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Two photon decay widths of non-standard Higgs bosons Bates, Ross Taylor 1986-12-31

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TWO PHOTON DECAY WIDTHS OF NON-STANDARD HIGGS BOSONS by ROSS TAYLOR BATES M.Sc,  The University of B r i t i s h Columbia, 1982  B.Sc,  The University of Western Ontario, 1980  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Physics  We accept t h i s thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA June 1986  © Ross Taylor Bates, 1986  In p r e s e n t i n g requirements  this thesis  British  it  freely available  for  that  Columbia,  I agree that f o r reference  permission  scholarly  f u l f i l m e n t of the  f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y  of  agree  in partial  the Library  shall  and study.  I  f o r extensive  p u r p o s e s may  for  that  copying or publication  f i n a n c i a l gain  shall  Department o f  Physics  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date  of this  Itis thesis  n o t be a l l o w e d w i t h o u t my  permission.  Columbia  June 01, 1986  thesis  by t h e h e a d o f my  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood  further  copying of t h i s  be g r a n t e d  make  written  Abstract  T h i s t h e s i s examines the two photon decay w i d t h s o f n o n - s t a n d a r d Higgs bosons.  These w i d t h s a r e c a l c u l a t e d f o r Two-Higgs-Doublet models i n  g e n e r a l , and f o r t h e m i n i m a l broken supersymmetry model i n p a r t i c u l a r .  For  Two-Higgs-Doublet models a l a r g e enhancement o f t h e s e w i d t h s r e l a t i v e t o t h e s t a n d a r d model i s p o s s i b l e .  This i n t u r n leads to l a r g e r production rates  f o r t h e s p i n - 0 bosons i n ep and e e ~ c o l l i d e r s . +  However, we f i n d t h a t f o r  the m i n i m a l broken supersymmetry c a s e , a s e v e r e upper bound on t h i s enhancement i s imposed by t h e supersymmetry f e a t u r e s .  possible  We f i n d t h a t w h i l e  the H i g g s bosons o f the Two-Higgs-Doublet model c o u l d p o s s i b l y be produced a t r e a d i l y o b s e r v a b l e r a t e s w i t h the HERA c o l l i d e r , t h i s w i l l n o t be t h e case i n t h e m i n i m a l supersymmetry model.  Hence d e t e c t i o n o f t h e s e Higgs  bosons c o u l d p r o v i d e an e x p e r i m e n t a l t e s t of supersymmetry, w h i c h would r u l e out t h e m i n i m a l model.  iii  Table of Contents Abstract L i s t of Tables L i s t of Figures Acknowledgement  i i iv v vi  '.  CHAPTER I - INTRODUCTION 1.1 The Standard Model 1.2 Why Alternative Models? 1.3 Fundamental Scalars 1.4 Supersymmetry 1.5 Thesis Overview  1 4 6 9 11 13  CHAPTER II - TWO-HIGGS-DOUBLET MODEL 2.1 The Model 2.2 Standard Model 2y-Decay Width 2.3 Two-Higgs-Doublet Model 2y-Decay Widths  18 19 28 29  CHAPTER I I I - MINIMAL BROKEN SUPERSYMMETRY MODEL 3.1 The Model 3.2 One Loop Calculation of X° •»• yy 3.3 Pseudoscalar Widths of X° + YY 3.4 Scalar Widths of X° + yy  33 34 53 57 62  CHAPTER IV - NON-STANDARD SPIN-0 BOSON PRODUCTION 4.1 The Calculation 4.2 Numerical Results and Discussion  70 71 77  CHAPTER V - SUMMARY AND CONCLUSIONS  87  Bibliography  92  APPENDIX A - SOME ACCELERATOR PROPERTIES  94  APPENDIX B.l B.2 B.3 B.4  B - EVALUATION OF FEYNMAN DIAGRAMS Scalar Higgs 2y-Decay v i a Fermion Loops Scalar Higgs 2y-Decay v i a Scalar Loops Scalar Higgs 2y-Decay v i a Gauge Boson Loops Pseudoscalar Higgs 2y-Decay v i a Fermion Loops  APPENDIX C - EVALUATION OF LOOP INTEGRALS  95 95 97 98 101 '.  103  APPENDIX D - PROPERTIES OF THE FUNCTION 1(A)  106  APPENDIX E - FEYNMAN RULES FOR MINIMAL BROKEN SUPERSYMMETRY  108  APPENDIX F - EQUIVALENT PHOTON APPROXIMATION  113  APPENDIX G - MONTE CARLO INTEGRATION ROUTINE  116  APPENDIX H - GLOSSARY  123  iv  List of Tables  I II III  Two-Higgs-Doublet Model Vertices Supersymmetric Field Content Accelerator Properties  27 35 94  L i s t of Figures  1  One Loop Contributions to the 2y-Decay of the Scalar  52  2  One Loop Contributions to the 2y-Decay of the Pseudoscalar  56  3  Pseudoscalar 2y-Decay Width f o r Case A  60  4  Pseudoscalar 2y-Decay Width f o r Case B  61  5  Scalar 2y-Decay Width f o r Case A  65  6  Scalar 2y-Decay Width f o r Case B  66  7  Scalar 2y-Decay Width f o r Best Case A  68  8  Feynman Diagrams f o r the Reaction eq •»• eqH°  72  9  Standard Model a vs /s f o r ep -»• eH°X  78  10  Cross Sections ff vs ^  79  11  Cross Sections a vs M„ f o r ep •*• eH°X f o r /s=l TeV  f o r ep + eH°X f o r /s=320 GeV  80  £1  12  Production Cross Sections f o r e e ~ •*• e e~H°  82  13  Rapidity D i s t r i b u t i o n s  83  14  Fermion Loop Contribution to Scalar 2y-Decay  96  15  Scalar Loop Contribution to Scalar 2y-Decay  97  16  Gauge Boson Loop Contribution to Scalar 2y-Decay  99  17  Fermion Loop Contribution to Pseudoscalar 2y-Decay  101  18  Plot of the Function XI(X) vs X  107  19  Feynman Rules f o r Scalar H° Couplings  109  20  Feynman Rules f o r Photon Couplings  110  21  Feynman Rules f o r Chargino-Scalar H° Couplings  22  Feynman Rules f o r Chargino-Pseudoscalar H° Couplings  112  23  Photon Quark Subprocess  114  +  +  Ill  Acknowledgement  I wish to thank my research supervisor, Dr. John Ng, whose guidance made this work possible.  I would also l i k e to thank Dr. Pat Kalynlak,  who collaborated with us on the supersymmetry c a l c u l a t i o n s , f o r her contribution.  F i n a l l y I would l i k e to thank my departmental supervisor,  Dr. Nathan Weiss. F i n a n c i a l assistance from the Natural Sciences and Engineering Research Council and from the University of B r i t i s h Columbia i s g r a t e f u l l y acknowledged.  1  I.  INTRODUCTION  At present the basic b u i l d i n g blocks of matter are thought to be the twelve spin-1/2 fermions known as quarks and leptons.  Four fundamental  forces are responsible f o r the interactions which describe t h e i r behaviour. These forces are the f a m i l i a r gravity and electromagnetism; the strong force which binds n u c l e i together; nuclear decays.  and the weak force responsible f o r c e r t a i n  The s i x leptons do not interact v i a the strong force.  They  consist of the electron, muon and tau p a r t i c l e s which carry e l e c t r i c charge, along with three neutrinos which do not.  The s i x quarks also carry e l e c t r i c  charge, and i n addition they have a "colour" charge through which the strong force acts. The  four fundamental forces can each be described by an underlying  invariance or symmetry of nature. quantity.  Such a symmetry implies a conserved  Many of our physical laws are based on t h i s p r i n c i p l e .  Theoretical models known as gauge theories, which are based on underlying symmetry groups, have been very successful i n describing three of the basic interactions.  The exception  adequately described  i s gravity, which has not as yet been  by such a gauge theory.  are very small and may be neglected  However the e f f e c t s of gravity  at the scale where elementary p a r t i c l e  physics i s currently studied. A general feature of these gauge theories i s that the forces between the basic fermions are mediated by the exchange of a new p a r t i c l e c a l l e d a gauge boson. underlying  These gauge bosons must be massless i n order to preserve the  symmetry of the gauge theory.  electromagnetic familiar.  The gauge theory for the  force, known as quantum electrodynamics (QED), i s the most  Here the gauge boson Is the massless photon, which i s exchanged  2  between e l e c t r i c a l l y charged p a r t i c l e s .  In the gauge theory of the strong  force, known as quantum chromodynamics (QCD), interactions are also mediated by exchange p a r t i c l e s .  In t h i s case massless gluons are exchanged between  colour charged quarks.  Early attempts to extend this highly successful  approach to the weak force postulated that i t must be mediated by the exchange of what are now known as W bosons.  However, the existence of a  massless W boson was not consistent with experiment.  The observed weakness  and very short range of the weak force could only be explained i f the W boson was very massive.  Hence these f i r s t attempts to describe the weak  force by a gauge theory were unsuccessful. The observation that the W boson must also carry e l e c t r i c charge suggested to some that perhaps the weak and electromagnetic and the same.  forces were one  The d i s p a r i t y i n t h e i r observed strengths and ranges could be  explained by the d i f f e r e n t masses of the photon and W boson exchange particles.  The f i r s t models to attempt to unify these two forces also  predicted the existence of another massive exchange p a r t i c l e , c a l l e d the Z boson, which c a r r i e s no e l e c t r i c charge. were not at that time observed.  However, such neutral currents  The need for massive exchange bosons i n  these models destroyed the underlying symmetries that one o r i g i n a l l y wished to incorporate.  This l e d to divergent results when higher order  calculations were done.  Such problems frustrated these subsequent attempts  at describing the weak force. The s o l u t i o n to these early t h e o r e t i c a l problems was the phenomenon of spontaneous symmetry breaking.  This refers to the fact that although a  theory may contain a given symmetry, the vacuum or ground state of the system described by the theory need not respect that symmetry. example i s that of a ferromagnet.  A simple  In general i t s spins are randomly aligned  3  and the theory possesses a r o t a t i o n a l symmetry.  In the ground state however  the random spins a l l a l i g n In one chosen d i r e c t i o n , and hence the r o t a t i o n a l symmetry of the theory i s "spontaneously broken" by the ground state. It can be shown that whenever such a symmetry i s spontaneously broken, a massless p a r t i c l e known as a Goldstone boson must r e s u l t .  For the  ferromagnet the Goldstone boson corresponds to long range spin waves. In the present gauge theory of the weak interaction, a new fundamental scalar c a l l e d a Higgs p a r t i c l e i s introduced.  The ground state of this new  matter f i e l d i s such that the o r i g i n a l symmetry of the theory i s spontaneously broken.  In this case the massless Goldstone boson appears not  as a physical p a r t i c l e , but instead as the longitudinal component of the massless gauge boson.  In this way masses can be generated for the gauge  bosons without destroying the underlying symmetry of the o r i g i n a l theory. This technique i s known as the Higgs mechanism and i t solves the t h e o r e t i c a l problems of the weak model, giving f i n i t e or renormalizable results f o r higher order c a l c u l a t i o n s . At  this point there existed a w e l l behaved gauge theory which u n i f i e d  the weak and electromagnetic forces.  The model predicted masses f o r the W  and Z exchange gauge bosons, implied the existence of neutral currents, and also a new fundamental Higgs scalar.  Both neutral currents and the gauge  bosons themselves [1,2] have subsequently been observed, i n very good agreement with prediction. The phenomenological  Only the Higgs scalar remains to be discovered.  successes of t h i s electroweak theory have been such  that, together with the theory of the strong force (QCD), i t i s now accepted as the standard model of elementary p a r t i c l e physics. Thus f a r , a l l the experimental tests of the standard model have proven successful, and i t i s now thought to be correct for energies up to at least the order of 100 GeV.  4  1.1 The Standard Model The subsequent chapters of t h i s thesis a l l begin with the i m p l i c i t assumption  that the reader i s f a m i l i a r with the standard model.  As the  currently accepted theory, i t i s the basis against which any new physics must necessarily be compared.  A detailed description of the standard model  can be found i n most modern textbooks on p a r t i c l e physics.  Consequently  only a b r i e f summary of the main features w i l l be presented here.  As the  strong i n t e r a c t i o n has no d i r e c t bearing on our r e s u l t s , only the electroweak aspects of the model are described. In the standard model the fundamental p a r t i c l e s of matter consist of twelve spin-1/2 fermions. e l e c t r i c charge -e. particles.  There are three massive leptons which carry  They are known as the electron, muon and tau (e~,u~,T~)  Associated with each one of these charged leptons i s a massive,  e l e c t r i c a l l y neutral lepton (  v  » y» V  e  v T  ) c a l l e d a neutrino.  fermions are massive p a r t i c l e s c a l l e d quarks.  The remaining s i x  Three of them carry e l e c t r i c  charge (2/3)e, and are known as the up, charm and top (u,c,t) quarks.  The  other three carry e l e c t r i c charge (-l/3)e and are c a l l e d the down, strange and bottom (d,s,b) quarks.  The standard model c l a s s i f i e s these twelve  fermions Into three generations or f a m i l i e s .  The structure of each family  i s very s i m i l a r , and consists of one charged lepton, one neutrino, one charge 2/3 quark and one charge -1/3  quark.  Since these quarks and leptons are spin-1/2 fermions, they may be i n either of two h e l i c i t y states, namely left-handed or right-handed.  Hence we  can decompose their wave functions into l e f t and right components.  There  exists a symmetry i n the standard model which involves only the left-handed components of the fermions, and i t i s convenient to group them into pairs or doublets as shown below.  Thus the three families are written as  5  where the subscript L denotes the left-handed component.  The corresponding  right-handed components are treated separately as i n d i v i d u a l s i n g l e t s . Experimentally only left-handed neutrinos are observed.  Hence the  right-handed neutrino singlets are not included i n the standard model. Having introduced the fundamental fermions, we turn now exchange bosons which mediate t h e i r electroweak i n t e r a c t i o n s . four of these gauge bosons i n the standard model. massive.  to the spin-1 There are  Three of them are quite  They consist of the e l e c t r i c a l l y neutral Z boson, as well as a  pair of W bosons which carry opposite charges of ±e. boson i s the f a m i l i a r massless photon. boson described below, t h i s completes  The fourth exchange  With the exception of the Higgs the description of the p a r t i c l e  content i n the standard electroweak model. The gauge theory of the standard model i s based upon what i s t e c h n i c a l l y known as the SU(2)xU(l) symmetry group.  A requirement of any  gauge theory i s that Its fermions and exchange bosons must be massless i n order to preserve the underlying symmetry.  Thus i n order to generate masses  for the p a r t i c l e s described above, the basic gauge theory must be supplemented by the introduction of one or more fundamental spin-0 s c a l a r s . As described e a r l i e r , these so c a l l e d Higgs f i e l d s are responsible f o r the spontaneous breaking of the underlying symmetry. standard model this i s done i n the simplest way, doublet of Higgs s c a l a r s .  In the case of the through the addition of one  The Higgs f i e l d i s said to acquire a non-zero  vacuum expectation value (VEV), meaning that Its ground state does not respect the same symmetry as the theory describing i t , and hence the  6  SU(2)xU(l) symmetry of the standard model i s spontaneously broken.  One  neutral Higgs boson and three Goldstone bosons result from t h i s breaking. The l a t t e r are then absorbed v i a the previously described Higgs mechanism as the longitudinal components of the W and Z bosons. generated f o r the exchange bosons.  In this way masses are  Through their so c a l l e d Yukawa  interactions with the Higgs f i e l d s , the basic fermions can also acquire a mass.  Unfortunately the model makes no prediction f o r the mass of the  physical Higgs boson i t s e l f , and i t has not been found by experiment.  To  date however, the addition of these fundamental scalars has been the only successful method of generating masses within gauge theories.  Hence the  Higgs sector i s a necessary and very important part of the standard model, and indeed of any gauge theory description of p a r t i c l e physics. This concludes our look at the main features of the standard model. Most of the technical d e t a i l s have been suppressed f o r s i m p l i c i t y , and the interested reader i s referred to the l i t e r a t u r e .  We have introduced the  p a r t i c l e content of the model, and stressed the importance of the Higgs sector.  These should be s u f f i c i e n t background f o r a general understanding  of the standard model aspects of the thesis.  Other features and s p e c i f i c  d e t a i l s of the standard model are discussed as they a r i s e .  1 . 2 Why Alternative Models? Despite a l l of i t s successes, there are s t i l l some untested and l i t t l e understood aspects of the standard model; most notable i s the a l l important Higgs sector needed f o r spontaneous symmetry breaking.  The model does not  predict a mass f o r the fundamental Higgs scalar, which has yet to be detected.  Also as w i l l be discussed, there are many technical reasons f o r  expecting new physics beyond the standard model.  7  There are several arguments [ 3 ] which suggest the need f o r improving upon the standard model.  Numerous d e t a i l s about the structure of the model,  and the values of i t s roughly 2 0 free parameters a l l need to be  explained.  Also the standard model i s not asymptotically free, meaning that i t w i l l become strongly i n t e r a c t i n g at some larger energy scale, where perturbation methods w i l l break down.  Why  this breakdown occurs i s discussed i n more  d e t a i l below. In p a r t i c l e physics, calculations are performed primarily using perturbation techniques.  An unknown quantity i s expanded i n a power series  of some small parameter.  In t h i s way  each successive term i n the expansion  serves as a small correction to the previous term.  The c o e f f i c i e n t s of the  expansion are then evaluated term by term to the desired accuracy approximation.  of the  However, i t i s a general property of gauge theories that the  c o e f f i c i e n t s of the higher order terms i n the perturbative expansion can often contain undesirable i n f i n i t i e s .  Fortunately, for what are known as  renormalizable gauge theories, these troublesome i n f i n i t i e s can be eliminated simply by a r e d e f i n i t i o n of parameters.  The sources of these  i n f i n i t i e s are quantities which diverge for large energy. In general these divergences into the new  are cut off at some scale A, and the i n f i n i t e piece absorbed parameter d e f i n i t i o n s .  again convergent.  The perturbation series i s then once  This so c a l l e d renormalization scheme w i l l of course only  make sense i f the cutoff scale A i s larger than the energy scale of the process we are interested i n .  If this i s not the case, then the  infinities  cannot be eliminated and perturbation methods w i l l break down. The scale A used i n the renormalization scheme above i s not an a r b i t r a r y parameter.  There must be some physical quantity i n the  which fixes the scale.  theory  In general this i s taken to be the mass of the  8  h e a v i e s t p a r t i c l e i n the theory, so t h a t p e r t u r b a t i o n techniques valid  over as l a r g e an energy range as p o s s i b l e .  will  Thus f o r e n e r g i e s  than t h i s mass, p e r t u r b a t i o n methods break down and  be  greater  the gauge theory  can  make no q u a n t i t a t i v e p r e d i c t i o n s . I n the s t a n d a r d model the h e a v i e s t p o s s i b l e p a r t i c l e i s the Higgs boson.  I t s mass  v a r i e s as  = 2Xv .  The  2  parameter X i s a measure of  the s t r e n g t h of the Higgs s e l f - I n t e r a c t i o n c o u p l i n g . vacuum e x p e c t a t i o n v a l u e  (VEV)  as d i s c u s s e d above.  c a l c u l a t i o n s X Is used as an expansion parameter.  p r e d i c t i o n s of the s t a n d a r d  for  In c e r t a i n p e r t u r b a t i o n Therefore  i t must  model which a r e o b t a i n e d  the Higgs mass i s s i m p l y  The  VEV  perturbatively.  parameter v i n the  experimentally  Thus  the s c a l e at which the e l e c t r o w e a k symmetry I s  and weak f o r c e s become e q u a l . to be v = 246  GeV.  the  This i s established  Combining these  r e s u l t s f o r X and v,  f i n d s t h a t the mass of the Higgs boson i s expected to be l e s s than the of 1 TeV.  and  expression  In o t h e r words i t i s the energy a t which the s t r e n g t h s of  electromagnetic  be  T h i s c o n t r a d i c t s the many s u c c e s s f u l  can e s t a b l i s h an upper bound on X.  broken.  from  the Higgs s e c t o r would become s t r o n g l y i n t e r a c t i n g  p e r t u r b a t i o n theory would f a i l .  we  q u a n t i t y v i s the  t h a t the Higgs f i e l d a c q u i r e s  spontaneous symmetry b r e a k i n g ,  s m a l l s i n c e otherwise  The  one order  T h i s then s e t s the s c a l e at which the r e n o r m a l l z a t i o n scheme, and  hence p e r t u r b a t i o n methods, w i l l break down i n the standard  model.  P r e s e n t l y the h i g h e s t a t t a i n a b l e e n e r g i e s which have been t e s t e d a r e the o r d e r of 100  GeV,  and  the r e s u l t s have been c o n s i s t e n t w i t h  p r e d i c t i o n s of the standard accelerators range.  model.  Soon a new  generation  of  the breakdown i n the s t a n d a r d  model, and  should  begin  t h i s i s one  the  particle  (see appendix A) w i l l be o p e r a t i n g at e n e r g i e s up  T h i s i s e x a c t l y the r e g i o n where we  to the  TeV  to see evidence  reason  on  f o r expecting  of  9  new physics to be observed i n these machines.  Hence there i s now  an  immediate need to develop alternative models, and establish a theoretical framework with which to describe t h i s expected new physics. Despite i t s possible breakdown at higher energies, the standard model has had great phenomenological  success to date.  This then suggests that i t  i s v a l i d only as an e f f e c t i v e low energy description of some more fundamental theory.  This new  theory should be based on some larger  symmetry, which when broken at low energy, results i n the standard model. Evidence for such a theory would manifest i t s e l f at a higher energy scale i n the form of new physics.  At presently available energies however, the  experimental data i s consistent with the standard model.  Hence any  attempt  to formulate a new underlying theory must at this point be guided by purely t h e o r e t i c a l motivations. begin by examining  These are discussed i n the next two sections.  We  some of the technical problems which occur i n the c r u c i a l  Higgs sector.  1.3 Fundamental Scalars The existence of a fundamental scalar Higgs p a r t i c l e i s an essential component of the standard model, and indeed of any gauge theory with massive exchange bosons.  It i s the Higgs p a r t i c l e which induces the spontaneous  symmetry breaking needed to generate the gauge boson mass. motivation f o r fundamental scalars i s very strong.  Thus the  Nevertheless there are  s t i l l many technical d i f f i c u l t i e s associated with these scalars. The above 1 TeV bound on the standard model Higgs mass leads to our f i r s t problem with scalars; namely understanding why light.  the scalar f i e l d i s so  More s p e c i f i c a l l y , the question i s why i s the electroweak breaking  scale v so small?  One would wish i n developing a new fundamental theory  that we could also incorporate the u n i f i c a t i o n of the strong and interactions, and possibly even gravity. force (grand u n i f i c a t i o n 1 0  12  electroweak  The scales at which the strong  TeV) or gravity (Planck mass 1 0  16  TeV) become  comparable to the electroweak force are very large compared to the Higgs mass (<1  TeV).  Understanding  how to relate these very large energy scales  to the much smaller scale at which the electromagnetic and weak forces are u n i f i e d i s known as the so c a l l e d naturalness problem.  The "natural" value  for the scalar boson mass should be the same as the mass scale f o r the fundamental theory. understood  The d i s p a r i t y of these scales i n the theory could be  i f there were some mechanism, such as an approximate symmetry,  which ensured that the scalar mass parameters are very small.  However, what  such a mechanism could be i s very d i f f i c u l t to determine for these fundamental scalar p a r t i c l e s . The solution to the naturalness problem proposed above leads immediately  to a related d i f f i c u l t y known as gauge hierarchy.  Although  o r i g i n a l l y synonymous with "naturalness", the term "gauge hierarchy" i s now used i n the l i t e r a t u r e to refer to the following s p e c i f i c aspect of the problem.  Even i f a mechanism could be found which ensures a small scalar  mass at lowest order of perturbation theory, the higher order corrections to the scalar mass can be very large.  Thus the naturalness problem reappears  via these corrections. The mass parameters must be chosen at each order i n perturbation theory with i n c r e d i b l e accuracy to avoid t h i s . tuning i s not a very s a t i s f a c t o r y way  Such f i n e  to solve the naturalness problem.  These are two of the p r i n c i p a l d i f f i c u l t i e s associated with fundamental scalars which w i l l a r i s e i n trying to construct alternative theories to the standard model.  One approach has been to avoid these problems by  eliminating fundamental scalars altogether.  Composite models and  11  Technicolour theories [4] try to treat the scalars as extended objects constructed from more basic fermions.  However, these attempts have not been  very successful thus f a r . The other approach i s to r e t a i n the fundamental scalars which work so well i n the standard model, and try to solve the problems discussed above.  Indeed t h i s w i l l prove to be possible by  employing a higher symmetry which eliminates the naturalness and gauge hierarchy problems.  This i s the elegant supersymmetry approach.  Of a l l the  possible alternatives to the standard model, supersymmetry i s now  the  leading candidate.  1.4  Supersymmetry What i s d i f f e r e n t about supersymmetry models i s that they incorporate  both fermions and bosons Into the gauge theory on an equal basis.  Each  boson (fermion) has a superpartner fermion (boson) of equal mass, and they d i f f e r only by their spin quantum number.  Gauge theories which are  supersymmetric [3,5] must remain invariant under transformations between these superpartners.  In order to be phenomenologically  acceptable,  supersymmetric gauge models must contain the usual standard model quarks, leptons and gauge bosons.  The superpartners to these p a r t i c l e s are known  respectively as squarks, sleptons and gauginos.  In addition i t i s necessary  i n these models that at least two Higgs f i e l d s be employed i n order to generate d i f f e r e n t masses f o r up and down type quarks.  Thus the minimal  supersymmetric extension of the standard model i s a two Higgs doublet model. The d e t a i l s of these models w i l l be examined i n l a t e r chapters. point we merely wish to i l l u s t r a t e some of the reasons why  At this  supersymmetry i s  the leading a l t e r n a t i v e to the standard model. As a candidate theory for an alternative to the standard model,  supersymmetry has many d e s i r a b l e f e a t u r e s . s c a l a r s are no l o n g e r supplemental  To b e g i n w i t h the fundamental  t o , but r a t h e r a r e now  a n a t u r a l p a r t of  the gauge t h e o r y , on the same f o o t i n g as the b a s i c f e r m i o n s .  Also  certain  d i v e r g e n t q u a n t i t i e s , which a r i s e i n the h i g h e r o r d e r c o r r e c t i o n s to the s c a l a r masses, now  c a n c e l a t each o r d e r i n p e r t u r b a t i o n t h e o r y .  This  c a n c e l l a t i o n occurs because the d i v e r g e n t c o n t r i b u t i o n from each p a r t i c l e i s e x a c t l y c a n c e l l e d by the c o n t r i b u t i o n from i t s s u p e r p a r t n e r .  Hence the  h i g h e r o r d e r c o r r e c t i o n s remain s m a l l and do not cause a l a r g e s c a l a r mass. T h i s e l i m i n a t e s t e c h n i c a l problems such as gauge h i e r a r c h y . supersymmetry models are much b e t t e r behaved.  Furthermore the n a t u r a l n e s s  problem can e a s i l y be s o l v e d by supersymmetry. difficult  Thus  In g e n e r a l i t has  been  to i d e n t i f y any mechanism which c o u l d e n f o r c e a s m a l l s c a l a r mass.  However i t i s w e l l known t h a t imposing z e r o mass f o r f e r m i o n s . to a massless  an exact c h i r a l symmetry e n f o r c e s a  In supersymmetry a massless  scalar partner.  fermion leads n a t u r a l l y  Hence the n a t u r a l n e s s problem can be s o l v e d  f o r supersymmetric models w i t h approximate c h i r a l  symmetries.  Perhaps the most i n t r i g u i n g a s p e c t of these supersymmetry models i s t h a t l o c a l supersymmetry t r a n s f o r m a t i o n s a r e r e l a t e d to transformations. supersymmetry, and of n a t u r e ' s  space-time  There then e x i s t s the p o t e n t i a l to couple g r a v i t y w i t h t h i s would a l l o w f o r the p o s s i b l e u n i f i c a t i o n of a l l f o u r  fundamental f o r c e s i n t o one  theory.  ( F o r a review of  such  s u p e r g r a v i t y models, see r e f e r e n c e [ 3 ] ) .  This e x c i t i n g p o s s i b i l i t y , along  w i t h the s o l u t i o n s of the n a t u r a l n e s s and  gauge h i e r a r c h y problems, are  main reasons At  f o r examining supersymmetry i n more d e t a i l .  t h i s p o i n t i n time, i t must be noted  supersymmetry are p u r e l y t h e o r e t i c a l . seen t o d a t e .  the  No  t h a t these m o t i v a t i o n s f o r  experimental  evidence has  Indeed, i f i t e x i s t s then the supersymmetry must be  been "softly"  broken a t low e n e r g i e s . although  The  d e s c r i p t i o n s o f t l y r e f e r s to the f a c t t h a t  the supersymmetry must be broken,  one wants t o p r e s e r v e  the  c a n c e l l a t i o n of the v a r i o u s d i v e r g e n t q u a n t i t i e s i n the s c a l a r mass corrections.  The  c a n c e l l a t i o n w i l l no l o n g e r be e x a c t , but i f the b r e a k i n g  i s s o f t enough o n l y f i n i t e p a r t s w i l l  remain.  T h i s b r e a k i n g of the  supersymmetry must occur s i n c e no s u p e r p a r t n e r s have been observed particles. Higgs  f o r known  A l s o an exact c h i r a l supersymmetry i m p l i e s a zero mass f o r the  s c a l a r , which i s i n c o n s i s t e n t w i t h the s t a n d a r d model.  Still  the  p o t e n t i a l e x i s t s f o r supersymmetry t o be the fundamental t h e o r y which w i l l reduce  t o the s t a n d a r d model a t low  The  energy.  s c a l e a t which supersymmetry i s broken w i l l  e s t a b l i s h the s i z e of  the s c a l a r mass, and hence should be r e l a t e d to the electroweak s c a l e of v = 246  GeV.  The b r e a k i n g of the supersymmetry would l i f t  degeneracy i n the masses of s u p e r p a r t n e r s , and d i f f e r e n c e s s h o u l d a l s o be on t h i s same s c a l e . known p a r t i c l e s , one would expect b e f o r e the TeV the new  energy  accelerators.  breaking  range.  the  t h e i r r e s u l t i n g mass Thus g i v e n the masses of the  to see evidence of supersymmetry a t o r  T h i s i s e x a c t l y the r e g i o n t o be s t u d i e d In  Thus the supersymmetry h y p o t h e s i s i s one which can  be  t e s t e d i n the v e r y near f u t u r e .  1.5  T h e s i s Overview Regardless  of whether or not supersymmetry i s a c o r r e c t approach,  c l e a r t h a t the fundamental s c a l a r s w i l l  p l a y a key r o l e i n c o n s t r u c t i n g  a l t e r n a t i v e s to the s t a n d a r d model.  wish t o l e a r n more about t h i s  important  Higgs  non-standard form of new  s e c t o r , and  s p i n - 0 Higgs  We  consequently  bosons.  p h y s i c s to expect  this thesis w i l l  focus on  i t is  these  A d d i t i o n a l l y we would l i k e t o know what  i n the new  particle accelerators.  The  dominant modes o f p r o d u c i n g Higgs bosons, and t h e r a t e s a t w h i c h they s h o u l d be observed i n t h e s e machines a r e t h e n examined  f o r c e r t a i n models.  G i v e n a l l o f t h e m o t i v a t i o n p r e v i o u s l y d i s c u s s e d , i t s h o u l d n o t be s u r p r i s i n g t h a t t h e s p e c i f i c model chosen f o r s t u d y i s one o f m i n i m a l broken supersymmetry.  As s t a t e d e a r l i e r , t h i s c h o i c e i s a s p e c i f i c example o f a  model w i t h two H i g g s d o u b l e t s . We would l i k e t o be a b l e t o d i s t i n g u i s h between t h e f e a t u r e s o f t h i s supersymmetry  model w h i c h a r i s e from e i t h e r t h e  new s u p e r p a r t i c l e c o n t e n t , o r because t h e r e a r e a d d i t i o n a l Higgs present.  Hence a more g e n e r a l Two-Higgs-Doublet  w i l l f i r s t be examined. literature.  supersymmetry  Both o f t h e s e models c a n be found i n t h e  What i s new i n t h i s t h e s i s a r e t h e c a l c u l a t i o n s t o w h i c h t h e  models a r e a p p l i e d .  These c a l c u l a t i o n s and t h e v a r i o u s r e s u l t s o b t a i n e d a r e  d i s c u s s e d i n more d e t a i l In  model w i t h no  fields  below.  g e n e r a l t h e Higgs boson i n t e r a c t s w i t h o t h e r p a r t i c l e s w i t h a  s t r e n g t h p r o p o r t i o n a l t o t h e i r mass.  Hence t h e dominant decay mode o f t h e  Higgs boson w i l l be i n t o t h e h e a v i e s t p a r t i c l e s a l l o w e d by energy conservation.  I f I t s mass i s g r e a t e r than 160 GeV/c , t h e Higgs t h e n decays 2  i n t o a p a i r o f gauge bosons.  This i s p o t e n t i a l l y undesirable since there  can be r e l a t i v e l y l a r g e backgrounds a s s o c i a t e d w i t h such gauge boson A l s o , fewer heavy H i g g s bosons c o u l d be o b t a i n e d t o b e g i n w i t h .  pairs.  Smaller  mass H i g g s bosons a r e more l i k e l y t o be produced w i t h t h e l i m i t e d e n e r g i e s a v a i l a b l e a t t h e new c o l l i d e r s (see a p p e n d i x A ) .  F o r t h e s e reasons i t was  d e c i d e d t o s t u d y Higgs bosons i n t h e i n t e r m e d i a t e mass range from 40 t o 160 G e V / c . 2  However, I t i s q u i t e s t r a i g h t f o r w a r d t o e x t e n d t h e a n a l y s i s f o r  l a r g e r Higgs masses. In  t h e i n t e r m e d i a t e mass range t h e p r i m a r y decay mode o f t h e Higgs  boson i s i n t o a q u a r k - a n t i q u a r k p a i r , which s u b s e q u e n t l y w i l l form two  hadronic j e t s .  Such a signal would be lost i n the much larger jet  backgrounds of hadronic c o l l i d e r s , and hence we s h a l l only consider e e ~ and +  ep machines.  The discussion to follow i s s i m i l a r f o r both these types of  machines, and hence f o r the moment we r e s t r i c t ourselves to e e ~ c o l l i d e r s . +  I f production of the Higgs boson HO i s possible at the SLC or LEP c o l l i d e r s , then i t would proceed v i a reactions such as  e  +  e" -»• Z° H°  e  +  e  (1.5.1)  •»• jZ }\- H°  -  (1.5.2)  +  These are the usual processes used i n standard model Higgs searches, and they produce Higgs bosons at an e s s e n t i a l l y unobservable rate. l i t t l e change i s found for the non-standard models studied.  Generally,  The exception  i s f o r a s p e c i f i c case of equation (1.5.2), which i s the reaction  e  + e  +  e  + - 0 e  H  (1.5.3)  Of the many processes which contribute to this reaction, one w i l l be of particular interest.  I t i s known as the two photon fusion mechanism [6].  This mechanism i s e s s e n t i a l l y the same as the process shown i n figure 8b. The Higgs boson i s produced during the exchange of a photon between the c o l l i d i n g p a r t i c l e s .  The detailed reasons as to why this  mechanism i s such an important process w i l l be given i n chapter IV. The point Is that although this two photon fusion mechanism i s not an important one i n the standard model, i t can be enhanced substantially and actually dominate i n models with more than one Higgs doublet.  This idea i s one which  has not been e x p l o r e d i n the l i t e r a t u r e , and i s the c e n t r a l new  f e a t u r e upon  which the t h e s i s i s b u i l t . F o r the c l a s s of models s t u d i e d , the s i z e o f the Higgs boson t o two photon i n t e r a c t i o n can be g r e a t l y i n c r e a s e d over what i t i s i n the s t a n d a r d model.  T h i s l a r g e enhancement i s what causes the two photon  mechanism t o dominate  Higgs boson p r o d u c t i o n .  The a c t u a l  fusion  Higgs-photon  i n t e r a c t i o n i s b e s t s t u d i e d by examining the two photon decay widths o f the Higgs boson.  P r o d u c t i o n c r o s s s e c t i o n s f o r the two photon f u s i o n mechanism  can then be e x p r e s s e d d i r e c t l y i n terms of these w i d t h s . will  initially  examine o n l y the Higgs 2 y - d e c a y w i d t h s .  F o r t h i s r e a s o n we L a t e r these r e s u l t s  a r e used to e s t i m a t e the v a r i o u s p r o d u c t i o n c r o s s s e c t i o n s and r a t e s . p o i n t s are a l l d i s c u s s e d i n more d e t a i l as they a r i s e i n the  thesis.  F i n a l l y the t h e s i s c o n t e n t s a r e b r i e f l y o u t l i n e d below. begins w i t h a d e s c r i p t i o n of the Two-Higgs-Doublet decay widths f o r the non-standard Higgs bosons calculated.  model.  These  Chapter I I  The two  photon  of t h i s model a r e then  I n c h a p t e r I I I the minimal broken supersymmetry model i s  i n t r o d u c e d , and a g a i n the Higgs 2 y - d e c a y widths a r e e v a l u a t e d . the two models, we  Comparing  are a b l e t o d i s t i n g u i s h which f e a t u r e s of the  supersymmetry model a r i s e from the new  s u p e r p a r t i c l e content.  The  c a l c u l a t i o n of the Higgs 2 y - d e c a y widths f o r these models i s a new and i n each case we  d i s c u s s how  l a r g e an enhancement r e l a t i v e t o the  s t a n d a r d model width i s p o s s i b l e . s e c t i o n s f o r the Higgs bosons  Chapter IV examines the p r o d u c t i o n c r o s s  i n e e~ +  and ep c o l l i d e r s .  two photon f u s i o n mechanism a r e d i s c u s s e d , as w e l l as how to the 2 y - d e c a y widths p r e v i o u s l y c a l c u l a t e d . are a l s o d e s c r i b e d .  result,  The d e t a i l s of the one can r e l a t e i t  The n u m e r i c a l procedures used  Based on t h i s a n a l y s i s , we make a p r e d i c t i o n of what  r a t e s t o expect f o r Higgs boson p r o d u c t i o n i n the new  particle  accelerators.  A comparison w i t h the a c t u a l experiment would then determine i f the expected new  p h y s i c s i s c o n s i s t e n t w i t h a supersymmetric  description.  Lastly  the  many r e s u l t s and c o n c l u s i o n s a r e summarized i n the d i s c u s s i o n o f c h a p t e r V.  II.  TWO-HIGGS-DOUBLET MODEL  T h i s model i s a simple e x t e n s i o n o f the standard model, w i t h two d o u b l e t s o f Higgs important  f i e l d s r a t h e r than one.  A knowledge o f t h i s model w i l l be  i n the next chapter f o r comparison  w i t h the minimal  broken  supersymmetry model, which i s a s p e c i f i c example o f a Two-Higgs-Doublet model.  T h i s a l l o w s one t o be a b l e t o d i s t i n g u i s h between the f e a t u r e s o f  the supersymmetric a d d i t i o n a l Higgs  model which a r i s e from e i t h e r the supersymmetry o r the f i e l d content.  Although  they a r e reviewed  extensively i n  the l i t e r a t u r e , the d e t a i l s o f the Two-Higgs-Doublet model w i l l be presented below i n o r d e r t o f a m i l i a r i z e the r e a d e r w i t h the g e n e r a l f e a t u r e s o f the model and t o e s t a b l i s h n o t a t i o n . The  p a r t i c l e content o f the Two-Higgs-Doublet model d i f f e r s from the  s t a n d a r d model o n l y i n the Higgs  sector.  As d i s c u s s e d below, i n s t e a d of  j u s t one n e u t r a l s c a l a r p a r t i c l e t h e r e a r e a p a i r o f charged Higgs, a n e u t r a l p s e u d o s c a l a r and two n e u t r a l s c a l a r s . however, i s t h a t the Higgs  The key r e s u l t  t o note  t o f e r m i o n c o u p l i n g s d i f f e r from those i n the  s t a n d a r d model by f a c t o r s o f tana, which i s the r a t i o of the vacuum e x p e c t a t i o n v a l u e s o f the two Higgs one,  fields.  I f tana i s v e r y d i f f e r e n t  then t h e r e i s the p o s s i b i l i t y o f g r e a t l y enhancing  c o u p l i n g s , w i t h important In the second  fermion  t o 2y decay p r o c e s s .  s e c t i o n of t h i s c h a p t e r i s p r e s e n t e d the c a l c u l a t i o n f o r  the s t a n d a r d model Higgs the l i t e r a t u r e .  consequences f o r the Higgs  these  2y-decay w i d t h .  These r e s u l t s can a l s o be found i n  They serve t o p r o v i d e some background and e s t a b l i s h t h e  p a t t e r n o f t h e s i m i l a r c a l c u l a t i o n f o r the Two-Higgs-Doublet model. first  from  r e s u l t t o note w i l l be t h a t I n the standard model the Higgs  decays i n t o two photons predominantly  v i a gauge boson l o o p s .  The  boson  19  More i m p o r t a n t l y the c o n t r i b u t i o n of the two  photon decay p r o c e s s i s shown  to be a n e g l i g i b l e p a r t of the t o t a l Higgs decay w i d t h . F i n a l l y the c a l c u l a t i o n of the Higgs the Two-Higgs-Doublet model.  Although  2y-decay width i s presented f o r  the c a l c u l a t i o n i s v e r y s i m i l a r to  t h a t f o r the standard model, the r e s u l t s a r e new c h a p t e r IV.  S p e c i f i c a l l y the enhanced Higgs  and w i l l be needed i n  to f e r m i o n c o u p l i n g s In the  model l e a d to a much l a r g e r 2y-decay w i d t h .  Maximum p o s s i b l e v a l u e s f o r the  enhancement f a c t o r tana are taken from the l i t e r a t u r e . process i s c o n s i d e r a b l y more important  than i t was  Thus the 2y-decay  i n the standard model.  The  consequences of t h i s r e s u l t w i l l be d i s c u s s e d f u r t h e r i n c h a p t e r  2.1  The  Model  R e c a l l i n g the standard electroweak model [ 7 ] , we  see t h a t one of the  s i m p l e s t e x t e n s i o n s to i t i s t o i n c l u d e an e x t r a d o u b l e t of Higgs One  IV.  need o n l y c o n s i d e r the changes i n the Higgs  scalars.  sector, since a l l other  f e a t u r e s i n Two-Higgs-Doublet models w i l l remain the same as i n the standard model.  The  two Higgs  s c a l a r s are d e s c r i b e d by complex f i e l d o p e r a t o r s $ :  (2.1.1)  which both have hypercharge charge.  The  charge  +1,  conjugate  where the s u p e r s c r i p t s denote  f i e l d s ^2^a'  m a t r i x , have the o p p o s i t e hypercharge two  w  to $ .  n  e  r  e  T  2  *"  8  t  n  e  u  s  u  electric a  l  Pauli  The vacuum i s c h a r a c t e r i z e d by  vacuum e x p e c t a t i o n v a l u e s (VEV's) f o r these f i e l d  operators  (2.1.2)  It i s convenient to define the rotated f i e l d s *' = *2  =  ^ c o s a + 4> sina ~ i $  s  l  n  c  t  +  *  (2.1.3a) (2.1.3b)  c 0 8 a 2  so that  <*p = 0  (2.1.4)  and tana » b/a  (2.1.5)  Then the f i e l d $^ can be considered as the "true" Higgs doublet, as i n the standard model.  The gauge transformation U ( E ) takes us to the unitary  gauge, where the f i v e physical f i e l d s which are mass eigenstates can be identified.  Assuming the physical f i e l d s have zero VEV's, we can write  (2.1.6a)  • A  —•  U(£)$: = 1  where v =a +b . 2  is  2  2  c  b  \  +  i  i  j  ;  (2.1.6b)  The two scalar f i e l d s are <(> and n, the pseudoscalar f i e l d  and the charged Higgs are x*•  The three degrees of freedom not  accounted f o r by the physical f i e l d s are the usual would-be Goldstone bosons,  which have been absorbed v i a the Higgs mechanism as the  longitudinal components of the gauge bosons W* and Z°. Transitions between l i k e charged quarks of d i f f e r e n t families by so c a l l e d flavour changing neutral currents are suppressed i n the standard  model, i n agreement w i t h o b s e r v a t i o n . The s c a l a r s o f t h e Two-Higgs-Doublet model w i l l i n g e n e r a l a l l o w such c u r r e n t s , and t h e s e must somehow be suppressed.  Glashow and Weinberg have shown [8] t h a t t h i s can o n l y be  a c c o m p l i s h e d by h a v i n g quarks o f t h e same charge c o u p l e t o o n l y one Higgs field.  T h i s i s done by demanding t h a t t h e L a g r a n g i a n remain i n v a r i a n t under  $  2  ~*2  '  d  R —* " R  $  1 ~~* ~*1  '  U  R —*  where u ,d a r e u,d-type R R  (2.1.7a)  d  - U  R  (2.1.7b)  r i g h t handed q u a r k s .  T h i s symmetry w i l l  restrict  the a l l o w e d Yukawa c o u p l i n g s , as d i s c u s s e d l a t e r i n t h i s s e c t i o n .  For  supersymmetry models ( s e e c h a p t e r I I I ) , t h i s r e s t r i c t i o n o c c u r s automatically.  The most 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 p o t e n t i a l i s g i v e n by [ 9 ]  V($ ,$ ) = - P f * 1  +  2  1  *  " "|*2 *2 +  1  +  x  i<*i\>  2  +  2 *2  X  (  2  t $  ) 2  (2.1.8) + X ($ 3  with ^ » u  + 1  $ )(* 1  t 2  * ) + X !* * +  2  4  > 0 f ° spontaneous  u  r  2  w i t h r e s p e c t t o *^ and $  *l [  l  +  *2 [^2  +  +  f  - u  ll*l I  2 X  2+  2 X 2  ' * 2 I"  where t h e f i e l d s *  2  X 3  +  X 3  1  F + (X /2)[(* $ )2 + (* * )2] +  2  5  t  1  symmetry b r e a k i n g .  2  2  1  Minimizing the p o t e n t i a l  leads to the conditions  I*2 ^  I*1  +  +  *2 t 4*l *2 +  X  +  *l t 4*2 *l +  X  +  +  +  X  5*2 *ll +  5*l\l  X  a r e e v a l u a t e d a t t h e i r VEV's.  =  =  °  °  (2.L9a)  ('' > 2  1  9b  These c o n d i t i o n s can then  22  be solved f o r the VEV's  (2X y 2 2) = — (4X^2 - A )  2  A  §  p  (2.1.10a)  2  , £  (2X.y 2 - Ay 2 ) ^ (4X X -A2)  2  -  X  (2.1.10b)  2  where A=X3+X4+X5. Expressing the potential i n terms of the rotated f i e l d s of equation (2.1.3), we find that  V ( * ' , * ' ) = - (* » )[v cos o + y s i n a ] ,+  1  2  1  - (*  , + 2  2  2  2  1  2  2  $ ')[p sin a + u cos2 ] 2  2  2  2  1  + (l/2)(  + (*  ,  1  2  a  - p 2)[$ ' $ - + *  2  t  P l  2  1  2  , t 2  * '] 1  * ' ) [ x c o s a + X sin *a + (A/4)sin 2a]  , t  2  1  l t  1  J  i  2  2  + ( * ' * ' ) [ X s i n ' a + X -cos^a + (A/4)sin 2a] +  2  2  t  2  (2.1.11)  2  1  + [ ( * ' * ' ) 2 + (• » # «)2](i/4)[(X +X -A)sin22a + 2X ] +  +  1  2  2  1  1  2  5  + ( $ ' $ ' ) ( $ ' c J ' ) ( l / 2 ) [ ( X + X - A ) s i n 2 2 a + 2X ] t  t  1  + (*  1  2  1  2  3  » ' +  * * ')[-X cos a+X sin a+(A/2)cos2a]sin2a  + (^'^'K^' *^ +  $ $ )[-X sin2a+X cos a-(A/2)cos2a]sin2a  , t 1  * )(#  2  ,  1  , t 1  2  1  + IV V +  ,t  2  2  1  ,+  2  1  ,  2  1  1  F(l/2)[ (X +X -A)sin 2a 2  1  2  2  2  2  + 2xJ  Choosing the unitary gauge and substituting from equation (2.1.6) w i l l give the  scalar p o t e n t i a l i n terms of the physical f i e l d s .  V = - (v /2)[y cos a+u sin a] + 2  2  2  +  (v' /4)[x cos' a+X sin' a+(A/4)sin 2o]  2  2  1  t  2  t  t  i  2  2  v«|>(sin2a)[y -y +v {-X cos a+X sin a+(A/2)cos2a}]/2 2  2  1  2  2  2  2  1  2  - vn[y^cos a+y sin a-v {x^cos *a+X sin a+(A/4)sin 2a} ] 2  2  2  1  lt  2  + X X " [ - V sin a-y +  2  2  2  2  x  cos a+(v /4) { (X +X ~ A ) s i n 2a+2X }]  2  2  2  2  2  L  2  3  - t [y sin a+y cos a-(v /4){(X +X -A)sin 2a+2(X +X -X )}]/2 2  2  2  2  1  - n [y c o s a + y 2  2  2  1  2  2  1  -  2  2  2 2  2  3  4  5  s i n a - 3 v {A cos^a+X^iir+a-KAM )sin 2a} ] /2 2  2  2  (2.1.12)  <t> [y sin a+y cos a-(v /4){3(X +X -A)sin 2a+2A}] 2  2  2  2  1  2  2  2  2  1  2  + n<t>[y -y +3v {-X cos a+X sin a+(A/2)cos2a}] 2  2  ]  + nx x'v[( +  2  2  2  x  2  ]L  + 1  2  (sin2a)/2  X -A)sin 2a+2X ]/2 2  2  3  + 4>x X~v[-X sin a+X cos a-(A/2)cos2a] sin2a +  2  1  2  2  + (3 neutral scalar terms) + (4 scalar terms)  The terms l i n e a r In n and < J > can be eliminated using the conditions i n equation (2.1.9).  The actual mass eigenstates w i l l i n general be mixtures  of the two neutral scalars, and the diagonalization of the n,<|> f i e l d s i s achieved through a r o t a t i o n given by  d> » T  T  4cos6 + nsine m m  H='-?sin8  m  + ncose  m  (2.1.13a) (2.1.13b)  The mixing angle 6  can be expressed i n terms of the \± parameters by  m  [(X a -X  b ) ( a - b ) + 2Aa b ] T~Pi— 2v [(X a -X b ) +a b A ] 2  sin 9  = i -  2  m  2  2  2  2  2  2  2  1  2  2  2  2  2  2  (2.1.14)  1 / 2  2  With 6 as a parameter, the mass eigenstates 4> and n are now orthogonal, m The masses of the spin-0 f i e l d s s i m p l i f y to  M~ 2  M  2 x  M  2  W  iji  ~  =  X a  + X b  =  -v (X +X )/2  =  -X v  2  L  2  2  + [(X  2 i a  -X b ) +a b A ] 2  2  2  2  2  (2.1.15a)  1 / 2  2  (2.1.15b)  2  4  A  5  2  5  (2.1.15c)  The parameters X^ and X^ can be chosen to be negative without loss of generality and hence equations (2.1.15) do not pose a consistency problem. The Lagrangian describing the interactions of  and *  2  with the gauge  bosons i s given by  ^g  =  (Vi> > *i) ta  P  +  (  D y  *2  ) + ( D l l $  2  )  (2.1.16)  where  D  and T  a  = - 3  - i(g'/2)B  are the P a u l i matrices,  -i(g/2)  T  A*  (2.1.17)  The gauge boson sector i s the same as i n the standard model with  A A A B  1  = (W  2  = i(W  y 3  y  y  +  + W ")//2  (2.1.18a)  - W ~)//2  (2.1.18b)  +  =  v  y  sinG A + cos9 Z w y wy  (2.1.18c)  = cos9 A - sine Z w y wy  (2.1.18d)  One can rewrite equation (2.1.16) i n terms of the f i e l d s T),4>,X,I|> v i a the same procedure used f o r the scalar p o t e n t i a l .  g  This gives the result  = (g /2)vcos8 W+^Wri - (g /2)vsin6 W^Wcfr + e A A x X ~ m y m y u 2  2  - leA [(9 x )x" ~ X 0 U  i i  +  +  2  y  +  X ) ] + others  (2.1.19)  -  where only terms i n the Higgs sector which w i l l contribute to the two photon decay width are e x p l i c i t l y shown. Either one of the two Higgs doublets can be used f o r the lepton Yukawa term.  Choosing *]_, one finds that the allowed Yukawa interactions must  take the form  under the discrete symmetry of equation (2.1.7).  Only one quark-lepton  family w i l l be important f o r the two photon decay width analysis. quark mixings and family labels w i l l be omitted.  Thus the  As before one can express  e q u a t i o n (2.1.20) i n terms of the p h y s i c a l f i e l d s , and we  oCy = y^[vcosaee+neecos(8+a)-c|>eesin(8+a)-ii|>ey  +y  2  f i n d that  ,-esina] //2  [ v s i n a d ^ d + n d < i s i n ( e + a ) + ^ d d c o s (9+a )-iii)d" d cosa+h. c. ] 1/2 R  L  R  L  R  L  R  + y 3 [ v c o s a u + n u ^ u c o s ( 8 +a ) -<|>u^UgSin ( 6 +a ) - i i j i u ^ ^ s i n a + h . c . ] //2 u  L  R  R  + others  (2.1.21)  where a g a i n o n l y those terms which c o n t r i b u t e to the two  photon decay  width are e x p l i c i t l y shown. T h i s completes Doublet model.  the d e s c r i p t i o n of the Higgs s e c t o r f o r the Two-Higgs-  Some of the important  summarized i n T a b l e I .  couplings obtained i n t h i s s e c t i o n are  Note t h a t the H i g g s - f e r m i o n  c o u p l i n g s d i f f e r most  s i g n i f i c a n t l y from those i n the standard model by f a c t o r s of tana ( c o t a ) , w e l l as to a l e s s e r e x t e n t due  t o the mixing a n g l e 8 » m  as  Thus f o r l a r g e  v a l u e s of tana ( c o t a ) these c o u p l i n g s can be enhanced r e l a t i v e to the standard model.  In s e c t i o n 2.3  t h i s w i l l be d i s c u s s e d f u r t h e r .  As a f i n a l p o i n t i t should be noted s e c t i o n were performed  t h a t a l l the c a l c u l a t i o n s i n t h i s  i n u n i t a r y gauge.  repeat the d e r i v a t i o n f o r a g e n e r a l gauge. the supersymmetry models i n c h a p t e r I I I .  I t i s q u i t e s t r a i g h t f o r w a r d to Indeed t h i s i s what i s done f o r However, I n the Two-Higgs-Doublet  case, the u n i t a r y gauge r e s u l t s w i l l be s u f f i c i e n t discussion.  f o r the remainder  of the  Table I  Vertex  -  Two-Higgs-Doublet Model V e r t i c e s  H°  -im eeX  cos(9 T  v -im  uuX  y  u  y  W+W~X  d  g w  cos(6 7  u  =  -im ddX  u  e  2  y  m cosa  +a)  d  sin(6 -y  e  +a)  s i n ( 6 -hx) m  d  —  =  m cosa  sina g cosO w m  m  m cosa  u  cos(9 y  -y iYj-tana e 5 J  cosa sin(6  -y  +a)  m  +a) -y  u  +a) y iY cota d  d -g  sina  w  iYctana 5  sin9  5  m  Yukawa and gauge c o u p l i n g s of s c a l a r s and p s e u d o s c a l a r to fermions and W-bosons f o r the Two-Higgs-Doublet model. The mixing angle o f the two VEV's i s a and the m i x i n g a n g l e between s c a l a r s i s 9 . The s t a n d a r d model v e r t i c e s f o r H° a r e shown f o r comparison. m  2.2 Standard Model 2y-Decay Width The standard model's r e s u l t s w i l l f i r s t be summarized, since the Two-Higgs-Doublet model i s similar to i t i n so many ways. benchmark f o r the subsequent discussions of non-standard  This serves as a  spin-0 boson  decays. In the standard model, three classes of diagrams contribute to the 2y-decay width of the Higgs boson; namely fermion loops, gauge boson loops and scalar loops.  This separation i s f o r l a t e r convenience  since the  standard model has no physical charged scalars, so the scalar loops consist only of would-be Goldstone bosons.  I f one writes the gauge invariant  amplitude M f o r H° — • y(k^) + y(k£) as  M = A  where e^ and e  W e i  e  V 2  [g  - (k^k^/CV^)]  y v  (2.2.1)  are p o l a r i z a t i o n vectors of the two photons, then the  2  structure function A i s given by  A = [ie gM /(8 r M )][A +A +A ]  (2.2.2a)  A  (2.2.2b)  2  2  H2  1  w  w  f  s  where  A  w  = 3X - 2X (2-3X )I(X ) w w w w  f  = - I (e c X )[2+(4X -l)I(X )]  (2.2.2c)  = (1/2) + X I(X )  (2.2.2d)  2  A S  f  f  f  f  f  W W  with X=m /MH and the function I(X) i s given i n appendix D. 2  The subscript  2  on X indicates the loop p a r t i c l e mass, i n this case either M  w  or mf.  Here the charges of the fermions are e^e. The quantities A A^,A w>  g  correspond to contributions from gauge boson loops, fermion loops and scalar  loops r e s p e c t i v e l y .  The sum  i n e q u a t i o n (2.2.2c) i s t a k e n over a l l charged  f e r m i o n s p e c i e s w i t h t h e c o l o u r f a c t o r Cf=3 (1) f o r quarks  (leptons).  The  c a l c u l a t i o n of the r e s u l t s i n e q u a t i o n (2.2.2) I s p r e s e n t e d i n appendix  B,  and can a l s o be found i n r e f e r e n c e [ 1 0 ] . The  two photon decay w i d t h o f the s t a n d a r d Higgs boson i s t h e n g i v e n by r(H°~+YY) = |Ap/(16irM )  (2.2.3)  H  The gauge boson l o o p g i v e s the l a r g e s t a m p l i t u d e and i s r o u g h l y a f a c t o r 5 l a r g e r t h a n the next c o n t r i b u t i o n due t o the t - q u a r k l o o p . contributions  interfere destructively.  s c a l a r loop are unimportant. 2  r(H°—»-YY)  the  say between 40 t o  e q u a t i o n (2.2.3) by » 10" M 5  w i t h MJI measured i n u n i t s o f G e V / c . 2  w i d t h I s o n l y about 10 keV  The o t h e r f e r m i o n l o o p s and  F o r a l a r g e range of Mg,  160 GeV/c , one can approximate  These two  H  keV  (2.2.4)  Hence the s t a n d a r d model two  ( i . e . a b r a n c h i n g r a t i o of l e s s t h a n  photon  0.1%).  2.3 Two-Higgs-Doublet Model 2Y~Decay Widths The model d e s c r i b e d i n s e c t i o n 2.1 has one p s e u d o s c a l a r and two n e u t r a l Higgs bosons.  scalar  T h i s s e c t i o n w i l l d i s c u s s the p o s s i b i l i t y t h a t one  more of t h e s e non-standard  s p i n - 0 bosons has a two photon decay w i d t h  which  i s g r e a t l y enhanced r e l a t i v e t o the s t a n d a r d model. R e c a l l the v e r t i c e s g i v e n i n T a b l e I .  The magnitude and s i g n of the  s c a l a r c o u p l i n g t o charged Higgs i s h i g h l y model dependent and w i l l  be  d i s c u s s e d below.  the  As e x p e c t e d the p s e u d o s c a l a r i|> i s not a f f e c t e d by  m i x i n g parameter 0 . m  I t does n o t c o u p l e t o the W-bosons o r t o the  Higgs bosons, but o n l y t o f e r m i o n s . the l e a s t model dependent.  or  charged  Hence the p s e u d o s c a l a r decay w i d t h i s  I n g e n e r a l the s c a l a r c o u p l i n g s t o the W-boson  30  are s m a l l e r than I n the s t a n d a r d model. l o o p s f o r any p o s s i b l e enhancement. f a c t produces the c o n s t r a i n t  X b -X 2  2  I f a l l the X^'s satisfied.  fermion  F o r s i m p l i c i t y we take 6 =0, which i n m  equation  2 i a  Thus one must l o o k t o the  = (b -a )(X +X +X )/2 2  (2.3.1)  2  3  4  5  a r e o f the same o r d e r , t h i s e q u a t i o n can be n a t u r a l l y  With t h i s c h o i c e o f 9  m  the c o u p l i n g s f o r one o f the  scalars,  n, become i d e n t i c a l t o those o f the s t a n d a r d model Higgs boson, g i v i n g same width as d i s c u s s e d i n the l a s t s e c t i o n .  the  The o t h e r s c a l a r , <j>, now does  not couple a t a l l t o the W-boson, e l i m i n a t i n g the d e s t r u c t i v e i n t e r f e r e n c e and  l e a v i n g o n l y the f e r m i o n l o o p  amplitude.  There are two ways t o enhance the c o n t r i b u t i o n o f the fermion l o o p s . From T a b l e I i t can be seen t h a t i f tana ( c o t a ) i s l a r g e then the l e p t o n and u-type  quark loops w i l l be enhanced ( d e c r e a s e d ) , and the d-type quark l o o p s  decreased bosons.  (enhanced).  T h i s i s t r u e f o r both the s c a l a r and p s e u d o s c a l a r  From the s t a n d a r d model I t was found  t h a t o n l y the t-quark  loop  made a s i g n i f i c a n t c o n t r i b u t i o n , and thus i t i s the l o g i c a l one t o t r y and enhance.  H e r e a f t e r the d i s c u s s i o n w i l l  w i t h l a r g e tana enhancement. s i m i l a r except The now  f o c u s on the Two-Higgs-Doublet model  The r e s u l t s f o r models w i t h l a r g e c o t a w i l l be  t h a t the width being enhanced i s much s m a l l e r t o b e g i n w i t h .  dominant c o n t r i b u t i o n f o r both the s c a l a r <|> and the  comes from the t-quark  loop enhanced by tana.  To f i n d  pseudoscalar  the maximum  enhancement a l l o w e d by the model, bounds on the magnitude o f the enhancement f a c t o r can be determined of the charged Higgs K°-K° mass d i f f e r e n c e charged Higgs mass  by low energy  phenomenology.  i n Bhabha s c a t t e r i n g , muon decays [14] g i v e an upper l i m i t .  The v i r t u a l  [11,12,13] and the  f o r tana as a f u n c t i o n o f the  The approximate bound t a n a < 2M^/m  from r e f e r e n c e [14] where m  2  c  c  effects  i s the charm quark mass.  i s taken  L i m i t s on  can be o b t a i n e d from c o n s i d e r i n g charged Higgs e f f e c t s i n t h e  W and Z-boson p r o p a g a t o r s .  For M  »  M~,M~  t h e gauge bosons i s p r o p o r t i o n a l t o M^.  t h e change i n t h e mass r a t i o o f  A  Tl  X  Specifically  = 1.2 TeV/c  g i v e a 5% change i n t h e p-parameter [15] where p=M /M cos 6 . 2  w i t h i n the allowed experimental e r r o r [1,2].  W  2  2  L  Thus we f i n d  2  will  This i s  W  tana < 40  (2.3.2)  I n t h e Two-Higgs-Doublet model t h e r e i s an a d d i t i o n a l c o n t r i b u t i o n t o t h e 2y-decay w i d t h of t h e H i g g s s c a l a r s coming from t h e charged Higgs s c a l a r loops.  The Feynman diagrams f o r t h i s p r o c e s s a r e t h e same as t h o s e i n  s e c t i o n B.2 o f appendix B, except t h e l o o p p a r t i c l e I s a charged Higgs x r a t h e r t h a n a would-be G o l d s t o n e boson.  +  From e q u a t i o n (2.1.12) t h e r e l e v a n t  s c a l a r c o u p l i n g t o charged H i g g s terms o f t h e L a g r a n g i a n a r e  Xx"<i> —• v [ - X s i n a + X c o s a - ( A / 2 ) c o s 2 a ] s l n 2 a  (2.3.3a)  + -,j, —  (2.3.3b)  2  +  x  x  1  2  2  v[(X +X -A)sin 2a+2X ]/2 2  y  1  2  3  with 6 0.  The magnitude and s i g n o f t h e s e c o u p l i n g s i s n o t d e t e r m i n e d  by t h e o r y .  Above i t has been argued t h a t  =  m  X^ and X^ a r e o f t h e o r d e r o f u n i t y . X j / s a r e o f t h e same o r d e r .  < 1.2 TeV/c  2  w h i c h means t h a t  I t i s n a t u r a l t o assume t h a t a l l t h e  T h i s argument I s by no means r i g o r o u s b u t i s  s u p p o r t e d by p a r t i a l wave u n i t a r i t y p l u s p e r t u r b a t i o n t h e o r y [16,17] w h i c h g i v e s a s i m i l a r bound on M^.  W i t h t h e s e c a v e a t s I t c a n be s a i d t h a t t h e  c o u p l i n g s o f e q u a t i o n (2.3.3) a r e n o t enhanced.  Hence t h e s c a l a r x~loops  g i v e a n e g l i g i b l e c o n t r i b u t i o n t o t h e two photon decay w i d t h . F o r l a r g e tana t h e two photon decay w i d t h o f t h e Higgs s c a l a r i s t h e n dominated by t h e t - q u a r k l o o p and i s g i v e n by  r ( * + YY) = tan a|(-ie gm ./6n M )[2+(4X - 1 ) I ( X j ] P/(16TTM~) 2  2  2  2  t where X =m /M~ . 2  2  t  w  t  (2.3.4)  cp  t  This result i s obtained using the same techniques,  described i n appendix B, as were used f o r the standard model c a l c u l a t i o n . The pseudoscalar fermion loop c a l c u l a t i o n i s performed i n section B.4 of appendix B.  Again the t-quark loop dominated with the result  r ( * * YY) « tan a|(-ie gm /6ir M )I(A ) 2  2  2  2  t  w  where X =m /M . t t i|> 2  2  p/(16irM^)  (2.3.5)  These widths are indeed greatly enhanced over the standard The other scalar width r(n •»• yy) i s the same  model result f o r large tana. as the standard model r e s u l t .  In general 8 *0, and the 2y-decay widths of m  the scalars cp and n l i e somewhere between the two extremes of the standard model r e s u l t [equation (2.2.3)] and the best case r e s u l t [equation (2.3.4)]. This concludes the chapter on the Two-Higgs-Doublet model.  The  reader  has been introduced to the general features of the model, and t h i s knowledge w i l l be useful background f o r the discussion of the results to be presented i n the remaining chapters.  Also i l l u s t r a t e d were the methods needed to  calculate the standard model Higgs 2y-decay width.  These c a l c u l a t i o n a l  methods can be found i n the l i t e r a t u r e , and the Two-Higgs-Doublet model i t s e l f has been extensively reviewed.  The only new result has been to  perform the Higgs 2y-decay width c a l c u l a t i o n for the Two-Higgs-Doublet model, obtaining equations (2.3.4) and (2.3.5) given above.  For large  values of the enhancement factor tana, these widths are much larger than i s the case f o r the standard model.  Hence the two photon decay process i s much  more important i n Two-Higgs-Doublet models. w i l l be discussed at length i n chapter IV.  The consequences of t h i s result  III.  MINIMAL BROKEN SUPERSYMMETRY MODEL  This chapter also looks at the two photon decay widths of non-standard spin-0 bosons [18]. In this case however, the model i s one of minimal broken supersymmetry, which i s a s p e c i f i c example of a Two-Higgs-Doublet model.  The motivation f o r supersymmetry has been discussed i n the  introduction.  As was the case i n the l a s t chapter, the minimal broken  supersymmetry model i t s e l f can be found i n the l i t e r a t u r e .  Again the  d e t a i l s are presented below f o r the readers e d i f i c a t i o n and to establish notation.  The remainder of the chapter i s devoted to the c a l c u l a t i o n of the  Higgs 2y-decay widths for this model.  These are a l l new r e s u l t s , and t h e i r  significance i s further discussed i n the next chapter. The p a r t i c l e content of the minimal broken supersymmetry model i s quite similar to that of the Two-Higgs-Doublet model, and i s discussed i n more d e t a i l below.  The main difference i s that each p a r t i c l e i s now accompanied  by a superpartner which d i f f e r s by one half a unit of quantum spin.  We w i l l  discover l a t e r i n this chapter that these new superparticles do not s i g n i f i c a n t l y a f f e c t the two photon decay widths of the Higgs bosons. In fact they s l i g h t l y reduce the widths through destructive interference with the usual p a r t i c l e contributions.  Once again i t i s the additional Higgs  f i e l d content, leading to an enhancement of the Higgs to fermion couplings, which has the largest effect on the 2y-decay widths.  However, unlike the  Two-Higgs-Doublet model, we find that supersymmetry imposes a new constraint on the maximum possible value of the enhancement factor tana.  This new  constraint has important consequences, which w i l l be discussed i n the next chapter.  3.1 The Model Deriving the actual supersymmetry Lagrangian i s too complicated to present here. below.  The results i n component f i e l d notation w i l l be summarized  The component f i e l d content needed for the minimal  model Is l i s t e d  i n table II [18].  usual SU(2) t r i p l e t  supersymmetric  I t includes, i n the gauge sector, the  of vector bosons, V  a y  (a=l,2,3), and the U ( l )  y  vector boson Vy' along with t h e i r corresponding fermionic partners represented by two component spinors X  a  (a=l,2,3) and X' respectively.  The matter sector contains a left-handed SU(2) lepton doublet of two component fermions L  i  (1=1,2) along with a two component SU(2) singlet  S i m i l a r l y , f o r the quark sector there i s a doublet Q* (1=1,2) and c c s i n g l e t s , u^ and d^.  SU(2) doublet and singlet are L  and e^ respectively.  T  two  The scalar partners of the quarks are denoted  and Uj^.dj^ for the SU(2) doublet and s i n g l e t s , respectively.  c e .  as  ~I  The slepton  As i n the Two-Higgs-  Doublet model, only one quark lepton family w i l l be of i n t e r e s t .  Thus the  quark mixings and family labels are omitted, although these can be included straightforwardly. Higgs m u l t l p l e t s .  The matter sector i s completed with the addition of Three sets of Higgs f i e l d ,  break the SU(2)xU(l) symmetry [19].  , H^ and N are employed to  At least two scalar SU(2) doublets are  necessary to give mass to both the up- and down-type quarks.  With the  additional constraints present i n supersymmetry models, the Higgs doublets alone are no longer s u f f i c i e n t to break the SU(2)xU(l) symmetry.  Although  not necessarily present, the addition of an extra Higgs f i e l d N Is the simplest way to remedy t h i s problem.  Enlargement of the Higgs sector to  include an SU(2) and U ( l ) singlet f i e l d N allows f o r discussion of the supersymmetric  l i m i t , with the gauge symmetry broken to U ( l )  # e m  these scalars are a l l accompanied by the fermionic partners \J»^ ,  Finally and iJ>.. T  T a b l e I I - Supersymmetrlc  Gauge Bosons  Gauginos  Field  Content  SU(2)  V» Leptons  Sleptons L =(v,e-)j  e  c L  Quarks Q =(u,d) L  L  < Higgs Bosons  H  -1  0  2  1/2  1/3  0  -4/3  0  2/3  1/2  -1  Squarks  "L  .i H  1/2  <*R Higgsinos  H  1  H  2  Hj_  H  1/2 2  F i e l d c o n t e n t o f the minimal supersymmetrlc S U ( 2 ) x U ( l ) model w i t h one family. SU(2) gauge bosons c a r r y the l a b e l a=l,2,3 and the matter f i e l d s have the SU(2) index 1=1,2. The l a s t two columns g i v e the SU(2) r e p r e s e n t a t i o n s and the U ( l ) hypercharges of the r e s p e c t i v e f i e l d s . The s u p e r s c r i p t c i n d i c a t e s charge c o n j u g a t i o n .  36  The component f i e l d L a g r a n g i a n has been e x t e n s i v e l y reviewed and i s p r e s e n t e d below.  The t o t a l  i n t e r a c t i o n Lagrangian, «£^ » nt  i n t o a supersymmetric p i e c e , «Cgg> supersymmetry,  ot  c e  a n <  *  a  p i e c e which s o f t l y  The supersymmetric p a r t i s i n v a r i a n t  , . n  [3,5,20] Is divided  breaks under  SOD  t r a n s f o r m a t i o n s between bosons and f e r m i o n s .  Thus one has  which i s c o n s t r u c t e d out o f t h e f i e l d s  i n table I I .  listed  A d e r i v a t i o n of  of gg from s u p e r f i e l d f o r m a l i s m can be o b t a i n e d i n r e f e r e n c e [ 2 1 ] . For  clarity  and completeness  gg w i l l be p r e s e n t e d i n s e v e r a l  pieces.  The i n t e r a c t i o n s o f t h e gauge m u l t i p l e t s among themselves and t h e matter fields  a r e d e s c r i b e d by c£ . gauge  T h i s i s g i v e n by  J  ^gauge  =  iS^CxVJ  " ^Vj) + T i ' ^ ^ A - X'V,)  - i g T ^ U A ^ A j - 0 A*) ] -  lV^j}  U  A j  - ^ f V j y j A * * ^ - O A*)A] - i y * o^*} y  f  . . ra ii, b„c + ige , X a X V abc y  (3.1.2)  In the above equation, A denotes the scalar f i e l d s and i> represents generically the Majorana spinor f i e l d s of table I I . the scalar f i e l d A i s y  and that of the matter fermion f i e l d i s y .  A  A  f  over a l l scalar f i e l d s A i s i m p l i c i t . Also o =(l,o)  a=l,2,3 and i , j = l , 2 .  The U(l) hypercharge of sum  The SU(2) generators are T*j where  where o denotes the three P a u l i  v  matrices. The Yukawa interactions between the fermions and the scalar bosons are described by a second piece, e £ .  Also included are the scalar-fermion  y  Higgs f i e l d interactions since they are the supersymmetric partner Interactions to the Yukawa ones.  ^Y  =  ^j^V"'-  + 2Re[(h  +  e  J  1;J  H  e  J  IVi/VVij^RF  +  k  i k  R  d  i k  R  F + eijh^Jq^+h.c. +  h  where f , h<j and h e  H  H J N ) ( f e L e + h e Q d ) * ] + 2Re[ ( h e ^ N ) ( h ^ Q ^ ) * ]  ij u 2Q V - h  el l F V R k  E l j  + |f e HjL e  f2  +  Explicitly  C  u  +  I d lV u 2\> ( h  H  h  H  ±  IVij l^ H  F  +  are the Yukawa couplings.  J  F  +  I V i j ^ R  I V i j  F (3a  -  3)  gives  In the absence of  supersymmetry breaking terms, the fermions and corresponding have degenerate masses.  ^  F  As usual  r i s e to quark (squark) and lepton (slepton) masses.  1  sfermions w i l l  The t h i r d piece, ^-s»  l s  superpotential and i t Is given by [5]  t n e  |N F +  = h (|H*F + |H| 2  Ihc^jHJ+.p  F - 2|HJ PIHJ F + J L T 2 [4|niX F -  +  - r A l H ^ F  -ZlHiPI^P] +4|(jJ*i: F  2|H*  FIQJ p  -2|QjF|L F  1  1  (3-1.4)  + i^r + IHJ r + IQ r + it r 1  1  + Ig' [ 14 F 2  where |A* f* = (A*" A * )  2  -  |Hj  F  + ilQ  1  F  - ||u  F + f |d F  R  R  for the scalar f i e l d A.  - |L  F+  2|e  p]2  R  F i n a l l y f o r completeness  the usual gauge f i x i n g Lagrangian  2 OF  "  " 2t  I  /  3  ™<-4<h>  +  " I F [ V'"  ^  +  y  -  iC I H  <  H  < H  i  i 1 l) H  >  >  "  <  H  i  >  t  I  H  2  i)  and Fadeev-Popov (FP) ghost Lagrangian have to be added.  I  2  < 3  K 5 )  The simple case  of equal VEV's for the two Higgs doublets w i l l be s u f f i c i e n t to i l l u s t r a t e the important features of the FP Lagrangian.  In the expression below the  ghost f i e l d s are C ,C ,C ; the would-be Goldstone bosons are G~,G°; the ±  y  Z  usual gauge bosons are W~ ,A ,Z ; and the Higgs scalar f i e l d i s H. W  W  V  Thus  *  F  P  -  - C> C  - CVC_  2  +  - c  - 5M2(c|c +C^C_)  ig^ca^c^)^  - O  w  - (lgSM /2)G0(cJc w  - C  +  Z  w  - ig5[O cJ)sin9 y  ^  3 C  5M2C  -  2  Z  t z  C /co 26 8  z  W  - (?gM /2)[(C^C +C^C_)H + C ^ H / c o s Z e J  +  +  V  +  c^)w- ](c sine  + c cose )  u  j j  Y  w  z  + O cJ)cos6 ](C_W jj  +  w  + C VT )  +lJ  +  - C +cJ - 5 M s i n e ( c | G + + c 8  w  (3.1.6)  U  w  w  V  )  ^  + (g£M /2cos6 )[C^(C,G +C G+) - ( 2 c o s 6 - l ) ( c T G + C G ) C , ] v v Z + w + Z -  K  2  +  t  _  J  Jl  - igd(9  C?)C. - (9 C ) C ] ( A s i n e f  y  The ' t Hooft-Feynman gauge w i l l be chosen.  - Z cos6 ) y  Combining a l l the p i e c e s  t o g e t h e r , the supersymmetrlc s t a n d a r d model L a g r a n g i a n i s j u s t  ^SS  " *gauge  +  * Y  +  +  ^GF  +  ^FP  ( 3  ' ' 1  7 )  The above i n t e r a c t i o n L a g r a n g i a n i s g l o b a l l y supersymmetrlc. To be p h e n o m e n o l o g i c a l l y r e a l i s t i c , the supersymmetry must be broken.  T h i s can be  a c h i e v e d by s o f t b r e a k i n g terms [22] which a r e thought t o be induced by s u p e r g r a v i t y a t the s c a l e of the P l a n c k mass, M . p  The e f f e c t i v e low  energy ( i . e . below the Planck mass) L a g r a n g i a n t h a t breaks supersymmetry can be w r i t t e n as [23,24]  -  I m^A*A - m A  3 / 2  ( h ( Z ) + h.c.)  (3.1.8)  where  h(Z) = (A-3)g(Z) + \ | | Z A  (3.1.9a)  A  and  g = (hE  +  As b e f o r e the sum S\S  gaugino masses m' parameters.  ±  .H^N  V i j  H  ^ R  +  sN)  +  V ^ H j L ^  V i j ^ L ^ R  +  over A r e p r e s e n t s a sum  +  h  ' C  ( 3  over a l l s c a l a r f i e l d s .  ' ' 1  9 b )  The  rV and m as w e l l as the g r a v i t i n o mass ^/2 m  E q u a t i o n (3.1.9a) c o n t a i n s terms t h a t s p i l t  the masses of the sfermions and f e r m i o n s .  a  r  e  ^  r e e  the degeneracy  Because o f the s i m p l i c i t y and  added a t t r a c t i o n of having a s t r u c t u r e c l o s e to the unbroken model, I t i s u s u a l [23] t o take the parameter  in the  supersymmetric  i n e q u a t i o n (3.1.9a) to be  A=3. It to  I s o f t e n argued  be a l l equal to the  t h a t the gaugino  1113/2  a t  t  n  e  and the s c a l a r masses can be taken  Planck s c a l e .  However, t h e r e are  many u n c a l c u l a b l e e f f e c t s I n v o l v i n g g r a v i t o n s i n the h i g h energy t h e o r y and it  i s not c l e a r t h a t a common mass can s t i l l  Hence the m^'s  are d i f f e r e n t i n g e n e r a l .  be maintained a t low e n e r g i e s .  H e r e a f t e r , these parameters  will  Thus m^,  carry s u b s c r i p t s denoting t h e i r p a r t i c l e species.  , m^ , e t c .  w i l l be t h e bare mass terms o f t h e s c a l a r f i e l d s N, H^, H^, e t c . respectively. The gauge symmetry b r e a k i n g i s a c h i e v e d by l e t t i n g t h e t h r e e s e t s o f Higgs f i e l d s , H^, H  and N develop vacuum e x p e c t a t i o n v a l u e s (VEV's),  2  given  by  (3.1.10a)  (3.1.10b)  <N>  =  3 72 v  (3.1.10c)  As was done i n t h e Two-Higgs-Doublet model, a s e t o f c o n s t r a i n t e q u a t i o n s on the VEV's can then be o b t a i n e d by m i n i m i z i n g t h e s c a l a r p o t e n t i a l c o n t a i n e d i n e q u a t i o n s (3.1.4) and ( 3 . 1 . 8 ) .  m  72«3 hv v 2  /2  h v  v  + m  3  72»3/2 i 3  3 / 2  +  (|h v +s) V l  v  2  »H 2 v  x  +  2  They a r e  + T^+fv ) 2  - 0  (3.1.11a)  + | [ h V j ( v | + v 2 )+2sv ] + g p v ^ v j - v , ) - 0 2  2  |[hv (v2+v2)+2sv ] - g ? v ( v * - v | ) = 0 2  1  2  (3.1.11b)  (3.1.11c)  where  = v  2  + v  = g  2  + g'  (3.1.12)  2  and  P  In and  the l i m i t  that v i = V 2 * 0 ,  (3.1.13)  2  t a k i n g the d i f f e r e n c e o f e q u a t i o n s  (3.1.11b)  (3.1.11c) g i v e s  (m  - m ^)v  2  2  1  = 0  (3.1.14)  Hence i t i s n e c e s s a r y f o r the 'bare' masses of H^ and H£ t o be e q u a l i f they a r e t o develop the same VEV. m^ =111^ =m^=m^j2 * in this limit  8  The s p e c i a l case where v i = V 2 = a and  s o l v e d i n r e f e r e n c e [ 2 3 ] . A p a r t i c u l a r l y simple  solution  i s g i v e n by  /2 ~h  =  /  2  m„  ^ h  . [  1  (3.1.15a)  "3/2  - - 5 l i _ ) _o  "3/2  1  /  2  (3.1.15b)  Next t h e phenomenologically more i n t e r e s t i n g case o f v ^ * V 2 i s investigated. the f o l l o w i n g  F o r t h i s case, the c o n s t r a i n t e q u a t i o n s can be r e c a s t forms  75 3/2 3 m  ~ Z^ l 2  hv  v  v  +  |(  h  v  i  v  +  2  s  2  )  =  —\—Y~ V  f^v  + ipv2)(v2-v|) + m  2  v -m  2  V  7I 3/2 3 m  h v  into  +  \ <. i h  hv  '  v +2s 2  2  2  i 2 v  (3.1.16a)  2 1 _ V  v  2  =0  (3.1.16b)  V  ~^F" H (ln  +  v  v  H 3) " 1 2  + m  + h 2 v  0  (3.1.16c)  which a r e u s e f u l i n s i m p l i f y i n g the mass m a t r i x f o r the Higgs bosons. E q u a t i o n s (3.1.16) s i m p l i f y f o r the case m^ =111^ =m i n the non-degenerate VEV region.  In p a r t i c u l a r e q u a t i o n (3.1.16b) becomes  h v 2  The  2  + 2m  2  t h r e e Higgs f i e l d s H^,  eigenstates  -- \pv  a  n  (3.1.16b')  2  d N a r e not the p h y s i c a l mass  and a d i a g o n a l l z a t i o n has t o be performed.  There a r e s i x  n e u t r a l s p i n - 0 f i e l d s g i v e n by the r e a l and Imaginary p a r t s o f the t h r e e  Higgs f i e l d s ;  explicitly  v H  =  they a r e g i v e n by  + ReH° + iImH°  ( _i  L_ fl  ?  N  = ( H;  =  ,  K)  ,  (3.1.17b)  l  ( v + ReN + ilmN)  (3.1.17c)  3  The s u p e r s c r i p t s on the H - f i e l d s a r e SU(2) i n d i c e s . form the f o l l o w i n g  (3.1.17a)  v_ + ReH° + iImH° ^) fl  1  H  I  Two charged  scalars  combinations  H  = ^ H *  +  + v H**)  (3.1.18)  2  ~ v( 2 2 " 1 1 ^  G +  V  H  V  (3.1.19)  H  w i t h H~ and G~ g i v e n by t h e c o n j u g a t e s o f e q u a t i o n s (3.1.18) and (3.1.19). The p h y s i c a l charged Higgs f i e l d s  a r e H*.  The G* a r e the would-be  Goldstone bosons which e n t e r i n the g a u g e - f i x i n g c o n d i t i o n s f o r t h e W-bosons g i v e n by  8 W  W+  V  8 W V  W  = ^ v G 25 = 4f v G 25  where 5 I s t h e gauge f i x i n g parameter.  +  (3.1.20a)  -  (3.1.20b)  Noting that M = g 2 v 2 / 4 , w 2  equations  (3.1.20) a r e seen t o be the u s u a l gauge c o n d i t i o n s f o r the s t a n d a r d model.  45  F o r the ' t Hooft-Feynman gauge 5=1.  The combinations i n e q u a t i o n s  (3.1.18-19) can be shown t o d i a g o n a l i z e the charged s c a l a r mass m a t r i x when the c o n s t r a i n t e q u a t i o n s (3.1.16) a r e used. mass ^  The u n p h y s i c a l bosons G* have  i n the ' t Hooft-Feynman gauge as e x p e c t e d .  The mass o f the  p h y s i c a l charged s c a l a r s i s g i v e n by  M  = h v  2  2  +  + m  2  + m  2  H~  1  f o r the g e n e r a l case o f v i * V 2 « v a l u e s f o r m^  and m^  .  + M  2  2  (3.1.21)  2  W  T h i s e q u a t i o n f u r t h e r l i m i t s the a l l o w e d  One example i s the case where  v  ^* 2 v  a n <  *  •  U s i n g e q u a t i o n (3.1.16b') g i v e s  M  2  = - r g' v 2  +  (3.1.21')  2  H"  and hence i f the e f f e c t i v e L a g r a n g i a n i s not t o g i v e u n p h y s i c a l masses t o the charged Higgs bosons,  then m^ *m^  i n the r e g i o n where  v  j *  v 2  '  I t I s i n s t r u c t i v e to c o n s i d e r the case of degenerate VEV's, i . e . v  l  = v  2*  F u r t h e r s i m p l i f i c a t i o n i s made by the c h o i c e o f  "fl  - V  =  m  3/2  and the s o l u t i o n of e q u a t i o n (3.1.15a).  M  2 ±  = 4^3^ +  ( 3  Then we  M  2  - ' 1  2 2  >  obtain  (3.1.23)  Thus the s i m p l e s t s o l u t i o n s l e a d t o the c o n c l u s i o n t h a t the charged  Higgs  boson I s h e a v i e r than t h e W-boson q u i t e independent o f the c o u p l i n g parameters i n the s c a l a r p o t e n t i a l . not be t r u e i n g e n e r a l f o r j * v  and m^ *m^  v  The  I t must be emphasized t h a t t h i s need  2  .  s i x n e u t r a l s p i n - 0 bosons c o n s i s t o f t h r e e s c a l a r s and t h r e e  pseudoscalars.  One o f t h e p s e u d o s c a l a r s  i s the would-be Goldstone  boson  which g i v e s mass to the Z° , and i t i s g i v e n by  G° = £ ( v I m H ° - v^mHO)  (3.1.24a)  2  Orthogonal  t o G^ i s a p s e u d o s c a l a r  h  and a t h i r d p s e u d o s c a l a r and  4  =  v ( i v  ImN.  I m H  h^; e x p l i c i t l y w r i t t e n as  2  +  v  2  I m H  l)  I n t h i s b a s i s G° decouples  (3.1.24b)  from t h e o t h e r two  o n l y p l a y s the r o l e i n Z° gauge f i x i n g , i . e . terms l i k e G°h^ and G°ImN  are r o t a t e d away. is s t i l l  However, the mass m a t r i x of the two remaining  not d i a g o n a l .  As u s u a l the d i a g o n a l i z a t i o n i s a c h i e v e d  0~ bosons by a  r o t a t i o n , l e a d i n g t o the two p h y s i c a l p s e u d o s c a l a r s , H° and H^, below.  The  mixing  H° = h° cosx - ImN s i n x  (3.1.25a)  H° = h° s i n x + ImN cosx  (3.1.25b)  angle x can be o b t a i n e d i n terms of the s c a l a r  potential  parameters v i a  t a n 2x =  6h m... v ^= /2 {h2 2 (4^ v  +  -^)}  (3.1.26)  In t h e degenerate VEV case w i t h m^ =111^  = m  = m N  3/2  a n c  * °l 8  u t  *-  o n  (3.1.15a),  this  reduces t o  /2 h M tan 2x =  (3.1.26') 8  m  3/2  The p s e u d o s c a l a r masses a r e g i v e n by  M  £  " (  h 2 v  3  + m  H  -t^^v ) cos x + ( ^ v - ^ ) 2  1  +  ^ 3/2 m  2  v  c  (3.1.27a)  o  s  x  s  *  1  1  y^"  which  3/2  v  2  2  n x  = (^v^+mfj +m2j + jh v ) n m  sin x  2  2  sin x  2  2  ( j ^ v  +  2  ^ )  cos x 2  (3.1.27b)  2  c  o  s  x  s  i  n  x  s a t i s f y t h e sum r u l e  «l  + M  2  = h^v^+v ) + m 2  + m  2  2  + m  (3.1.28)  2  The remaining degrees o f freedom a r e t h e s c a l a r ( 0 ) f i e l d s R e H ° , ReH^ and +  ReN which a g a i n have a n o n - d i a g o n a l mass m a t r i x . e i g e n s t a t e s s h a l l be denoted by  The p h y s i c a l mass  with eigenvalues  (where 1=1,2,3), and  they a r e o b t a i n e d from t h e above by a u n i t a r y t r a n s f o r m a t i o n  H  i  =  hi  R  e  H  j  (3.1.29)  In  e q u a t i o n (3.1.29) i t i s understood  In  g e n e r a l the elements  bare s c a l a r masses.  of U a r e c o m p l i c a t e d f u n c t i o n s of the VEV's and t h e  These w i l l not be examined here as they a r e not  particularly illuminating.  However, the i n t e r e s t i n g sum  I  M  1=1  should be n o t e d . parameters.  t h a t ReN i s t o be s u b s t i t u t e d f o r j=3.  =  2  M  2  +  M  P h e n o m e n o l o g l c a l l y the U-JJ can be t r e a t e d as f r e e  From the above d i s c u s s i o n one would expect t h a t these s p l n - 0 or m w  In  (3.1.30)  2  1  bosons have masses of the o r d e r of M  parameters  rule  are w i l d l y  „ , u n l e s s the t r a n s f o r m a t i o n 3/2  0 /  different.  a d d i t i o n t o the m i x i n g i n the Higgs s e c t o r , the s c a l a r fermions  a l s o mix to a c e r t a i n e x t e n t .  will  A t t e n t i o n i s f o c u s e d i n p a r t i c u l a r on the  s c a l a r t - q u a r k s , s i n c e they w i l l be the o n l y r e l e v a n t ones c o n t r i b u t i n g t o the 2y-decay width c a l c u l a t i o n .  A m i x i n g between the d i s t i n c t  states t  T  and  l_t  t  R  a r i s e s from the l a s t  The mass m a t r i x f o r t  /  term In e q u a t i o n (3.1.9b) when H  2  develops a VEV.  and t.. i s g i v e n by  T  hv -R  m  " 3/2 1m  ~  +  v 2  \ 3^ (3.1.31a)  h v  \  m  t( 3/2 m  +  ~  l V  2  4  3^  /  where  4-  and  m  2  L  \  g' (v2- 2) +  = — g 24  2  V  , 2  m2  R  (v -v ) + m 1 2 BL 2  2  2  (3.1.31b)  (3.1.31c)  w i t h m^ b e i n g the f e r m i o n t-quark mass and nig J^^L) in  *»gg «  I  general 1 1 1 ^ * 1 1 1 ^ .  n  B  t  n  ^are  e  m  a  s  appearing  s  The m i x i n g a n g l e 9 between t ^ and t  R  can be  deduced from e q u a t i o n (3.1.31) g i v i n g  16m  . m 29 = ^=-^ 8(m2 -m2 ) + ( ' 2 _ 2 )  tan  BR  In V  the symmetrical case o f v^=v  1* 2  tan  BL  V  - 1  a  n  d  m  BR* BL*  [m  fc  / ^/2^'  20 < m^ < 50 GeV/c than M » w  simplicity  ^  m  P  2  ° r  e  l  n  i  e t  m  i  a  ^  n  a  e  r  s  v  m  g  and j £ m2  2  BR 3/2 = m  d  a  t  a  f  r  o  m  t  g  =m2  (3.1.32) ( v  2_ 2) v  j L then 0=TT/4.  n  e  n  C  E  R  ® * N  and i t i s p o s s i b l e t h a t  t  2  °^  s  5  t  n  e  o r  I n g e n e r a l however &  e r  ] indicates c  a  n  a  ^  e  °f that w t :  *-  m e s  heavier  Thus 8 i s g e n e r a l l y q u i t e s m a l l even f o r t h e s c a l a r t - q u a r k s . t h i s s m a l l m i x i n g i s n e g l e c t e d , and the s c a l a r quarks  t  T  and t ^  Li  are  For  K.  t r e a t e d as mass e i g e n s t a t e s . There i s y e t a t h i r d s e t o f mixed s t a t e s t h a t a r e important i n t h e  2y-decay w i d t h c a l c u l a t i o n .  These a r e t h e s t a t e s formed  the W-gauginos and charged H i g g s i n o s .  from t h e m i x i n g o f  In the L a g r a n g i a n  [see e q . (3.1.2)] + the charged gauginos and H i g g s i n o s a r e r e p r e s e n t e d by Majorana s p i n o r s X and tp* and i j i , r e s p e c t i v e l y w i t h 1 2 2  H  X" = ( X  1  + iX )//2 2  A g a i n they a r e not the p h y s i c a l mass e i g e n s t a t e s .  (3.1.33)  These p h y s i c a l s t a t e s a r e  c o n s t r u c t e d e x p l i c i t l y as f o l l o w s :  IX+ c o s *  +  + iji^  sin<|> \ +  (3.1.34a)  V  i X ~ cos<J>  +  sin* H  l  IX"" sine)) - i p * + H  cos*  1  Xo  =  w  -IX~  2  + (3.1.34b)  sin*  + ip2 l  cos*  H  N o t i c e t h a t t h e r e a r e two s e p a r a t e m i x i n g angles <f>+ and <j>_. One can read o f f d i r e c t l y terms i n v o l v i n g the W-gauginos  from e q u a t i o n s  (3.1.2) and (3.1.8) the mass  and charged H i g g s i n o s .  Diagonalizationi s  a c h i e v e d u s i n g e q u a t i o n (3.1.34), which then g i v e s the mixing angles *+ and  the masses  respectively.  °f the two p h y s i c a l c h a r g i n o s t a t e s x^  2  a  n  d  x  2  In terms of the parameters a p p e a r i n g i n the L a g r a n g i a n ,  these  angles a r e g i v e n by  s  i  n  2  K  =  [d+sin2a) w  1 / 2  ±  { +  (l-sin2 ~  s  i  n  2  2  a  1  1 / 2 a )  >  1  /  2  w where  tan a =  v  L  / v  2  (3.1.35')  T h i s r e s u l t agrees w i t h t h a t p r e s e n t e d i n a d i f f e r e n t form i n r e f e r e n c e [19].  F o r v^ »  or  »  v^ the a n g l e s become  -=i-2.  s i n 24. =  (3.1.36a) w  and 4_ = 0  (3.1.36b)  On the o t h e r hand w i t h e q u a l VEV's t h e a n g l e s become e q u a l 4 =4 =4 and +  e q u a t i o n (3.1.35) reduces t o  sin  2  24 = (1 +  The c a l c u l a t i o n a l s o y i e l d s the masses  (3.1.37)  4M' w  and K^.  These a r e w r i t t e n  e x p l i c i t l y as  5  i,2  - 7J w M  [( * ° + i i - ) 1+  in2  1 7 2 1  (1  Sin2a +  1|-) ] 1/2  <-- > 3  1  38  T h i s completes the d i s c u s s i o n on the p h y s i c a l s t a t e s which w i l l appear i n the  2y-decay w i d t h c a l c u l a t i o n f o r the minimal broken supersymmetric  model.  The d e t a i l e d Feynman r u l e s which a r e o b t a i n e d from the L a g r a n g i a n a r e g i v e n i n appendix E .  F i g u r e 1 - One  X  X  X  X  x  x  Loop C o n t r i b u t i o n s to the 2y-Decay of the S c a l a r  X  X  X  XX  X  X  X  X  X  X  X  X  •  •  I  I  I  •  •  i  l  l  •  •  I  I  I  x  x  x  x  x  x  X  X  The diagrams are grouped i n t o s e p a r a t e l y gauge i n v a r i a n t s e t s . ( I ) gauge boson, would-be Goldstone boson, and ghost loops ( I I ) would-be Goldstone boson l o o p s ( I I I ) p h y s i c a l charged Higgs boson loops (IV) c h a r g i n o s l o o p s (V) f e r m i o n loops (VI) s c a l a r - f e r m i o n loops  53  3.2  One Loop C a l c u l a t i o n of X  •»• yy  u  The Feynman r u l e s l i s t e d  i n Appendix E a r e used t o c a l c u l a t e the m a t r i x  elements which c o n t r i b u t e t o the two photon decay widths of the s p i n - 0 bosons, denoted by X° , i n the one-loop a p p r o x i m a t i o n . for  The i n t e r n a l l o o p s  a l l of the s c a l a r H^, decays c o n s i s t of fermion and s c a l a r f e r m i o n s ,  gauge bosons and gauginos, and p h y s i c a l charged Higgs bosons. Goldstone boson and Fadeev-Popov  The would-be  ghost l o o p s are a l s o i n c l u d e d .  The  one-loop c o n t r i b u t i o n to the two photon decays o f the s c a l a r H^ a r e d i s p l a y e d i n s i x s e t s of diagrams i n f i g u r e 1. invariant.  Each s e t i s s e p a r a t e l y gauge  Set 1 i s the gauge boson l o o p c o n t r i b u t i o n , denoted by a j , w  and i n c l u d e s mixed would-be Goldstone bosons-gauge  bosons and  Fadeev-Popov  Set 2 i s denoted by ajQ and c o n s i s t s o n l y of f u l l would-be  ghosts.  Goldstone boson l o o p s .  Both of these s e t s have the same s t r u c t u r e as i n  s t a n d a r d model Higgs boson to two photon decays [6].  In a d d i t i o n t h e r e a r e  c o n t r i b u t i o n s from loops c o n t a i n i n g the p h y s i c a l charged Higgs boson, H , and the c h a r g i n o s , a  J  the  and a ,  Jx  l,2  x^ (1 1»2). =  respectively.  3  and  4,  c o n t r i b u t i n g amounts  I f the Yukawa c o u p l i n g s a r e n o n - v a n i s h i n g then  f e r m i o n l o o p s o f s e t 5 w i l l g i v e the a j f . F i n a l l y s e t 6 shows the  scalar-fermlon contribution j » a  r  pieces. right all  These a r e s e t s  which c o n t a i n s both gauge and Yukawa  As noted i n the p r e v i o u s s e c t i o n , the s m a l l m i x i n g between l e f t and  types of s c a l a r - f e r m i o n s has been n e g l e c t e d f o r s i m p l i c i t y .  the c o n t r i b u t i o n s , the m a t r i x elements f o r the s c a l a r H°  decays i n t o two photons w i t h p o l a r i z a t i o n v e c t o r s e^ and e below.  2  Combining  (j=l,2,3) are presented  The d e t a i l s of the c a l c u l a t i o n are g i v e n by combining the Feynman  r u l e s of appendix E w i t h the c a l c u l a t l o n a l t e c h n i q u e s o f appendix B. r e s u l t s are  The  l  i  ( 4 w  =  a  j  +  J * G  3  + A  j  J  +  j  ie gM  A  J +a. +a. +a. ) N X H f V J  X  l  J  a J  *  [6+(-8+12X )I(X )] [ w w  2  ie gM T(4ir) M w 2  j G b  [l 2X I(X )][ +  w  2  2/2  1  3  2  V  1  +v 3  U 2  2  +vU_, ] 2  j  ie gM^ 2  1  ™ ^ * * ^ ^  k'x ) +  2ie gM  v U. ,+v,U,  2 23 11J  2  I  ]  j  v  v  w  ^ " i -  •  1  w  W  ^  a  1  vU  2  3  J  1  v U  u  (4ir)  W  e e  U  2  2  with  y V  3  I  +  V < V H ( I -  2  Zh^v.D,,  4  V  2ige e m c v 2  2  2  f  a  - -  L-L-L-  I f  J t  (4ir) M  f  w g2(v,D,-vO  2  )  ~  w  where  = —  IL . f o r f=d-fermions l  V  1  (sfermions)  3  ^ — U^j f o r f=u-fermions  (sfermions)  Furthermore,  f L  N  and  =  I * f n  2  f  f  f  2ie e c 2  [2+(4X - l ) I ( X ) ] V  2  ~ e sin 9 f 2  £  X =  w  ,  = e sin 9 f w 2  R n^/M  2  £  ~  The s u b s c r i p t o f the A's corresponds t o the mass of the i n t e r n a l particle.  Here n^ i s +1  denote sin<p , cos<p +  (-1)  f o r up (down) type s f e r m i o n s .  respectively  +  [see eq. ( 3 . 1 . 3 4 ) ] .  i s 3 f o r quarks and 1 f o r l e p t o n s .  N  -  1  +  +  The c o l o u r f a c t o r c^ is  V  ]  P»q  (3.2.11)  where p and q are the photon momenta, o f the f i r s t respectively,  Also s , c  The gauge i n v a r i a n t q u a n t i t y Ny  = [ g yv  yv  loop  and second photon  and  I ( X ) = |*dx i  l n [ l - J x(l-x)]  The f u n c t i o n AI(A) has a v e r y weak dependence T h i s dependence  (3.2.12)  on A f o r v a l u e s o f A »  1/4.  i s shown i n appendix D.  S i m i l a r l y the m a t r i x elements f o r the two photon decays of the p s e u d o s c a l a r s H^ and H^ a r e computed.  The diagrams which c o n t r i b u t e a r e  d i s p l a y e d i n f i g u r e 2, and c o n s i s t o n l y of f e r m i o n and gaugino l o o p s . m a t r i x elements f o r the p s e u d o s c a l a r H^  v  4/2 W  ±  t  ^Xl  h  4  k  where and n  f  - -  Z f  (3.2.13)  2  l  l  )  2  W + ^ ^ k  +  v  4e e gm c  v  (4n) M  V  2  2  2  w  i s cosx ( s i n x ) f o r k=4 ( 5 ) .  ( 3  ' 2  1 4 )  n  i—LJ. (_L)  =  k  2  e gM^  2  a _  P  ps y v  2  = (  (k=4,5) are  N Ve  k x+a,K f _)  a. = (a, +a, k kXj^  The  2  I(A,)n, F  (3.2.15)  K  (3.2.16)  56  (I) chargino loops  ( I I ) f e r m i o n loops  The  above g i v e s the g e n e r a l amplitudes  boson t o two photon decays component f i e l d supersymmetric parameters next  i n broken  techniques. limit  f o r s c a l a r or pseudoscalar  supersymmetric  (Note t h a t these remain  [21]).  theory, c a l c u l a t e d with n o n - v a n i s h i n g i n the  O b v i o u s l y these amplitudes  c o n t a i n many unknown  such as m i x i n g a n g l e s and masses o f unseen p a r t i c l e s .  two s e c t i o n s a r e some r e a s o n a b l e s i m p l i f y i n g assumptions,  e s t i m a t e o f the w i d t h s . of these widths  In the  and an  A l s o examined c a r e f u l l y i s the a l l o w e d enhancement  i n broken  supersymmetric  3.3 P s e u d o s c a l a r Widths o f X° The p s e u d o s c a l a r widths  gauge t h e o r i e s .  yy o f X° •»• yy can now be c a l c u l a t e d  r e s u l t s p r e s e n t e d i n the l a s t  section.  range f o r the many parameters  a p p e a r i n g i n the a m p l i t u d e s .  However, f i r s t  2  from the  one must e s t a b l i s h a  cases f o r the r a t i o o f the two VEV's, v^ and v , a r e denoted f o r v^ »  Higgs  The two extreme as case A  v^ and case B f o r v^ = v^.  In case A, the mixings  o f the c h a r g i n o s a r e g i v e n by e q u a t i o n  S u b s t i t u t i n g i n t o equations  (3.2.15) and (3.2.16) one f i n d s t h e dominant  c o n t r i b u t i o n s come from x^ and the t-quark;  /2 e g 2  a  = x  l  a  4*  decays  L  I(X 2  x  2 = - ^- — M 3TT 2  thus f o r H^  M  ) cosx sin<p, l m  W  2  I ( X J cosx  2  t h a t t h e r e l a t i v e phases of a  w  r e a s o n why the mixing between  Hence, one e x p e c t s cosx = 1//2.  and a t  (3.3.2)  t  From e q u a t i o n (3.1.26) t h e r e i s no apparent the p s e u d o s c a l a r s s h o u l d by s m a l l .  (3.3.1)  +  tana — M  8  fc  (3.1.35).  x  Notice  a r e d e s t r u c t i v e s i n c e d>, I s i n the  +  first  quadrant.  The w i d t h f o r H? decays i s then o b t a i n e d t o be  a M cos x w 3  r C H j + Y Y )  -  2  2  lU )] t  (4ir) sin 0 2  2  v  M, w 4  w  2  (3.3.3)  w  The width o f t h e H° decay i s o b t a i n e d from the above by s u b s t i t u t i n g for  cosx and M. by 4  At  first  sinx  M. c  5  s i g h t , bounds on the magnitude o f tana can be determined by  low energy phenomenology j u s t as i n the Two-Higgs-Doublet  model.  Again t h i s  i s a c h i e v e d by examining c o n s t r a i n t s from Bhabha s c a t t e r i n g , muon decays [11,12,13] and the K°-K° mass d i f f e r e n c e  [14].  Taken  t o g e t h e r t h e s e gave  tana < 40  Moreover,  (3.3.4a)  one c o u l d a l s o argue f o r the much more s t r i n g e n t bound o f  tana < 12  (3.3.4b)  from t h e o r e t i c a l reasons which take i n t o account p a r t i a l wave u n i t a r i t y  plus  perturbation theory [26]. In the  supersymmetry  t h e r e I s a f u r t h e r c o n s t r a i n t on t a n a .  r e l a t i o n between the masses, M^ and M , 2  and t a n a .  T h i s i s due t o  Now r e c a l l  from  e q u a t i o n (3.1.37) t h a t  g  -  [ (1  + S  in2a + - ^ - ) w  1  /  2  ± (l-sin2a + - ^ - ) w  1  /  2  ]  (3.3.5)  For the case of i n t e r e s t , i t i s e a s i l y proven that tana  =  2M  /  2  As can be seen In equation (3.3.5), x  (3.3.6)  *-  8  2  It must have a mass greater than 20 GeV/c experimental data [27].  w  2  2  <  8^  and M^ > 20 GeV/c . 2  / \  w  e e~ +  < M, < 7 M . 1 w  (3.3.4c)  Hence i n order to remain i n case A, x^  must have a mass no more than ~7 times M^. would be v2 M  i n order to agree with  This then puts  tana for M =80 Gev/c  l i g h t e r of the two charginos.  t n e  The consistent range f o r  Thus the upper bound becomes * v  tana  <  5.7  (3.3.4d)  Figure 3 displays the width of H^ decaying into two photons as a function of i t s mass, f o r the range of allowed x i masses, with the mixing angle x chosen to be T T / 4 .  It i s seen that this width i s t y p i c a l l y of the  order of 60 keV or less for an intermediate mass pseudoscalar. The width i s dominated i n this case by the t-quark loop contribution. The dependence of tana v i a equation (3.3.4c) on the x^ mass i s the reason for the large range of possible widths for a given pseudoscalar mass.  As  the pseudoscalar mass approaches and then crosses the threshold f o r decay into a pair of r e a l t-quarks (m =40 GeV/c ), the r i s e i n the width 2  t  respectively increases sharply and then slows abruptly.  In general this  type of behaviour w i l l occur whenever the threshold f o r pair production of a p a r t i c l e which makes an important loop contribution to the decay width i s crossed.  Similar results hold for H° decays.  F i g u r e 3 - P s e u d o s c a l a r 2y-Decay Width f o r Case A  Case A ( v » v > : 1  2  two photon decay width as a f u n c t i o n of mass f o r the  p s e u d o s c a l a r H° (k=4,5) w i t h mixing angle x=ir/4, f o r the range of a l l o w e d X,  masses.  K.  F i g u r e 4 - P s e u d o s c a l a r 2y-Decay Width f o r Case B  m  Case B (v^=V2): p s e u d o s c a l a r H°  Xi  masses.  in  two  in  photon decay width as a f u n c t i o n of mass f o r the  ( k = 4 , 5 ) w i t h m i x i n g angle X=TT/4, f o r the range of a l l o w e d  62  Next examine case B.  With a = ir/4 one o b t a i n s a g a i n <J>=<|>_=<j> and e x p l i c i t l y +  < > f = \  sin  i  [  _ 1  (1  ]  (3.3.7)  ± - L ]  (3.3.8)  +_*!_) 1/2 4 2 w M  The masses o f the c h a r g i n o s a r e g i v e n by  = J  M  (l -^)l/2  M  2  +  4M  W  and they a r e t h e o r d e r o f M^.  2M  2  w  w  Now t h e two photon widths o f  and  are  e a s i l y o b t a i n e d t o be  a M cos x 3  r(HO-YY) =  2  M  2  [  ( 4 i r ) 2 s i n e M. w 4  —  M.  1  K  X  M w  2  )  v  +  1  —  K  , m  2  X  M w  )  v  sin24 -  =  2  (3.3.9)  M w 2  and s i m i l a r l y s u b s t i t u t i n g s i n x f o r c o s x and M,. f o r 2  K X . ) ]  2  t o g e t T(H°->-YT)'  2  These widths as a f u n c t i o n of t h e i r mass, f o r t h e range o f a l l o w e d c h a r g i n o masses, a r e p l o t t e d  i n f i g u r e 4.  Neither  X  l o o p c o n t r i b u t i o n s dominate,  s i n c e t h e r e i s no tana enhancement.  n  o  r  t  n  e  2  Consequently t h e w i d t h i s much s m a l l e r than i n case A, i . e . l e s s  t-quark  than  25 keV. The upper bound curve i n f i g u r e 4 corresponds t o both c h a r g i n o s h a v i n g mass M , and hence i s s h a r p l y peaked W  3.4  near the t h r e s h o l d a t 2M « W  S c a l a r Widths of X° ->• YY It  i s now s t r a i g h t f o r w a r d t o c a r r y out the same a n a l y s i s f o r the two  photon widths o f the s c a l a r Higgs bosons H° ( j = l , 2 , 3 ) . c o n s i d e r o n l y H^ decays.  For definiteness  I t I s c l e a r t h a t t h e same a n a l y s i s c a n be pushed  through almost v e r b a t i m f o r H° and H ° . 2  3  J u s t as i n t h e case o f t h e  63  p s e u d o s c a l a r s t h e r e i s no apparent r e a s o n f o r the m i x i n g  between these  s c a l a r s t o be s m a l l .  For s i m p l i c i t y , assume t h a t they are a l l a p p r o x i m a t e l y  equal, i . e .  3  «U~  2  -U" j=U=l//3 f o r j = l , 2 , 3 .  As seen i n f i g u r e 1 t h e r e a r e many more i n t e r n a l l o o p c o n t r i b u t i o n s compared to p s e u d o s c a l a r s ; hence, more f r e e parameters i n t e r n a l masses appear.  i n the form of  I t has a l r e a d y been noted t h a t the  combination  XI(X) does not v a r y a g r e a t d e a l over a wide range of v a l u e s f o r X.  Thus,  one does not expect the two photon widths t o be too s e n s i t i v e to the v a l u e s chosen f o r these masses. Observe t h a t the amplitude due dominated  t o f e r m i o n l o o p s o f e q u a t i o n (3.2.6) i s  by the t-quark f o r both cases A and B.  T h i s i s due  to the mass o f  the t-quark b e i n g much l a r g e r than o t h e r fermions i n the minimal 3 quark lepton families universe. presence of the  factor.  F o r case A f u r t h e r enhancement i s due N o t i c e t h a t the s c a l a r - f e r m i o n l o o p  c o n t r i b u t i o n of e q u a t i o n (3.2.7) i s dominated cases A and B.  to the  by the s c a l a r - t o p f o r both  To the e x t e n t t h a t X~I(X~) i s i n s e n s i t i v e t o the c h o i c e o f  s c a l a r f e r m i o n mass, the term i n v o l v i n g N ^ j R  over a l l s c a l a r f e r m i o n t y p e s .  w i l l g i v e zero when summed  The remaining Yukawa term i s p r o p o r t i o n a l t o  the square of the c o r r e s p o n d i n g f e r m i o n mass, and hence the s c a l a r - t o p dominates.  A g a i n i n case A t h e r e i s f u r t h e r enhancement by the V  factor.  I n c o r p o r a t i n g the above c o n s i d e r a t i o n s , one f i n d s t h a t f o r case A the s c a l a r decays have  < ° + rt) -  r  H  |a  w +  a  G +  a  X i +  a  x 2 +  a  H +  a  t +  a~  p  (3.4.1)  where  ie gM 2  a  w  ( 4 l f )  »  -  [6 + (-8+12X )I(X )]U w w  (3.4.2)  ie gM 2  ar G  ^  I X  . ( 4TT )  2  1  +  2  I  < *  > ]  W  (3.4.3)  U  W  - 2 / 2 i e gM^ 2  a  [2 + ( 4 X , - l ) I ( X i )]U sine? +  v X  2-  (4ir)  2  2  2ie M  2h v  2  (4ir )  (3.4.4)  2  2  gM  z  w Sige !!! — 3(4ir) M 2  2  [2 + (4X.-1)I(X,.)]U  2  16ie gm 2  C  tana  (3.4.6)  C  w  2  ^ [ l + 2X~I(X~)]U tana  a~ 3(4TT) M 2  C  (3.4.7)  T  w  T h i s w i d t h as a f u n c t i o n o f the s c a l a r mass i s d i s p l a y e d i n f i g u r e 5. standard  model s c a l a r w i d t h i s a l s o shown f o r comparison.  The  The major  c o n t r i b u t i o n s t o the s c a l a r w i d t h i n t h i s mass range a r e the t-quark l o o p , W-gauge boson l o o p , and t o a s m a l l e r extent  the c h a r g i n o  loops.  For larger  s c a l a r masses, the s c a l a r - t o p and charged Higgs l o o p s w i l l a l s o c o n t r i b u t e , but o n l y near o r above t h e i r t h r e s h o l d s .  D e s t r u c t i v e i n t e r f e r e n c e between  the t-quark and W-gauge boson loops r e s u l t s i n a g e n e r a l l y s m a l l e r w i d t h than i n the s t a n d a r d  model which by comparison i s dominated o n l y by the  W-gauge boson l o o p .  A l l the curves  160  I n f i g u r e 5 r i s e s h a r p l y near  GeV/c , which corresponds t o the t h r e s h o l d f o r W-gauge boson p a i r s . 2  F i g u r e 5 - S c a l a r 2y-Decay Width f o r Case A  Case A ( v ^ » v ) : 2  Two photon decay w i d t h as a f u n c t i o n o f mass f o r the  s c a l a r H° ( j = l , 2 , 3 ) w i t h m i x i n g a n g l e s a l l o w e d x^ masses.  y=U y=U j=l//3, f o r the range o f 2  3  The broken curve shows the s t a n d a r d model Higgs  w i d t h f o r comparison.  boson  F i g u r e 6 - S c a l a r 2y-Decay Width f o r Case B  Case B ( v ^ = V 2 > :  Two photon decay w i d t h as a f u n c t i o n of mass f o r the s c a l a r  H° ( j = l , 2 , 3 ) w i t h m i x i n g a n g l e s ! j U  masses. comparison.  = U  2j  = U  3j  = 1  ^»  f  o  r  t  h  e  r  a  n  8  e  o  f  allowed  The broken curve shows t h e s t a n d a r d model Higgs boson w i d t h f o r  67  S i m i l a r l y f o r case B one  has  ie gM 2  a  = w  ( 4 i r )  [6 +  (-8+12X )I(X )lu/5 w w  [X"  + 2I(X  2  (3.4.8)  ie gM 2  a  -  G  1  )]u/2  (3.4.9)  (4TTV -2/2ie gM^ 2  =  a X\  [2 + ( 4 X , - 1 ) I ( X ) ] U sin24 2  2  2ie M  2h v,  2  3  H  =  (3.4.10)  1  (4TT)2  -rrS (4TT)  2  [1  +  2  z  8ige m 2  X  H  I  (  -2 ^  V]KI  r  i  +  — >  <-->  u  3  gM  4  n  w  2  — 3(4TT) M  [2 +  (4X -l)I(X )]u/2 t  (3.4.12)  t  2  16ie  w  gm - [ l + 2X~I(X~)]u/2  2  2  3(4TT) M  (3.4.13)  2  Substituting  w  these i n t o e q u a t i o n (3.4.1) g i v e s the width r(H°  a g a i n d i s p l a y e d as a f u n c t i o n o f s c a l a r mass I n f i g u r e 6. standard model w i d t h i s shown f o r comparison.  The  + YY) which i s  Once a g a i n the  discussion i s s i m i l a r to  t h a t f o r case A, except t h a t the t-quark l o o p i s not an important c o n t r i b u t i o n h e r e , s i n c e t h e r e i s no tana enhancement. w i d t h i s a b i t l a r g e r , and dominated gaugino the  the  m o s t l y by the W-gauge boson l o o p .  The  l o o p s i n t e r f e r e d e s t r u c t i v e l y w i t h the W-gauge boson l o o p and hence  s c a l a r width I s s t i l l Finally  all  Consequently  s m a l l e r than In the s t a n d a r d model.  note t h a t thus f a r o n l y the example where the mixings U^^  a p p r o x i m a t e l y e q u a l has been used.  possibility,  Now  c o n s i d e r the b e s t case  where the r e l a t i v e phases between mixings i s such t h a t  the  are  F i g u r e 7 - S c a l a r 2y-Decay Width f o r Best Case A  CVJ  1^-  CvJ  (A9M)J  Case A ( v ^ » V 2 ) :  Two photon decay w i d t h as a f u n c t i o n o f mass f o r the  s c a l a r H° ( j = l , 2 , 3 ) w i t h m i x i n g angles 1 ^ j = - U j = U j = l / / 3 , 2  allowed  masses.  3  f o r the range  The broken curve shows the s t a n d a r d model Higgs bos  w i d t h f o r comparison.  dominant loop  contributions  interfere constructively.  i n case B s i n c e the gauge boson and combinations of case A one  can  constructive  gaugino l o o p s c o n t a i n  the  w i t h an o v e r a l l r e l a t i v e minus s i g n . greatly increase  rather  than d e s t r u c t i v e  case s c e n a r i o  enhanced r e l a t i v e to the  same  This w i l l  i n t e r f e r e n c e between the t-quark l o o p s .  Is p l o t t e d In f i g u r e 7 and  possible  However, f o r  the width i f U^_. - " " ^ j *  c o n t r i b u t o r s , namely the gauge boson and f o r t h i s best  T h i s i s not  The  two  main  s c a l a r width  i t i s Indeed  s t a n d a r d model w i d t h , a l t h o u g h not  give  by a  now  great  amount. This  concludes c h a p t e r I I I on the minimal broken supersymmetry model.  The  model, which i s a l s o d e s c r i b e d  and  then a p p l i e d  bosons.  The  i n the  to the c a l c u l a t i o n of the  content d i d not  First  introduced  and  we  remind  s i g n i f i c a n t l y a l t e r the Higgs 2y-decay w i d t h s ,  the  generally  enhancement of the Higgs to f e r m i o n c o u p l i n g s ,  a d d i t i o n a l Higgs d o u b l e t , which l e d to a g r e a t l y i n c r e a s e d Unlike  the  the  the a d d i t i o n a l s u p e r p a r t i c l e  a s m a l l d e c l i n e through d e s t r u c t i v e i n t e r f e r e n c e e f f e c t s .  a g a i n i t was  width.  first  2y-decay widths of the Higgs  r e s u l t s of t h i s c a l c u l a t i o n are a l l new,  reader of some important h i g h l i g h t s .  causing  l i t e r a t u r e , was  due  Once to  the  Higgs 2y-decay  case f o r the Two-Higgs-Doublet model however,  supersymmetry imposes a much more s e v e r e bound on t h i s p o s s i b l e enhancement. The  next c h a p t e r w i l l d i s c u s s  the e f f e c t s of t h i s new  c o n s t r a i n t , as w e l l  the  s i g n i f i c a n c e of the o t h e r r e s u l t s o b t a i n e d thus f a r .  as  IV.  NON-STANDARD SPIN-0 BOSON PRODUCTION  The r e s u l t s of the l a s t  two c h a p t e r s w i l l now  be used t o determine  p r o d u c t i o n c r o s s s e c t i o n s f o r the s p i n - 0 bosons i n ep and e e +  colliders.  -  These c r o s s s e c t i o n s can be e x p r e s s e d i n terms of the Higgs boson to two photon decay widths p r e v i o u s l y c a l c u l a t e d . enough,  I f i t s 2y-decay w i d t h i s l a r g e  the dominant p r o d u c t i o n mode i n these c o l l i d e r s f o r the Higgs boson  w i l l be v i a the two photon f u s i o n mechanism. s p e c i f i c s o f how d e t a i l below.  T h i s mechanism and the  t o c a l c u l a t e the p r o d u c t i o n c r o s s s e c t i o n s a r e d i s c u s s e d i n  The p o i n t i s t h a t the Higgs boson p r o d u c t i o n r a t e s can be  d i r e c t l y r e l a t e d to t h e i r 2y-decay w i d t h s , i f the widths a r e l a r g e T h i s i s indeed the case f o r the supersymmetry  and Two-Higgs-Doublet  p r e v i o u s l y i n t r o d u c e d , i f the enhancement f a c t o r tana Is l a r g e . boson p r o d u c t i o n r a t e s a r e c a l c u l a t e d f o r each of these models e e~ colliders.  These new  +  comparison  enough. models  Spin-0 i n ep and  r e s u l t s a r e p r e s e n t e d f o r d i s c u s s i o n and  below.  D e t e c t i o n o f the Higgs bosons i n these c o l l i d e r s i s a c h i e v e d by o b s e r v i n g a peak In the i n v a r i a n t mass d i s t r i b u t i o n of t h e i r decay p r o d u c t s . For  the Higgs mass range s t u d i e d , these decay p r o d u c t s w i l l  c o n s i s t of two  h a d r o n i c j e t s of p a r t i c l e s , which form from the o r i g i n a l p a i r of quarks t h a t the  Higgs p r e d o m i n a n t l y decays i n t o .  e e ~ machines. +  U n f o r t u n a t e l y we w i l l  SLC c o l l i d e r w i l l scenario.  Hence the c l e a n e s t s i g n a l w i l l  be f o r  f i n d t h a t the p r o d u c t i o n r a t e s at the  be too low f o r o b s e r v a t i o n even i n the most  optimistic  Thus we must t u r n to the h i g h e r l u m i n o s i t y ep machines, a t the  expense of l a r g e r backgrounds and p o t e n t i a l problems i n d i s t i n g u i s h i n g decay j e t s from the i n i t i a l  beam j e t s .  N e v e r t h e l e s s we w i l l  find  promising  r e s u l t s f o r the Two-Higgs-Doublet model, w i t h p r o d u c t i o n r a t e s i n the b e s t  case which a r e r e a d i l y o b s e r v a b l e a t the HERA c o l l i d e r . broken supersymmetry model however, we w i l l are u n o b s e r v a b l e ,  F o r the minimal  f i n d t h a t once a g a i n the r a t e s  as a r e s u l t o f the more severe c o n s t r a i n t s on the  enhancement f a c t o r tana;  The s i g n i f i c a n c e o f these r e s u l t s i s d i s c u s s e d  below.  4.1 The C a l c u l a t i o n T h i s s e c t i o n s t u d i e s the p r o d u c t i o n o f s c a l a r (S°) and p s e u d o s c a l a r (P°) s p i n - 0 bosons i n e l e c t r o n - p r o t o n and e e ~ c o l l i d e r s . +  The ep  semi-inclusive reactions studied are  e + p — • e + S° + X  (4.1.1a)  e + p —  (4.1.1b)  e + P  where X denotes any h a d r o n i c s t a t e s . the c o l l i s i o n i n e q u a t i o n (4.1.1).  u  + X  The quark-parton model i s assumed f o r The e l e c t r o n - q u a r k s c a t t e r i n g  subprocesses a r e  eU)  + Q(q)  — e(JJ')  + S° (h) + Q(q')  (4.2.2a)  e(Jl)  + Q(q) — •  e(j^')  + P° (h) + Q(q')  (4.2.2b)  where Q denotes e i t h e r the u-type  o r d-type quark i n the p r o t o n .  In  e q u a t i o n (4.2.2) the 4-momenta o f the v a r i o u s p a r t i c l e s a r e g i v e n i n t h e i r respective  parentheses.  I t i s w e l l known t h a t the p r o d u c t i o n r a t e f o r the s t a n d a r d model Higgs boson i s v e r y s m a l l [28] f o r an ep c o l l i d e r such as HERA. understood  by examining  ep c o l l i s i o n s , Higgs  coupling.  On  be  the p r o d u c t i o n mechanism f o r r e a c t i o n (4.2.2).  boson p r o d u c t i o n proceeds  d e p i c t e d i n f i g u r e 8. Z-propagators,  T h i s can  The  In  v i a the t - c h a n n e l diagrams  process of f i g u r e 8a i s suppressed  by the  two  although t h i s i s p a r t l y compensated by the l a r g e H°ZZ the o t h e r hand the p r o c e s s i n f i g u r e 8b has no  s u p p r e s s i o n but i s enhanced by the double  such  photon exchange p o l e s .  However,  the H°YY v e r t e x i s of h i g h e r o r d e r thereby r e n d e r i n g t h i s amplitude In most cases the standard model amplitude of f i g u r e 8b, but I t s t i l l  small.  of f i g u r e 8a Is l a r g e r than t h a t  r e s u l t s i n an extremely  s m a l l p r o d u c t i o n r a t e as  d i s c u s s e d below. The  s i t u a t i o n i s more o p t i m i s t i c f o r Two-Higgs-Doublet models.  As  seen  i n the p r e v i o u s c h a p t e r s , both the S°YY and P°YY v e r t i c e s can be enhanced substantially.  F o r the s c a l a r t h i s makes the photon exchange  dominate over the Z-exchange one to lowest  order.  s i n c e the S°ZZ c o u p l i n g remains unchanged  The p s e u d o s c a l a r w i l l be produced  exchange amplitude.  Thus one  amplitude  can express  o n l y through  the photon  the p r o d u c t i o n c r o s s s e c t i o n s of  these spin-0 bosons In terms of t h e i r 2Y~decay w i d t h s . I n the same view one in e e +  -  scattering.  above two  The  should a l s o c o n s i d e r the p r o d u c t i o n of S° and  purpose i s t o compare the r e l a t i v e s t r e n g t h s of the  types of c o l l i d e r s f o r s c a l a r and  pseudoscalar production.  r e a c t i o n s s t u d i e d here are ones s i m i l a r t o e q u a t i o n (4.1.1);  e  e  + e" — •  +  +  +  P°  e  - —•  e  e  +  +  + e~ + S°  + - + 0 e  P  The  namely  (4.1.3a) (4.1.3b)  Being i n t e r e s t e d i n cases where the S°YY and/or P°YY v e r t i c e s a r e enhanced, one a g a i n c o n c e n t r a t e s on the two  photon p r o d u c t i o n mechanism.  same as f i g u r e 8b w i t h quark l i n e s b e i n g r e p l a c e d by e added c o m p l i c a t i o n f o r the e e ~ +  s-channel  +  T h i s i s the  lines.  There i s an  r e a c t i o n not p r e s e n t f o r the ep c a s e .  The  e q u i v a l e n t of the diagram i n f i g u r e 8b s h o u l d be i n c l u d e d .  The  r e s u l t i n g d e s t r u c t i v e i n t e r f e r e n c e w i t h the more dominant t - c h a n n e l exchange graph  causes a s m a l l c o r r e c t i o n .  purposes  T h i s c o r r e c t i o n can be n e g l e c t e d f o r the  of o b t a i n i n g o r d e r of magnitude e s t i m a t e s f o r the p r o d u c t i o n c r o s s  section.  Hence o n l y the dominant t - c h a n n e l process w i l l  Otherwise  the c a l c u l a t i o n i s the same as f o r the ep case, except  does not need t o c o n v o l u t e over p a r t 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 s f o r the p r o c e s s e s i n e q u a t i o n  photon spectrum  used  i s g i v e n by  that  one  distributions.  were c a l c u l a t e d u s i n g the e q u i v a l e n t photon a p p r o x i m a t i o n The  be r e t a i n e d .  (EPA)  (4.1.1) [29 3 0 ] .  [31]  s  6  V  T h i s method r e l a t e s the photon-quark c r o s s s e c t i o n (see f i g u r e 8b) e l e c t r o n - q u a r k c r o s s s e c t i o n f o r the subprocesses  i n equation  D e t a i l s of the c a l c u l a t i o n are d e s c r i b e d i n appendix F.  The  i s u s e f u l i n t h a t e q u a t i o n (4.1.4) can s o l v e d a n a l y t i c a l l y .  (4.1.2). approximation The EPA  been demonstrated to be good to w i t h i n t e n p e r c e n t i n resonance In e e ~ +  c o l l i s i o n s , and  collisions.  i t i s expected  t o the  has  production  t o be of the same accuracy i n ep  As a check the p r o d u c t i o n c r o s s s e c t i o n s were a l s o  calculated  d i r e c t l y u s i n g the Monte-Carlo method (see appendix G), which e v a l u a t e s the Integrals numerically.  The  slow In ep c o l l i s i o n s due  convergence of the Monte-Carlo r o u t i n e i s v e r y  t o the L o r e n t z boost between the l a b and  the  cm frames.  T h i s boost prevents t h e use o f importance  which c o n c e n t r a t e the e f f o r t photon p o l e s . two  sampling  o f the Monte-Carlo r o u t i n e near  N e v e r t h e l e s s the r e s u l t s agree  techniques,  the important  t o w i t h i n the a c c u r a c y o f the  methods. The  r e s u l t s o f the EPA c a l c u l a t i o n f o r the c r o s s s e c t i o n s o f the  subprocess  e q u a t i o n (4.1.2) a r e g i v e n below.  A  _,  a  (3) = 4 a e 2 £ r ( S ° + yy)ln{-^r){ " e 2  Ms  M ln(^ )[p2+2p-3-(2+2p+p2/2)lnp] q  + ( p / 4 ) l n p + (2p +4p-6)ln(l-p) 2  2  2  F o r the s c a l a r one o b t a i n s  2  0  r  - (2.5p +4p-5)lnp 2  (4.1.5)  + (25-24p-p )/4 + ( p + 4 p + 4 ) [ - L i ( l ) + L i ( p ) + ( l n p ) / 2 ] } 2  2  where PHM2Q/§, and s E ( l + q ) quark charges function.  i  2  s  2  the (cm e n e r g y )  2  f o r the subprocess.  a r e g i v e n by ee^ and L i ( x ) = - ^ d t l n ( l - t ) / t  i s the d i l o g a r i t h m  X  F o r the p s e u d o s c a l a r one o b t a i n s the s l i g h t l y  The  different  form  g i v e n by  + (l+p+p /4)ln p 2  2  + ( 2 p + 4 p - 6 ) l n ( l - p ) + (6-4p-7p /4)lnp 2  2  + (47-28p-19p )/8 + ( p + 4 p + 4 ) [ - L i ( l ) + L i ( p ) + ( l n p ) / 2 ] } 2  2  2  (4.1.6)  The  2y-decay widths  previous chapters.  i n e q u a t i o n s (4.1.5) and The  quark-parton  (4.1.6) were c a l c u l a t e d i n the  model i s then used  to e s t i m a t e  the  c r o s s s e c t i o n f o r the p h y s i c a l p r o c e s s e s of e q u a t i o n (4.1.1) by c o n v o l u t i n g over the quark d i s t r i b u t i o n f u n c t i o n s f q ( x ) .  a  (s) -  J  dx E f (x) a  Explicitly  (xs)  where t h i s l a s t i n t e g r a t i o n i s done n u m e r i c a l l y .  (4.1.7)  The  s p e c i f i c quark  d i s t r i b u t i o n s used were  — 4Q f ( x ) = 2.2x  9 R (1-x)  u  — 49 f ( x ) = 1.25x  °  (4.1.8a)  T R (l-x) '°  f (x) = f (x) = 0.27x- (l-x) 1  u  (4.1.8b)  J  d  (4.1.8c)  8 , 1  d  which are taken from r e f e r e n c e [ 3 2 ] . S i m i l a r l y the r e s u l t s f o r the e e ~ +  e a s i l y obtained.  c o l l i s i o n s of e q u a t i o n (4.1.3) a r e  They are simply g i v e n by e q u a t i o n s  w i t h s r e p l a c e d by s, the (cm e n e r g y )  2  of the e e ~ +  (4.1.5) and  system.  (4.1.6)  Of course  there  i s no need t o c o n v o l u t e over p a r t o n d i s t r i b u t i o n f u n c t i o n s , as the r e s u l t i s a l r e a d y i n i t s f i n a l form. Monte-Carlo check The  importance  A g a i n the r e s u l t s agree w i t h the  For t h i s case the l a b and  sampling  numerical  cm frames are the same.  t e c h n i q u e s mentioned above can t h e r e f o r e used,  and  convergence of the Monte-Carlo r o u t i n e i s q u i t e r a p i d . Thus the p r o d u c t i o n c r o s s s e c t i o n s of the spin-0 bosons a r e expressed  i n terms of t h e i r 2y-decay widths  f o r both ep and  e e~ +  now colliders.  4.2  N u m e r i c a l R e s u l t s and The  Discussion  photon-exchange p r o d u c t i o n c r o s s s e c t i o n as a f u n c t i o n of /s i s  g i v e n In f i g u r e 9 f o r the standard accordance w i t h p r e v i o u s distressingly energies.  model Higgs boson i n an ep c o l l i d e r .  c a l c u l a t i o n s [28]  t h i s cross s e c t i o n i s  s m a l l , t y p i c a l l y on the o r d e r of I O *  The  -1  lower curve  i n d i c a t e d i s the c r o s s s e c t i o n due  obscured  S i m i l a r curves  0  cm  2  for collider  i n f i g u r e 10 d e p i c t s the same c r o s s s e c t i o n as a  f u n c t i o n of the Higgs boson mass f o r /s=320 GeV,  alone.  In  to the two  f o r / s = l TeV  a p p r o p r i a t e f o r HERA.  Also  Z-boson f u s i o n mechanism  are shown i n f i g u r e 11.  Although  somewhat i n f i g u r e 10 by the e f f e c t s of phase space, t h e r e i s a  r i s e i n the photon exchange c r o s s s e c t i o n f o r l a r g e MJJ, which i s v e r y apparent i n f i g u r e 11.  T h i s r i s e i s due  I ( X ) as d i s c u s s e d i n appendix D.  The  comes from the W-boson l o o p , and t h r e s h o l d a t MH=2M . w  Standard  f o r l i g h t Higgs. if two  biggest  standard model c o n t r i b u t i o n the  model Higgs boson p r o d u c t i o n i n ep Z mechanism.  At /s=320 GeV  the  cross  o r d e r of magnitude too s m a l l f o r o b s e r v a t i o n even  F o r / s = l TeV  the two  Z mechanism becomes j u s t  the same l u m i n o s i t y can be m a i n t a i n e d , photon p r o c e s s  of the f u n c t i o n  hence the c r o s s s e c t i o n r i s e s near  c o l l i s i o n i s dominated by the two s e c t i o n i s a t l e a s t one  to the behaviour  i s too low.  and  l a r g e enough  the p r o d u c t i o n r a t e f o r the  Thus f o r the s t a n d a r d model, the p r e d i c t i o n  f o r the p r o d u c t i o n o f Higgs bosons i s t h a t ep c o l l i d e r s w i l l not be a b l e t o observe them.  S i m i l a r r e s u l t s hold i n e e ~  w e l l known and  the s t a n d a r d  comparison w i t h  +  machines.  These c o n c l u s i o n s  are  model r e s u l t s have o n l y been shown f o r  the more i n t e r e s t i n g Two-Higgs-Doublet model.  P l o t s of the enhanced c r o s s s e c t i o n s as a f u n c t i o n o f mass, u s i n g tana=40, a r e g i v e n f o r the s p i n - 0 bosons of the Two-Higgs-Doublet model i n figures  10 and  11.  Again  t h e r e i s a peak i n the c r o s s s e c t i o n s o f f i g u r e  11  Photon exchange p r o d u c t i o n c r o s s s e c t i o n w i t h  MH=40,150  GeV/c . 2  The dash-dot (dashed) l i n e i s f o r s t a n d a r d model photon (Z-boson) exchange. The s o l i d (broken) l i n e i s f o r Two-Higgs-Doublet model photon exchange s c a l a r ( p s e u d o s c a l a r ) p r o d u c t i o n w i t h tana=40.  The dash-dot (dashed) l i n e i s f o r s t a n d a r d model photon (Z-boson) exchange. The s o l i d (broken) l i n e i s f o r Two-Higgs-Doublet model photon exchange s c a l a r ( p s e u d o s c a l a r ) p r o d u c t i o n w i t h tana=40.  due t o the b e h a v i o u r o f 1 ( A ) .  However i n t h i s case the t - q u a r k  dominates so t h a t the t h r e s h o l d o c c u r s f o r Mn=2m . t  loop  I n the s c a l a r  case  t h e peak i s l e s s pronounced and i s d i s p l a c e d a t l a r g e r Mg due t o the f a c t o r (4A-1) which m u l t i p l i e s 1(A) i n e q u a t i o n ( 2 . 3 . 4 ) .  Similar  behaviour  i n f i g u r e 10 i s somewhat obscured s i n c e the more r e s t r i c t i v e phase space dominates the shape of the c r o s s s e c t i o n .  F o r a range of Higgs mass, the  enhanced photon exchange c r o s s s e c t i o n s a r e much l a r g e r than the unchanged Z-boson f u s i o n mechanism by r o u g h l y an o r d e r of magnitude. r a t e i s about t h r e e times t h a t of the s c a l a r . s e c t i o n s f o r a Two-Higgs-Doublet model may r e a s o n a b l y l a r g e c r o s s s e c t i o n s (up t o 1 0 energies.  Hence one may  The  pseudoscalar  A l t h o u g h the a c t u a l c r o s s  f a l l below the bounds shown, cm )  - 3 7  2  a r e p o s s i b l e even a t HERA  be a b l e t o observe Higgs boson p r o d u c t i o n i n ep  c o l l i s i o n s w i t h i n the c o n t e x t of the Two-Higgs-Doublet model. P l o t s of the enhanced (tana=40) photon exchange c r o s s s e c t i o n s a r e g i v e n i n f i g u r e 12 f o r the Two-Higgs-Doublet model bosons produced i n e e ~ +  collisions.  The v a r i a t i o n o f c r o s s s e c t i o n w i t h /s f o r MJJ=60 GeV/c , and 2  the c r o s s s e c t i o n v e r s u s Mg f o r /s=150 GeV were chosen t o f a c i l i t a t e comparison reference [33]. above.  are d i s p l a y e d .  These v a l u e s  w i t h the s t a n d a r d model graphs g i v e n i n  The b e h a v i o u r i s s i m i l a r t o t h a t found f o r ep s c a t t e r i n g  The peak i n the a v s . MJJ d i s t r i b u t i o n i s s h a r p e r s i n c e t h e r e i s no  smearing by the p a r t o n d i s t r i b u t i o n s .  The enhanced c r o s s s e c t i o n i s r o u g h l y  an o r d e r of magnitude l a r g e r than f o r the s t a n d a r d model one  [33]  which  makes i t j u s t o b s e r v a b l e as d i s c u s s e d below. As i n the s t a n d a r d model, the i n t e r m e d i a t e mass s c a l a r o r p s e u d o s c a l a r bosons w i l l decay p r i m a r i l y i n t o a p a i r o f heavy q u a r k s , l e a d i n g t o a s i g n a l of two j e t s .  I n f i g u r e 13 the r a p i d i t y d i s t r i b u t i o n s o f the Higgs s c a l a r <p  f o r ep and e e ~ photon exchange mechanisms are g i v e n . +  The  pseudoscalar  Figure 12 - Production Cross Sections f o r e e~ -»• e e~H° +  100  110  120 V  s  (  G  130 e  V  140  +  150  )  Production cross sections as a function of (a) /s f o r M =60 GeV/c (b) Mj for /s=150 GeV. The s o l i d (broken) l i n e i s for Two-Higgs-Doublet model photon exchange scalar (pseudoscalar) production with tana=40. 2  H  83  F i g u r e 13 - R a p i d i t y  0  Distributions  1  '  1  -.75  -.25  1  Y  .25  L_  .75  S c a l a r boson r a p i d i t y d i s t r i b u t i o n i n the cm frame f o r (a) ep •*• eH°X w i t h /s=320 GeV and 1^=40 GeV/c (b) e e " -»• e e " H ° w i t h /s=150 GeV and M =60 GeV. The n o r m a l i z a t i o n i s a r b i t r a r y . 2  +  +  R  r a p i d i t y d i s t r i b u t i o n s are very s i m i l a r . r a p i d i t y i s b r o a d l y peaked about z e r o .  In both cases the cm frame I t i s s t r a i g h t f o r w a r d to obtain  these d i s t r i b u t i o n s from the EPA c a l c u l a t i o n o f appendix F, o r d i r e c t l y the Monte-Carlo c a l c u l a t i o n . and  the cm frame c o i n c i d e .  F o r the e e ~ + e e ~ H ° p r o c e s s , the l a b frame +  The two Higgs  +  decay j e t s s h o u l d t h e r e f o r e stand  out w e l l away from the beam a x i s , p r o v i d i n g a good s i g n a l .  However i n t h e  ep •*• eH°X p r o c e s s , t h e cm frame has a l a r g e v e l o c i t y i n the l a b frame. r e s u l t i n g L o r e n t z boost w i l l  from  shift  The  the s c a l e on the r a p i d i t y d i s t r i b u t i o n i n  f i g u r e 13a by r o u g h l y -1.6 a t HERA.  Hence i n the l a b o r a t o r y frame the Higgs  r a p i d i t y w i l l peak a t l e s s than 10° from the beam a x i s , and a t l e a s t one o f the decay j e t s may be d i f f i c u l t  t o d i s t i n g u i s h from the beam j e t s .  to r e s o l v e t h i s problem, much h i g h e r event  In order  r a t e s may be needed f o r ep  c o l l i d e r s than f o r e e ~ machines. +  The  ep event  r a t e s d i s c u s s e d below a r e f o r HERA assuming /s=320 GeV and  an i n t e g r a t e d l u m i n o s i t y over one y e a r ' s running of 1 . 8 9 x l 0 e e~  cm . -2  The  r a t e s a r e f o r SLC assuming /s=100 GeV and an i n t e g r a t e d l u m i n o s i t y over  +  one  39  year o f 9 . 4 5 x l 0  3 7  cm . -2  In the s t a n d a r d model, Higgs boson p r o d u c t i o n  i s dominated by the Z-boson exchange mechanism and the event year a r e too s m a l l t o be observed.  On the o t h e r hand the c r o s s s e c t i o n s f o r  the Two-Higgs-Doublet model can be q u i t e s u b s t a n t i a l . p o s s i b i l i t y of enhancing  r a t e s o f <3 per  T h i s i s due t o the  the Yukawa c o u p l i n g of the t-quark  i n t h i s model.  Without f o l d i n g i n d e t e c t i o n e f f i c i e n c y , the upper bound e s t i m a t e s on the event  r a t e s i n ep c o l l i d e r s a r e c a l c u l a t e d t o be <65 f o r the s c a l a r boson  and <176 f o r the p s e u d o s c a l a r .  I n e e ~ c o l l i d e r s the c o r r e s p o n d i n g +  event  r a t e s a r e <2 f o r the s c a l a r and <13 f o r the p s e u d o s c a l a r . Although  the c l e a n e r s i g n a l may be found w i t h e e ~ machines, the upper  bound p r o d u c t i o n r a t e s f o r SLC a r e not l a r g e enough.  +  A higher luminosity  e e~ machine, perhaps L E P I I , would be u s e f u l i n p r o v i d i n g both l a r g e r r a t e s and a c l e a n s i g n a l .  On t h e o t h e r hand t h e ep p r o d u c t i o n r a t e s a r e a l r e a d y  q u i t e l a r g e , e s p e c i a l l y f o r the p s e u d o s c a l a r boson. concluded  Hence i t can be  t h a t i t may be p o s s i b l e t o d e t e c t non-standard  s p i n - 0 bosons i n ep  c o l l i d e r s such as HERA. The  r e s u l t s f o r the minimal  broken supersymmetry model a r e n o t as  p r o m i s i n g as f o r Two-Higgs-Doublet models i n g e n e r a l .  The q u a l i t a t i v e  r e s u l t s a r e s i m i l a r , but the more s t r i n g e n t bound on tana i n e q u a t i o n (3.3.4d) g i v e s a r e d u c t i o n i n the p r o d u c t i o n c r o s s s e c t i o n s by r o u g h l y a f a c t o r o f 50.  Thus the best p o s s i b l e event  r a t e s f o r one years r u n n i n g  will  be <3 f o r the p s e u d o s c a l a r i n ep c o l l i s i o n s , and even s m a l l e r r a t e s occur f o r the o t h e r c a s e s .  These r a t e s a r e much too low, and hence i t w i l l not be  p o s s i b l e t o d e t e c t the non-standard  Higgs bosons o f t h e minimal  broken  supersymmetry model w i t h t h i s method. T h i s concludes  chapter IV.  p r o d u c t i o n r a t e s o f Higgs Doublet  and minimal  The new r e s u l t s o b t a i n e d were t h e  bosons i n ep and e e ~ c o l l i d e r s f o r the Two-Higgs+  broken supersymmetry models.  I t was found  machines c o u l d not produce o b s e r v a b l e q u a n t i t i e s o f the Higgs hence the emphasis s h i f t e d  t o ep c o l l i d e r s .  Here i t was found  that e e ~ +  bosons and that f o r the  Two-Higgs-Doublet model, r e a d i l y o b s e r v a b l e r a t e s on the o r d e r o f 100 events per year were p o s s i b l e f o r the HERA c o l l i d e r .  F o r the minimal  supersymmetry model however, p r o d u c t i o n r a t e s were once a g a i n due  t o the s m a l l e r tana enhancement f a c t o r .  conclude difficult  broken unobservable  From these r e s u l t s we may  t h a t the p r o d u c t i o n o f Higgs bosons i n supersymmetry models i s as as i n the s t a n d a r d model.  p o t e n t i a l experimental  More i m p o r t a n t l y we now have a  t e s t o f supersymmetry.  I f Higgs bosons a r e d e t e c t e d  at HERA as allowed by Two-Higgs-Doublet models, then we can r u l e out the  minimal broken supersymmetry model. be r u l e d out i s d i s c u s s e d i n the next  How  g e n e r a l a supersymmetry model c o u l d  chapter.  V.  The  two  calculated  SUMMARY AND  photon decay widths  f o r the two  CONCLUSIONS  of non-standard  s p i n - 0 bosons were  d o u b l e t s e x t e n s i o n of the s t a n d a r d  SU(2)xU(l)  electroweak model, and  f o r the minimal  r e s u l t s were then used  to o b t a i n p r o d u c t i o n r a t e s f o r these spin-0 bosons i n  ep and  e e~ +  broken supersymmetry model.  The  c o l l i d e r s , f o r the i n t e r m e d i a t e mass range (40-160 G e V / c ) 2  studied. F e a t u r e s of the Two-Higgs-Doublet model i n c l u d e the s c a l a r  and  p s e u d o s c a l a r bosons which are p r e s e n t In a d d i t i o n t o the u s u a l s t a n d a r d model Higgs  scalar.  A l s o t h e r e i s the p o s s i b i l i t y of enhanced  fermion  c o u p l i n g s to these bosons r e l a t i v e to the s t a n d a r d model, i f the r a t i o of the two VEV's, t a n a , i s l a r g e . lies  i n the charged  The  l a r g e s t u n c e r t a i n t y of the  calculation  Higgs boson l o o p s (see f i g u r e 15), but f o r t u n a t e l y  their  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 parameters i n the s c a l a r p o t e n t i a l are not large.  I f not, one  instability. by enhancing t-quark  encounters  L a r g e r 2y-decay widths  than i n the s t a n d a r d model are p o s s i b l e  the f e r m i o n l o o p s (see f i g u r e 14).  In the best case  the  l o o p i s enhanced by the upper bound v a l u e of tana=40, and i t  dominates the o t h e r p r o c e s s e s . one  e i t h e r s t r o n g l y i n t e r a c t i n g s c a l a r s or vacuum  of the s c a l a r Higgs bosons.  T h i s occurs f o r both the p s e u d o s c a l a r  The o t h e r s c a l a r would behave as i n the  s t a n d a r d model but w i t h the gauge boson loop suppressed, s m a l l e r width  i s of l i t t l e  u s i n g o n l y phenomenological  and  interest.  The  and hence i t s  bound of tana < 40 was  c o n s t r a i n t s on the mixing of the two  obtained VEV's.  P e r t u r b a t i v e p a r t i a l wave u n i t a r i t y c o n s t r a i n t s [ 2 6 ] , which would l e a d t o the more s t r i n g e n t bound of tana < 12, were not used.  I f one adopts  then the n u m e r i c a l r e s u l t s d i s c u s s e d s h o u l d be s c a l e d a c c o r d i n g l y .  this  88  The minimal broken supersymmetry model i s a s p e c i f i c example o f a TwoHiggs-Doublet model, w i t h some added t h e o r e t i c a l m o t i v a t i o n . it  too has a l l o f the f e a t u r e s j u s t d e s c r i b e d .  Consequently  I n a d d i t i o n t h e r e a r e new  f e a t u r e s a s s o c i a t e d w i t h the u n d e r l y i n g a l b e i t broken  supersymmetry.  S p e c i f i c a l l y t h e charged gauginos and s c a l a r - f e r m i o n s w i l l g i v e a d d i t i o n a l loop c o n t r i b u t i o n s t o the two photon decay w i d t h s . range s t u d i e d  (40-160 G e V / c ) , 2  I n the i n t e r m e d i a t e mass  i t was found t h a t these a d d i t i o n a l  c o n t r i b u t i o n s a r e not v e r y l a r g e and i n f a c t they i n g e n e r a l reduce t h e width by i n t e r f e r i n g d e s t r u c t i v e l y w i t h the u s u a l c o n t r i b u t i o n s .  One  important new f e a t u r e a r i s i n g from the supersymmetry i s the upper bound imposed  on tana by t h e a l l o w e d range o f the gaugino masses.  The upper  limit  of tana < 5.7 i n the minimal broken supersymmetry model i s l e s s than h a l f o f the most r e s t r i c t i v e bound f o r a g e n e r a l Two-Higgs-Doublet s m a l l e r than the phenomenological  bound.  model, and much  The w i d t h v a r i e s as t a n a f o r 2  l a r g e tana, so that the l a r g e s t p o s s i b l e width f o r t h i s model i s from 4 t o 50 times s m a l l e r than one might  have hoped f o r .  A more o p t i m i s t i c  situation  can a r i s e i f the mixings between the s c a l a r s , coming from the b r e a k i n g o f the supersymmetry, have phases  such t h a t c o n s t r u c t i v e i n t e r f e r e n c e o c c u r s  between the W-gauge boson and t-quark l o o p s .  I n t h i s case the s c a l a r width  i s enhanced t o p a r t i a l l y compensate f o r the s m a l l e r tana f a c t o r .  Even  best case p o s s i b i l i t y a l l o w s an enhancement of the s c a l a r w i d t h o f l e s s an o r d e r o f magnitude over the s t a n d a r d model w i d t h .  this than  F o r the p s e u d o s c a l a r  t h i s best case p o s s i b i l i t y does n o t o c c u r s i n c e the r e l a t i v e phases a r e f i x e d , and hence the p s e u d o s c a l a r w i d t h i s d e f i n i t e l y  smaller.  Hence, i t  can be concluded t h a t the supersymmetry imposes a much lower upper bound on the p o s s i b l e tana enhancement o f the two photon decay widths than do TwoHiggs-Doublet models i n g e n e r a l .  Widths o f the o r d e r o f 100 keV a r e the  best one  can hope f o r , f o r both s c a l a r s and p s e u d o s c a l a r s , i n the  broken supersymmetric The  gauge t h e o r y .  p r o d u c t i o n c r o s s s e c t i o n s i n ep and  f o r the non-standard widths.  e e~ +  occur f o r e e ~ +  were too low t o be observed  colliders.  found  photon decay  t h a t the c l e a n e s t  U n f o r t u n a t e l y the p r o d u c t i o n r a t e s  at the SLC  c o l l i d e r f o r a l l cases s t u d i e d .  T h e r e f o r e one must c o n c e n t r a t e on the ep The  c o l l i d e r s were c a l c u l a t e d  spin-0 bosons i n terms of t h e i r two  From the r a p i d i t y d i s t r i b u t i o n s i t was  signal w i l l  minimal  results.  ep upper bound p r o d u c t i o n r a t e s f o r the Two-Higgs-Doublet model are  v e r y l a r g e , being of the o r d e r of 100 concludes  t h a t i t may  i n ep c o l l i d e r s .  events per year a t HERA.  be p o s s i b l e t o d e t e c t these non-standard  The ep r e s u l t s f o r the minimal  are more d i s a p p o i n t i n g .  Hence Higgs  one bosons  broken supersymmetry model  The much s t r i c t e r upper bound on the p o s s i b l e  enhancement In t h i s model l e a d s t o v e r y low p r o d u c t i o n r a t e s f o r the non-standard  spin-0 bosons, which a g a i n are not o b s e r v a b l e .  o b s e r v a b l e p r o d u c t i o n of non-standard  Higgs  models w i l l not be p o s s i b l e a t HERA.  The  Higgs  Thus the  bosons f o r minimal  two  supersymmetry  photon decay widths  of  p a r t i c l e s i n supersymmetry models are o f l i t t l e more importance  i s the case f o r the s t a n d a r d model.  T h i s n e g a t i v e c o n c l u s i o n f o r the  these than two  photon p r o c e s s does not of course prevent supersymmetry from m a n i f e s t i n g itself  i n other processes.  I t should be noted produced  t h a t I f non-standard  s p i n - 0 Higgs  bosons a r e  at HERA by t h i s mechanism, as allowed by g e n e r a l Two-Higgs-Doublet  models, then the minimal r u l e d out.  Thus we  broken supersymmetry model would d e f i n i t e l y  have a p o s s i b l e e x p e r i m e n t a l  Such an o c c u r r e n c e would not however c o m p l e t e l y have o n l y c o n s i d e r e d the minimal  be  t e s t of supersymmetry. r u l e out supersymmetry.  model w i t h t h r e e f a m i l i e s of quarks  and  We  leptons.  I f t h e r e a r e more than t h r e e f a m i l i e s , some o f the a d d i t i o n a l  heavy fermions c o u l d c o n t r i b u t e t o the 2y-decay w i d t h as much as the t-quark does. not  Thus the p r e d i c t e d p r o d u c t i o n  occur  i n the standard  r a t e s c o u l d be much l a r g e r .  model u n l e s s  a d d i t i o n a l f a m i l i e s (>35) a r e used.  However, f o r supersymmetry models the  Thus we c o u l d s t i l l  the same t o the 2y-decay w i d t h .  have a supersymmetric model, a l t h o u g h a more  complicated  than the minimal model. T h i s concludes the p r e s e n t a t i o n o f t h i s t h e s i s .  two  r a t e by a f a c t o r o f  T h i s i s because the t-quark, the new heavy u-type quark, and the new  heavy l e p t o n would each c o n t r i b u t e r o u g h l y  one  will  I n c r e d i b l y l a r g e numbers o f  a d d i t i o n of o n l y one e x t r a f a m i l y l e a d s t o an i n c r e a s e d nine.  This  We have examined the  photon decay widths o f non-standard spin-0 bosons f o r Two-Higgs-Doublet  models i n g e n e r a l ,  and the minimal broken supersymmetry model i n p a r t i c u l a r .  While the models themselves a r e e s t a b l i s h e d i n t h e l i t e r a t u r e ,  their  a p p l i c a t i o n t o the Higgs 2y-decay w i d t h and the subsequent c a l c u l a t i o n o f the v a r i o u s Higgs p r o d u c t i o n  rates a l l represent  models w i t h a d d i t i o n a l Higgs d o u b l e t s , spin-0 boson t o fermion values  (tana) i s l a r g e .  couplings  new r e s u l t s .  I n these  t h e r e i s the p o s s i b i l i t y f o r enhanced  i f the r a t i o o f the two vacuum e x p e c t a t i o n  T h i s i n t u r n l e a d s t o enhanced p r o d u c t i o n  o f these  bosons v i a the two photon f u s i o n mechanism a t r a t e s which c o u l d r e a d i l y be observed a t t h e HERA c o l l i d e r . s u p e r p a r t i c l e content 2y-decay p r o c e s s .  I n supersymmetry models t h e new  w i l l g i v e r i s e t o a d d i t i o n a l c o n t r i b u t i o n s t o the  We found t h a t these a r e not very  l a r g e and t h a t  their  e f f e c t i s t o s l i g h t l y reduce the w i d t h v i a d e s t r u c t i v e i n t e r f e r e n c e w i t h t h e usual c o n t r i b u t i o n s .  The important new r e s u l t a r i s i n g from supersymmetry i s  t h a t i t imposes a much s m a l l e r upper bound on the p o s s i b l e tana of the fermion  couplings  enhancement  than do Two-Higgs-Doublet models i n g e n e r a l .  The  o r i g i n of t h i s a d d i t i o n a l c o n s t r a i n t l i e s i n the e x p e r i m e n t a l l y lower l i m i t s  established  f o r the mass of the supersymmetric gaugino p a r t i c l e s .  Hence  even f o r the best case p o s s i b i l i t y , t h e Higgs bosons o f the minimal 3-family supersymmetry  model cannot be produced a t o b s e r v a b l e r a t e s .  Only  supersymmetry  models w i t h a d d i t i o n a l g e n e r a t i o n s o f heavy f e r m i o n s can  produce Higgs bosons a t r a t e s which c o u l d be o b s e r v a b l e a t HERA. we have a p o s s i b l e e x p e r i m e n t a l t e s t o f supersymmetry of Higgs bosons a t HERA c o u l d r u l e out t h e minimal  Therefore  i n t h a t the d e t e c t i o n  model.  92  BIBLIOGRAPHY  1.  G. A r n i s o n e t a l . , UA1 C o l l a b o r a t i o n , Phys. L e t t . 122B, 103 ( 1 9 8 3 ) ; i b i d , 126B, 398 ( 1 9 8 3 ) .  2.  M. Banner e t a l . , UA2 C o l l a b o r a t i o n , Phys. 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Thesis, University of B r i t i s h Columbia, (1984),  Sobol, The Monte Carlo Method, MIR Publishers, Moscow (1975).  APPENDIX A - SOME ACCELERATOR PROPERTIES  Table I I I - A c c e l e r a t o r Properties  Location  Collider  Type  SLAC  SLC  e e~  3xl0  3 0  100  CERN  LEP I  e e~  6xl0  3 1  100  CERN  LEP I I  e e~  2xl0  3 2  170  DESY  HERA  ep  6x10  3 1  320  U.S.A.  SSC  pp  10  +  +  +  Luminosity  (cm~ s ) 2  3 3  - 1  / s (GeV)  40,000  APPENDIX B - EVALUATION OF FEYNMAN DIAGRAMS  C o n s i d e r a s p i n - 0 Higgs p a r t i c l e d e c a y i n g i n t o two photons w i t h 4-momenta k^.k^  a  n  d  polarization vectors e^Ckj^.e^k^) respectively.  One  would a p r i o r i expect the m a t r i x element f o r t h i s p r o c e s s t o have the gauge invariant  forms  *  v v  =A[g  y v  (B.la)  -kjkj/kj.kj]  lct 2e k  / k  (B.lb)  i' 2 k  f o r a s c a l a r or p s e u d o s c a l a r Higgs p a r t i c l e r e s p e c t i v e l y .  These terms a r e  the o n l y t e n s o r s one can make from the two independent momenta ( ^ ^ 2 )  which  do not v a n i s h when the s c a l a r product i s taken w i t h the p o l a r i z a t i o n vectors.  The c o n t r i b u t i o n s of the d i f f e r e n t Feynman diagrams t o t h i s m a t r i x  element a r e shown below. Each s e t i s s e p a r a t e l y gauge i n v a r i a n t as i s demonstrated.  Although  the d e t a i l s of the c o u p l i n g s t r e n g t h s w i l l depend on the model s t u d i e d , the methods of c a l c u l a t i o n i l l u s t r a t e d a r e the same f o r the s t a n d a r d model, Higgs-Doublet models and supersymmetric models.  Consequently o n l y  one  r e p r e s e n t a t i v e example i s e v a l u a t e d f o r each s e t o f Feynman diagrams. s t a n d a r d model Feynman r u l e s needed  Two-  The  can be found i n r e f e r e n c e [ 1 0 ] .  B . l S c a l a r Higgs 2y-Decay v i a Fermion Loops The c o n t r i b u t i o n of the f e r m i o n l o o p i n f i g u r e 14 i s demonstrated f o r the s t a n d a r d model.  Such a c o n t r i b u t i o n w i l l a r i s e from quarks and  i n a l l the models s t u d i e d , as w e l l as from c h a r g i n o l o o p s f o r  leptons  supersymmetry.  96  f \ w w  F i g u r e 14 - Fermion Loop C o n t r i b u t i o n t o S c a l a r 2y-Decay  The m a t r i x element, o b t a i n e d u s i n g the s t a n d a r d model Feynman r u l e s , i s  M  wv.  J_  d ^ _  (2ir)  T  r  ( ^ ! ! f ) ( J _ ) ( _  [  w  H  e  y ) ( _ i _ )  e  1  n  C-  x  where h=kj+k2 i s the Higgs  i  f  n  f  V K^^  l e  v  4-momentum and k ^ * ^  ]  (B.i.i)  are the photon 4-momenta.  E v a l u a t i n g the t r a c e r e s u l t s i n  r — <  -2ge e m . = 4 c 2  M  2  2  w  r - >v 2c  + gyvr[- „a C™  +  y  A  v  -  2  i  k  +  c  i  cywkX  +  + (nr|-k .h)C ] 1  0  where the loop i n t e g r a l s CQ,C^,C2 a r e g i v e n i n appendix C.  }  (B.1.2)  S u b s t i t u t i n g the  r e s u l t s from appendix C, and r e t a i n i n g o n l y those terms c o n s i s t e n t w i t h the form of e q u a t i o n ( B . l ) g i v e s  k rv,ky.  2ige 2m2 2  e  M  [- 2 - ( 4 X - 1 ) I _ f  16TT M  1  *  2  w where X=m2/M2 and 1 ^  ] [  g  "  (B.1.3)  v  k..k.  1 2 i s g i v e n by e q u a t i o n ( C . l ) .  An a d d i t i o n a l f a c t o r of 2  Is i n c l u d e d i n e q u a t i o n (B.1.3), a r i s i n g from the c r o s s e d diagram where i d e n t i c a l photons are i n t e r c h a n g e d . uv g  Note that the c o e f f i c i e n t s of both the  v u and k^k2/k^»k2 terms a r e i d e n t i c a l , e x p l i c i t l y d e m o n s t r a t i n g the gauge  i n v a r i a n c e o f the m a t r i x element. B.2  S c a l a r Higgs 2y-Decay v i a S c a l a r Loops The c o n t r i b u t i o n of the s c a l a r loops i n f i g u r e 15 i s demonstrated f o r  the s t a n d a r d model i n the ' t Hooft-Feynman  gauge.  l o o p p a r t i c l e i s the would-be G o l d s t o n e boson.  In t h i s case the s c a l a r  In Two-Higgs-Doublet models  t h i s s e t of Feynman diagrams a r i s e s f o r both would-be G o l d s t o n e bosons and p h y s i c a l charged Higgs bosons.  S i m i l a r c o n t r i b u t i o n s occur f o r  supersymmetry models, a l o n g w i t h s c a l a r - f e r m l o n  loops.  F i g u r e 15 - S c a l a r Loop C o n t r i b u t i o n t o S c a l a r  2y-Decay  The m a t r i x elements, o b t a i n e d from t h e s t a n d a r d model Feynman r u l e s , a r e  S  (  3  (2*)*  )  (q-k.)2- 2  1  1  M  -igM* x ( i — ) ( — ^ C - T T ) (q-h) -M 2M q -M 2  2  2  2  W W  M  • 5  s(b) S Q t > ;  L  /^ (2 r)'  V  (q-h) -M2  t  1  2M  2  q -M2 2  w  R e w r i t i n g i n terms o f t h e loop i n t e g r a l s C Q , C ^ , C 2 , C  M  s(a)  =  T Swf W "" ^  Combining  w  "  ^  4  c  l  C  k  l  "  2  C  i 2  "  k  2  C  l  k  l  +  (B.2.I.)  B (. 2 . 1 W  M l — J ^ )  (21«VH  1  w  k  i(  2 k  from appendix C g i v e s  l  + k  2>  V c  0l  < ' ' > B  '  2  2 a  < -- > B  2  2b  these two m a t r i x elements w i t h the r e s u l t s o f appendix C,  r e t a i n i n g terms o f the form i n e q u a t i o n ( B . l ) and i n c l u d i n g a f a c t o r o f 2 for  the i d e n t i c a l photons  ie2gM2 M  s  V 8  =  gives  >  k  , (1 + 2 X I J 16ir M w  where X=M^/M2.  2  1  (g  ! V  V  J  K  l2 k  )  CB.2.3)  A g a i n note t h e e x p l i c i t gauge i n v a r i a n t form o f eq. (B.2.3)  As d i s c u s s e d i n appendix D, t h e f a c t o r (1+2XI_^) i s s m a l l f o r X »  1/4 and  consequently t h e s c a l a r l o o p c o n t r i b u t i o n i s u s u a l l y s m a l l as w e l l .  B.3  S c a l a r Higgs 2y-Decay  v i a Gauge Boson  Loops  The diagrams I n f i g u r e 16 a r e t h e l a r g e s t s e t t o c o n t r i b u t e to t h e s c a l a r Higgs decay w i d t h .  The l o o p p a r t i c l e s i n c l u d e W-gauge bosons,  99  F i g u r e 16 - Gauge Boson Loop C o n t r i b u t i o n  to S c a l a r 2y-Decay  100  would-be Goldstone bosons and the Fadeev-Popov g h o s t s . arises i na l l in  Such a c o n t r i b u t i o n  the models s t u d i e d , and i s demonstrated f o r the standard model  ' t Hooft-Feynman gauge.  As was done i n the p r e v i o u s s e c t i o n s , one  o b t a i n s the m a t r i x elements g i v e n by  -  -e  2  «JlOC--9cX-^ 5  8  +  + g  = - 3e gM  g  2  M  ^  M  w  yv _ (d)  6  2  (B.3.1a)  Q  (B.3.1b)  + 3  C  y  - 4C /  + g (C°  V  y V  2  w „yv _ __u.v . _v,u . ..y.v, - ^ [ C ^ " 5CjkJ + c j k j + 2 ^ C  a  + 2  C l  .k )]  (B.3.1C)  2  r  = - e2g 3 M  2  (f)  (- C  U V  ^ C  s  e gM M  b  l  g M  + g  M  K  C'  y V  = - f f j L [-  c )  V  - C^O^+kj) - 2 ^ - ^ - Hi »V. C )]  (2C^  y V  2 a  + (4k +2k )-C 1  2  1  Q  - UL^VJCQ)]  (B.3.1d)  (B.3.1e)  Q  2  2 ^  ^  C  (B.3.1f)  Q  --  (B  ! 8)-- T [- ! - M V  !  £  = f?JL  [2c  2C  wv _  M?* - - e g>l g Ci; w 2  M j  V j  )  = - e  2  g M  w  P  V  [ C  2 c  V  y v k  2C  _  4 c  v y k  +  4  k  v y j k  c  (  C"  2  V  - c K ]  ie gM  >  l  h  )  Substituting  f o r m , w i t h X=M^/M , a s 2  k X  2  Z [6 + <-8+12X)I_ ][g  wv  1  2  3  (B.3.1J)  f r o m a p p e n d i x C g i v e s t h e e x p e c t e d gauge I n v a r i a n t  16TT  >  LG)  (B.3.1I)  where o n l y t e r m s o f t h e f o r m i n e q u a t i o n ( B . l ) h a v e b e e n k e p t .  M'  B  3  -  (B.3.2) 1  2  101  B.4 P s e u d o s c a l a r Higgs 2y-Decay v i a Fermion Loops The f e r m i o n l o o p c o n t r i b u t i o n of f i g u r e 17 i s demonstrated f o r the TwoHiggs-Doublet model.  S i m i l a r c o n t r i b u t i o n s a r i s e i n supersymmetry models.  The m a t r i x element i s g i v e n by  M  f- *  /^  [(^)(-^V K qrz -)(-iee y ) (2TT ) f l f (Fh=m7K- f 5^ T r  y  q m  1  v  i  1  m  q fc  m  X  where y^ i s the c o u p l i n g Evaluating  f  1  y  Y  (B.4.1)  s t r e n g t h of the p s e u d o s c a l a r to two f e r m i o n s .  the t r a c e r e s u l t s i n  yvaB  (B.4.2)  r  F i g u r e 17 - Fermion Loop C o n t r i b u t i o n t o P s e u d o s c a l a r 2y-Decay  102  S u b s t i t u t i n g f o r the loop i n t e g r a l s C Q J ^ photons  a  n  d  m u l t i p l y i n g by 2 f o r i d e n t i c a l  gives  e e y .m, 2  f'  which i s the form  =  „  2  " ^  -  1  E  k  la 2B k  / k  l* 2 k  ( B  - ' 4  3 )  expected.  T h i s concludes the d e m o n s t r a t i o n o f the e v a l u a t i o n o f the Feynman diagrams  which c o n t r i b u t e t o the two photon decay w i d t h o f Higgs  bosons.  The r e l a t i v e weights due t o the c o u p l i n g s t r e n g t h s w i l l v a r y f o r d i f f e r e n t models, but the b a s i c s t r u c t u r e as shown i n e q u a t i o n s (B.1.2), (B.3.1) and (B.4.2) i s the same.  (B.2.2),  103  APPENDIX C - EVALUATION OF LOOP INTEGRALS  The one-loop Feynman diagrams which c o n t r i b u t e t o the Higgs two photon decay width c o n t a i n s e v e r a l i n t e g r a l s which a r e e v a l u a t e d  i n t h i s appendix.  A useful definition i s  i (XM n  - {dx x  n  m[  ^tyTx)  (  where X=m /M2 i s the r a t i o o f the l o o p p a r t i c l e mass squared 2  C  over  D  the Higgs  mass squared. First  c o n s i d e r the i n t e g r a l  c  =  q (2TT )**  j  (C.2)  i  a  0  [q2- 2][( -k )2- 2] ( -h)2- 2] m  q  1  m  [  q  m  where h=k^+k2 i s the Higgs 4-moraentum and k^.k^ a r e the photon  4-momenta.  Expanding w i t h Feynman parameters g i v e s  C U  = /  fdxfdyfdz r(3) 6(l-x-y-z) (2^)** ° ° [ (q2- 2)x+(q2-2q.k -m )y+(q -2q.h+M2- 2) ]3 6  2  m  Performing  ^  2  1  m  z  the i n t e g r a l over the l o o p v a r i a b l e q g i v e s the r e s u l t  C  0 U  =  — fixfdy - 16TV M2 0 t» [ x ( l - x - y ) - x] 6  (  1  X  y  (C.4)  )  2  F i n a l l y i n t e g r a t i n g over y and u s i n g d e f i n i t i o n  c  o °  — 16ir2M2  T  (C.l) gives  (C5)  -l 1  Second c o n s i d e r the  C  =/  y  1  Again  ** (2iry* d  integral  £ [q2-m2][(q2- 2)- 2][( -h)2- 2]  q  k  m  expanding w i t h Feyman parameters and  q  (C.6)  m  e v a l u a t i n g the i n t e g r a l over  the  loop parameter q g i v e s  =  v C  i  — fdx/dy  16TT M2  =  (C.7)  0 0  2  1  [x(l-x-y)-X]  Then i n t e g r a t i n g over y and u s i n g d e f i n i t i o n ( C . l ) r e s u l t s i n  C  i ^ij =  T h i r d c o n s i d e r the  c  yv  [  k  i  (  I  o  - -i> z  (  c  -  8  )  integral  _  =;  _a_g  _  _  ( ( J # 9 )  [q-m2][(q-k)2-m2][(q-h)2-m2]  (2TT )**  2  1  which i s expanded u s i n g Feynman parameters as b e f o r e .  U n l i k e the  previous  uv two  cases, C  i s not f i n i t e and  2  be r e g u l a r i z e d . dimensional  v  v  r e g u l a r i z a t i o n , and  2  = —4T  i  yields  [yk +(l-x-y)h ][yk^+(l-x-y)h ] U  U  V  9(1-x-y)  W<*  16TT M2 0 0  [x(l-x-y)-X]  2  +  the loop v a r i a b l e q must  T h i s i s performed i n d=4-e dimensions u s i n g the method of  1 c  the i n t e g r a l over  tfr iV*  7  {  A+  l n [  x-x(i-x-  y )  ]  i  ( c  -  1 0 )  105  where  A = ^+ij)(l)+ln(4iru /m )  by  regularization.  2  the  c  y v  The  —  and  2  u Is the  a r b i t r a r y mass s c a l e  i n t e g r a l over y then  [ (~ \ + I  introduced  gives  " M ^ X k J k i J + kjk*) ]  Q  (C.ll)  16TT M2 2  +  i  M  —  [ A + 1 - I  Q  + 2XI_  ] +  X  (terms -  kjk^.kjkl*)  pv When c o n t r a c t i n g 4-e  i n d i c e s on  dimensions so  that  g  i t must be  =4-e.  W  remembered t h a t  the m e t r i c i s i n  Thus  P  2 p 7T7 t 16*  c  =  A +  X I  2  Next c o n s i d e r  the  c  .  = j  _iHl_  I  (2TT) *  [q -m ] [ ( q - h ) - 2 ] 2  - j J * i - r (2*)* *  A g a i n the  *  1 2 )  d  x  r t)  d  (C.13)  2  2  m  Expanding w i t h Feynman parameters  .  ( c  integral  4  c  -i 1  gives  n i ) 6 i-x[(q2- 2) ( 2_ .h+M2- 2)y]2  v  (  m  i n t e g r a l over the  C  x f  q  y )  2 q  loop parameter must be  = — — 16*2  [ A - I  (C14)  m  n  regularized  ]  giving  (C.15)  u  Similarly  C - - / - ^ (2TT )*•  1 tq - 2][(q-k )2-m2] 2  m  1  =  _J_A 16ir  2  (c.16)  106  APPENDIX D - PROPERTIES OF THE  FUNCTION  I(X)  T h i s appendix d e s c r i b e s the f u n c t i o n I ( X ) where  1(A) = /dx  I  X-x(l-x) ln[ X  (D.l)  Note t h a t I ( X ) i s j u s t I _ ^ as d e f i n e d i n e q u a t i o n ( C . l ) .  This function  appears i n a l l of the m a t r i x elements f o r s p i n - 0 boson to two widths.  Evaluating equation (D.l) gives  K X )  2[sin-l(^ )]  2  r  (D.2)  =  J . 9 1 ? r 1+/1-4X  1  The  f u n c t i o n XI(X)  imaginary  the r e a l and  r  i s p l o t t e d i n f i g u r e 18.  dependence on X f o r v a l u e s of X » The  photon decay  1/4,  p a r t i s zero f o r X > 1/4, imaginary  and  The  t h e r e i s a sharp peak a t X=l/4.  to the t h r e s h o l d to produce a p a i r of p a r t i c l e vertex.  t h i s case the peak a t X=l/4 i s  e q u a t i o n (D.2)  Hence  one  and hence i n  suppressed.]  i s w e l l below t h r e s h o l d , i . e . X » l / 4 ,  can be approximated  In making simple comparisons.  by I ( X ) =  +  then  ° ( ^ ~ ) which i s u s e f u l 2  In p a r t i c u l a r 1 + 2XI(X) = 0 ( X  _ 1  ) , and  hence  the s c a l a r l o o p s i n e q u a t i o n (B.2.3) w i l l make s m a l l c o n t r i b u t i o n s below threshold.  real  behaviour whenever a t h r e s h o l d i s c r o s s e d .  [ I n e q u a t i o n (B.1.3) I ( X ) i s m u l t i p l i e d by a f a c t o r of 4X-1,  P r o v i d i n g t h a t one  Both  0.  r a t h e r than v i r t u a l p a r t i c l e s at the H i g g s — l o o p should see t h i s peaking  r e a l p a r t shows a v e r y weak  and peaks between 0 < X < 1/4.  p a r t s go to zero as X  P h y s i c a l l y X=l/4 corresponds  1+/1-4X-I  F i g u r e 18 - P l o t o f the F u n c t i o n XI(X) vs X  The  solid  (broken) curve shows the r e a l ( i m a g i n a r y ) p a r t .  108  APPENDIX E - FEYNMAN RULES FOR  The  e f f e c t i v e Lagrangian  MINIMAL BROKEN SUPERSYMMETRY  f o r broken supersymmetric  s t a n d a r d models i s  g i v e n i n e q u a t i o n (3.1.2)-(3.1.8) i n component f i e l d s . content of the t h e o r y i s d i s p l a y e d i n t a b l e I I . fields,  Due  the ghost  to  the FP  field.  The  i s thus the same as t h a t of the s t a n d a r d model  to the mixing of the s c a l a r s a l l three f i e l d s H°  to  The  [34] i s chosen.  symmetry i s f i x e d to be the ' t Hooft-Feynman gauge.  The Fadeev-Popov (FP) ghost [35].  field  In working w i t h component  the u s u a l Wess-Zumino gauge of supersymmetry  gauge of the S U ( 2 ) x U ( l )  The component  As expected  the p s e u d o s c a l a r s H^  ( j = l , 2 , 3 ) couple  and H^ do not  couple  ghosts. r e l e v a n t c o u p l i n g s f o r the c a l c u l a t i o n of the amplitudes  are g i v e n i n d i f f e r e n t  s e t s below.  The  f i r s t s e t i n v o l v e s the  c o u p l i n g s t o f e r m i o n s , s f e r m i o n s , charged Higgs  T h i s i s d i s p l a y e d i n f i g u r e 19.  The  their  i n the two  Higgs  Demanding t h a t t h i s gauge i n v a r i a n c e h o l d s i n supersymmetry  i s a r e a s o n a b l e c o n d i t i o n , which g r e a t l y s i m p l i f i e d (and c o n s e q u e n t l y couplings.  and  s e t of diagrams i n  f i g u r e 16 i s gauge i n v a r i a n t i n the s t a n d a r d model and d o u b l e t model.  +  scalar  ± ± bosons, G , as w e l l as the gauge bosons W and  companion would-be Goldstone the FP g h o s t s .  bosons H  of X° -»• "YY  the H+H^O v e r t e x ) .  the  ff^GH^  vertex  F i g u r e 20 g i v e s a l l the photon  The mixed s t a t e s of charged H l g g s i n o s and W-gauginos, x-^ and  have c o u p l i n g s t o the Higgs  s c a l a r s g i v e n i n f i g u r e 21.  x>  Sewing t o g e t h e r the  v e r t i c e s g i v e n below g i v e s the f u l l s e t of Feynman diagrams d i s p l a y e d i n f i g u r e 1 f o r H°  to two  photon  decays.  For the p s e u d o s c a l a r s the c o u p l i n g s are s i m p l e r . r e l e v a n t ; namely x^  a t l  represented i n figure  d x  2  22.  a n <  2  Only two  * f e r m i o n c o u p l i n g s are i n v o l v e d and  are these are  109  F i g u r e 19 - Feynman Rules f o r S c a l a r H° C o u p l i n g s  -im V f  H  w  2M,  H-  G'  -IgM  6"  2M  „--" f tn. R / L  *  "  2  i  h  w  ig^tv.U-.-v.u,.) ~ ' ' " Nl..R/L 2C0S 9, w 2  '  V M .+ V U . rij 2 2j  -igM +igM  H  2  2  V  fl  w2 , ,l  m  "f  2  v'  . .j 2 2j U  + V  U  2 " M  H  t  v U +v U I Ij 2 2j  2  '  vU + v U . I IJ 2 2j  .» 6  Here j=l,2,3 and the d e f i n i t i o n s on page 54 a r e used.  V s  110  F i g u r e 20 - Feynman R u l e s f o r Photon C o u p l i n g s  "  A / W W * ; :  'i  ~  to  i  *L-  lee.tp + p L  L/R  P  x  o  vwwv<^ Y  ±iep  8  +  W*  / w \ A ^ 7 P  |  P  W 3  . X  Kin  W"  - [ P,-P2 xV• P2" 3 >' »'X* 3" l *' X ^ ,e  2 , e  {  (  )  V  P  )  5  (p  ?  P  ^  ,  0  H  .  -MP-P^  ( G 1  *  f9^  V v  £  A A A  <C^ J  "  l  e  e  f  r  / *  F i g u r e 2 1 - Feynman Rules f o r C h a r g i n o - S c a l a r H° C o u p l i n g s  <T Hj  +  +  2 i  I  — —<x H. J  "7^r(s-c u.,+s c_u )  1  e  A  7r  2  2  V  ^  ( s  c +  - .j u  + s  - ^ c  )  112  F i g u r e 22 - Feynman Rules f o r C h a r g i n o - P s e u d o s c a l a r 11° C o u p l i n g s  H  X  2  g  X s ^ ( V . S . C . + V . S . C J T ;  H  f  _T7  f  f  H  Here k=4,5 and n = + l f  fermions.  (~1) f o r up (down) type  A l s o n, =cosx ( s i n x ) f o r k=4 ( 5 ) . k  113  APPENDIX F ~ EQUIVALENT PHOTON APPROXIMATION  The e q u i v a l e n t photon a p p r o x i m a t i o n (EPA) [29,30] i s used to s i m p l i f y the c a l c u l a t i o n of the c r o s s s e c t i o n f o r the e l e c t r o n - q u a r k process i n f i g u r e 8b.  scattering  I t i n v o l v e s t r e a t i n g one of the exchange photons as a  p a r t o n - l i k e o b j e c t , c a l c u l a t i n g the photon-quark c r o s s s e c t i o n and then c o n v o l u t i n g over a photon spectrum d i s t r i b u t i o n to o b t a i n the e l e c t r o n - q u a r k cross  section. The photon-quark " s u b p r o c e s s " i s d e p i c t e d i n f i g u r e 23.  This leads to  a m a t r i x element g i v e n by  _ M = u(  )[iee Y ]u( P  q i  P l  "ig )(— ^-)M  a V  E  (-p ,-p +q )£ 2  1  1  C T  (p ) 2  (F.l)  where M ° ( k ^ , k ) i s the Higgs t o 2y decay w i d t h m a t r i x element, and V  2  i s the p o l a r i z a t i o n v e c t o r of the Incoming photon. variables w i l l  be used.  The s t r u c t u r e of M  .  (g  CTV  The u s u a l  e a  (P2)  Mandelstam  ( k ^ , k ) must be o f the form 2  kV M  ^  V  =F  (F.2.)  a V  for a s c a l a r Higgs, or  k M  (k  l f  k ) - F [e 2  k  - j ^ - )  f o r a p s e u d o s c a l a r , as d e s c r i b e d e a r l i e r i n appendix B.  (F.2b)  114  F i g u r e 23 - Photon Quark Subprocess  Substituting of the  these i n t o e q u a t i o n ( F . l ) l e a d s to photon quark c r o s s s e c t i o n s  form  ae F (scalar) = — 3 — 4s 2  a  2  2  sCs-M ) 2s (s-M ) { ln[ — ] + — (s-M )ln[ — ] m 2  2  2  2  2  2  q M H  28(8-4) — 2 -  }  (F.3a)  and ae F (pseudoscalar) = — ^ — 4s 2  a Y  q  2s + —  respectively.  s(s-M )  2  2  J Xn[ 2  2  1 m M^ 2  q  s(s-M ) (s-Mplnt — ] 2  2  3(s-M ) — 2  2  }  These are then c o n v o l u t e d w i t h the photon spectrum  (F.3b)  in  e q u a t i o n (4.1.4) t o g i v e the e l e c t r o n - q u a r k s c a t t e r i n g c r o s s s e c t i o n s . The r e s u l t s are shown In e q u a t i o n s  (4.1.5) and  (4.1.6).  I d e n t i c a l s t e p s a r e taken to o b t a i n e l e c t r o n - e l e c t r o n c r o s s s e c t i o n s w i t h t h i s method, and may  be needed.  i n t e g r a l s may electron-quark  f o r t h a t matter any 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 which  The main advantage i n u s i n g the EPA be done a n a l y t i c a l l y . s c a t t e r i n g process,  t r e a t e d with EPA.  method i s t h a t a l l  I t should be noted t h a t f o r the the e l e c t r o n ' s photon must be the  one  T h i s i s to a v o i d the subsequent c o m p l i c a t i o n s which a r i s e  when the quark i s c o n v o l u t e d  over the u s u a l p a r t o n  distributions.  APPENDIX G - MONTE CARLO INTEGRATION ROUTINE  T h i s appendix c o n t a i n s the FORTRAN code f o r n u m e r i c a l l y c a l c u l a t i n g a g e n e r a l 2 to N body s c a t t e r i n g c r o s s s e c t i o n . program f i r s t  developed  i n reference  [36].  I t i s a g e n e r a l i z a t i o n of a  A l l of the i n t e g r a t i o n r o u t i n e s  used were m o d i f i e d v e r s i o n s of t h i s program.  The  "amplitude"  which must be  s u p p l i e d f o r the g i v e n p r o c e s s , r e f e r s to the m a t r i x element squared. 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 are e a s i l y o b t a i n e d by u s i n g the BIN  subroutine,  which s t o r e s the d e s i r e d v a r i a b l e w i t h the a p p r o p r i a t e weight f o r each event.  The  chapter 4. reference  convergence of the r o u t i n e was  where noted  in  A g e n e r a l d e s c r i p t i o n of the Monte-Carlo method can be found  in  [37].  q u i t e good except  _.  c  C C C C C C C C  PROGRAM MONTE.FOR FILE FILE FILE FILE FILE FILE  5 6 74 75 76 77  INPUT FROM SOURCE OUTPUT TO SINK ANSWER EVERY 1000 P T S FOR DISPLAY STORE L A T E S T ANSWER FOR RESTART STORE MASSES FOR RESTART MISCELLANEOUS FOR D I S P L A Y  c  CC  DECLARE  VARIABLES  c  C C C C C C C C C C C C C C C C C C C C C C C C C C C  I I . J d . K K ARE ABBREVIATIONS FOR OFTEN USED INTEGER EXPRESSIONS NPTS I S THE NUMBER OF POINTS TO BE USED IN THE MONTE CARLO N I S THE NUMBER OF P A R T I C L E S I N THE FINAL STATE SEED I S A PARAMETER NEEDED BY RANDOM NUMBER GENERATOR GGUBFS P I S THE I N I T I A L P A R T I C L E MOMENTUM I N THE LAB CM FRAME M ( I ) I S THE MASS OF P A R T I C L E I M 2 ( I ) I S THE SQUARE OF M ( I ) MSUM(I) IS THE SUM OF M ( J ) FOR J = I TO N M X ( I ) I S THE MASS OF THE VIRTUAL P A R T I C L E ABOUT TO DECAY INTO M ( I ) AND MX(I+1) M X 2 ( I ) I S THE SQUARE OF M X ( I ) THE MATRIX B ( 4 , 4 , I ) BOOSTS THE M X ( I ) CM FRAME ONE BACK LAMBDA(I ) I S THE MAGNITUDE OF THE MOMENTUM OF P A R T I C L E I IN THE M X ( I ) CM FRAME STOT I S THE CM ENERGY SQUARED OF THE PROCESS IN LAB CM FRAME X1.X2 ARE THE USUAL PARTON MOMENTUM FRACTIONS S I S THE CM ENERGY SQUARED OF THE SUBPROCESS V . X I ARE V E L O C I T Y AND RAPIDITY OF ONE FRAME W.R.T. ANOTHER THETA,PHI ARE THE USUAL ANGLES K 4 V ( 4 , I ) IS THE MOMENTUM 4-VECT0R CF P A R T I C L E I L K 4 V ( 4 , I ) IS K 4 V ( 4 . I ) AFTER BOOSTING TO LAB FRAME DV1 I S A DUMMY 4-VECTOR USED FOR PROGRAMMING EASE A I S THE SUBPROCESS AMPLITUDE S U P P L I E D BY THE USER W I S THE ELEMENT OF X - S E C T I O N CALCULATED ON EACH LOOP PASS SUMW I S THE SUM OF THE ELEMENTS W FOR A L L LOOP PASSES UAC I S THE JACOBIAN FACTOR FROM THE INTEGRALS INTEGRAL IS THE FINAL ANSWER  c  INTEGER REAL*8 REAL*8 REAL*8 REAL*8 COMMON C CC c  C  11,JJ,KK,NPTS.START,N,RAT M(9),M2(9),MSUM(9),MX(9),MX2(9),B(4,4,9),LAMBDA(8) SEED,P,DUMMY,STOT,X1,X2,S,V,XI.COSTHETA,THETA,PHI,PI K4V(4,9),LK4V(4,9),DV1(4) A.W,SUMW,JAC.FLUX,FACTOR,INTEGRAL SEED  PROGRAM  SETUP  _ NEW C A L C U L A T I O N OR RESTART, NUMBER OF EVENTS  c  12 15  17  WRITE(6,12) F O R M A T C NEW C A L C U L A T I O N ( T Y P E 0 ) OR RESTART READ(5,15)Jd FORMAT(H) I F ( J J . N E . O . A N D . J J . N E . 1 ) THEN WRITE(6,17) F O R M A T C YOU MUST TYPE O OR 1') STOP  (TYPE  1) ?')  118  18 19  END IF WRITE(6,18) FORMATC ENTER NUMBER OF EVENTS DESIRED READ(5,19)NPTS FORMAT(17)  (17)')  c  C  NEW: READ ENTRIES THROUGH TERMINAL  c  21 22 23  26  29  IF(JJ.EQ.O) THEN START=1 SUMW=0.0 SEED=12345.0 WRITE(6.21 ) FORMATC ENTER P (F15.8)') READ(5,22)P F0RMAT(F15.8) WRITE(6,23) FORMATC ENTER NUMBER (3-9) OF PARTICLES READ(5,15)N IF(N.LT.3.0R.N.GT.9) STOP DO 29 1 = 1 ,N WRITE(6.26)I FORMAT(' ENTER M(',11,') (F15.8)') READ(5.22)M(I) M2(I)=M(I)**2 WRITE(76,32)M(I),M2(I) CONTINUE END IF  c  C  RESTART: READ ENTRIES FROM FILES  c  31  32 33  I'F(JJ.EO.I) THEN READ(75,31)START,N,SEED.SUMW,P,DUMMY FORMAT(I7,I2.4D18.10) START = START+ 1 IF(START.GE.NPTS) STOP DO 33 I=1.N READ(76,32)M(I),M2(I) F0RMAT(2D18.10) CONTINUE END IF --  C  C  INITIALIZE  -  VARIABLES  c  41 42  43  DO 42 1=1,N MSUM(I)=0.0 DO 41 J=I,N MSUM(I)=MSUM(I)+M(J) CONTINUE KK=N-1 II=3*N-4 ST0T=4.0*P*P IF((2.0*P) .LE.MSUM(O) THEN WRITE(6,43) FORMATC NOT ENOUGH ENERGY FOR REACTION') STOP END IF PI=3.141592654 MX(N)=M(N) MX2(N)=M2(N)  (11)')  119  48  WRITE(6,48) FORMAT(' BEGINNING  MAIN MONTE CARLO LOOP',/) -BEGIN MAIN MONTE CARLO LOOP  C CC  .  c  DO  999  c  C  IJ=START,NPTS _  GENERATE X I , X 2 AND  _  CHECK  —  I F ENOUGH ENERGY  ---  c  50  C C C C  65 C C  X1 = 1 .0 X2=1.0 S=X1*X2*ST0T MX(1 ) = S Q R T ( S ) MX2(1)=S I F ( M X ( 1 ) . L E . M S U M ( 1 ) ) GO TO 50 GENERATE VIRTUAL P A R T I C L E MASSES DO 65 I=2.KK M X ( I ) = ( M X ( I - 1)-MSUM(I- 1 ) ) * G G U B F S ( S E E D )+MSUM( I ) MX2(I)=MX(I)**2 CONTINUE FIND BOOST MATRIX FROM SUBPROCESS CM TO LAB FRAME  c  V=(X1-X2)/(X1+X2) XI=LOG((1.0+V)/(1.0-V))/2.0 C0STHETA=2.0*GGUBFS(SEED)-1.0 THETA=AC0S(C0STHETA) PHI=2.0*PI*GGUBFS(SEED) CALL B 0 0 S T ( B ( 1 , 1 . 1 ) , X I , T H E T A , P H I ) C C • C-79  C C  L E T THE P A R T I C L E S DECAY, GET THE BOOST MATRICES AND 4-VECTORS DO 79 1=1,KK CALL D E C A Y ( M X ( I ) , M X ( I + 1 ) , M ( I ) , K 4 V ( 1 , I ) , B ( 1 . 1 , 1 + 1 ) ) K4V(1,N)=MX(KK)-K4V(1,KK) K4V(2,N)=-K4V(2,KK) K4V(3,N)=-K4V(3,KK) K4V(4,N)=-K4V(4,KK) BOOST 4-MOMENTA TO LAB FRAME  c  DO  81  B3 84 88  88 1=1,N DO 81 K=1,4 DV1 ( K ) = K 4 V ( K , I ) DO 84 JK=1.I d=I-JK+1 I F ( I . E O . N ) d=I-JK I F ( d . E O . O ) GO TO 84 CALL M U L T ( B ( 1 , 1 . d ) , D V 1 , L K 4 V ( 1 . I ) ) DO 83 K=1,4 DV1(K)=LK4V(K,I) CONTINUE CONTINUE C A L L AMPLITUDE - - MUST BE LINKED TO, OR A=1 .0  PART OF THE  PROGRAM  C  C A L C U L A T E PHASE  SPACE DENSITY AND ELEMENT OF INTEGRAL  c  92  93  C C  W=1 .0 DO 92 1=1,KK LAMBDA(I)=-4.0*M2(I)*MX2(I+1)+(MX2(I)-M2(I)-MX2(1+1))**2 LAMBDA(I)=SQRT(LAMBDA(I))/(2.0*MX(I)) W=W*LAMBDA(I) CONTINUE <JAC=(4 .0*PI )**KK DO 9 3 I=1,KK-1 dAC=dAC*(MX(I)-MSUM(I)) FLUX=1.0/(2.0*S) FACTOR=SORT(S)*(2.0*PI)**II FACT0R=FACT0R*(2.0**N) W=W*A*JAC*FLUX/FACTOR SUMW=SUMW+W ----WRITE OUT THE F I R S T T E N EVENTS  c  101  102 103 104 105 _ J C C  I F ( I J . L E . 1 0 . 0 R . W . L E . 0 ) THEN WRITE(77,101) FORMATC Id I LK4V( 1 , I ) ' , 10X , ' 2 ' , 14X , ' 3 ' , 14X , ' 4 ' ) DO 103 1=1,N WRITE(77,102)Id,I,LK4V(1,1),LK4V(2,I),LK4V(3,I),LK4V(4, I ) FORMAT(17,1X,12,4D15.5) CONTINUE WRITE(77,104) FORMATC Id A W') WRITE(77,105)Id,A,W F0RMAT(I7,2D12.4./) END I F STORE  ANSWER  EVERY  1000 EVENTS FOR RESTART  I F SYSTEM  c  112  RAT=M0D(Id,1000) I F ( R A T . E O . O ) THEN WRITE(74.31)Id.N,SEED.SUMW,P,SUMW/FLOAT(Id) 0PEN(UNIT=75) WRITE(75,31)Id,N,SEED,SUMW,P,SUMW/FLOAT(Id) CL0SE(UNIT=75) WRITE(6,112)Id F O R M A T C F I N I S H E D ',17,' P O I N T S ' ) END I F  c  CC C--999  END MAIN  MONTE CARLO  LOOP  CONTINUE  c  CC  C A L C U L A T E THE INTEGRAL AND OUTPUT  c  113 114 C C c  INTEGRAL=SUMW/FLOAT(NPTS) WRITE(77.113)NPTS,P F O R M A T C NPTS= ' . I 7 . 4 X . ' P= '.D15.6) WRITE(77,114)INTEGRAL F O R M A T C X-SECTION= '.D15.6) END --MUST LINK TO RTNS.OBd FOR SUBROUTINES  .  CRASHES  c  RTNS.FOR CONTAINS MONTE-CARLO SUBROUTINES  C c  SUBROUTINE FOR  DECAY2(M1,M2.M3,BM2,BM3)  TWO CHAIN  DECAY OF M1 INTO M2.M3 FIND BOOST MATRICES  BACK  REAL*8 M 1 , M 2 , M 3 . V 2 ( 4 ) , V 3 ( 4 ) , B M 2 ( 4 , 4 ) , B M 3 ( 4 , 4 ) REAL*8 C O S T H E T A . T H E T A . P H I . V , S E E D . P I , X I COMMON SEED EXTERNAL GGUBFS.BOOST PI=3.141592654 V3(1)=(M1*M1+M3*M3-M2*M2)/(2.0*M1) V3(4)=DS0RT(V3(1)*V3(1)-M3*M3) V2(1)=M1-V3( 1 ) V2(4)=-V3(4) V=V2(4)/V2(1 ) I F ( V . E O . - I . O ) STOP •XI=DLOG( ( 1 .0+V)/( 1 .O-V) )/2.0 COSTHETA = 2.0*GGUBFS(SEED )-1 .0 THETA=DACOS(COSTHETA) PHI=2.0*PI*GGUBFS(SEED) CALL BOOST(BM2,XI,THETA,PHI) V=V3(4)/V3( 1 ) I F ( V . E Q . - I . O ) STOP XI=DL0G((1.0+V)/(1.O-V))/2.0 C0STHETA=2.0*GGUBFS(SEED)-1.O THETA=DACOS(COSTHETA) PHI=2.0*PI*GGUBFS(SEED) CALL B00ST(BM3,XI.THETA.PHI) RETURN END SUBROUTINE B 0 0 S T ( B . X I , T H E T A , P H I ) REAL *8 B ( 4 , 4 ) . X I . THETA, PHI B( 1 . 1 )•= D C 0 S H ( X I ) B(1.2)= -DSINH(XI) * DSIN(THETA) B ( 1 , 3 ) = 0. B(1,4)= DSINH(XI)*DC0S(THETA) B ( 2 , 1 ) = O. B(2,2)= DCOS(PHI)*DCOS(THETA) B(2,3)= -DSIN(PHI) B(2,4)= DSIN(THETA)*DCOS(PHI) B ( 3 , 1 ) = 0. B(3,2)= DSIN(PHI)*DC0S(THETA) B(3,3)= DCOS(PHI) B(3,4)= DSIN(THETA)*DSIN(PHI) B ( 4 . 1 )= D S I N H ( X I ) B(4,2)«= -DC0SH(XI ) * D S I N ( T H E T A ) B ( 4 , 3 ) = 0. B(4,4)= DC0SH(XI)*DC0S(THETA) RETURN END SUBROUTINE B I N ( F , A R , I N F , S U P , W) C L A S S E S F INTO ONE OF 1C1 BINS EETWEEN INF AND SUP AND PUT IT INTO ARRAY AR  1 1  c  REAL*8 F . A R ( 1 0 1 ) , I N F , S U P . W INTEGER POS POS = D I N T O O O . * ( F - I N F ) / ( S U P I F (POS .GT. 101) POS = 101 IF (POS . L T . 1) P0S=1 AR(POS) = AR(POS) + W RETURN END  - INF)) + 1  SUBROUTINE S C A L P 3 ( V 1 . V 2 , V S V ) REAL*8 V 1 ( 4 ) , V 2 ( 4 ) , VSV VSV = V 1 ( 2 ) * V 2 ( 2 ) + V 1 ( 3 ) * V 2 ( 3 ) + V 1 ( 4 ) * V 2 ( 4 ) RETURN END SUBROUTINE  MULT(B,V1,V2)  c  C C  C A L C U L A T E S THE PRODUCT BETWEEN THE MATRIX VECTOR V1 AND PUTS RESULT INTO V2  B AND  c  301 300  REAL*8 B ( 4 , 4 ) , V 1 ( 4 ) , V 2 ( 4 ) , PH DO 3 0 0 1=1,4 PH=0. DO 301 J=1,4 PH = B ( I , d ) * V 1 ( d ) + PH V 2 ( I ) = PH CONTINUE RETURN END SUBROUTINE  C-C C C  DECAY(M1,M2,M3,V3,M)  FOR DECAY OF M1 INTO M2 AND M3, CALCULATE THE 4-VECT0R V3 OF P A R T I C L E 3 IN M1 REST FRAME, THEN CALCULATE BOOST MATRIX FROM M2 TO M1 REST FRAME ___  c  REAL*8 M1,M2,M3,V3(4),M(4,4),COSTHETA.THETA,PHI.V.SEED.PI ,XI COMMON SEED EXTERNAL GGUBFS,BOOST PI= 3.141592G54 V 3 ( 1 ) = (M1*M1+M3*M3-M2*M2)/(2.*M1) V 3 ( 2 ) = O. V 3 ( 3 ) = 0. V 3 ( 4 ) = D S 0 R T ( V 3 ( 1 ) * V 3 ( 1 ) - M3*M3) PX2 = - V 3 ( 4 ) EX2 = M1 - V 3 ( 1 ) COMPOSE BOOST MATRIX  BETWEEN QUARKS CM AND X CM  V = PX2/EX2 I F ( V .NE. - 1 . ) XI= D L 0 G ( ( 1 . + V ) / ( 1 . I F ( V .EO. - 1 . ) RETURN COSTHETA = 2 . * G G U B F S ( S E E D ) - 1 . THETA = DACOS(COSTHETA) PHI = 2. * P I * G G U B F S ( S E E D ) C A L L B 0 0 S T ( M , X I , THETA, P H I ) RETURN END  SUBROUTINE  - V))/2.  SCALP(V1,V2.S)  TAKE THE SCALAR PRODUCT OF THE TWO 4-VECT0RS V1 AND V2 AND PUT THE RESULT INTO S REAL*8 V 1 ( 4 ) , V 2 ( 4 ) , S S = V1(1)*V2(1) - V1(2)*V2(2) - V1(3)*V2(3) - V1(4)*V2(4) RETURN END  1 1  123  APPENDIX H ~ GLOSSARY c h a r g i n g : T h i s charged s u p e r p a r t i c l e i s a f e r m i o n . I t s mass e i g e n s t a t e i s a c t u a l l y a mixture of the s u p e r p a r t n e r s f o r the W-boson and the H i g g s . c h i r a l symmetry: A symmetry which p r e s e r v e s the handedness of fermions. T e c h n i c a l l y t h i s means a Y 5 i n v a r i a n c e .  massless  electroweak b r e a k i n g s c a l e ; Energy s c a l e a t which the e l e c t r o m a g n e t i c and weak f o r c e s a r e u n i f i e d . E x p e r i m e n t a l l y e s t a b l i s h e d as b e i n g = 2 4 6 GeV. Fadeev-Popov g h o s t s : Mathematical c o n s t r u c t s known as ghosts are i n t r o d u c e d i n a d d i t i o n t o the p h y s i c a l f i e l d s t o p r e s e r v e gauge i n v a r i a n c e . These a r e simply a convenient t e c h n i c a l i n v e n t i o n which a l l o w us to express the complex mathematics of a p h y s i c a l p r o c e s s i n a simple g r a p h i c a l form. Goldstone boson: Massless p a r t i c l e which r e s u l t s whenever a g l o b a l symmetry i s spontaneously broken.  continuous  Higgs mechanism: Process through which gauge bosons a c q u i r e mass, where a Goldstone boson becomes the l o n g i t u d i n a l component of the gauge boson. l o o p s , l o o p diagram, loop p a r t i c l e : P h y s i c a l p r o c e s s e s can be r e p r e s e n t e d g r a p h i c a l l y , w i t h p a r t i c l e s r e p r e s e n t e d by a l i n e and t h e i r i n t e r a c t i o n s by a v e r t e x . The diagrams f o r h i g h e r o r d e r processes w i l l i n v o l v e loops formed by the p a r t i c l e s . one-loop approximation: Used w i t h the g r a p h i c a l r e p r e s e n t a t i o n of a p h y s i c a l p r o c e s s , where o n l y diagrams w i t h one loop a r e i n c l u d e d . r e n o r m a l i z a t i o n : Procedure i n which d i v e r g e n t q u a n t i t i e s t h a t a r i s e from h i g h e r o r d e r c o r r e c t i o n s are absorbed i n t o a r e d e f i n i t i o n of parameters, thereby making a p e r t u r b a t i v e expansion convergent. spontaneous symmetry b r e a k i n g : The ground s t a t e of the system does not r e s p e c t the same symmetry as the L a g r a n g i a n which i s used t o d e s c r i b e i t . s u p e r p a r t i c l e s , s u p e r p a r t n e r s : The supersymmetric p a r t n e r s to the u s u a l s t a n d a r d model p a r t i c l e s , which d i f f e r by 1/2 i n t e g e r u n i t of quantum s p i n . two photon f u s i o n mechanism: Any p r o c e s s i n which the f i n a l s t a t e i s produced by the i n t e r a c t i o n of two i n i t i a l photons. In t h i s case used t o d e s c r i b e a method of producing Higgs bosons d u r i n g the exchange of a photon between c o l l i d i n g p a r t i c l e s . vacuum e x p e c t a t i o n v a l u e (VEV): The v a l u e which the s c a l a r f i e l d i t s ground s t a t e .  acquires i n  would-be Goldstone boson: T h i s r e f e r s to the degree of freedom which would n o r m a l l y be a Goldstone boson, but i s i n s t e a d absorbed v i a the Higgs mechanism as the l o n g i t u d i n a l component of a gauge boson. Yukawa i n t e r a c t i o n : The i n t e r a c t i o n between s c a l a r s ( o r p s e u d o s c a l a r s ) fermions. In our case i t i s the Higgs boson-fermion i n t e r a c t i o n .  and  

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