G2SH*G1SP2*02SP2))/S 0665 M48 — 4 . *S**2*P1SP2*MK2 0666 C +S**2*PlSP2*MH2-4. *S**2*MK2**2-S**2*MK2*MH2-< 0667 C 4. *S**2*MK2)*(HSP2+HSP1)-2. *8**2*HSP2*HSP1+8. *S*P1SP2*MK2*( 0668 C -01SH+ 0669 C G2SH-ClSP2+G2SP2>+2.•S*PlSP2*MH2*(GlSP2-02SP2)+4. *S*P1SP2*G1SH*( 0670 C 01SH-02SH)+8. *S*MK2**2*(-01SP1+02SP1)+2.*S*MK2*MH2*(01SP1-G2SP1) 0671 C +4. *S*MK2*(-HSP2*01SH+HSP2*02SH-2.*HSP2*GlSP2+2. »HSP2*C2SP2-G1SH 0672 C **2+GlSH*02SH-2. *01SH*C1SP1-2.*GlSH*GlSP2+2. *GlSH*028P2+2. *G2SH* 0673 C 01SP1-4. *01SPl»GlSP2+2.*01SPl*C2SP2+2.»G1SP2*G2SP1)+2. *S»MH2*<2. 0674 C *GlSPl*01SP2-01SPl*02SP2-01SP2*02SPl)+4. *S*(-HSP2*HSP1*01SP2+ 0675 C HSP2*HSP1*02SP2-HSP2*01SH*G1SP1+HSP2*G2SH*G1SP1-HSP1*01SH*G1SP2+ 0676 C HSP1*01SH*C2SP2) 0677 M48-M48+B. *P 1SP2* < -C1 SH**2*02SP2-01 SH*C2SH*C 1SP2+018H* 0678 C 02SH»C2SP2+G2SH**2*01SP2)+8.«MK2*(01SH»*2*G2SP2+G1SH»G2SH*C1SP2- 0679 C GlSH*G2SH»02SP2+2. *G18H*GlSPl»02SP2+2.*G1SH»C18P2*02SP1+2. »01SH* 0680 C 01SP2*02SP2-2. *01SH*028P2**2-02SH««2»GlSP2-2. •02SH*G1SP1«G2SP2+ 0681 C 2. *C2SH*CISP2**2-2. *C2SH*ClSP2»C2SPl-2. *G2SH*G1SP2*G2SP2 0682 C+6. *01SP1 0683 C *01SP2*02SP2-2. »01SPl*02SP2*»2+2. »01SP2**2*G2SPl-6. *G1SP2*02SP1» 0684 C G2SP2)+4. »MH2*(-3. *01SP1*01SP2*02SP2+G1SP1*02SP2*«2-G1SP2**2* 140 0665 C 02SP1+3. »01SP2«02SPl»02SP2>+8. *<HSP2»C1SH»01SP1#02SP2+HSP2»C1SH» 0686 C 01SP2*G2SP1-HSP2*C2SH»C1SP1*C2SP2-HSP2»C2SH»C1SP2*G2SP1+HSP1* 0687 C 01 SH»C 1 SP2»G2SP2-HSP 1 *C 1 SH*G2SP2»»2+HSP 1 #C2SH*G 1 SP2**2-HSP 1 »G2SH 0688 C •01SP2*Q2SP2> 0689 M48-M48/S 0690 M5S "(8. »S**3*MK2*»2 0691 C +8. #S**3»MK2*<C1SP2+C2SP1)+B.•S*«3*C1SP2*C2BP1+16. 0692 C *S««2*P1SP2«MK2*«2+16. •S»*2»P1SP2*MK2#<01SP2+G2SP1 > + 16. «S*»2« 0693 C P18P2*C1SP2»C2SP1-16. »S»»2*MK2»»3-(16. »S»»2*MK2*»2)•(01SP1+C2SP2 0694 C > + 16. •S*«2*t1K2*(-01SPl*01SP2-2.•01SPl»02SPl-2. •C1SP2*02SP2-G2SP1 0695 C •G2SP2)-(32. •S»*2*G1SP2*C2SP1)»<ClSPl+C2SP2>-<64. •S*PlSP2«liK2)«< 0696 C 01SPl*02SPl+01SP2*C2SP2>-<64. *S»P1SP2*C1SP2*G2SP1 >»<01SP1+G2SP2> 0697 C +64. *S«MK2»*2«<GlSPl»G2SPl+01SP2*C2SP2>+32. »S*MK2»C2. *G1SP1»«2» 0698 C G2SP1+01SP1*01SP2»G2SP2+01SP1*G2SP1»G2SP2-01SP2#*2*G2SPI-01SP2» 0699 C C2SPl**2+2. *GlSP2*G2SP2**2)+32. *S*G1SP2*G2SP1*<2. *01SPl*«2+3. • 0700 C 01SPl*G2SP2-GlSP2*G2SPl+2.*G2SP2**2>+256.•P1SP2»018P1#01SP2* 0701 C G2SP1*02SP2-2S6. *MK2*01SP1*G1SP2«G2SP1»02SP2+12B. •G1SP2*G2SP1«<- 0702 C C1SP1»»2»02SP2+01SP1»G1SP2»02SP1-01SP1•02SP2«»2+G1SP2*G2SP1• 0703 C G2SP2)>/S**2 0704 M56 -8. #S»*3*MK2»»2 0705 C +4. *S»«3*MK2# < HSP2+2. »02SP1)+4.»S»*3»HSP2*02SP1 + 0706 C 16. »S««2»P1SP2»MK2««2+16 •S««2«P1SP2*MK2«<H8P2+028P1>+16. •S**2* 0707 C P1SP2*HSP2*02SP1-16. •S««2»MK2«»3+8. •S**2*MK2»«2»(-HSP2-HSP1-C1SH 0708 C +02SH-2. *01SPl-2: »01SP2)+8.«S»»2»MK2»<-HSP2*G1SP1-HSP2*G2SP2- 0709 C HSPl»G28Pl-01BH«C2SP2+Q2SH«GlSP2+02SH*02SPl-4. •C1SP1*C2SP1)+B. *S 0710 C *«2«G2SPl*(-2. #HSP2*01SP1-HSP2*G2SP2-G1SH*02SP2+02SH«G1SP2) + 16. • 0711 C S*PlSP2*MK2»<-01SH*02SP2-G2SH»01SP2-4. *018P1»02SP1) + 16. *S»P1SP2* 0712 C 02SPl#<-4. »HSP2«01SP1-01SH»G2SP2-02SH»C1SP2>+16. »S»MK2*»2*(C1SH* 0713 C 02SP2+C2SH»01SP2+4. •01BPl*02SPl-2. •GlSPl»02SP2-2. •01SP2»C2SPl+4. 0714 C •G1SP2*G2SP2) 0715 M56-M56+16. «S«MK2«(2.*HSP2»01SP1«G2SP1-HSP2*G1SP1*G2SP2- 0716 C HSP2*ClSP2*G2SPl+2. «HSPl*01SPl*G2SPl+2. »HSPl»ClSP2*02SP2+2. •CISH 0717 C »C1SP1*G2SP1+01SH»G1SP1»C2SP2+G1SH*G2SP1*#2+G1SH»G2SP2«»2-G2SH* 0718 C 01SPl»ClSP2-C2SH*ClBPl»C2SPl-C2SH#ClSP2*C2SP2+4. •01SP1«»2*C2SP1 + 0719 C 4. •GlSPl«01SP2»Q2SPl-2. •01SPl«02SPl«02SP2-2. •G1SP2*G2SP1«»2> 0720 C +16. • 0721 C S«C2SP1•(2. »HSP2»C1SP1»»2-2. •HSP2»C1SP2*G2SP1+2. »HSP1*C1SP2* 0722 C G2SP2+2. «01SH*01SPl*02SP2+ClSH*G2SP2*»2-2. •02SH»01SP1»Q1SP2-02SH 0723 C *GlSP2»C2SP2)+64. »PlSP2»GlSPl»02SPl»<018H»C2SP2+C2SH»GlSP2>+64.# 0724 C MK2»GlSPl*C2SPl«(-01SH*C2SP2-G2SH»ClSP2+2. *01SPl*G2SP2+2. *G1SP2» 0725 C G2SP1-4. *01SP2*C2SP2)+32. »C2SP1»<2. *HSP2*01SPl#*2*02SP2+2. *HSP2» 0726 C 01SPl»GlSP2*G2SPl-4.»HSP1*G1SPl*GlSP2*G2SP2-2. »G1SH*G1SP1**2* 0727 C G2SP2-GlSH»0lSPl*G2SP2»*2-01SH*GlSP2*G2SPl*G2SP2+2. •G2SH*G1SP1•* 0728 C 2»G1SP2+Q2SH*G1SP1«01SP2*G2SP2+02SH«G1SP2**2*02SP1> 0729 M56«M56/S**2 0730 M57 — 4 . «S»«2*P1SP2*MK2 0731 C +2. «S*«2#PlSP2»G29H-4. »S«*2«MK2»»2+2. »S**2»MK2 0732 C »(-HSP2-HSPl-G2SH-2.*01SP2-2.»02SPl)-(2. •8»»2>»<HSP2»02SP1+HSP1» 0733 C GlSP2)+4. •S«PlSP2*»2»(-01SH+G2SH>+4. •S*PlSP2»MK2*(GlSH-G2SH+2. * 0734 C 01SP1-2. »01SP2-2. *G2SPl+2. »G2SP2)+4. •S»P1SP2«(HSP2»01SP1-HSP2* 0735 C G2SP1-HSP1*01SP2+HSP1«02SP2-02SH*G1SP1-02SH»62SP2)+4. »S«MK2» <- 0736 C 01SH«01SP2+ClSH»C2SP2+02SH»01SPl+02SH»GlSP2+2. »01SPl*G2SPl+2. » 0737 C ClSPl»02SP2+2. *01SP2»02SPl+2. *01SP2»02SP2) 0738 M57-M57+4. *S*(HSP2*G1SP1* 0739 C 02SP1+HSP 2*G2SP1«G2SP2+HSP1«01SP1*01SP 2+HSP1»01SP2*02SP2-G1SH* 0740 CGISP2*C2SP1+G1SH#02SP1*G2SP2+G2SH«G1SP1*01SP2-G2SH*G1SP1»C2SP2)+ 0741 C 8. •P1SP2»(©1SH*01SP1»02SP2+01SH»C1SP2*C2SP1-C2SH»G1SP1»G2SP2 141 0742 C -02SH 0743 C •Q1SP2*Q2SP1>+8. *MK2*<-G1SH»G1SP1*Q2SP2-Q1SH*01SP2*02SP1+G2SH* 0744 C 01SPl»02SP2+C2SH*ClSP2»02SPl-2.»C1SPl**2*02SP2-2. *C1SP1*C1SP2« 0745 C 02SP1+2. «01SPl*QlSP2*02SP2+2.*01SPl«02SPl*02SP2-2. *01SP1*02SP2«» 0746 C 2+2. »01SP2**2*02SPl+2. »01SP2*C2SPl#*2-2. #01SP2*02SPl*02SP2>+8. *( 0747 C -HSP2»01SPl»«2»02SP2-HSP2»GiSPl*ClSP2»G2SPl+HSP2*01SPl*G2SPl« 0748 C 02SP2+HSP2»C1SP2»C2SP1*#2+HSP1 »01 SP 1 «Q 1SP2*02SP2-H8P1*G1SP1* 0749 C 02SP2««2+HSP1#01SP2»#2«02SP1-HSP1*©1SP2*G2SP1*G2SP2+G1SH*G1SP1» 0750 C 01SP2*C2SP1-01SH*CIBP1»C2SP1*C2SP2+G1SH»G1SP2*G2SP1*C2SP2-G1SH* 0751 C G2SP1»C2SP2»*2-02SH»G1SP1*«2»C1SP2+02SH»C1SP1••2«02SP2-C2SH# 0752 C 01SP1*C1SP2«G2SP2+02SH»C1SP1*G2SP2»«2> 0753 M57-M57/S 0754 H58 — 4 . *B«*2*P1BP2»MK2 0755 C +2. «S»#2#PlSP2*01SH-4. »S*»2*MK2««2+2. »S*«2«MK2 0756 C •<-HSP2-HSPl-01SH-2. »01SP2-2.*02SPl)-(2. •S«-»2>*<HSP2»C2SP1+HSP1* 0757 C 01SP2>+4. *S*PlSP2«»2»(01SH-02SH)+4. •S»PlSP2«t1K2»(-01SH+02SH+2. • 0758 C G1SP1-2. *GlSP2-2. #C2SPl+2.*©2SP2>+4. #S*P1SP2»<HSP2»C1SP1-HSP2* 0759 C 02SPl-HSPl«01SP2+HSPl»02SP2-01SH«01SPl-01SH»©2SP2>+4. «S»MK2»( 0760 C 01SH*02SPl+018H*C2SP2+©2SH*01SPl-02SH*02SPl+2. »0ISP 1*02BP1+2. • 0761 C 01SPl»G2SP2+2. •01SP2»G2SPl+2 *018P2*G2SP2)+4. *S»<HSP2*01SPI» 0762 C 02SP1+HSP2»02SP1•02SP2+HSP1*0ISP 1*01SP2+HSP1•©1SP2*G2SP2-G1SH* 0763 C 01SP1*02SP2+01SH«C2SP1*02SP2+G2SH*C1SP1*C1SP2-02SH*01SP2*G2SP1>+ 0764 C 8. *P1SP2«(-01SH*01SP1»G2SP2-G1SH*C1SP2»C28P1*02SH»Q1SP1*G2SP2+ 0765 C 02SH»01SP2«G2SP1> 0766 M58-M58+8. *riK2«<01SH*GlSPl*G2SP2+01SH*GlBP2#G2SPl-02SH 0767 C »GlSPl#02SP2-02SH»01SP2»G2SPl-2.*ClSPl«»2»G2SP2-2. »G1SP1*G1SP2» 0768 C G2SP1+2. *GlSPl*01SP2*G2SP2+2.•01SPl*G2SPl«G2SP2-2. «G1SP1*G2SP2*« 0769 C 2+2. »ClSP2»»2*G2SPl+2 •GlSP2*G2SPl**2-2. *GlSP2«02SPl*02SP2)+8. •( 0770 C -HSP2»01SP1•»2*©2SP2-HSP2»01SP1»G1SP2*G2SP1+HSP2»G1SP1»C2SP1• 0771 C 02SP2+HSP2*G1SP2»G2SP1••2+HSP1«01SP1*G1SP2*G2SP2-HSP1*01SP1• 0772 C G2SP2««2+HSP1*01SP2**2*G2SP1-HSP1»C1SP2«G2SP1*02SP2+G1SH*G1SP1•* 0773 C 2*G2SP2-G1SH«G1SP1*G2SP1•G2SP2+01SH*G1SP1*G2SP2**2-G1SH*G2SP1* 0774 C G2SP2**2-02SH*C1SP1••2»G1SP2+G2SH»C1SP1»01SP2»G2SP 1-G2SH*G1SP1• 0775 C 01SP2*G2SP2+G2SH»G18P2*G2SP1»02SP2) 0776 M5B-M58/S 0777 M66 -<-32. »S»»2»MK2**3 0778 C -8. •S*»2«MK2«»2#MH2+32.•S»»2*MK2**2« <-HSP2+G2SH+ 0779 C C2SP2)-8. »S«*2»MK2«MH2»02SP2+16. »S«»2»HK2»(HSP2»02SH+2. •G2SH* 0780 C G2SP1+2. »C2SPl»G2SP2>-8.*S»»2»MH2»C2SP1»C2SP2+16. »S»»2*HSP2»G2SH 0781 C *02SPl+64. •S»MK2**2*(-01SH*G2SPl-02SH*GlSPl+2. *0ISP1*02SP1-G1SP1 0782 C •C2SP2-G1SP2»G2SP1>+16. *S*MK2*MH2*<2.•G1SP1«G2SP1+018P1*02SP2+ 0783 C G1SP2»C2SP1)+32. #S*MK2*<-HSP2*GlSH*02SPl-HSP2*G2SH*01SPl+4. #H8P2 0784 C »01SPl»02SPl-4. »C2SH*01SPl»Q2SPl-4.»01SPl»C2SPl»Q2SP2>+32. »S*MH2 0785 C *01SPl*G2SPl«G2SP2-64. •S»HSP2»G2SH»G1SP1»02SP1+256. •MK2*01SP1« 0786 C 02SPl«(01SH«G2SPl+G2SH»GlSPl+01SPl»G2SP2+GlSP2»02SPl)-t64. »MH2* 0787 C G18P1*G28P1)*(01SP1#02SP2+G1SP2»G25P1) + 12B. •HSP2*01SP1«G2SP1«( 0788 C C1BH»C2SP1+02SH»01SP1))/S»«2 0789 (167 --4. •S»»2*P1SP2»WK2+S««2»P1SPZ*MH2 0790 C -4. •S**2«MK2##2-B»*2»MK2#MH2-( 0791 C 4. •S**2«flK2)*(HSP2+HSPl)-2. *S»«2«HSP2»HSPl+8. •S»P1SP2»MK2*<G1SH- 0792 C 02SH+G1SP1-02SP1)+2. •S»P1SP2»MH2*<-Q1SP1+02SP1>+4. •S»P1SP2*C2SH» 0793 C <-ClSH+G2SH>+8. •S«MK2««2«<ClSP2-02SP2>+2. *S»HK2*HH2«<-01SP2+ 0794 C 02SP2>+4. »S»MK2*(HSPl*01SH-HSPl*02SH+2. »H6Pl»QlSPl-2. •HSP1*02SP1 0795 C +01SH»02SH+2. •018H»02SP2-G2SH**2+2. »02SH*ClSPl-2. •02SH*02SPl-2. * 0796 C 028H«G2SP2+2.#01SPl»02SP2+2.«01SP2»02SPl-4. •02SPl*02SP2>+2. *S« 0797 C MH2«<-01SPl*02SP2-01SP2«02SPl+2. «02SP1*02SP2> 079B M67-M67+4. *S»(HSP2«HSP1• 142 0799 C 01SP1-HSP2»HSP1*02SP1+HSP2«C2SH*C1SP1-HSP2»Q2SH«C2SP1+HSP1*C1SH* 0800 C C2SP2-HSPl*02SH«G2SP2>+8.*P1SP2«<Q16H*»2»G2SP1+G1SH»C2SH»G1SP1- 0801 C 01SH*G2SH»C2SP1-02SH»#2»01SP1>+B. »MK2*(-C1SH«*2»02SP1-C1SH«G2SH* 0802 C 01BPl+QlSH#02SH»G2SPl-2. *01SH«01SPl«G2SPl-2. •01SH*01SPl*02SP2-2. 0803 C «01SH»01SP2*C2SPl+2. »GlSH»02SPl*»2+Q2SH«#2«ClSPl-2. «02SH*01SP1*« 0804 C 2+2. »02SH»ClSPl*G2SPl+2.*G2SH»ClSPl»G2SP2+2. •02SH»ClSP2»C2SPl-2. 0805 C *01SPl#»2*02SP2-6. «01SPl*01SP2*G2SPl+6. *GlSPl»Q2SPl*G28P2+2. • 0806 C 01SP2»C2SPl*«2)+4.*MH2*(01SPl«»2«02SP2+3. *01SPl*01SP2«G2SPl-3. • 0807 C 01SP1*G2SP1»C2SP2-01SP2*C2SP1**2)+B. «<-HSP2*01SH*GlSPl«C2SPl+ 0808 C HSP2»01SH*G2SP1»»2-HSP2»G2SH»G1SP1##2+HSP2*G2SH#Q1SP1»G2SP1-HSP1 0809 C *G1SH*01SP1*C2SP2-HSP1*G1SH«G1SP2»02SP1+HSP1*02SH*01SP1*G2SP2+ 0810 C HSP1*G2SH*G1SP2*G2SP1) 0811 M67-M67/S 0812 M68 -(-4. «S»»2*P1SP2»HK2 0813 C +S«»2»PlSP2»MH2-4. •S»«2»MK2««2-S**2*MK2*MH2-< 0814 C 4. •S»#2«MK2>»<HSP2+HSP1) 0815 C -2. «S»«2»HSP2»HSPl+8.»S*P1SP2»MK2»CG1SP1- 0816 C G2SP1)+2. *S»P1SP2«MH2»<-G1SP1+C2SP1>+8. »S«MK2**2»<G1SH-C2SH+ 0817 C 01SP2-Q2SP2>+2. •S»MK2»MH2*<-QlSP2+Q2SP2>+4. «S*MK2«<HSP2*G1SH- 0818 C HSP2»G2SH+2. *HSPl*01SPl-2.*HSPl»G2SPl+2. *01SH»C2SPl+2. «C2SH* 0819 C 01SP1-4. •02SH»C2SPl+2. »G1SPl*G2SP2+2. *01SP2»02SP1-4. *G2SP1»C2SP2 0820 C )+2. «S»MH2*<-QlSPl«02SP2-01SP2»G2SPl+2. »Q2SPl*C2SP2>+4. #S*HSP2«< 0821 C HSP1*C1SP1-HSP1»G2SP1+G1SH*G2SP1+G2SH*G1SP1-2. #G2SH*G2SP1 > +16. « 0822 C MK2*<-3. *01SH*G1SP1»02SP1+C1SH»G2SP1**2-G2SH*G1SP1*«2+3. »G2SH« 0823 C GlSPl»G2SPl-01SPl**2*G2SP2-3.•ClSPl*01SP2»C2SPl+3. »G1SP1*G2SP1» 0824 C G2SP2+QlSP2*02SPl*«2>+4. »l1H2*CClSPl»«2*C2SP2+3. #G1SP1*C1SP2* 0825 C G2SP1-3. •GlSPl«02SPl*G2SP2-GlSP2*G2SPl**2)+8. *HSP2«<-3. »G1SH» 0826 C 01SPl*C2SPl+GlSH»G2SPl**2-C2SH*GlSPl**2+3. •02SH»G1SP1*G2SP1)>/S 0827 M77 »8. *S»P1SP2»MK2 0828 C -2. *S»PlSP2«MH2+8. •S*MK2**2+2.»S«MK2*MH2+8. *S»MK2«( 0829 C HSP2+HSP1)+4.*S»HSP2«HSP1+16.*MK2»(C1SH*01SP2-G1SH»C2SP2-G2SH» 0830 C O1SP2+G2SH»G2SP2+G1SP1*01SP2-G1SP1«G2SP2-C1SP2*G2SP1+G2SP1»C2SP2 0831 C )+4. •MH2»(-G15Pl»GlSP2+GlSPl»C2SP2+GlSP2»G2SPl-C2SPl»G2SP2>+8. • 0832 C HSP1*(01SH*01SP2-G1SH*02SP2-G2SH*G18P2+G2SH*02SP2> 0833 M7B -8. *S*P1SP2*MK2 0834 C -2. *S*P1SP2«MH2+B.*S«MK2*»2+2.»S«MK2«MH2+B. «S*MK2«< 0835 C HSP2+HSP1)+4. *S*HSP2»HSPl+4. »P1SP2«<-GlSH»«2+2. »C1SH»G2SH-C2SH*« 0836 C 2)+4. •MK2«(01SH*«2-2. *GlSH*G2SH+2. *G1SH*G1SP1+2. »GlSH*GlSP2-2. • 0837 C 01SH*02SPl-2. »GlSH*G2SP2+G2SH**2-2. «G2SH*GlSPl-2. *G2SH«GlSP2+2. « 0838 C 02SH«G2SPl+2. «02SH*02SP2+4.*GlSPl«GlSP2-4. «01SPl*G2SP2-4. •C1SP2* 0839 C G2SP1+4. *G2SP1«G2SP2)+4. «MH2»<-01SP1*G1SP2+G1SP1*G2SP2+G1SP2* 0840 C G2SPl-G2SPl»02SP2>+4.•<HSP2»G1SH»G1SP1-HSP2»C1SH*G2SP1-HSP2»C2SH 0841 C *C1SP1+HSP2*C2SH*C2SP1+HSP1»G1SH«G1SP2-HSP1*G1SH«G2SP2-HSP1#G2SH 0842 C *G1SP2+HSP1*02SH*G2SP2) 0843 MSB -8. *S*P1SP2*I1K2 0B44 C -2. »S»P1SP2*MH2+B. #S»MK2»»2+2. *S*MK2*MH2+B. #B»MK2»< 0845 C HSP2+HSP1)+4. «S*HSP2»HSP1 + 16.»MK2«(G1SH»G1SP1-G1SH*02SP1-G2SH* 0B46 C 019P1+G2SH*02SP1+G1SP1»C1SP2-G1SP1«G2SP2-G1SP2»G2SP1+02SP1»G2SP2 0847 C )+4. *MH2*(-018P1«G18P2+G1SP1*G2SP2+C1SP2*02SP1-G2SP1»02SP2)+B. • 0848 C HSP2*(01SH»01SP1-C1SH*02SP1-G2SH«G1SP1+02SH»02SP1> 0849 C 0850 C DO THE DIVISION BY THE PROPAGATORS 0851 C 0852 MUA - MU/(D1#D3*D1*D3) 0853 M12A - M12/<D1«D3*D3*D4> 0854 M13A • M13/(D1»D3«D2*D4) 0855 M14A - M14/<D1#D3»D2«D5) 143 0656 M15A - M13/<D1*D3*D5*D6> 0837 M16A - M16/(D1*D3*D1*D6) 0838 M17A - M17/<Dl*D3*D2*B> 0839 M1SA - M1B/<D1*D3*D1*6> 0B60 M22A - M22/<D3*D4*D3*D4) 0861 M23A - M23/<D3*D4*D2*D4> 0862 M24A - M24/<D3*D4*D2*D5) 0863 M25A - M25/<D3*D4*D5*D6> 0864 M26A - M26/<D3#D4*D1*D6> 0865 M27A - M27/<D3*D4*D2*S> 0866 M28A - M28/<D3*D4*D1*8> 0867 M33A - M33/(D2#D4*D2«D4) 0868 M34A - M34/<D2*D4*D2*D5> 0869 M33A « M35/<D2*D4*D3*D6) 0870 M36A - M36/<D2*D4*D1*D6> 0871 M37A « M37/(D2*D4*D2*S> 0872 M38A = M38/(D2»D4*D1*8> 0873 M44A • 1144/(D2*D5*D2*D5> 0874 M45A - M45/(D2*D5*D5*D6) 0875 M46A - M46/(D2*D5*D1»D6> 0876 H47A - M47/<D2»D5*D2*S) 0877 M48A - M48/(D2*D5*D1*8) 0878 M55A - M55/(D5*D6*D5*D6> 0879 MS6A - M56/(D5*D6«D1*D6) 0880 (157A - M57/(05*D6*D2*S> 0881 M58A - M58/(D5*D6*D1*S) 0882 M66A - M66/(D1*D6*D1*D6> 0883 M67A - M67/<D1*D6»D2*S> 0884 M68A - M68/<D1*D6*D1*S) 0885 M77A - M77/<D2»S*D2*S> 0886 M7BA - M7B/(D2»S*D1*S) 0887 M8BA - M88/<D1*S*D1*S) 08B8 0889 MAI - M i l A • M22A + M33A + M44A + M5SA + M66A + 0890 C 2. *<M12A + M13A • M23A + M45A • M46A + M36A) 0891 MA2 - 2. »(M14A • M13A • M16A + M24A + M25A • M26A 0892 C + M34A • M35A • M36A) 0893 MA3 • M77A • M88A + 2. *M78A 0894 MA4 - 2. *(M17A + M1BA + M27A + M28A + M37A + M38A) 0895 MAS - 2. *<M47A • M48A + M57A • MS8A • M67A • M6BA) 0B96 HO - 16. *MAl/3. -2. *MA2/3. • 12. *MA3 + 6. *MA4 -6. *MA5 0897 RETURN 089B END 1 44 APPENDIX F - QUARK AND GLUON DISTRIBUTION PARAMETRIZATIONS One of the sources of u n c e r t a i n t y of r a t e p r e d i c t i o n s in pp or pp i n t e r a c t i o n s i s the shape of the parton d i s t r i b u t i o n employed to convolve over parton momenta. The quark d i s t r i b u t i o n can be d i r e c t l y measured from lepton-proton i n t e r a c t i o n s ( F i e l d and Feynman, 1977), because they can i n t e r a c t through the el e c t r o m a g n e t i c f o r c e . The disc r e p a n c y between the p o s s i b l e quark d i s t r i b u t i o n s a r i s e s from experimental u n c e r t a i n t y , and the d i f f e r e n t p a r a m e t r i z a t i o n s used to f i t d ata. Gluons d i s t r i b u t i o n s on the other hand, are only i n d i r e c t l y probed i n hadron-hadron i n t e r a c t i o n s . The u n c e r t a i n t y on them i s much l a r g e r , as i s r e f l e c t e d i n the d i f f e r e n c e s i n c r o s s s e c t i o n s they give r i s e to ( F i g . 22 and 23) . For quarks, we used the f o l l o w i n g d i s t r i b u t i o n s ( P e i e r l s et a l . , 1977) V? au) = z.n ( i - *) 3 ( F - 1 a ) ( F . l b ) and (Barger and P h i l l i p s , 1974) Vx woo = Q57Y0-xf + 0.%ICl-if + Oill ( h ( F ' 2 a ) vrd(^= o.ov 0-xl)3+ 0.20k G-xWaK/-**) 7 (p.2b> 145 For gluons, we used the simple ansatz (Brodsky and F a r r a r , 1973) 7(GM = 3 ( F . 3 ) and the s c a l e - v i o l a t i n g d i s t r i b u t i o n (Baier et a l . , 1980) ^ . -.13/0 + O.36X,1 1.7+/•«/) XGcx) = (2.01-l.Kp + /.2?yo (M) (F.4) with and Q = 5 GeV The c h o i c e s f o r the gluon d i s t r i b u t i o n s are motivated by the f a c t that they represent two extreme e x p e c t a t i o n s on the a c t u a l gluon d i s t r i b u t i o n s . 146 APPENDIX G - HADRON-HADRON COLLIDERS Here i s a t a b l e of the three planned or e x i s t i n g hadron-hadron c o l l i d e r s , and the estimated v a l u e s of t h e i r c m . energy, l u m i n o s i t y and corresp o n d i n g event r a t e s f o r a referenc e c r o s s s e c t i n of 1 Pi c o b a r n . A l s o l i s t e d i s t h e i r s t a r t i n g year of o p e r a t i o n . There are two values of l u m i n o s i t y l i s t e d f o r the SSC, as i t i s not decided as yet i f i t w i l l be a proton-proton or p r o t o n - a n t i p r o t o n c o l l i d e r . C o l l i d e r year c m . energy Luminosity rate/1 Pb. (TeV) cm s _ SPS 1980 0.54 10 1 event/115 day FERMILAB 1986 2 10 1 event/11.5day SSC PP PP 1995? 10 to 40 1 event/1 1.5day 1 event/17 min

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