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Two photon decay widths of non-standard Higgs bosons 1986

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TWO PHOTON DECAY WIDTHS OF NON-STANDARD HIGGS BOSONS by ROSS TAYLOR BATES M.Sc, The University of British 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 this 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 t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Physics The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date June 01, 1986 A b s t r a c t This t h e s i s examines the two photon decay widths of non-standard Higgs bosons. These widths are c a l c u l a t e d f o r Two-Higgs-Doublet models i n gene r a l , and f o r the minimal 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 of these widths r e l a t i v e to the standard 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 r a t e s f o r the spin - 0 bosons i n ep and e + e ~ c o l l i d e r s . However, we f i n d that f o r the minimal broken supersymmetry case, a severe upper bound on t h i s p o s s i b l e enhancement i s imposed by the supersymmetry f e a t u r e s . We f i n d that w h i l e the Higgs bosons of the Two-Higgs-Doublet model could p o s s i b l y be produced at r e a d i l y observable 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 not be the case i n the minimal supersymmetry model. Hence d e t e c t i o n of these Higgs bosons could provide an experimental t e s t of supersymmetry, which would r u l e out the minimal model. i i i Table of Contents Abstract i i List of Tables iv List of Figures v Acknowledgement '. v i CHAPTER I - INTRODUCTION 1 1.1 The Standard Model 4 1.2 Why Alternative Models? 6 1.3 Fundamental Scalars 9 1.4 Supersymmetry 11 1.5 Thesis Overview 13 CHAPTER II - TWO-HIGGS-DOUBLET MODEL 18 2.1 The Model 19 2.2 Standard Model 2y-Decay Width 28 2.3 Two-Higgs-Doublet Model 2y-Decay Widths 29 CHAPTER III - MINIMAL BROKEN SUPERSYMMETRY MODEL 33 3.1 The Model 34 3.2 One Loop Calculation of X° •»• yy 53 3.3 Pseudoscalar Widths of X° + YY 57 3.4 Scalar Widths of X° + yy 62 CHAPTER IV - NON-STANDARD SPIN-0 BOSON PRODUCTION 70 4.1 The Calculation 71 4.2 Numerical Results and Discussion 77 CHAPTER V - SUMMARY AND CONCLUSIONS 87 Bibliography 92 APPENDIX A - SOME ACCELERATOR PROPERTIES 94 APPENDIX B - EVALUATION OF FEYNMAN DIAGRAMS 95 B.l Scalar Higgs 2y-Decay via Fermion Loops 95 B.2 Scalar Higgs 2y-Decay via Scalar Loops 97 B.3 Scalar Higgs 2y-Decay via Gauge Boson Loops 98 B.4 Pseudoscalar Higgs 2y-Decay via Fermion Loops 101 APPENDIX C - EVALUATION OF LOOP INTEGRALS '. 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 i v List of Tables I Two-Higgs-Doublet Model Vertices 27 II Supersymmetric Field Content 35 III Accelerator Properties 94 List 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 for Case A 60 4 Pseudoscalar 2y-Decay Width for Case B 61 5 Scalar 2y-Decay Width for Case A 65 6 Scalar 2y-Decay Width for Case B 66 7 Scalar 2y-Decay Width for Best Case A 68 8 Feynman Diagrams for the Reaction eq •»• eqH° 72 9 Standard Model a vs /s for ep -»• eH°X 78 10 Cross Sections ff vs ̂  for ep + eH°X for /s=320 GeV 79 11 Cross Sections a vs M„ for ep •*• eH°X for /s=l TeV 80 £1 12 Production Cross Sections for e +e~ •*• e+e~H° 82 13 Rapidity Distributions 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 for Scalar H° Couplings 109 20 Feynman Rules for Photon Couplings 110 21 Feynman Rules for Chargino-Scalar H° Couplings I l l 22 Feynman Rules for Chargino-Pseudoscalar H° Couplings 112 23 Photon Quark Subprocess 114 Acknowledgement I wish to thank my research supervisor, Dr. John Ng, whose guidance made this work possible. I would also like to thank Dr. Pat Kalynlak, who collaborated with us on the supersymmetry calculations, for her contribution. Finally I would like to thank my departmental supervisor, Dr. Nathan Weiss. Financial assistance from the Natural Sciences and Engineering Research Council and from the University of British Columbia i s gratefully acknowledged. 1 I. INTRODUCTION At present the basic building blocks of matter are thought to be the twelve spin-1/2 fermions known as quarks and leptons. Four fundamental forces are responsible for the interactions which describe their behaviour. These forces are the familiar gravity and electromagnetism; the strong force which binds nuclei together; and the weak force responsible for certain nuclear decays. The six leptons do not interact via the strong force. They consist of the electron, muon and tau particles which carry electric charge, along with three neutrinos which do not. The six quarks also carry electric charge, and in 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. Such a symmetry implies a conserved quantity. Many of our physical laws are based on this principle. Theoretical models known as gauge theories, which are based on underlying symmetry groups, have been very successful in describing three of the basic interactions. The exception i s gravity, which has not as yet been adequately described by such a gauge theory. However the effects of gravity are very small and may be neglected at the scale where elementary particle physics i s currently studied. A general feature of these gauge theories is that the forces between the basic fermions are mediated by the exchange of a new particle called a gauge boson. These gauge bosons must be massless in order to preserve the underlying symmetry of the gauge theory. The gauge theory for the electromagnetic force, known as quantum electrodynamics (QED), i s the most familiar. Here the gauge boson Is the massless photon, which i s exchanged 2 between elec t r i c a l l y charged particles. In the gauge theory of the strong force, known as quantum chromodynamics (QCD), interactions are also mediated by exchange particles. In this 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 electric charge suggested to some that perhaps the weak and electromagnetic forces were one and the same. The disparity in their observed strengths and ranges could be explained by the different 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 particle, called the Z boson, which carries no electric charge. However, such neutral currents were not at that time observed. The need for massive exchange bosons in these models destroyed the underlying symmetries that one originally wished to incorporate. This led to divergent results when higher order calculations were done. Such problems frustrated these subsequent attempts at describing the weak force. The solution to these early theoretical 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. A simple example i s that of a ferromagnet. In general i t s spins are randomly aligned 3 and the theory possesses a rotational symmetry. In the ground state however the random spins a l l align In one chosen direction, and hence the rotational 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 particle known as a Goldstone boson must result. 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 called a Higgs particle is introduced. The ground state of this new matter f i e l d i s such that the original symmetry of the theory i s spontaneously broken. In this case the massless Goldstone boson appears not as a physical particle, 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 original theory. This technique i s known as the Higgs mechanism and i t solves the theoretical problems of the weak model, giving f i n i t e or renormalizable results for higher order calculations. At this point there existed a well behaved gauge theory which unified the weak and electromagnetic forces. The model predicted masses for 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. Only the Higgs scalar remains to be discovered. The phenomenological successes of this 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 particle physics. Thus far, 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 this thesis a l l begin with the implicit assumption that the reader is familiar with the standard model. As the currently accepted theory, i t is the basis against which any new physics must necessarily be compared. A detailed description of the standard model can be found in most modern textbooks on particle physics. Consequently only a brief summary of the main features w i l l be presented here. As the strong interaction has no direct bearing on our results, only the electroweak aspects of the model are described. In the standard model the fundamental particles of matter consist of twelve spin-1/2 fermions. There are three massive leptons which carry electric charge -e. They are known as the electron, muon and tau (e~,u~,T~) particles. Associated with each one of these charged leptons i s a massive, el e c t r i c a l l y neutral lepton ( v e» Vy» v T) called a neutrino. The remaining six fermions are massive particles called quarks. Three of them carry electric charge (2/3)e, and are known as the up, charm and top (u,c,t) quarks. The other three carry electric charge (-l/3)e and are called the down, strange and bottom (d,s,b) quarks. The standard model classifies these twelve fermions Into three generations or families. The structure of each family is very similar, 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 in 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 in the standard model which involves only the left-handed components of the fermions, and i t is 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 individual singlets. Experimentally only left-handed neutrinos are observed. Hence the right-handed neutrino singlets are not included in the standard model. Having introduced the fundamental fermions, we turn now to the spin-1 exchange bosons which mediate their electroweak interactions. There are four of these gauge bosons in the standard model. Three of them are quite massive. They consist of the el 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. The fourth exchange boson is the familiar massless photon. With the exception of the Higgs boson described below, this completes the description of the particle content in the standard electroweak model. The gauge theory of the standard model is based upon what is technically known as the SU(2)xU(l) symmetry group. A requirement of any gauge theory is that Its fermions and exchange bosons must be massless in order to preserve the underlying symmetry. Thus in order to generate masses for the particles described above, the basic gauge theory must be supplemented by the introduction of one or more fundamental spin-0 scalars. As described earlier, these so called Higgs fields are responsible for the spontaneous breaking of the underlying symmetry. In the case of the standard model this i s done in the simplest way, through the addition of one doublet of Higgs scalars. 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 this breaking. The latter are then absorbed via the previously described Higgs mechanism as the longitudinal components of the W and Z bosons. In this way masses are generated for the exchange bosons. Through their so called Yukawa interactions with the Higgs fields, the basic fermions can also acquire a mass. Unfortunately the model makes no prediction for 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 particle physics. This concludes our look at the main features of the standard model. Most of the technical details have been suppressed for simplicity, and the interested reader is referred to the literature. We have introduced the particle content of the model, and stressed the importance of the Higgs sector. These should be sufficient background for a general understanding of the standard model aspects of the thesis. Other features and specific details of the standard model are discussed as they arise. 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 is the a l l important Higgs sector needed for spontaneous symmetry breaking. The model does not predict a mass for the fundamental Higgs scalar, which has yet to be detected. Also as w i l l be discussed, there are many technical reasons for expecting new physics beyond the standard model. 7 There are several arguments [ 3 ] which suggest the need for improving upon the standard model. Numerous details 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 is not asymptotically free, meaning that i t w i l l become strongly interacting at some larger energy scale, where perturbation methods w i l l break down. Why this breakdown occurs is discussed in more detail below. In particle physics, calculations are performed primarily using perturbation techniques. An unknown quantity i s expanded in a power series of some small parameter. In this way each successive term in the expansion serves as a small correction to the previous term. The coefficients of the expansion are then evaluated term by term to the desired accuracy of the approximation. However, i t is a general property of gauge theories that the coefficients of the higher order terms in 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 redefinition 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 are cut off at some scale A, and the i n f i n i t e piece absorbed into the new parameter definitions. The perturbation series i s then once again convergent. This so called renormalization scheme w i l l of course only make sense i f the cutoff scale A is larger than the energy scale of the process we are interested i n . If this i s not the case, then the i n f i n i t i e s cannot be eliminated and perturbation methods w i l l break down. The scale A used in the renormalization scheme above i s not an arbitrary parameter. There must be some physical quantity in the theory which fixes the scale. In general this i s taken to be the mass of the 8 heaviest p a r t i c l e i n the theory, so that perturbation techniques w i l l be v a l i d over as large an energy range as possible. Thus f o r energies greater than t h i s mass, perturbation methods break down and the gauge theory can make no quantitative p r e d i c t i o n s . In the standard model the heaviest possible p a r t i c l e i s the Higgs boson. I t s mass varies as = 2Xv 2. The parameter X i s a measure of the strength of the Higgs s e l f - I n t e r a c t i o n coupling. The quantity v i s the vacuum expectation value (VEV) that the Higgs f i e l d acquires from spontaneous symmetry breaking, as discussed above. In c e r t a i n perturbation c a l c u l a t i o n s X Is used as an expansion parameter. Therefore i t must be small since otherwise the Higgs sector would become strongly i n t e r a c t i n g and perturbation theory would f a i l . This contradicts the many successful predictions of the standard model which are obtained perturbatively. Thus we can e s t a b l i s h an upper bound on X. The VEV parameter v i n the expression f o r the Higgs mass i s simply the scale at which the electroweak symmetry Is broken. In other words i t i s the energy at which the strengths of the electromagnetic and weak forces become equal. This i s established experimentally to be v = 246 GeV. Combining these r e s u l t s f or X and v, one finds that the mass of the Higgs boson i s expected to be less than the order of 1 TeV. This then sets the scale at which the renormallzation scheme, and hence perturbation methods, w i l l break down i n the standard model. Presently the highest attainable energies which have been tested are on the order of 100 GeV, and the r e s u l t s have been consistent with the predictions of the standard model. Soon a new generation of p a r t i c l e accelerators (see appendix A) w i l l be operating at energies up to the TeV range. This i s exactly the region where we should begin to see evidence of the breakdown i n the standard model, and t h i s i s one reason for expecting 9 new physics to be observed in these machines. Hence there i s now an immediate need to develop alternative models, and establish a theoretical framework with which to describe this 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 valid only as an effective 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 in the standard model. Evidence for such a theory would manifest i t s e l f at a higher energy scale in the form of new physics. At presently available energies however, the experimental data is consistent with the standard model. Hence any attempt to formulate a new underlying theory must at this point be guided by purely theoretical motivations. These are discussed in the next two sections. We begin by examining some of the technical problems which occur in the crucial Higgs sector. 1.3 Fundamental Scalars The existence of a fundamental scalar Higgs particle i s an essential component of the standard model, and indeed of any gauge theory with massive exchange bosons. It is the Higgs particle which induces the spontaneous symmetry breaking needed to generate the gauge boson mass. Thus the motivation for fundamental scalars i s very strong. 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 the scalar f i e l d i s so light. More specifically, the question is why is the electroweak breaking scale v so small? One would wish in developing a new fundamental theory that we could also incorporate the unification of the strong and electroweak interactions, and possibly even gravity. The scales at which the strong force (grand unification 1 0 1 2 TeV) or gravity (Planck mass 1 0 1 6 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 unified is known as the so called naturalness problem. The "natural" value for the scalar boson mass should be the same as the mass scale for the fundamental theory. The disparity of these scales in the theory could be understood 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 is very d i f f i c u l t to determine for these fundamental scalar particles. 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 originally synonymous with "naturalness", the term "gauge hierarchy" is now used in the literature to refer to the following specific 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 in perturbation theory with incredible accuracy to avoid this. Such fine tuning is not a very satisfactory way to solve the naturalness problem. These are two of the principal d i f f i c u l t i e s associated with fundamental scalars which w i l l arise in 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 far. The other approach i s to retain the fundamental scalars which work so well in the standard model, and try to solve the problems discussed above. Indeed this w i l l prove to be possible by employing a higher symmetry which eliminates the naturalness and gauge hierarchy problems. This is the elegant supersymmetry approach. Of a l l the possible alternatives to the standard model, supersymmetry is now the leading candidate. 1.4 Supersymmetry What is different 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 diffe 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 particles are known respectively as squarks, sleptons and gauginos. In addition i t is necessary in these models that at least two Higgs fields be employed in order to generate different masses for up and down type quarks. Thus the minimal supersymmetric extension of the standard model is a two Higgs doublet model. The details of these models w i l l be examined in later chapters. At this point we merely wish to il l u s t r a t e some of the reasons why supersymmetry i s the leading alternative to the standard model. As a candidate theory for an alternative to the standard model, supersymmetry has many desirable features. To begin with the fundamental scalars are no longer supplemental to, but rather are now a natural part of the gauge theory, on the same footing as the basic fermions. Also c e r t a i n divergent q u a n t i t i e s , which a r i s e i n the higher order corrections to the sc a l a r masses, now cancel at each order i n perturbation theory. This c a n c e l l a t i o n occurs because the divergent contribution from each p a r t i c l e i s exactly cancelled by the c o n t r i b u t i o n from i t s superpartner. Hence the higher order corrections remain small and do not cause a large scalar mass. This eliminates t e c h n i c a l problems such as gauge hierarchy. Thus supersymmetry models are much better behaved. Furthermore the naturalness problem can e a s i l y be solved by supersymmetry. In general i t has been d i f f i c u l t to i d e n t i f y any mechanism which could enforce a small s c a l a r mass. However i t i s well known that imposing an exact c h i r a l symmetry enforces a zero mass f o r fermions. In supersymmetry a massless fermion leads n a t u r a l l y to a massless s c a l a r partner. Hence the naturalness problem can be solved f o r supersymmetric models with approximate c h i r a l symmetries. Perhaps the most i n t r i g u i n g aspect of these supersymmetry models i s that l o c a l supersymmetry transformations are re l a t e d to space-time transformations. 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 with supersymmetry, and t h i s would allow for the possible u n i f i c a t i o n of a l l four of nature's fundamental forces i n t o one theory. (For a review of such supergravity models, see reference [3]). This e x c i t i n g p o s s i b i l i t y , along with the solutions of the naturalness and gauge hierarchy problems, are the main reasons for examining supersymmetry i n more d e t a i l . At t h i s point i n time, i t must be noted that these motivations f o r supersymmetry are purely t h e o r e t i c a l . No experimental evidence has been seen to date. Indeed, i f i t e x i s t s then the supersymmetry must be " s o f t l y " broken at low energies. 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 that although the supersymmetry must be broken, one wants to preserve the c a n c e l l a t i o n of the various divergent quantities i n the s c a l a r mass corr e c t i o n s . The c a n c e l l a t i o n w i l l no longer be exact, but i f the breaking i s s o f t enough only f i n i t e parts w i l l remain. This breaking of the supersymmetry must occur since no superpartners have been observed f o r known p a r t i c l e s . Also an exact c h i r a l supersymmetry implies a zero mass for the Higgs s c a l a r , which i s inconsistent with the standard model. S t i l l the p o t e n t i a l e x i s t s f or supersymmetry to be the fundamental theory which w i l l reduce to the standard model at low energy. The scale at 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 re l a t e d to the electroweak breaking scale of v = 246 GeV. The breaking of the supersymmetry would l i f t the degeneracy i n the masses of superpartners, and t h e i r r e s u l t i n g mass differences should also be on t h i s same scale. Thus given the masses of the known p a r t i c l e s , one would expect to see evidence of supersymmetry at or before the TeV energy range. This i s exactly the region to be studied In the new a c c e l e r a t o r s . Thus the supersymmetry hypothesis i s one which can be tested i n the very near future. 1.5 Thesis Overview Regardless of whether or not supersymmetry i s a correct approach, i t i s c l e a r that the fundamental scalars w i l l play a key r o l e i n constructing a l t e r n a t i v e s to the standard model. We wish to l e a r n more about t h i s important Higgs sector, and consequently t h i s thesis w i l l focus on these non-standard spin-0 Higgs bosons. A d d i t i o n a l l y we would l i k e to know what form of new physics to expect i n the new p a r t i c l e a c celerators. The dominant modes of producing Higgs bosons, and the r a t e s at which they should be observed i n these machines are then examined f o r c e r t a i n models. Given a l l of the mo t i v a t i o n p r e v i o u s l y discussed, i t should not be s u r p r i s i n g that the s p e c i f i c model chosen f o r study i s one of minimal broken supersymmetry. As st a t e d e a r l i e r , t h i s choice i s a s p e c i f i c example of a model w i t h two Higgs doublets. We would l i k e to be able to d i s t i n g u i s h between the feat u r e s of t h i s supersymmetry model which a r i s e from e i t h e r the new s u p e r p a r t i c l e content, or because there are a d d i t i o n a l Higgs f i e l d s present. Hence a more general Two-Higgs-Doublet model w i t h no supersymmetry w i l l f i r s t be examined. Both of these models can be found i n the l i t e r a t u r e . What i s new i n t h i s t h e s i s are the c a l c u l a t i o n s t o which the models are a p p l i e d . These c a l c u l a t i o n s and the various r e s u l t s obtained are discussed i n more d e t a i l below. In general the Higgs boson i n t e r a c t s w i t h other p a r t i c l e s w i t h a streng t h p r o p o r t i o n a l to t h e i r mass. Hence the dominant decay mode of the Higgs boson w i l l be i n t o the heaviest p a r t i c l e s allowed by energy conservation. I f I t s mass i s gr e a t e r than 160 GeV/c 2, the Higgs then decays i n t o a p a i r of gauge bosons. This i s p o t e n t i a l l y u n d e s i r a b l e s i n c e 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 p a i r s . A l s o , fewer heavy Higgs bosons could be obtained to begin w i t h . Smaller mass Higgs bosons are more l i k e l y to be produced w i t h the l i m i t e d energies a v a i l a b l e at the new c o l l i d e r s (see appendix A). For these reasons i t was decided to study Higgs bosons i n the intermediate mass range from 40 to 160 GeV/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 to extend the a n a l y s i s f o r l a r g e r Higgs masses. In the intermediate mass range the primary decay mode of the Higgs boson i s i n t o a quark-antiquark p a i r , which subsequently w i l l form two hadronic jets. Such a signal would be lost in the much larger jet backgrounds of hadronic colliders, and hence we shall only consider e +e~ and ep machines. The discussion to follow i s similar for both these types of machines, and hence for the moment we restrict ourselves to e +e~ colliders. If production of the Higgs boson HO i s possible at the SLC or LEP colliders, then i t would proceed via reactions such as e + e" -»• Z° H° (1.5.1) e + e - •»• jZ + }\- H° (1.5.2) These are the usual processes used in standard model Higgs searches, and they produce Higgs bosons at an essentially unobservable rate. Generally, l i t t l e change is found for the non-standard models studied. The exception is for a specific case of equation (1.5.2), which i s the reaction e+ e - + e+ e - H0 (1.5.3) Of the many processes which contribute to this reaction, one w i l l be of particular interest. It i s known as the two photon fusion mechanism [6]. This mechanism is essentially the same as the process shown in figure 8b. The Higgs boson i s produced during the exchange of a photon between the colliding particles. The detailed reasons as to why this mechanism is such an important process w i l l be given in chapter IV. The point Is that although this two photon fusion mechanism is not an important one in the standard model, i t can be enhanced substantially and actually dominate in models with more than one Higgs doublet. This idea i s one which has not been explored 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 feature upon which the thesis i s b u i l t . For the class of models studied, the s i z e of the Higgs boson to two photon i n t e r a c t i o n can be greatly increased over what i t i s i n the standard model. This large enhancement i s what causes the two photon fusion mechanism to dominate Higgs boson production. The actual Higgs-photon i n t e r a c t i o n i s best studied by examining the two photon decay widths of the Higgs boson. Production cross sections f or the two photon fusion mechanism can then be expressed d i r e c t l y i n terms of these widths. For t h i s reason we w i l l i n i t i a l l y examine only the Higgs 2y-decay widths. Later these r e s u l t s are used to estimate the various production cross sections and rates. These points are a l l discussed i n more d e t a i l as they a r i s e i n the t h e s i s . F i n a l l y the thesis contents are b r i e f l y outlined below. Chapter II begins with a d e s c r i p t i o n of the Two-Higgs-Doublet model. The two photon decay widths f o r the non-standard Higgs bosons of t h i s model are then c a l c u l a t e d . In chapter III the minimal broken supersymmetry model i s introduced, and again the Higgs 2y-decay widths are evaluated. Comparing the two models, we are able to d i s t i n g u i s h which features 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 2y-decay widths for these models i s a new r e s u l t , and i n each case we discuss how large an enhancement r e l a t i v e to the standard model width i s possible. Chapter IV examines the production cross sections f o r the Higgs bosons i n e +e~ and ep c o l l i d e r s . The d e t a i l s of the two photon fusion mechanism are discussed, as well as how one can r e l a t e i t to the 2y-decay widths previously c a l c u l a t e d . The numerical procedures used are also described. 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 rates to expect f o r Higgs boson production i n the new p a r t i c l e a c c e l e r a t o r s . A comparison with the actual experiment would then determine i f the expected new physics i s consistent with a supersymmetric d e s c r i p t i o n . L a s t l y the many r e s u l t s and conclusions are summarized i n the discussion of chapter V. I I . TWO-HIGGS-DOUBLET MODEL This model i s a simple extension of the standard model, with two doublets of Higgs f i e l d s rather than one. A knowledge of t h i s model w i l l be important i n the next chapter f o r comparison with the minimal broken supersymmetry model, which i s a s p e c i f i c example of a Two-Higgs-Doublet model. This allows one to be able to d i s t i n g u i s h between the features of the supersymmetric model which a r i s e from e i t h e r the supersymmetry or the ad d i t i o n a l Higgs f i e l d content. Although they are reviewed extensively i n the l i t e r a t u r e , the d e t a i l s of the Two-Higgs-Doublet model w i l l be presented below i n order to f a m i l i a r i z e the reader with the general features of the model and to e s t a b l i s h notation. The p a r t i c l e content of the Two-Higgs-Doublet model d i f f e r s from the standard model only i n the Higgs sector. As discussed below, instead of just one neutral s c a l a r p a r t i c l e there are a pair of charged Higgs, a neutral pseudoscalar and two neutral s c a l a r s . The key r e s u l t to note however, i s that the Higgs to fermion couplings d i f f e r from those i n the standard model by factors of tana, which i s the r a t i o of the vacuum expectation values of the two Higgs f i e l d s . If tana i s very d i f f e r e n t from one, then there i s the p o s s i b i l i t y of greatly enhancing these fermion couplings, with important consequences f o r the Higgs to 2y decay process. In the second section of t h i s chapter i s presented the c a l c u l a t i o n f o r the standard model Higgs 2y-decay width. These r e s u l t s can also be found i n the l i t e r a t u r e . They serve to provide some background and e s t a b l i s h the pattern of the 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. The f i r s t r e s u l t to note w i l l be that In the standard model the Higgs boson decays i n t o two photons predominantly v i a gauge boson loops. 19 More importantly the contribution of the two photon decay process i s shown to be a n e g l i g i b l e part of the t o t a l Higgs decay width. F i n a l l y the c a l c u l a t i o n of the Higgs 2y-decay width i s presented f o r the Two-Higgs-Doublet model. Although the c a l c u l a t i o n i s very s i m i l a r to that f o r the standard model, the r e s u l t s are new and w i l l be needed i n chapter IV. S p e c i f i c a l l y the enhanced Higgs to fermion couplings In the model lead to a much larger 2y-decay width. Maximum possible values f o r the enhancement fa c t o r tana are taken from the l i t e r a t u r e . Thus the 2y-decay process i s considerably more important than i t was i n the standard model. The consequences of t h i s r e s u l t w i l l be discussed further i n chapter IV. 2.1 The Model R e c a l l i n g the standard electroweak model [ 7 ] , we see that one of the simplest extensions to i t i s to include an extra doublet of Higgs s c a l a r s . One need only consider the changes i n the Higgs sector, since a l l other features i n Two-Higgs-Doublet models w i l l remain the same as i n the standard model. The two Higgs scalars are described by complex f i e l d operators $ : (2.1.1) which both have hypercharge +1, where the superscripts denote e l e c t r i c charge. The charge conjugate f i e l d s ^2^a' w n e r e T2 *"8 t n e u s u a l P a u l i matrix, have the opposite hypercharge to $ . The vacuum i s characterized by two vacuum expectation values (VEV's) for these f i e l d operators (2.1.2) It is convenient to define the rotated fields *' = ^cosa + 4> sina (2.1.3a) so that *2 = ~ $ i s l n c t + * 2 c 0 8 a (2.1.3b) <*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 in the standard model. The gauge transformation U ( E ) takes us to the unitary gauge, where the five physical fields which are mass eigenstates can be identified. Assuming the physical fields have zero VEV's, we can write (2.1.6a) • A — • U ( £ ) $ : = \ (2.1.6b) 1 c b + i i j ; where v 2=a 2+b 2. The two scalar fields are <(> and n, the pseudoscalar f i e l d is and the charged Higgs are x*• The three degrees of freedom not accounted for by the physical fields are the usual would-be Goldstone bosons, which have been absorbed via the Higgs mechanism as the longitudinal components of the gauge bosons W* and Z°. Transitions between like charged quarks of different families by so called flavour changing neutral currents are suppressed in 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 of the Two-Higgs-Doublet model w i l l i n general allow such c u r r e n t s , and these must somehow be suppressed. Glashow and Weinberg have shown [8] that t h i s can only be accomplished by having quarks of the same charge couple to only one Higgs f i e l d . This i s done by demanding that the Lagrangian remain i n v a r i a n t under $ 2 ~*2 ' dR — * " dR (2.1.7a) $ 1 ~~* ~*1 ' UR — * - U R (2.1.7b) where u ,d are u,d-type r i g h t handed quarks. This symmetry w i l l r e s t r i c t R R the allowed Yukawa c o u p l i n g s , as discussed l a t e r i n t h i s s e c t i o n . For supersymmetry models (see chapter I I I ) , t h i s r e s t r i c t i o n occurs a u t o m a t i c a l l y . The most general renormalizable s c a l a r p o t e n t i a l i s given by [ 9 ] V ( $ 1 , $ 2 ) = - P f * 1 + * 1 " "|*2 +*2 + x i < * i \ > 2 + X 2 ( * 2 t $ 2 ) 2 (2.1.8) + X 3 ( $ 1 + $ 1 ) ( * 2 t * 2 ) + X 4!* 1 +* 2 F + ( X 5 / 2 ) [ ( * 1 + $ 2 ) 2 + (* 2 t* 1)2] w i t h u ^ » u 2 > 0 f ° r spontaneous symmetry breaking. M i n i m i z i n g the p o t e n t i a l w i t h respect to *^ and $ 2 leads to the c o n d i t i o n s * l + [ - u l + 2 X l l * l I2 + X 3 I * 2 ^ + * 2 + t X 4 * l + * 2 + X 5 * 2 + * l l = ° ( 2 . L 9 a ) * 2 f [ ^ 2 + 2 X 2'*2 I" + X 3 I * 1 + * l + t X 4 * 2 + * l + X5*l\l  = ° (2'1'9b> where the f i e l d s * are evaluated at t h e i r VEV's. These c o n d i t i o n s can then 22 be solved for the VEV's 2 (2X y 2 - A p 2 ) § = — - (2.1.10a) (4X^2 - A 2 ) , 2 (2X.y 2 - Ay 2 ) £ - ^ - (2.1.10b) (4X XX 2-A2) where A=X3+X4+X5. Expressing the potential in terms of the rotated fields of equation (2.1.3), we find that V(* 1',* 2') = - (*1,+»1,)[v12cos2o + y 2 2 s i n 2 a ] - ( * 2 , + $ 2 ' ) [ p 1 2 s i n 2 a + u 2 2cos2 a] + ( l / 2 ) ( P l 2 - p 22)[$ 1' t$ 2- + * 2 , t* 1'] + (* 1 , t* 1') 2[x icos l ta + X 2sin J*a + (A/4)sin 22a] + (* 2' +* 2') 2[X 1sin' ta + X -cos^a + (A/4)sin 22a] (2.1.11) + [ ( * 1 ' + * 2 ' ) 2 + (•2»+#1«)2](i/4)[(X1+X2-A)sin22a + 2X5] + ($ 1' t$ 1')($ 2' tcJ 2')(l/2)[(X 1+X 2-A)sin22a + 2X3] + (* 1 , t* 1 ,)(# 1 , t» 2' + * 2 , t* 1')[-X 1cos 2a+X 2sin 2a+(A/2)cos2a]sin2a + ( ^ ' ^ ' K ^ ' 1 * ^ + $ 2 , +$ 1 ,)[-X 1sin2a+X 2cos 2a-(A/2)cos2a]sin2a + IV+V F(l/2)[ (X 1+X 2-A)sin 22a + 2 x J Choosing the unitary gauge and substituting from equation (2.1.6) w i l l give the scalar potential in terms of the physical f i e l d s . V = - (v 2/2)[y 1 2cos 2a+u 2 2sin 2 a ] + (v't/4)[xicos'ta+X2sin' ta+(A/4)sin22o] + v«|>(sin2a)[y12-y22+v2{-X1cos2a+X2sin2a+(A/2)cos2a}]/2 - vn[y^cos 2a+y2sin 2a-v 2{x^cos1*a+X2sinlta+(A/4)sin22a} ] + X +X"[-V x 2 sin 2a-y 2 2 cos 2a+(v 2/4) { (XL+X 2~A)sin 2 2a+2X 3}] - t 2[y 1 2sin 2a+y 2 2cos 2a-(v 2/4){(X 1+X 2-A)sin 22a+2(X 3+X 4-X 5)}]/2 - n 2 [y 1 2cos 2 a+y 2 2 sin 2a-3v 2 {A cos^a+X^iir+a-KAM )sin 22a} ] /2 (2.1.12) - <t> 2[y 1 2sin 2a+y 2 2cos 2a-(v 2/4){3(X 1+X 2-A)sin 22a+2A}] + n<t>[y]2-y22+3v2{-X]Lcos2a+X2sin2a+(A/2)cos2a}] (sin2a)/2 + nx +x'v[( x 1 +X 2-A)sin 22a+2X 3]/2 + 4>x+X~v[-X1sin2a+X2cos2a-(A/2)cos2a] sin2a + (3 neutral scalar terms) + (4 scalar terms) The terms linear In n and <J> can be eliminated using the conditions i n equation (2.1.9). The actual mass eigenstates w i l l in general be mixtures of the two neutral scalars, and the diagonalization of the n,<|> fields i s achieved through a rotation given by d> » 4cos6 + nsine (2.1.13a) T T m m H='-?sin8 + ncose (2.1.13b) m m The mixing angle 6 m can be expressed in terms of the \± parameters by [(X a2-X b 2)(a 2-b 2) + 2Aa 2b 2] sin 29 = i - - T~Pi— (2.1.14) m 2 2v 2[(X 1a 2-X 2b 2) 2+a 2b 2A 2] 1 / 2 With 6 as a parameter, the mass eigenstates 4> and n are now orthogonal, m The masses of the spin-0 fields simplify to M2~ ~ = X La 2 + X 2b 2 + [ ( X i a 2 - X 2 b 2 ) 2 + a 2 b 2 A 2 ] 1 / 2 (2.1.15a) M 2 x = -v 2(X 4+X 5)/2 (2.1.15b) M2 = -X v 2 W iji A5 (2.1.15c) The parameters X̂  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 is given by ^ g = (Vi>ta>P*i) + ( D y * 2 ) + ( D l l $ 2 ) (2.1.16) where D = - 3 - i(g'/2)B -i(g/2) T A* (2.1.17) and T are the Pauli matrices, a The gauge boson sector is the same as in the standard model with A 1 = (W + + W ")//2 (2.1.18a) A 2 = i(W + - W ~)//2 y v y (2.1.18b) A 3 = y sinG A + cos9 Z w y w y (2.1.18c) B y = cos9 A - sine Z w y w y (2.1.18d) One can rewrite equation (2.1.16) in terms of the fields T),4>,X,I|> via the same procedure used for the scalar potential. This gives the result where only terms in the Higgs sector which w i l l contribute to the two photon decay width are exp l i c i t l y shown. Either one of the two Higgs doublets can be used for the lepton Yukawa term. Choosing *]_, one finds that the allowed Yukawa interactions must take the form = (g2/2)vcos8 W+̂ Wri - (g 2/2)vsin6 W^Wcfr + e 2A yA x + X ~ g m y m y u - l e A U [ ( 9 i i x + ) x " ~ X + 0 X - ) ] + others (2.1.19) under the discrete symmetry of equation (2.1.7). Only one quark-lepton family w i l l be important for the two photon decay width analysis. Thus the quark mixings and family labels w i l l be omitted. As before one can express equation (2.1.20) i n terms of the physical f i e l d s , and we f i n d that oCy = y^[vcosaee+neecos(8+a)-c|>eesin(8+a)-ii|>ey ,-esina] //2 +y 2 [ vsinad^d R+nd L<i Rsin(e+a )+^d Ld Rcos (9+a )-iii)d" Ld Rcosa+h. c. ] 1/2 +y3[vcosa u Lu R+nu^u Rcos ( 8 +a ) -<|>u^UgSin ( 6 +a ) - iijiu^^sina+h.c.] //2 + others (2.1.21) where again only those terms which contribute to the two photon decay width are e x p l i c i t l y shown. This completes the d e s c r i p t i o n of the Higgs sector f o r the Two-Higgs- Doublet model. Some of the important couplings obtained i n t h i s section are summarized i n Table I. Note that the Higgs-fermion couplings 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 factors of tana (cota), as well as to a l e s s e r extent due to the mixing angle 8 m» Thus f o r large values of tana (cota) these couplings can be enhanced r e l a t i v e to the standard model. In section 2.3 t h i s w i l l be discussed further. As a f i n a l point i t should be noted that a l l the c a l c u l a t i o n s i n t h i s s e ction were performed i n unitary gauge. I t i s quite straightforward to repeat the d e r i v a t i o n f or a general gauge. Indeed t h i s i s what i s done f o r the supersymmetry models i n chapter I I I . However, In the Two-Higgs-Doublet case, the unitary gauge r e s u l t s w i l l be s u f f i c i e n t f o r the remainder of the discussion. Table I - Two-Higgs-Doublet Model V e r t i c e s Vertex H° eeX uuX ddX -im v -im u y u = -im d y d = — cos(9 +a) m T e cosa cos(6 +a) m 7 u cosa sin(6 -hx) m -y sin(6 +a) m e cosa -y sin(6 +a) m u cosa cos(9 +a) m y y d sina d sina -y iYj-tana J e 5 -y iYctana u 5 y d i Y 5 c o t a W+W~X g w 2 g cosO w m -g sin9 w m Yukawa and gauge couplings of scalars and pseudoscalar to fermions and W-bosons f o r the Two-Higgs-Doublet model. The mixing angle of the two VEV's i s a and the mixing angle between scalars i s 9 m. The standard model v e r t i c e s f o r H° are shown f o r comparison. 2.2 Standard Model 2y-Decay Width The standard model's results w i l l f i r s t be summarized, since the Two-Higgs-Doublet model is similar to i t in so many ways. This serves as a benchmark for the subsequent discussions of non-standard 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 for later convenience since the standard model has no physical charged scalars, so the scalar loops consist only of would-be Goldstone bosons. If one writes the gauge invariant amplitude M for H° — • y(k^) + y(k£) as M = A e i W e 2 V [ g y v - ( k ^ k ^ / C V ^ ) ] (2.2.1) where e^ and e 2 are polarization vectors of the two photons, then the structure function A i s given by A = [ie 2gM H 2/(8 1r 2M w)][A w+A f+A s] (2.2.2a) where A = 3X - 2X (2-3X )I(X ) (2.2.2b) w w w w w A f = - I (e f 2c fX f)[2+(4X f-l)I(X f)] (2.2.2c) A = (1/2) + X I(X ) (2.2.2d) S W W with X=m2/MH2 and the function I(X) i s given in appendix D. The subscript on X indicates the loop particle mass, in this case either Mw or mf. Here the charges of the fermions are e^e. The quantities Aw>A^,Ag 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 equation (2.2.2c) i s taken over a l l charged fermion species w i t h the c o l o u r f a c t o r Cf=3 (1) f o r quarks ( l e p t o n s ) . The c a l c u l a t i o n of the r e s u l t s i n equation (2.2.2) Is presented i n appendix B, and can a l s o be found i n reference [10]. The two photon decay width of the standard Higgs boson i s then given by r(H°~+YY) = | A p/(16irM H) (2.2.3) The gauge boson loop gives the l a r g e s t amplitude and i s roughly a f a c t o r 5 l a r g e r than the next c o n t r i b u t i o n due to the t-quark loop. These two c o n t r i b u t i o n s i n t e r f e r e d e s t r u c t i v e l y . The other fermion loops and the s c a l a r loop are unimportant. For a l a r g e range of Mg, say between 40 to 160 GeV/c 2, one can approximate equation (2.2.3) by r(H°—»-YY) » 10" 5M H keV (2.2.4) w i t h MJI measured i n u n i t s of GeV/c 2. Hence the standard model two photon width I s only about 10 keV ( i . e . a branching r a t i o of l e s s than 0.1%). 2.3 Two-Higgs-Doublet Model 2Y~Decay Widths The model described i n s e c t i o n 2.1 has one pseudoscalar and two s c a l a r n e u t r a l Higgs bosons. This 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 that one or more of these non-standard spin-0 bosons has a two photon decay width which i s g r e a t l y enhanced r e l a t i v e to the standard model. R e c a l l the v e r t i c e s given i n Table I . The magnitude and s i g n of the s c a l a r c o u p l i n g to charged Higgs i s h i g h l y model dependent and w i l l be discussed below. As expected the pseudoscalar i|> i s not a f f e c t e d by the mixing parameter 0 m. I t does not couple to the W-bosons or to the charged Higgs bosons, but only to fermions. Hence the pseudoscalar decay width i s the l e a s t model dependent. In general the s c a l a r couplings to the W-boson 30 are smaller than In the standard model. Thus one must look to the fermion loops f o r any possible enhancement. For s i m p l i c i t y we take 6m=0, which i n fac t produces the constraint equation X 2 b 2 - X i a 2 = (b 2-a 2)(X 3+X 4+X 5)/2 (2.3.1) I f a l l the X^'s are of the same order, t h i s equation can be n a t u r a l l y s a t i s f i e d . With t h i s choice of 9 m the couplings f o r one of the s c a l a r s , n, become i d e n t i c a l to those of the standard model Higgs boson, g i v i n g the same width as discussed i n the l a s t section. The other s c a l a r , <j>, now does not couple at a l l to the W-boson, eliminating the destructive interference and leaving only the fermion loop amplitude. There are two ways to enhance the co n t r i b u t i o n of the fermion loops. From Table I i t can be seen that i f tana (cota) i s large then the lepton and u-type quark loops w i l l be enhanced (decreased), and the d-type quark loops decreased (enhanced). This i s true f o r both the s c a l a r and pseudoscalar bosons. From the standard model I t was found that only the t-quark loop made a s i g n i f i c a n t contribution, and thus i t i s the l o g i c a l one to try and enhance. Hereafter the discussion w i l l focus on the Two-Higgs-Doublet model with large tana enhancement. The r e s u l t s f o r models with large cota w i l l be si m i l a r except that the width being enhanced i s much smaller to begin with. The dominant cont r i b u t i o n f o r both the scalar <|> and the pseudoscalar now comes from the t-quark loop enhanced by tana. To fi n d the maximum enhancement allowed by the model, bounds on the magnitude of the enhancement factor can be determined by low energy phenomenology. The v i r t u a l e f f e c t s of the charged Higgs i n Bhabha s c a t t e r i n g , muon decays [11,12,13] and the K°-K° mass di f f e r e n c e [14] give an upper l i m i t f o r tana as a function of the charged Higgs mass . The approximate bound tan 2a < 2M^/mc i s taken from reference [14] where m c i s the charm quark mass. L i m i t s on can be obtained from c o n s i d e r i n g charged Higgs e f f e c t s i n the W and Z-boson propagators. For M » M~,M~ the change i n the mass r a t i o of X Tl A the gauge bosons i s p r o p o r t i o n a l to M^. S p e c i f i c a l l y = 1.2 TeV/c 2 w i l l g ive a 5% change i n the p-parameter [15] where p=M 2/M 2cos 26 . This i s W L W w i t h i n the allowed experimental e r r o r [1,2]. Thus we f i n d tana < 40 (2.3.2) In the Two-Higgs-Doublet model there 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 to the 2y-decay width of the Higgs s c a l a r s coming from the charged Higgs s c a l a r loops. The Feynman diagrams f o r t h i s process are the same as those i n s e c t i o n B.2 of appendix B, except the loop p a r t i c l e I s a charged Higgs x + r a t h e r than a would-be Goldstone boson. From equation (2.1.12) the r e l e v a n t s c a l a r c o u p l i n g to charged Higgs terms of the Lagrangian are X+x"<i> —• v[-X 1sin 2a+X 2cos 2a-(A/2)cos2a] sln2a (2.3.3a) x+ x-,j, — y v[(X 1+X 2-A)sin 22a+2X 3]/2 (2.3.3b) w i t h 6 m = 0 . The magnitude and s i g n of these couplings i s not determined by theory. Above i t has been argued that < 1.2 TeV/c 2 which means th a t X^ and X^ are of the order of u n i t y . I t i s n a t u r a l to assume that a l l the Xj/ s are of the same order. This argument Is by no means rig o r o u s but i s supported by p a r t i a l wave u n i t a r i t y plus p e r t u r b a t i o n theory [16,17] which gives a s i m i l a r bound on M^. With these caveats I t can be s a i d that the couplings of equation (2.3.3) are not enhanced. Hence the s c a l a r x~loops give a n e g l i g i b l e c o n t r i b u t i o n to the two photon decay width. For l a r g e tana the two photon decay width of the Higgs s c a l a r i s then dominated by the t-quark loop and i s given by r ( * + YY) = tan 2a|(-ie 2gm 2./6n 2M )[2+(4X -1)I(Xj] P/(16TTM~) (2.3.4) t w t t cp where Xt=m2/M~2. This result i s obtained using the same techniques, described in appendix B, as were used for the standard model calculation. The pseudoscalar fermion loop calculation i s performed in section B.4 of appendix B. Again the t-quark loop dominated with the result r ( * * YY) « tan 2a|(-ie 2gm 2/6ir 2M w)I (A t ) p/(16irM^) (2.3.5) where X =m2/M2. These widths are indeed greatly enhanced over the standard t t i|> model result for large tana. The other scalar width r(n •»• yy) i s the same as the standard model result. In general 8m*0, and the 2y-decay widths of the scalars cp and n l i e somewhere between the two extremes of the standard model result [equation (2.2.3)] and the best case result [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 this knowledge w i l l be useful background for the discussion of the results to be presented in the remaining chapters. Also illustrated were the methods needed to calculate the standard model Higgs 2y-decay width. These calculational methods can be found in the literature, 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 calculation 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 for the standard model. Hence the two photon decay process is much more important in Two-Higgs-Doublet models. The consequences of this result w i l l be discussed at length in chapter IV. I I I . 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 specific example of a Two-Higgs-Doublet model. The motivation for supersymmetry has been discussed in the introduction. As was the case in the last chapter, the minimal broken supersymmetry model i t s e l f can be found in the literature. Again the details are presented below for the readers edification and to establish notation. The remainder of the chapter i s devoted to the calculation of the Higgs 2y-decay widths for this model. These are a l l new results, and their significance is further discussed in the next chapter. The particle content of the minimal broken supersymmetry model is quite similar to that of the Two-Higgs-Doublet model, and i s discussed in more detail below. The main difference is that each particle is now accompanied by a superpartner which differs by one half a unit of quantum spin. We w i l l discover later in this chapter that these new superparticles do not significantly affect the two photon decay widths of the Higgs bosons. In fact they slightly reduce the widths through destructive interference with the usual particle 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 in the next chapter. 3.1 The Model Deriving the actual supersymmetry Lagrangian is too complicated to present here. The results in component f i e l d notation w i l l be summarized below. The component f i e l d content needed for the minimal supersymmetric model Is listed in table II [18]. It includes, in the gauge sector, the usual SU(2) t r i p l e t of vector bosons, V y a (a=l,2,3), and the U ( l ) y vector boson Vy' along with their 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 i c component fermions L (1=1,2) along with a two component SU(2) singlet e T. Similarly, for the quark sector there is a doublet Q* (1=1,2) and two c c ~I singlets, u^ and d^. The scalar partners of the quarks are denoted as and Uĵ .dj^ for the SU(2) doublet and singlets, respectively. The slepton SU(2) doublet and singlet are L and e^ respectively. As in the Two-Higgs- Doublet model, only one quark lepton family w i l l be of interest. Thus the quark mixings and family labels are omitted, although these can be included straightforwardly. The matter sector is completed with the addition of Higgs multlplets. Three sets of Higgs f i e l d , , Ĥ  and N are employed to break the SU(2)xU(l) symmetry [19]. 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 in supersymmetry models, the Higgs doublets alone are no longer sufficient 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 this problem. Enlargement of the Higgs sector to include an SU(2) and U(l) singlet f i e l d N allows for discussion of the supersymmetric limit, with the gauge symmetry broken to U ( l ) e m # Finally these scalars are a l l accompanied by the fermionic partners \J»̂  , and iJ>.T. Table I I - Supersymmetrlc F i e l d Content Gauge Bosons Gauginos SU(2) V» Leptons c e L Quarks Q L=(u,d) L "L < Higgs Bosons . i H H Sleptons L =(v,e-)j Squarks <*R Higgsinos H 1 Hj_ H 2 H 2 1/2 0 1/2 0 0 1/2 1/2 -1 2 1/3 -4/3 2/3 -1 F i e l d content of the minimal supersymmetrlc SU(2)xU(l) model with one family. SU(2) gauge bosons carry 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 give the SU(2) representations and the U ( l ) hypercharges of the respective f i e l d s . The superscript c indicates charge conjugation. 36 The component f i e l d Lagrangian has been extensively reviewed [3,5,20] and i s presented below. The t o t a l i n t e r a c t i o n Lagrangian, «£^ n t» Is divided into a supersymmetric piece, «Cgg> a n < * a piece which s o f t l y breaks supersymmetry, o t c e , n . The supersymmetric part i s invar i a n t under S O D transformations between bosons and fermions. Thus one has which i s constructed out of the f i e l d s l i s t e d i n table I I . A deri v a t i o n of of gg from s u p e r f i e l d formalism can be obtained i n reference [21]. For c l a r i t y and completeness gg w i l l be presented i n several pieces. The i n t e r a c t i o n s of the gauge mult i p l e t s among themselves and the matter f i e l d s are described by c£ . This i s given by J gauge ^gauge = iS^CxVJ " ̂ Vj) + T i ' ^ ^ A - X'V,) - i g T ^ U A ^ A j - 0 U A*) A j ] - l V ^ j } - ^ f V j y j A * * ^ - OyA*)A] - i y f * ô *} (3.1.2) . . ra ii, b„c + ige , X a X V abc y In the above equation, A denotes the scalar fields and i> represents generically the Majorana spinor fields of table II. The U(l) hypercharge of the scalar f i e l d A i s y A and that of the matter fermion f i e l d i s y f. A sum over a l l scalar fields A i s implicit. The SU(2) generators are T*j where a=l,2,3 and i,j=l,2. Also ov=(l,o) where o denotes the three Pauli matrices. The Yukawa interactions between the fermions and the scalar bosons are described by a second piece, e £ y . Also included are the scalar-fermion Higgs f i e l d interactions since they are the supersymmetric partner Interactions to the Yukawa ones. E x p l i c i t l y ^Y = ^ j ^ V " ' - + f 2 e l H l F V R + I V i / V V i j ^ R F + 2Re[(h E l jHJN)(f ee i kL ke R+h de i kQ kd R)*] + 2Re[ ( h e ± ^ N ) ( h ^ Q ^ ) * ] + |f ee 1 ; JHjL J F + eijh^Jq^+h.c. + I V i j H l ^ J F + I V i j 1 ^ F + e i j h u H 2 Q J V h - C - + I ( hd HlV hu H2\> F + I V i j ^ R F (3a-3) where f e , h<j and h u are the Yukawa couplings. As usual gives rise to quark (squark) and lepton (slepton) masses. In the absence of supersymmetry breaking terms, the fermions and corresponding sfermions w i l l have degenerate masses. The third piece, -̂s» l s t n e superpotential and i t Is given by [5] = h 2(|H*F + |H| |N F + I h c ^ j H J + . p + F - 2|HJ PIHJ F + J L T 2 [4|niX F - 2|H* FIQJ p - r A l H ^ F -ZlHiPÎ P] +4|(jJ*i:1F -2 |QjF |L 1F (3-1.4) + i^r + IHJ r + IQ1 r + it1 r + Ig'2[ 14 F - |Hj F + i l Q 1 F - ||u R F + f |d R F - |L F + 2|eR p]2 where |A* f* = (A*" A*) 2 for the scalar f i e l d A. Finally for completeness the usual gauge fixing Lagrangian 2 OF " " 2 t I 3 / + ™<-4<h> - < H i > 1 H l ) I 2 " I F [ V'" + ^ y i C H I < H i > " < H i > t H i ) I 2 < 3 - K 5 ) and Fadeev-Popov (FP) ghost Lagrangian have to be added. The simple case of equal VEV's for the two Higgs doublets w i l l be sufficient to il l u s t r a t e the important features of the FP Lagrangian. In the expression below the ghost fields are C ,C ,C ; the would-be Goldstone bosons are G~,G°; the ± y Z usual gauge bosons are W~W,AW,ZV; and the Higgs scalar f i e l d i s H. Thus * F P - - C > 2 C + - C V C _ - c V ^ - C Z + 3 2 C Z - 5 M 2 C z t C z / c o 8 2 6 W - 5 M 2 ( c | c + + C ^ C _ ) - (?gM w/2)[(C^C ++C^C_)H + C ^ H / c o s Z e J + i g ^ c a ^ c ^ ) ^ - O j j c ^ ) w - u ] ( c Y s i n e w + c z c o s e w ) - i g 5 [ O y c J ) s i n 9 w + O j j cJ)cos6 w](C_W + l J + C +VT U) (3.1.6) - ( l g S M w / 2 ) G 0 ( c J c + - C+cJ - 8 5 M w s i n e w ( c | G + + c V ) ^ + (g£M /2cos6 )[C^(C,G -+C G+) - (2cos 26 - l ) ( c T G + + C t G _ ) C , ] K v vJl Z + - w + - Z J - i g d ( 9 C?)C. - (9 C f)C ] ( A y s i n e - Z ycos6 ) The 't Hooft-Feynman gauge w i l l be chosen. Combining a l l the pieces together, the supersymmetrlc standard model Lagrangian i s just ^ S S " *gauge + * Y + + ^GF + ^ F P ( 3 ' 1 ' 7 ) The above i n t e r a c t i o n Lagrangian i s g l o b a l l y supersymmetrlc. To be phenomenologically r e a l i s t i c , the supersymmetry must be broken. This can be achieved by soft breaking terms [22] which are thought to be induced by supergravity at the scale of the Planck mass, Mp. The e f f e c t i v e low energy ( i . e . below the Planck mass) Lagrangian that breaks supersymmetry can be written as [23,24] - I m^A*A - m 3 / 2 ( h ( Z ) + h.c.) (3.1.8) A where h(Z) = (A-3)g(Z) + \ | | Z A (3.1.9a) A and g = ( h E ± .H^N +sN) + V ^ H j L ^ + V i j H ^ R + V i j ^ L ^ R + h' C- ( 3 ' 1 ' 9 b ) As before the sum over A represents a sum over a l l scalar f i e l d s . The S\S rV gaugino masses m' and m as well as the g r a v i t i n o mass m^/2 a r e ^ r e e parameters. Equation (3.1.9a) contains terms that s p i l t the degeneracy i n the masses of the sfermions and fermions. Because of the s i m p l i c i t y and the added a t t r a c t i o n of having a structure close to the unbroken supersymmetric model, I t i s usual [23] to take the parameter i n equation (3.1.9a) to be A=3. It Is often argued that the gaugino and the scalar masses can be taken to be a l l equal to the 1113/2 a t t n e Planck scale. However, there are many uncalculable e f f e c t s Involving gravitons i n the high energy theory and i t i s not clear that a common mass can s t i l l be maintained at low energies. Hence the m^'s are d i f f e r e n t i n general. Hereafter, these parameters w i l l c a r r y s u b s c r i p t s denoting t h e i r p a r t i c l e s p e c i e s . Thus m^, , m^ , e t c . w i l l be the bare mass terms of the s c a l a r f i e l d s N, H^, H^, e t c . r e s p e c t i v e l y . The gauge symmetry breaking i s achieved by l e t t i n g the three sets of Higgs f i e l d s , H^, H 2 and N develop vacuum e x p e c t a t i o n values (VEV's), given by <N> = v3 72 (3.1.10a) (3.1.10b) (3.1.10c) As was done i n the Two-Higgs-Doublet model, a set of c o n s t r a i n t equations on the VEV's can then be obtained by minimizing the s c a l a r p o t e n t i a l contained i n equations (3.1.4) and (3.1.8). They are m 3 / 2 ( | h V l v 2 + s ) + T ^ + f v 2 ) - 0 (3.1.11a) 7 2«3 / 2hv 2 v 3 + m2 v x + | [ hVj ( v|+v2 )+2sv 2] + g p v ^ v j - v 2 , ) - 0 (3.1.11b) 7 2»3 / 2 h vi v3 + »H v2 + |[hv 2(v2+v2)+2sv 1] - g ? v 2 ( v * - v | ) = 0 (3.1.11c) where = v 2 + v 2 (3.1.12) and P = g 2 + g' 2 (3.1.13) In the l i m i t that v i = V 2 * 0 , taking the differ e n c e of equations (3.1.11b) and (3.1.11c) gives (m2 - m 2^)v 1 = 0 (3.1.14) Hence i t i s necessary f o r the 'bare' masses of H^ and H£ to be equal i f they are to develop the same VEV. The s p e c i a l case where vi=V2=a and m̂  =111̂  =m^=m^j2 * 8 solved i n reference [23]. A p a r t i c u l a r l y simple s o l u t i o n i n t h i s l i m i t i s given by /2 ~h "3/2 (3.1.15a) m„ = / 2 ^ . [ 1 - - 5 l i _ ) 1 / 2 h _o "3/2 (3.1.15b) Next the phenomenologically more i n t e r e s t i n g case of v ^ * V 2 i s investigated. For t h i s case, the constraint equations can be recast i n t o the following forms 75m3/2hv3 ~ Z^vlv2 + | ( h v i v 2 + 2 s ) = —\—Y~ v i v 2 (3.1.16a) V 2 _ V 1 f^v 2 + ipv2)(v2-v|) + m2 v 2 - m2 v 2 = 0 (3.1.16b) V V 7I m3/2 h v3 + \h<.hviv2+2s' + ~^F"(lnH + m H + h 2 v 3 ) " 0 (3.1.16c) v 1 2 which are useful i n s i m p l i f y i n g the mass matrix f o r the Higgs bosons. Equations (3.1.16) s i m p l i f y f o r the case m̂  =111̂  =m i n the non-degenerate VEV region. In p a r t i c u l a r equation (3.1.16b) becomes h 2v 2 + 2m2 - - \ p v 2 (3.1.16b') The three Higgs f i e l d s H^, a n d N are not the phys i c a l mass eigenstates and a diagon a l l z a t i o n has to be performed. There are s i x neutral spin-0 f i e l d s given by the r e a l and Imaginary parts of the three Higgs f i e l d s ; e x p l i c i t l y they are given by v + ReH° + iImH° H = ( _ i L_ I , K) (3.1.17a) f l 1 v_ + ReH° + iImH° H ? = ( H; , l- ^ ) (3.1.17b) f l N = ( v 3 + ReN + ilmN) (3.1.17c) The superscripts on the H - f i e l d s are SU(2) i n d i c e s . Two charged scalars form the following combinations H + = ^ H * + v2H**) (3.1.18) G + ~ v ( V 2 H 2 " V 1 H 1 ^ (3.1.19) with H~ and G~ given by the conjugates of equations (3.1.18) and (3.1.19). The phys i c a l charged Higgs f i e l d s are H*. The G* are the would-be Goldstone bosons which enter i n the gauge-fixing conditions f o r the W-bosons given by 8 WW+ = ̂  v G + (3.1.20a) V 25 8 WW = 4f v G - (3.1.20b) V 25 where 5 Is the gauge f i x i n g parameter. Noting that M 2 = g 2 v 2 / 4 , equations w (3.1.20) are seen to be the usual gauge conditions f o r the standard model. 45 For the 't Hooft-Feynman gauge 5=1. The combinations i n equations (3.1.18-19) can be shown to diagonalize the charged scalar mass matrix when the constraint equations (3.1.16) are used. The unphysical bosons G* have mass ^ i n the 't Hooft-Feynman gauge as expected. The mass of the physical charged scalars i s given by M2 + = h 2 v 2 + m2 + m2 + M2 (3.1.21) H~ 1 2 W f o r the general case of vi*V2« This equation further l i m i t s the allowed values f o r m̂  and m̂  . One example i s the case where v ^ * v 2 a n < * • Using equation (3.1.16b') gives M2 + = - r g' 2v 2 (3.1.21') H" and hence i f the e f f e c t i v e Lagrangian i s not to give unphysical masses to the charged Higgs bosons, then m^ *m̂  i n the region where v j * v 2 ' I t Is i n s t r u c t i v e to consider the case of degenerate VEV's, i . e . v l = v 2 * Further s i m p l i f i c a t i o n i s made by the choice of " f l - V = m3/2 ( 3- 1' 2 2> and the s o l u t i o n of equation (3.1.15a). Then we obtain M 2 ± = 4 ^ 3 ^ + M2 (3.1.23) Thus the simplest solutions lead to the conclusion that the charged Higgs boson Is heavier than the W-boson quite independent of the coupling parameters i n the sc a l a r p o t e n t i a l . It must be emphasized that t h i s need not be true i n general f o r v j * v 2 and m̂  *m̂  . The s i x neutral spin-0 bosons consist of three scalars and three pseudoscalars. One of the pseudoscalars i s the would-be Goldstone boson which gives mass to the Z° , and i t i s given by G° = £ (v 2ImH° - v^mHO) (3.1.24a) Orthogonal to Ĝ  i s a pseudoscalar h^; e x p l i c i t l y written as h4 = v ( v i I m H 2 + v 2 I m H l ) (3.1.24b) and a t h i r d pseudoscalar ImN. In t h i s basis G° decouples from the other two and only plays 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 rotated away. However, the mass matrix of the two remaining 0~ bosons i s s t i l l not diagonal. As usual the diag o n a l i z a t i o n i s achieved by a ro t a t i o n , leading to the two phys i c a l pseudoscalars, H° and H^, below. H° = h° cosx - ImN sinx (3.1.25a) H° = h° sinx + ImN cosx (3.1.25b) The mixing angle x can be obtained i n terms of the sc a l a r p o t e n t i a l parameters v i a 6h m... v tan 2x = ^= (3.1.26) /2 {h2v2 + ( 4 ^ - ^ ) } In the degenerate VEV case with m̂  =111̂  = m N = m 3 / 2 a n c * 8 ° l u t * - o n (3.1.15a), t h i s reduces to /2 h M tan 2x = (3.1.26') 8 m3/2 The pseudoscalar masses are given by M £ " ( h 2 v 3 + m H - t ^ ^ v 2 ) cos 2x + ( ^ v 2 - ^ ) s i n 2 x 1 2 (3.1.27a) + ^ m 3 / 2 v c o s x s * n x cos 2x = (^v^+mfj +m2j +1jh2v2) s i n 2 x + ( j ^ v 2 ^ ) 1 2 (3.1.27b) ŷ " n m 3 / 2 v c o s x s i n x which s a t i s f y the sum rule «l + M2 = h^v^+v 2) + m2 + m2 + m2 (3.1.28) The remaining degrees of freedom are the sca l a r ( 0 +) f i e l d s ReH°, ReH^ and ReN which again have a non-diagonal mass matrix. The physical mass eigenstates s h a l l be denoted by with eigenvalues (where 1=1,2,3), and they are obtained from the above by a unitary transformation H i = hi R e H j (3.1.29) In equation (3.1.29) i t i s understood that ReN i s to be substituted f o r j=3. In general the elements of U are complicated functions of the VEV's and the bare s c a l a r masses. These w i l l not be examined here as they are not p a r t i c u l a r l y i l l u m i n a t i n g . However, the i n t e r e s t i n g sum rule I M2 = M2 + M2 1=1 1 (3.1.30) should be noted. Phenomenologlcally the U-JJ can be treated as free parameters. From the above discussion one would expect that these spln-0 bosons have masses of the order of M or m 0 / „ , unless the transformation w 3/2 parameters are w i l d l y d i f f e r e n t . In a d d i t i o n to the mixing i n the Higgs sector, the sc a l a r fermions w i l l also mix to a c e r t a i n extent. Attention i s focused i n p a r t i c u l a r on the scalar t-quarks, since they w i l l be the only relevant ones contributing to the 2y-decay width c a l c u l a t i o n . A mixing 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 term In equation (3.1.9b) when H 2 develops a VEV. The mass matrix f o r t T and t.. i s given by / -R hv \ m" 1 m3/2 + ~ 2 v3^ h v l \ m t ( m 3 / 2 + ~ 2 V3^ 4 (3.1.31a) / where 4 - \ g' 2(v2- V2) + m2R (3.1.31b) and m2 = — g , 2 ( v 2 - v 2 ) + m2 (3.1.31c) L 24 1 2 BL with m̂  being the fermion t-quark mass and nig Ĵ L̂) t n e ^are m a s s appearing i n *»gg B« I n general 1 1 1 ^ * 1 1 1 ^ . The mixing angle 9 between t ^ and t R can be deduced from equation (3.1.31) giving 16m . m tan 29 = ^=-^ (3.1.32) 8(m2 B R-m2 B L) + ( g'2_ g2) ( v2_ v2 ) In the symmetrical case of v^=v 2 and m 2j£ = m 2jL then 0=TT/4. In general however V1* V2 a n d mBR* mBL* ^ ° n e t a ^ e s mBR = m3/2 t n e n ® * s °^ t n e o r & e r °f t a n - 1 [mfc / ^/2^' P r e l i m i n a r v d a t a f r o m C E R N t 2 5 ] indicates that 20 < m̂  < 50 GeV/c 2 and i t i s possible that c a n a ^ e w t :*- m e s heavier than M w» Thus 8 i s generally quite small even f o r the scalar t-quarks. For s i m p l i c i t y t h i s small mixing i s neglected, and the sca l a r quarks t T and t^ Li K. are treated as mass eigenstates. There i s yet a t h i r d set of mixed states that are important i n the 2y-decay width c a l c u l a t i o n . These are the states formed from the mixing of the W-gauginos and charged Higgsinos. In the Lagrangian [see eq. (3.1.2)] + the charged gauginos and Higgsinos are represented by Majorana spinors X and tp* and i j i 2 , res p e c t i v e l y with 1 H2 X" = (X 1 + i X 2 ) / / 2 (3.1.33) Again they are not the physi c a l mass eigenstates. These physical states are constructed e x p l i c i t l y as follows: V IX+ c o s * + + i j i ^ sin<|>+\ iX~ cos<J> + si n * H l (3.1.34a) X o = IX"1" sine)) - i p * cos* + H 2 + w -IX~ s i n * + ip2 cos* H l (3.1.34b) Notice that there are two separate mixing angles <f>+ and <j>_. One can read o f f d i r e c t l y from equations (3.1.2) and (3.1.8) the mass terms i n v o l v i n g the W-gauginos and charged Higgsinos. Diagonalization i s achieved using equation (3.1.34), which then gives the mixing angles *+ and the masses 2 °f the two p h y s i c a l chargino states x̂  a n d x2 r e s p e c t i v e l y . In terms of the parameters appearing i n the Lagrangian, these angles are given by s i n 2 K = [ d + s i n 2 a ) 1 / 2 ± ( l - s i n 2 a ) 1 / 2 1 w { + ~ s i n 2 2 a > 1 / 2 w where tan a = v L / v 2 (3.1.35') This r e s u l t agrees with that presented i n a d i f f e r e n t form i n reference [19]. For v^ » or » v^ the angles become s i n 24. = -=i-2. (3.1.36a) w and 4_ = 0 (3.1.36b) On the other hand with equal VEV's the angles become equal 4+=4 =4 and equation (3.1.35) reduces to s i n 2 24 = (1 + (3.1.37) 4M' w The c a l c u l a t i o n also y i e l d s the masses and K^. These are written e x p l i c i t l y as 5 i , 2 - 7JMw [(1+*in2° + i i - ) 1 7 2 1 (1-Sin2a +1|-) 1 / 2] <3-1-38> This completes the discussion on the physical states which w i l l appear i n the 2y-decay width 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 rules which are obtained from the Lagrangian are given i n appendix E. Figure 1 - One Loop Contributions to the 2y-Decay of the Scalar X X X X X X X X X X 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 X The diagrams are grouped into separately gauge invariant sets. (I) gauge boson, would-be Goldstone boson, and ghost loops (II) would-be Goldstone boson loops (III) physical charged Higgs boson loops (IV) charginos loops (V) fermion loops (VI) scalar-fermion loops 53 3.2 One Loop C a l c u l a t i o n of X u •»• yy The Feynman rules l i s t e d i n Appendix E are used to c a l c u l a t e the matrix elements which contribute to the two photon decay widths of the spin - 0 bosons, denoted by X° , i n the one-loop approximation. The i n t e r n a l loops for a l l of the scalar Ĥ , decays consist of fermion and scalar fermions, gauge bosons and gauginos, and phy s i c a l charged Higgs bosons. The would-be Goldstone boson and Fadeev-Popov ghost loops are also included. The one-loop c o n t r i b u t i o n to the two photon decays of the s c a l a r H^ are displayed i n s i x sets of diagrams i n figure 1. Each set i s separately gauge in v a r i a n t . Set 1 i s the gauge boson loop contribution, denoted by a j w , and includes mixed would-be Goldstone bosons-gauge bosons and Fadeev-Popov ghosts. Set 2 i s denoted by ajQ and consists only of f u l l would-be Goldstone boson loops. Both of these sets have the same structure as i n standard model Higgs boson to two photon decays [6]. In add i t i o n there are contributions from loops containing the physical charged Higgs boson, H , and the charginos, x̂  (1=1»2). These are sets 3 and 4, contributing amounts a and a, r e s p e c t i v e l y . If the Yukawa couplings are non-vanishing then J J x l ,2 the fermion loops of set 5 w i l l give the a j f . F i n a l l y set 6 shows the scalar-fermlon c o n t r i b u t i o n a j r » which contains both gauge and Yukawa pieces. As noted i n the previous section, the small mixing between l e f t and ri g h t types of scalar-fermions has been neglected f or s i m p l i c i t y . Combining a l l the contributions, the matrix elements f o r the s c a l a r H° (j=l,2,3) decays into two photons with p o l a r i z a t i o n vectors e^ and e 2 are presented below. The d e t a i l s of the c a l c u l a t i o n are given by combining the Feynman rules of appendix E with the c a l c u l a t l o n a l techniques of appendix B. The r e s u l t s are l i = ( a 4 + 3 J + A * + A J +a. +a. +a. ) N y V e 1 e 2 j jw JG j X l J X 2 JH 3 f V U V i e 2 g M u v U +v U with a [6+(-8+12X )I(X )] [ 1 1 3 2 j ] J W (4ir) 2 w w v ie 2gM 2 v U +vU_, * j G T3- [ l + 2 X w I ( X w ) ] [ 1 1 3 2 2 j ] J b (4ir) 2M W w v w 2/2 ie 2gM^ a ^ ^ " i - 1 ™ ^ * * ^ ^ 2ie 2gM v U. ,+v,U,4 Zh^v.D,, • I I + 2 V < V H ( I - k'x11JV 2 23 ) + 2 i g e 2 e 2 m 2 c f v a - - I L-L-L- [2+(4X - l ) I ( X f ) ] V f J t f (4ir) 2M f f w 2 i e 2 e 2 c g2(v,D,-vO ) ~ ~ w where = — IL . f o r f=d-fermions (sfermions) V l 1 3 ^— U^j f o r f=u-fermions (sfermions) Furthermore, Nf = I n * ~ e £ s i n 2 9 , = e £ s i n 2 9 L 2 f f w R f w and X = n^/M2 The subscript of the A's corresponds to the mass of the i n t e r n a l loop p a r t i c l e . Here n^ i s +1 (-1) for up (down) type sfermions. Also s + , c + denote sin<p+, cos<p+ r e s p e c t i v e l y [see eq. (3.1.34)]. The colour f a c t o r c^ i s 3 f o r quarks and 1 f o r leptons. The gauge invariant quantity Ny V i s N = [ g - ] (3.2.11) yv 1 yv P»q where p and q are the photon momenta, of the f i r s t and second photon re s p e c t i v e l y , and I(X) = |*dx i l n [ l - J x(l-x)] (3.2.12) The function AI(A) has a very weak dependence on A for values of A » 1/4. This dependence i s shown i n appendix D. S i m i l a r l y the matrix elements f o r the two photon decays of the pseudoscalars H^ and H^ are computed. The diagrams which contribute are displayed i n figure 2, and consist only of fermion and gaugino loops. The matrix elements for the pseudoscalar H^ (k=4,5) are a. = (a, +a, +a, _) N PVe 2 (3.2.13) k v kXj^ k x 2 K f ps y v 4/2 e2gM^ W ± t h ^ X l = ( 4 l l ) 2 v + W + ^ ^ k ( 3 ' 2 - 1 4 ) 4e 2e 2gm 2c v n a _ - - Z i—LJ. (_L) I(A,)n, (3.2.15) k f f (4n) 2M w V2 F K where = (3.2.16) and n k i s cosx (sinx) f or k=4 (5). 56 (I) chargino loops (II) fermion loops The above gives the general amplitudes f o r scalar or pseudoscalar Higgs boson to two photon decays i n broken supersymmetric theory, calculated with component f i e l d techniques. (Note that these remain non-vanishing i n the supersymmetric l i m i t [21]). Obviously these amplitudes contain many unknown parameters such as mixing angles and masses of unseen p a r t i c l e s . In the next two sections are some reasonable s i m p l i f y i n g assumptions, and an estimate of the widths. Also examined c a r e f u l l y i s the allowed enhancement of these widths i n broken supersymmetric gauge theories. 3.3 Pseudoscalar Widths of X° yy The pseudoscalar widths of X° •»• yy can now be calculated from the re s u l t s presented i n the l a s t section. However, f i r s t one must e s t a b l i s h a range f o r the many parameters appearing i n the amplitudes. The two extreme cases f o r the r a t i o of the two VEV's, v^ and v 2 , are denoted as case A for v^ » v^ and case B f o r v^ = v^. In case A, the mixings of the charginos are given by equation (3.1.35). Substituting into equations (3.2.15) and (3.2.16) one finds the dominant contributions come from x^ and the t-quark; thus f o r H^ decays /2 e 2 g ML a = I(X ) cosx sin<p, (3.3.1) x l 4* 2 x l + 2 m 2 a = - ^ - 8 — M tana — I ( X J cosx (3.3.2) fc 3TT 2 W M2 t w From equation (3.1.26) there i s no apparent reason why the mixing between the pseudoscalars should by small. Hence, one expects cosx = 1//2. Notice that the r e l a t i v e phases of a and a are destructive since d>, Is i n the t x + f i r s t quadrant. The width f o r H? decays i s then obtained to be a 3M 2 cos 2x r C H j + Y Y ) - w l U t ) ] 2 (3.3.3) ( 4 i r ) 2 s i n 2 0 M, v w 4 w w The width of the H° decay i s obtained from the above by s u b s t i t u t i n g sinx for cosx and M. by Mc. 4 5 At f i r s t s i g h t , bounds on the magnitude of tana can be determined by low energy phenomenology just as i n the Two-Higgs-Doublet model. Again t h i s i s achieved by examining constraints from Bhabha sc a t t e r i n g , muon decays [11,12,13] and the K°-K° mass differ e n c e [14]. Taken together these gave from t h e o r e t i c a l reasons which take into account p a r t i a l wave u n i t a r i t y plus perturbation theory [26]. In supersymmetry there Is a further constraint on tana. This i s due to the r e l a t i o n between the masses, M^ and M2, and tana. Now r e c a l l from equation (3.1.37) that tana < 40 (3.3.4a) Moreover, one could also argue f o r the much more stringent bound of tana < 12 (3.3.4b) g - [ ( 1 + S i n 2 a + - ^ - ) 1 / 2 ± ( l - s i n 2 a + - ^ - ) 1 / 2 ] (3.3.5) w w For the case of interest, i t i s easily proven that tana = 2M2 / (3.3.6) As can be seen In equation (3.3.5), x 2 *-8 t n e lighter of the two charginos. It must have a mass greater than 20 GeV/c2 in order to agree with e +e~ experimental data [27]. This then puts tana < 8 ^ / \ (3.3.4c) for Mw=80 Gev/c2 and M̂  > 20 GeV/c2. Hence in order to remain in case A, x^ must have a mass no more than ~7 times M̂ . The consistent range for would be v2 M < M, < 7 M . Thus the upper bound becomes w 1 w v* tana < 5.7 (3.3.4d) Figure 3 displays the width of Ĥ  decaying into two photons as a function of i t s mass, for the range of allowed x i masses, with the mixing angle x chosen to be TT/4. It i s seen that this width i s typically of the order of 60 keV or less for an intermediate mass pseudoscalar. The width i s dominated in this case by the t-quark loop contribution. The dependence of tana via 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 for decay into a pair of real t-quarks (mt=40 GeV/c2), the rise in the width respectively increases sharply and then slows abruptly. In general this type of behaviour w i l l occur whenever the threshold for pair production of a particle which makes an important loop contribution to the decay width is crossed. Similar results hold for H° decays. Figure 3 - Pseudoscalar 2y-Decay Width for Case A Case A ( v 1 » v 2 > : two photon decay width as a function of mass for the pseudoscalar H° (k=4,5) with mixing angle x=ir/4, f or the range of allowed K. X, masses. Figure 4 - Pseudoscalar 2y-Decay Width f o r Case B m in in Case B (v^=V2): two photon decay width as a function of mass for the pseudoscalar H° (k=4,5) with mixing angle X=TT/4, for the range of allowed Xi masses. 62 Next examine case B. With a = ir/4 one obtains again <J>+=<|>_=<j> and e x p l i c i t l y <f> = \ s i n _ 1 [ i ] (3.3.7) (1 +_*!_) 1/2 4 M2 w The masses of the charginos are given by M = MJ ( l + - ^ ) l / 2 ± - L ] (3.3.8) 2 W 4M2 2M w w and they are the order of M^. Now the two photon widths of and are e a s i l y obtained to be a 3M 2cos 2x M1 M. , m2 r(HO-YY) = [ — K X v ) + — K X v ) sin24 - = K X . ) ] 2 (3.3.9) (4ir)2sin 2e M. M 1 M 2 M2 w 4 w w w 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 2 x f o r cos 2x and M,. for to get T(H°->-YT)' These widths as a function of t h e i r mass, for the range of allowed chargino masses, are plotted i n figu r e 4. Neither X 2 n o r t n e t-quark loop contributions dominate, since there i s no tana enhancement. Consequently the width i s much smaller than i n case A, i . e . less than 25 keV. The upper bound curve i n figure 4 corresponds to both charginos having mass MW, and hence i s sharply peaked near the threshold at 2MW« 3.4 Scalar Widths of X° ->• YY It i s now straightforward to carry out the same analysis f o r the two photon widths of the scalar Higgs bosons H° ( j = l , 2 , 3 ) . For definiteness consider only H^ decays. I t Is clear that the same analysis can be pushed through almost verbatim f o r H° and H°. Just as i n the case of the 2 3 63 pseudoscalars there i s no apparent reason f o r the mixing between these scalars to be small. For s i m p l i c i t y , assume that they are a l l approximately equal, i . e . «U~2 -U"3j=U=l//3 for j=l,2,3. As seen i n fi g u r e 1 there are many more i n t e r n a l loop contributions compared to pseudoscalars; hence, more free parameters i n the form of i n t e r n a l masses appear. It has already been noted that the combination XI(X) does not vary a great deal over a wide range of values for X. Thus, one does not expect the two photon widths to be too s e n s i t i v e to the values chosen for these masses. Observe that the amplitude due to fermion loops of equation (3.2.6) i s dominated by the t-quark for both cases A and B. This i s due to the mass of the t-quark being much larger than other fermions i n the minimal 3 quark lepton f a m i l i e s universe. For case A further enhancement i s due to the presence of the f a c t o r . Notice that the scalar-fermion loop cont r i b u t i o n of equation (3.2.7) i s dominated by the scalar-top f or both cases A and B. To the extent that X~I(X~) i s i n s e n s i t i v e to the choice of scalar fermion mass, the term i n v o l v i n g N ^ R j w i l l give zero when summed over a l l scalar fermion types. The remaining Yukawa term i s proportional to the square of the corresponding fermion mass, and hence the scalar-top dominates. Again i n case A there i s further enhancement by the V f a c t o r . Incorporating the above considerations, one finds that f o r case A the sc a l a r decays have r<H° + rt) - | a w + a G + a X i + a x 2 + a H + a t + a ~ p (3.4.1) where a a a~ ie 2gM - [6 + (-8+12X )I(X )]U (3.4.2) w ( 4 l f ) » w w ie 2gM r ^ I X 1 + 2 I < * > ] U (3.4.3) G . ( 4TT ) 2 W W -2/2ie 2 gM̂ a v [2 + ( 4 X , - l ) I ( X i )]U sine? ( 3 . 4 . 4 ) X2- ( 4 i r ) 2 2 2 + 2ie 2M 2h 2v (4ir ) z gM w Sige2!!!2 — [2 + (4X.-1)I(X,.)]U tana (3.4.6) 3(4ir) 2M C C w 16ie 2gm 2 ^ [ l + 2X~I(X~)]U tana (3.4.7) 3(4TT) 2M C T w This width as a function of the scalar mass i s displayed i n fig u r e 5. The standard model scalar width i s also shown f o r comparison. The major contributions to the scalar width i n t h i s mass range are the t-quark loop, W-gauge boson loop, and to a smaller extent the chargino loops. For larger s c a l a r masses, the scalar-top and charged Higgs loops w i l l also contribute, but only near or above t h e i r thresholds. Destructive interference between the t-quark and W-gauge boson loops r e s u l t s i n a generally smaller width than i n the standard model which by comparison i s dominated only by the W-gauge boson loop. A l l the curves In figure 5 r i s e sharply near 160 GeV/c 2, which corresponds to the threshold for W-gauge boson p a i r s . Figure 5 - Scalar 2y-Decay Width f o r Case A Case A ( v ^ » v 2 ) : Two photon decay width as a function of mass for the scalar H° (j=l,2,3) with mixing angles y=U2y=U3j=l//3, for the range of allowed x^ masses. The broken curve shows the standard model Higgs boson width f o r comparison. Figure 6 - Scalar 2y-Decay Width f o r Case B Case B (v^=V2>: Two photon decay width as a function of mass for the scalar H° (j=l,2,3) with mixing angles U ! j = U 2 j = U 3 j = 1 ^ » f o r t h e r a n 8 e o f allowed masses. The broken curve shows the standard model Higgs boson width f or comparison. 67 S i m i l a r l y f o r case B one has ie 2gM a = [6 + (-8+12X )I(X )lu/5 (3.4.8) w ( 4 i r )2 w w - [X" 1 + 2I(X )]u/2 (3.4.9) aG a X ie 2gM (4TTV -2/2ie 2gM^ = [2 + (4X,-1)I(X 1)]U sin24 (3.4.10) \ (4TT)2 2 2 2ie 2M 2h 2v, 3H = -rrS[1 + 2 X H I ( V ] K I - 2 ^ r i + — > u <3-4-n> (4TT) z gM w 8ige 2m 2 — [2 + ( 4 X t - l ) I ( X t ) ] u / 2 (3.4.12) 3(4TT) 2M w 16ie 2 gm2 - [ l + 2X~I(X~)]u/2 (3.4.13) 3(4TT) 2M w Substituting these into equation (3.4.1) gives the width r(H° + YY) which i s again displayed as a function of s c a l a r mass In fig u r e 6. Once again the standard model width i s shown f o r comparison. The discussion i s s i m i l a r to that f o r case A, except that the t-quark loop i s not an important cont r i b u t i o n here, since there i s no tana enhancement. Consequently the width i s a b i t l a r g e r , and dominated mostly by the W-gauge boson loop. The gaugino loops i n t e r f e r e d e s t r u c t i v e l y with the W-gauge boson loop and hence the s c a l a r width Is s t i l l smaller than In the standard model. F i n a l l y note that thus f a r only the example where the mixings U^^ are a l l approximately equal has been used. Now consider the best case p o s s i b i l i t y , where the r e l a t i v e phases between mixings i s such that the Figure 7 - Scalar 2y-Decay Width f or Best Case A CVJ 1̂- CvJ (A9M)J Case A ( v ^ » V 2 ) : Two photon decay width as a function of mass for the sca l a r H° (j=l,2,3) with mixing angles 1^j=-U 2j=U 3j=l//3, f or the range allowed masses. The broken curve shows the standard model Higgs bos width f o r comparison. dominant loop contributions i n t e r f e r e c o n s t r u c t i v e l y . This i s not possible i n case B since the gauge boson and gaugino loops contain the same combinations of with an o v e r a l l r e l a t i v e minus sign. However, f o r case A one can greatly increase the width i f Û _. - " " ^ j * This w i l l give constructive rather than destructive interference between the two main contributors, namely the gauge boson and t-quark loops. The scalar width for t h i s best case scenario Is plotted In f i g u r e 7 and i t i s Indeed now enhanced r e l a t i v e to the standard model width, although not by a great amount. This concludes chapter III on the minimal broken supersymmetry model. The model, which i s also described i n the l i t e r a t u r e , was f i r s t introduced and then applied to the c a l c u l a t i o n of the 2y-decay widths of the Higgs bosons. The 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, and we remind the reader of some important h i g h l i g h t s . F i r s t the a d d i t i o n a l superparticle content did not s i g n i f i c a n t l y a l t e r the Higgs 2y-decay widths, generally causing a small decline through destructive interference e f f e c t s . Once again i t was the enhancement of the Higgs to fermion couplings, due to the a d d i t i o n a l Higgs doublet, which led to a greatly increased Higgs 2y-decay width. Unlike the case for the Two-Higgs-Doublet model however, supersymmetry imposes a much more severe bound on t h i s possible enhancement. The next chapter w i l l discuss the e f f e c t s of t h i s new constraint, as well as the s i g n i f i c a n c e of the other r e s u l t s obtained thus f a r . IV. NON-STANDARD SPIN-0 BOSON PRODUCTION The r e s u l t s of the l a s t two chapters w i l l now be used to determine production cross sections for the spin-0 bosons i n ep and e + e - c o l l i d e r s . These cross sections can be expressed i n terms of the Higgs boson to two photon decay widths previously c a l c u l a t e d . I f i t s 2y-decay width i s large enough, the dominant production 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 fusion mechanism. This mechanism and the s p e c i f i c s of how to c a l c u l a t e the production cross sections are discussed i n d e t a i l below. The point i s that the Higgs boson production rates 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 widths, i f the widths are large enough. This i s indeed the case for the supersymmetry and Two-Higgs-Doublet models previously introduced, i f the enhancement fa c t o r tana Is large. Spin-0 boson production rates are calculated for each of these models i n ep and e +e~ c o l l i d e r s . These new r e s u l t s are presented f or discussion and comparison below. Detection of the Higgs bosons i n these c o l l i d e r s i s achieved by observing a peak In the inva 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 products. For the Higgs mass range studied, these decay products w i l l consist of two hadronic jets 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 that the Higgs predominantly decays i n t o . Hence the cleanest s i g n a l w i l l be f o r e +e~ machines. Unfortunately we w i l l f i n d that the production rates at the SLC c o l l i d e r w i l l be too low f o r observation even i n the most o p t i m i s t i c scenario. Thus we must turn to the higher luminosity ep machines, at the expense of larger 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 jets from the i n i t i a l beam j e t s . Nevertheless we w i l l f i n d promising r e s u l t s f o r the Two-Higgs-Doublet model, with production rates i n the best case which are r e a d i l y observable at the HERA c o l l i d e r . For the minimal broken supersymmetry model however, we w i l l f i n d that once again the rates are unobservable, as a r e s u l t of the more severe constraints on the enhancement factor tana; The s i g n i f i c a n c e of these r e s u l t s i s discussed below. 4.1 The C a l c u l a t i o n This s e c t i o n studies the production of sca l a r (S°) and pseudoscalar (P°) spin-0 bosons i n electron-proton and e +e~ c o l l i d e r s . The ep semi-inclusive reactions studied are e + p — • e + S° + X e + p — e + P u + X (4.1.1a) (4.1.1b) where X denotes any hadronic states. The quark-parton model i s assumed f o r the c o l l i s i o n i n equation (4.1.1). The electron-quark s c a t t e r i n g subprocesses are eU) + Q(q) — e(JJ') + S° (h) + Q(q') (4.2.2a) e(Jl) + Q(q) — • e(ĵ ') + P° (h) + Q(q') (4.2.2b) where Q denotes ei t h e r the u-type or d-type quark i n the proton. In equation (4.2.2) the 4-momenta of the various p a r t i c l e s are given i n t h e i r respective parentheses.  I t i s well known that the production rate f o r the standard model Higgs boson i s very small [28] for an ep c o l l i d e r such as HERA. This can be understood by examining the production mechanism f o r r e a c t i o n (4.2.2). In ep c o l l i s i o n s , Higgs boson production proceeds v i a the t-channel diagrams depicted i n fi g u r e 8. The process of fi g u r e 8a i s suppressed by the two Z-propagators, although t h i s i s p a r t l y compensated by the large H°ZZ coupling. On the other hand the process i n fi g u r e 8b has no such suppression but i s enhanced by the double photon exchange poles. However, the H°YY vertex i s of higher order thereby rendering t h i s amplitude small. In most cases the standard model amplitude of figure 8a Is larger than that of f i g u r e 8b, but I t s t i l l r e s u l t s i n an extremely small production rate as discussed 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 or Two-Higgs-Doublet models. As seen i n the previous chapters, both the S°YY and P°YY v e r t i c e s can be enhanced s u b s t a n t i a l l y . For the scalar t h i s makes the photon exchange amplitude dominate over the Z-exchange one since the S°ZZ coupling remains unchanged to lowest order. The pseudoscalar w i l l be produced only through the photon exchange amplitude. Thus one can express the production cross sections of these spin-0 bosons In terms of t h e i r 2Y~decay widths. In the same view one should also consider the production of S° and P° i n e + e - s c a t t e r i n g . The purpose i s to compare the r e l a t i v e strengths of the above two types of c o l l i d e r s f o r scalar and pseudoscalar production. The reactions studied here are ones s i m i l a r to equation (4.1.1); namely e + + e" — • e + + e~ + S° (4.1.3a) e+ + e - — • e+ + e - + P0 (4.1.3b) Being interested i n cases where the S°YY and/or P°YY v e r t i c e s are enhanced, one again concentrates on the two photon production mechanism. This i s the same as f i g u r e 8b with quark l i n e s being replaced by e + l i n e s . There i s an added complication for the e +e~ reaction not present for the ep case. The s-channel equivalent of the diagram i n fi g u r e 8b should be included. The r e s u l t i n g destructive interference with the more dominant t-channel exchange graph causes a small c o r r e c t i o n . This c o r r e c t i o n can be neglected for the purposes of obtaining order of magnitude estimates for the production cross sec t i o n . Hence only the dominant t-channel process w i l l be retained. Otherwise the c a l c u l a t i o n i s the same as for the ep case, except that one does not need to convolute over parton d i s t r i b u t i o n s . The production cross sections for the processes i n equation (4.1.1) were calculated using the equivalent photon approximation (EPA) [29 30]. The photon spectrum used i s given by [31] s 6 V This method re l a t e s the photon-quark cross section (see fi g u r e 8b) to the electron-quark cross section f or the subprocesses i n equation (4.1.2). D e t a i l s of the c a l c u l a t i o n are described i n appendix F. The approximation i s useful i n that equation (4.1.4) can solved a n a l y t i c a l l y . The EPA has been demonstrated to be good to within ten percent i n resonance production In e +e~ c o l l i s i o n s , and i t i s expected to be of the same accuracy i n ep c o l l i s i o n s . As a check the production cross sections were also c a l c u l a t e d d i r e c t l y using the Monte-Carlo method (see appendix G), which evaluates the Integrals numerically. The convergence of the Monte-Carlo routine i s very slow In ep c o l l i s i o n s due to the Lorentz boost between the lab and the cm frames. This boost prevents the use of importance sampling techniques, which concentrate the e f f o r t of the Monte-Carlo routine near the important photon poles. Nevertheless the r e s u l t s agree to within the accuracy of the two methods. The r e s u l t s of the EPA c a l c u l a t i o n f o r the cross sections of the subprocess equation (4.1.2) are given below. For the scalar one obtains _, A M 2 0 a (3) = 4a 2e 2 M s£r(S° + yy)ln{-^r){ ln(^ r)[p2+2p-3-(2+2p+p2/2)lnp] " e q + ( p 2 / 4 ) l n 2 p + (2p 2+4p-6)ln(l-p) - (2.5p 2+4p-5)lnp (4.1.5) + (25-24p-p 2)/4 + (p 2+4p+4)[-Li(l)+Li(p)+(ln 2p)/2] } where PHM2Q/§, and s E ( l + q ) 2 i s the (cm energy) 2 for the subprocess. The quark charges are given by ee^ and L i ( x ) = - ^ X d t l n ( l - t ) / t i s the dilogarithm func t i o n . For the pseudoscalar one obtains the s l i g h t l y d i f f e r e n t form given by + (l+p+p 2/4)ln 2p + (2p 2+4p-6)ln(l-p) + (6-4p-7p 2/4)lnp (4.1.6) + (47-28p-19p 2)/8 + (p 2+4p+4)[-Li(l)+Li(p)+(ln 2p)/2] } The 2y-decay widths i n equations (4.1.5) and (4.1.6) were calculated i n the previous chapters. The quark-parton model i s then used to estimate the cross section f or the physical processes of equation (4.1.1) by convoluting over the quark d i s t r i b u t i o n functions f q ( x ) . E x p l i c i t l y a (s) - J dx Ef (x) a (xs) (4.1.7) where t h i s l a s t i n t e g r a t i o n i s done numerically. 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 9 R f u ( x ) = 2.2x (1-x) ° (4.1.8a) — 49 T R f d ( x ) = 1.25x (l-x) J'° (4.1.8b) f u ( x ) = f d ( x ) = 0 . 2 7 x - 1 ( l - x ) 8 , 1 (4.1.8c) which are taken from reference [32]. S i m i l a r l y the r e s u l t s for the e +e~ c o l l i s i o n s of equation (4.1.3) are e a s i l y obtained. They are simply given by equations (4.1.5) and (4.1.6) with s replaced by s, the (cm energy) 2 of the e +e~ system. Of course there i s no need to convolute over parton d i s t r i b u t i o n functions, as the r e s u l t i s already i n i t s f i n a l form. Again the r e s u l t s agree with the numerical Monte-Carlo check For t h i s case the lab and cm frames are the same. The importance sampling techniques mentioned above can therefore used, and convergence of the Monte-Carlo routine i s quite r a p i d . Thus the production cross sections of the spin-0 bosons are now expressed i n terms of t h e i r 2y-decay widths for both ep and e +e~ c o l l i d e r s . 4.2 Numerical Results and Discussion The photon-exchange production cross section as a function of /s i s given In f i g u r e 9 for the standard model Higgs boson i n an ep c o l l i d e r . In accordance with previous c a l c u l a t i o n s [28] t h i s cross section i s d i s t r e s s i n g l y small, t y p i c a l l y on the order of IO - 1* 0 cm2 for c o l l i d e r energies. The lower curve i n f i g u r e 10 depicts the same cross section as a function of the Higgs boson mass for /s=320 GeV, appropriate for HERA. Also ind i c a t e d i s the cross s e c t i o n due to the two Z-boson fusion mechanism alone. Similar curves for /s=l TeV are shown i n fi g u r e 11. Although obscured somewhat i n figure 10 by the e f f e c t s of phase space, there i s a r i s e i n the photon exchange cross section f o r large MJJ, which i s very apparent i n figure 11. This r i s e i s due to the behaviour of the function I(X) as discussed i n appendix D. The biggest standard model contribution comes from the W-boson loop, and hence the cross section r i s e s near the threshold at MH=2Mw. Standard model Higgs boson production i n ep c o l l i s i o n i s dominated by the two Z mechanism. At /s=320 GeV the cross section i s at l e a s t one order of magnitude too small for observation even f o r l i g h t Higgs. For /s=l TeV the two Z mechanism becomes just large enough i f the same luminosity can be maintained, and the production rate for the two photon process i s too low. Thus for the standard model, the p r e d i c t i o n f o r the production of Higgs bosons i s that ep c o l l i d e r s w i l l not be able to observe them. Similar r e s u l t s hold i n e +e~ machines. These conclusions are well known and the standard model r e s u l t s have only been shown f o r comparison with the more i n t e r e s t i n g Two-Higgs-Doublet model. Pl o t s of the enhanced cross sections as a function of mass, using tana=40, are given for the spin-0 bosons of the Two-Higgs-Doublet model i n figures 10 and 11. Again there i s a peak i n the cross sections of f i g u r e 11 Photon exchange production cross section with MH=40,150 GeV/c 2. The dash-dot (dashed) l i n e i s for standard 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 sca l a r (pseudoscalar) production with tana=40. The dash-dot (dashed) l i n e i s f o r standard model photon (Z-boson) exchange. 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. due to the behaviour of 1(A). However i n t h i s case the t-quark loop dominates so that the t h r e s h o l d occurs f o r Mn=2mt. In the s c a l a r case the peak i s l e s s pronounced and i s d i s p l a c e d at l a r g e r Mg due to the f a c t o r (4A-1) which m u l t i p l i e s 1(A) i n equation (2.3.4). S i m i l a r 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 cross s e c t i o n . For a range of Higgs mass, the enhanced photon exchange cross s e c t i o n s are much l a r g e r than the unchanged Z-boson f u s i o n mechanism by roughly an order of magnitude. The pseudoscalar r a t e i s about three times that of the s c a l a r . Although the a c t u a l cross s e c t i o n s f o r a Two-Higgs-Doublet model may f a l l below the bounds shown, reasonably l a r g e cross s e c t i o n s (up to 1 0 - 3 7 cm 2) are p o s s i b l e even at HERA energies. Hence one may be able to observe Higgs boson production i n ep c o l l i s i o n s w i t h i n the context of the Two-Higgs-Doublet model. P l o t s of the enhanced (tana=40) photon exchange cross s e c t i o n s are given i n f i g u r e 12 f o r the Two-Higgs-Doublet model bosons produced i n e +e~ c o l l i s i o n s . The v a r i a t i o n of cross s e c t i o n w i t h /s f o r MJJ=60 GeV/c 2, and the cross s e c t i o n versus Mg f o r /s=150 GeV are d i s p l a y e d . These values were chosen to f a c i l i t a t e comparison w i t h the standard model graphs given i n reference [33]. The behaviour i s s i m i l a r to that found f o r ep s c a t t e r i n g above. The peak i n the a vs. MJJ d i s t r i b u t i o n i s sharper s i n c e there i s no smearing by the parton d i s t r i b u t i o n s . The enhanced cross s e c t i o n i s roughly an order of magnitude l a r g e r than f o r the standard model one [33] which makes i t j u s t observable as discussed below. As i n the standard model, the intermediate mass s c a l a r or pseudoscalar bosons w i l l decay p r i m a r i l y i n t o a p a i r of heavy quarks, l e a d i n g to a s i g n a l of two j e t s . In 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 of the Higgs s c a l a r <p f o r ep and e +e~ photon exchange mechanisms are given. The pseudoscalar Figure 12 - Production Cross Sections for e +e~ -»• e+e~H° 100 110 120 130 140 150 V s ( G e V ) Production cross sections as a function of (a) /s for MH=60 GeV/c2 (b) Mj for /s=150 GeV. The solid (broken) line i s for Two-Higgs-Doublet model photon exchange scalar (pseudoscalar) production with tana=40. 83 Figure 13 - Rapidity D i s t r i b u t i o n s 0 1 ' 1 1 L_ -.75 -.25 .25 .75 Y Scalar 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 with /s=320 GeV and 1^=40 GeV/c 2 (b) e +e" -»• e +e"H° with /s=150 GeV and MR=60 GeV. The normalization i s a r b i t r a r y . 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 . In both cases the cm frame r a p i d i t y i s broadly peaked about zero. I t i s straightforward 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 of appendix F, or d i r e c t l y from the Monte-Carlo c a l c u l a t i o n . For the e +e~ + e +e~H° process, the lab frame and the cm frame coincide. The two Higgs decay jets should therefore stand out well away from the beam axis, providing a good s i g n a l . However i n the ep •*• eH°X process, the cm frame has a large v e l o c i t y i n the lab frame. The r e s u l t i n g Lorentz boost w i l l s h i f t the scale on the r a p i d i t y d i s t r i b u t i o n i n figur e 13a by roughly -1.6 at HERA. Hence i n the laboratory frame the Higgs r a p i d i t y w i l l peak at less than 10° from the beam axis, and at le a s t one of the decay jets may be d i f f i c u l t to d i s t i n g u i s h from the beam j e t s . In order to resolve t h i s problem, much higher event rates may be needed f or ep c o l l i d e r s than f o r e +e~ machines. The ep event rates discussed below are f o r HERA assuming /s=320 GeV and an integrated luminosity over one year's running of 1.89xl0 3 9 cm - 2. The e +e~ rates are f o r SLC assuming /s=100 GeV and an integrated luminosity over one year of 9.45xl0 3 7 cm - 2. In the standard model, Higgs boson production i s dominated by the Z-boson exchange mechanism and the event rates of <3 per year are too small to be observed. On the other hand the cross sections f o r the Two-Higgs-Doublet model can be quite s u b s t a n t i a l . This i s due to the p o s s i b i l i t y of enhancing the Yukawa coupling of the t-quark i n t h i s model. Without f o l d i n g i n detection e f f i c i e n c y , the upper bound estimates on the event rates i n ep c o l l i d e r s are calculated to be <65 for the scalar boson and <176 f o r the pseudoscalar. In e +e~ c o l l i d e r s the corresponding event rates are <2 for the scalar and <13 for the pseudoscalar. Although the cleaner s i g n a l may be found with e +e~ machines, the upper bound production rates for SLC are not large enough. A higher luminosity e e~ machine, perhaps LEPII, would be useful i n providing both larger rates and a clean s i g n a l . On the other hand the ep production rates are already quite large, e s p e c i a l l y f o r the pseudoscalar boson. Hence i t can be concluded that i t may be possible to detect non-standard spin-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 are not as promising as for Two-Higgs-Doublet models i n general. The q u a l i t a t i v e r e s u l t s are s i m i l a r , but the more stringent bound on tana i n equation (3.3.4d) gives a reduction i n the production cross sections by roughly a fac t o r of 50. Thus the best possible event rates f o r one years running w i l l be <3 for the pseudoscalar i n ep c o l l i s i o n s , and even smaller rates occur for the other cases. These rates are much too low, and hence i t w i l l not be possible to detect the non-standard Higgs bosons of the minimal broken supersymmetry model with t h i s method. This concludes chapter IV. The new r e s u l t s obtained were the production rates of Higgs bosons i n ep and e +e~ c o l l i d e r s f o r the Two-Higgs- Doublet and minimal broken supersymmetry models. It was found that e +e~ machines could not produce observable quantities of the Higgs bosons and hence the emphasis s h i f t e d to ep c o l l i d e r s . Here i t was found that f o r the Two-Higgs-Doublet model, r e a d i l y observable rates on the order of 100 events per year were possible f o r the HERA c o l l i d e r . For the minimal broken supersymmetry model however, production rates were once again unobservable due to the smaller tana enhancement f a c t o r . From these r e s u l t s we may conclude that the production of Higgs bosons i n supersymmetry models i s as d i f f i c u l t as i n the standard model. More importantly we now have a po t e n t i a l experimental test of supersymmetry. If Higgs bosons are detected at HERA as allowed by Two-Higgs-Doublet models, then we can ru l e out the minimal broken supersymmetry model. How general a supersymmetry model could be ruled out i s discussed i n the next chapter. V. SUMMARY AND CONCLUSIONS The two photon decay widths of non-standard spin-0 bosons were calculated for the two doublets extension of the standard SU(2)xU(l) electroweak model, and for the minimal broken supersymmetry model. The r e s u l t s were then used to obtain production rates for these spin-0 bosons i n ep and e +e~ c o l l i d e r s , f o r the intermediate mass range (40-160 GeV/c 2) studied. Features of the Two-Higgs-Doublet model include the scalar and pseudoscalar bosons which are present In addition to the usual standard model Higgs s c a l a r . Also there i s the p o s s i b i l i t y of enhanced fermion couplings to these bosons r e l a t i v e to the standard model, i f the r a t i o of the two VEV's, tana, i s large. The larges t uncertainty of the c a l c u l a t i o n l i e s i n the charged Higgs boson loops (see figure 15), but fortunately t h e i r c ontribution i s n e g l i g i b l e i f the parameters i n the scalar p o t e n t i a l are not large. I f not, one encounters e i t h e r strongly i n t e r a c t i n g scalars or vacuum i n s t a b i l i t y . Larger 2y-decay widths than i n the standard model are possible by enhancing the fermion loops (see figure 14). In the best case the t-quark loop i s enhanced by the upper bound value of tana=40, and i t dominates the other processes. This occurs for both the pseudoscalar and one of the s c a l a r Higgs bosons. The other scalar would behave as i n the standard model but with the gauge boson loop suppressed, and hence i t s smaller width i s of l i t t l e i n t e r e s t . The bound of tana < 40 was obtained using only phenomenological constraints on the mixing of the two VEV's. Perturbative p a r t i a l wave u n i t a r i t y constraints [26], which would lead to the more stringent bound of tana < 12, were not used. If one adopts t h i s then the numerical r e s u l t s discussed should be scaled accordingly. 88 The minimal broken supersymmetry model i s a s p e c i f i c example of a Two- Higgs-Doublet model, with some added t h e o r e t i c a l motivation. Consequently i t too has a l l of the features just described. In addition there are new features associated with the underlying a l b e i t broken supersymmetry. S p e c i f i c a l l y the charged gauginos and scalar-fermions w i l l give a d d i t i o n a l loop contributions to the two photon decay widths. In the intermediate mass range studied (40-160 GeV/c 2), i t was found that these a d d i t i o n a l contributions are not very large and i n fac t they i n general reduce the 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 with the usual contributions. One important new feature a r i s i n g from the supersymmetry i s the upper bound imposed on tana by the allowed range of the gaugino masses. The upper l i m i t of tana < 5.7 i n the minimal broken supersymmetry model i s less than half of the most r e s t r i c t i v e bound f or a general Two-Higgs-Doublet model, and much smaller than the phenomenological bound. The width varies as tan 2a for large tana, so that the largest possible width f o r t h i s model i s from 4 to 50 times smaller than one might have hoped f o r . A more o p t i m i s t i c s i t u a t i o n can a r i s e i f the mixings between the s c a l a r s , coming from the breaking of the supersymmetry, have phases such that constructive interference occurs between the W-gauge boson and t-quark loops. In t h i s case the scalar width i s enhanced to p a r t i a l l y compensate f or the smaller tana f a c t o r . Even t h i s best case p o s s i b i l i t y allows an enhancement of the sca l a r width of less than an order of magnitude over the standard model width. For the pseudoscalar t h i s best case p o s s i b i l i t y does not occur since the r e l a t i v e phases are fi x e d , and hence the pseudoscalar width i s d e f i n i t e l y smaller. Hence, i t can be concluded that the supersymmetry imposes a much lower upper bound on the possible tana enhancement of the two photon decay widths than do Two- Higgs-Doublet models i n general. Widths of the order of 100 keV are the best one can hope f o r , f o r both scalars and pseudoscalars, i n the minimal broken supersymmetric gauge theory. The production cross sections i n ep and e +e~ c o l l i d e r s were calculated for the non-standard spin-0 bosons i n terms of t h e i r two photon decay widths. 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 found that the cleanest s i g n a l w i l l occur for e +e~ c o l l i d e r s . Unfortunately the production rates were too low to be observed at the SLC c o l l i d e r f o r a l l cases studied. Therefore one must concentrate on the ep r e s u l t s . The ep upper bound production rates f o r the Two-Higgs-Doublet model are very large, being of the order of 100 events per year at HERA. Hence one concludes that i t may be possible to detect these non-standard Higgs bosons i n ep c o l l i d e r s . The ep r e s u l t s for the minimal broken supersymmetry model are more disappointing. The much s t r i c t e r upper bound on the possible enhancement In t h i s model leads to very low production rates f o r the non-standard spin-0 bosons, which again are not observable. Thus the observable production of non-standard Higgs bosons f o r minimal supersymmetry models w i l l not be possible at HERA. The two photon decay widths of these Higgs p a r t i c l e s i n supersymmetry models are of l i t t l e more importance than i s the case for the standard model. This negative conclusion f o r the two photon process does not of course prevent supersymmetry from manifesting i t s e l f i n other processes. It should be noted that If non-standard spin-0 Higgs bosons are produced at HERA by t h i s mechanism, as allowed by general Two-Higgs-Doublet models, then the minimal broken supersymmetry model would d e f i n i t e l y be ruled out. Thus we have a possible experimental test of supersymmetry. Such an occurrence would not however completely rule out supersymmetry. We have only considered the minimal model with three f a m i l i e s of quarks and leptons. If there are more than three f a m i l i e s , some of the a d d i t i o n a l heavy fermions could contribute to the 2y-decay width as much as the t-quark does. Thus the predicted production rates could be much l a r g e r . This w i l l not occur i n the standard model unless Incredibly large numbers of ad d i t i o n a l f a m i l i e s (>35) are used. However, f o r supersymmetry models the addition of only one extra family leads to an increased rate by a factor of nine. This i s because the t-quark, the new heavy u-type quark, and the new heavy lepton would each contribute roughly the same to the 2y-decay width. Thus we could s t i l l have a supersymmetric model, although a more complicated one than the minimal model. This concludes the presentation of t h i s t h e s i s . We have examined the two photon decay widths of non-standard spin-0 bosons f o r Two-Higgs-Doublet models i n general, and the minimal broken supersymmetry model i n p a r t i c u l a r . While the models themselves are established i n the l i t e r a t u r e , t h e i r a p p l i c a t i o n to the Higgs 2y-decay width and the subsequent c a l c u l a t i o n of the various Higgs production rates a l l represent new r e s u l t s . In these models with a d d i t i o n a l Higgs doublets, there i s the p o s s i b i l i t y f o r enhanced spin-0 boson to fermion couplings i f the r a t i o of the two vacuum expectation values (tana) i s large. This i n turn leads to enhanced production of these bosons v i a the two photon fusion mechanism at rates which could r e a d i l y be observed at the HERA c o l l i d e r . In supersymmetry models the new superpa r t i c l e content w i l l give r i s e to a d d i t i o n a l contributions to the 2y-decay process. We found that these are not very large and that t h e i r e f f e c t i s to s l i g h t l y reduce the width v i a destructive interference with the usual contributions. The important new r e s u l t a r i s i n g from supersymmetry i s that i t imposes a much smaller upper bound on the possible tana enhancement of the fermion couplings than do Two-Higgs-Doublet models i n general. The o r i g i n of t h i s a d d i t i o n a l constraint l i e s i n the experimentally established lower l i m i t s for 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 , the Higgs bosons of the minimal 3-family supersymmetry model cannot be produced at observable rates. Only supersymmetry models with a d d i t i o n a l generations of heavy fermions can produce Higgs bosons at rates which could be observable at HERA. Therefore we have a possible experimental test of supersymmetry i n that the detection of Higgs bosons at HERA could rule out the minimal model. 92 BIBLIOGRAPHY 1. G. A r n i s o n et a l . , UA1 C o l l a b o r a t i o n , Phys. L e t t . 122B, 103 (1983); i b i d , 126B, 398 (1983). 2. M. Banner et a l . , UA2 C o l l a b o r a t i o n , Phys. L e t t . 122B, 476 (1983); i b i d , 129B, 130 (1983). 3. N.P. N i l l e s , Phys. Rept. HOC, 1 (1984). 4. E. F a r h i and L. Susskind, Phys. 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Kamal, J.N. Ng and H.C. Lee, Phys. Rev. D24, 2842 (1981). 32. R. Baler, J. Engels and B. Peterson, Z. Phys. C6_, 309 (1980). 33. R. Bates and J.N. Ng, Phys. Rev. D32, 51 (1985). 34. J. Wess and B. Zumino, Phys. Lett. 49B, 52 (1974). 35. B. deWit and D.Z. Freedman, Phys. Rev. D12, 2286 (1975). 36. P. Zakarauskas, Ph.D. Thesis, University of British Columbia, (1984), unpublished. 37. I.M. Sobol, The Monte Carlo Method, MIR Publishers, Moscow (1975). APPENDIX A - SOME ACCELERATOR PROPERTIES Table I II - Accelerator Properties Location C o l l i d e r Type Luminosity (cm~ 2s - 1) /s (GeV) SLAC SLC e +e~ 3 x l 0 3 0 100 CERN LEP I e +e~ 6 x l 0 3 1 100 CERN LEP II e +e~ 2 x l 0 3 2 170 DESY HERA ep 6x10 3 1 320 U.S.A. SSC pp 1 0 3 3 40,000 APPENDIX B - EVALUATION OF FEYNMAN DIAGRAMS Consider a spin-0 Higgs p a r t i c l e decaying into two photons with 4-momenta k^.k^ a n d p o l a r i z a t i o n vectors e^Ckj^.e^k^) r e s p e c t i v e l y . One would a p r i o r i expect the matrix element f o r t h i s process to have the gauge invari a n t forms * v v = A [ g y v - k j k j / k j . k j ] lctk2e / k i ' k 2 (B.la) (B.lb) for a s c a l a r or pseudoscalar Higgs p a r t i c l e r e s p e c t i v e l y . These terms are the only tensors one can make from the two independent momenta ( ^ ^ 2 ) which do not vanish when the scalar product i s taken with the p o l a r i z a t i o n vectors. The contributions of the d i f f e r e n t Feynman diagrams to t h i s matrix element are shown below. Each set i s separately gauge invar i a n t as i s demonstrated. Although the d e t a i l s of the coupling strengths w i l l depend on the model studied, 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 are the same for the standard model, Two- Higgs-Doublet models and supersymmetric models. Consequently only one representative example i s evaluated f o r each set of Feynman diagrams. The standard model Feynman rules needed can be found i n reference [10]. B.l Scalar Higgs 2y-Decay v i a Fermion Loops The con t r i b u t i o n of the fermion loop i n fig u r e 14 i s demonstrated for the standard model. Such a contribution w i l l a r i s e from quarks and leptons i n a l l the models studied, as well as from chargino loops f o r supersymmetry. 96 f \ w w Figure 14 - Fermion Loop Contribution to Scalar 2y-Decay The matrix element, obtained using the standard model Feynman r u l e s , i s M w v . J _ d ^ _ T r [ ( ^ ! ! f ) ( J _ ) ( _ i e e y ) ( _ i _ ) (2ir) H w n f n 1 f x C - l e V v K ^ ^ ] ( B . i . i ) where h=kj+k2 i s the Higgs 4-momentum and k^*^ are the photon 4-momenta. Evaluating the trace r e s u l t s i n -2ge 2e 2m 2. Mr= — < 4 c r - 2 c> +v v - 2 k i c i + cywkX w yvr „a + g y [- C™ A + + (nr|-k 1.h)C 0] } (B.1.2) where the loop i n t e g r a l s CQ,C^,C2 are given i n appendix C. Substituting the r e s u l t s from appendix C, and r e t a i n i n g only those terms consistent with the form of equation (B.l) gives v, y M 2ige 2 e2m2 k r k . [- 2 - (4X-1)I_ 1 ] [ g " v (B.1.3) f 16TT 2M * k..k. w 1 2 where X=m2/M2 and 1 ^ i s given by equation ( C . l ) . An a d d i t i o n a l factor of 2 Is included i n equation (B.1.3), a r i s i n g from the crossed diagram where i d e n t i c a l photons are interchanged. Note that the c o e f f i c i e n t s of both the uv v u g and k^k2/k^»k2 terms are i d e n t i c a l , e x p l i c i t l y demonstrating the gauge invariance of the matrix element. The cont r i b u t i o n of the scalar loops i n figure 15 i s demonstrated for the standard model i n the 't Hooft-Feynman gauge. In t h i s case the scalar loop p a r t i c l e i s the would-be Goldstone boson. In Two-Higgs-Doublet models t h i s set of Feynman diagrams a r i s e s f o r both would-be Goldstone bosons and physical charged Higgs bosons. Similar contributions occur for supersymmetry models, along with scalar-fermlon loops. B.2 Scalar Higgs 2y-Decay v i a Scalar Loops Figure 15 - Scalar Loop Contribution to Scalar 2y-Decay The matrix elements, obtained from the standard model Feynman rules, are S ( 3 ) (2*)* 1 (q-k.)2- M2 1 -igM* x ( i — ) ( — ^ C - T T ) 1 (B.2.I.) (q-h) 2-M 2 2M q2-M2 W W w M s ( b ) • 5 / ̂ (21«VVH M l — J ^ ) (B.2.1W S Q t > ; L (2 1r)' t (q-h)2-M2 2Mw q2-M2 Rewriting i n terms of the loop i n t e g r a l s CQ , C ^ , C 2 , C from appendix C gives M s ( a ) = T S f W " 4 C l k l " 2 C i k 2 " 2 C l k l + k i ( 2 k l + k 2 > V c 0 l < B ' 2 ' 2 a > w " " ^ ^ c ' <B-2-2b> w Combining these two matrix elements with the r e s u l t s of appendix C, ret a i n i n g terms of the form i n equation (B.l) and including a fac t o r of 2 for the i d e n t i c a l photons gives ie2gM2 k > ! J M s V = , (1 + 2 X I J ( g V V - ) CB.2.3) 8 16ir2M 1 K l k2 w where X=M^/M2. Again note the e x p l i c i t gauge invariant form of eq. (B.2.3) As discussed i n appendix D, the fa c t o r (1+2XI_^) i s small f or X » 1/4 and consequently the scalar loop c o n t r i b u t i o n i s usually small as well. B.3 Scalar Higgs 2y-Decay v i a Gauge Boson Loops The diagrams In figu r e 16 are the largest set to contribute to the scal a r Higgs decay width. The loop p a r t i c l e s include W-gauge bosons, 99 Figure 16 - Gauge Boson Loop Contribution to Scalar 2y-Decay 100 would-be Goldstone bosons and the Fadeev-Popov ghosts. Such a contribution a r i s e s i n a l l the models studied, and i s demonstrated for the standard model i n 't Hooft-Feynman gauge. As was done i n the previous sections, one obtains the matrix elements given by - - e 2 8 « J l O C - - 9 c X - ^ + 5 V K + g y V ( 2 C ^ - C^ O^+kj) - 2 ^ - ^ - H i l » V . 2 C Q ) ] (B.3.1a) = - 3e2gMw g y V C' (B.3.1b) M ^ c ) = - f f j L [- + 3 C y - 4C V/ 2 + g y V ( C ° a + 2 C l . k 2 ) ] (B.3.1C) yv _ 6 g Mw r„yv _ __u.v . _v,u . ..y.v, M (d) - ^ [ C ^ " 5CjkJ + c j k j + 2 ^ C Q + g U V ( - C 2 a + (4k 1+2k 2)-C 1 - UL^VJCQ)] (B.3.1d) = - e2g M3 s ^ C Q (B.3.1e) e 2gM M 2 M ( f ) 2 ^ ^ C Q (B.3.1f) b ! V 8)-- !T £[- 2 C! V- 2 CM (B-3-LG) = f ? J L [ 2 cwv _ 2 c y k v _ 4 c v k y + 4 k v k y c j ( B > 3 > l h ) M?* - - e 2g>l g P V C" (B.3.1I) C i ; w M j V j ) = - e 2 g M w [ C 2 V - c K ] (B.3.1J) where o n l y terms o f the form i n e q u a t i o n ( B . l ) have been k e p t . S u b s t i t u t i n g from appendix C g i v e s the expe c t e d gauge I n v a r i a n t form, w i t h X=M^/M2, as M' ie 2gM k X Z [6 + <-8+12X)I_1][gwv - (B.3.2) 16TT2 1 2 101 B.4 Pseudoscalar Higgs 2y-Decay v i a Fermion Loops The fermion loop contribution of fig u r e 17 i s demonstrated for the Two- Higgs-Doublet model. Similar contributions a r i s e i n supersymmetry models. The matrix element i s given by Mf- * / ̂ T r [ ( ^ ) ( - ^ V y K i q r z m - ) ( - i e e f y v ) 1 (2TT ) q mf 1 q fcl mf 1 X (Fh=m7K-yfY5^ (B.4.1) where y^ i s the coupling strength of the pseudoscalar to two fermions. Evaluating the trace r e s u l t s i n yvaB (B.4.2) r Figure 17 - Fermion Loop Contribution to Pseudoscalar 2y-Decay 102 Substituting f o r the loop i n t e g r a l s C Q J ^ 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 photons gives e 2 e 2 y .m, „ f' = " ^ - 1 E k l a k 2 B / k l * k 2 ( B - 4 ' 3 ) which i s the form expected. This concludes the demonstration of the evaluation of the Feynman diagrams which contribute to the two photon decay width of Higgs bosons. The r e l a t i v e weights due to the coupling strengths w i l l vary f o r d i f f e r e n t models, but the basic structure as shown i n equations (B.1.2), (B.2.2), (B.3.1) and (B.4.2) i s the same. 103 APPENDIX C - EVALUATION OF LOOP INTEGRALS The one-loop Feynman diagrams which contribute to the Higgs two photon decay width contain several i n t e g r a l s which are evaluated i n t h i s appendix. A useful d e f i n i t i o n i s i n(XM - {dx x n m[ ^tyTx) ( C D where X=m2/M2 i s the r a t i o of the loop p a r t i c l e mass squared over the Higgs mass squared. F i r s t consider the i n t e g r a l c = j a q i (C.2) 0 (2TT )** [ q 2 - m 2 ] [ ( q - k 1 ) 2 - m 2 ] [ ( q - h ) 2 - m 2 ] where h=k^+k2 i s the Higgs 4-moraentum and k^.k^ are the photon 4-momenta. Expanding with Feynman parameters gives C = / fdxfdyfdz r(3) 6(l-x-y-z) ^ U (2^)** ° 6 ° [ (q2- m2)x+(q2-2q.k 1-m 2)y+(q 2-2q.h+M2- m2) z]3 Performing the i n t e g r a l over the loop v a r i a b l e q gives the r e s u l t C 0 = — f i x f d y 6 ( 1 - X - y ) (C.4) U 16TV2M2 0 t» [x(l-x-y) - x] F i n a l l y i n t e g r a t i n g over y and using d e f i n i t i o n (C.l) gives co — T-l ° 16ir2M2 1 ( C 5 ) Second consider the i n t e g r a l C y = / d**q £ (C.6) 1 (2iry* [q2-m2][(q2- k2)- m2][( q-h)2- m2] Again expanding with Feyman parameters and evaluating the i n t e g r a l over the loop parameter q gives C v = i — fdx/dy = (C.7) 1 16TT2M2 0 0 [x(l-x-y)-X] Then i n t e g r a t i n g over y and using d e f i n i t i o n (C.l) r e s u l t s i n C i = ^ i j [ k i ( I o - z-i> - ( c - 8 ) Third consider the i n t e g r a l cyv = ; _ _a_g _ _ ( ( J # 9 ) (2TT )** [q2-m2][(q-k1)2-m2][(q-h)2-m2] which i s expanded using Feynman parameters as before. Unlike the previous uv two cases, C 2 i s not f i n i t e and the i n t e g r a l over the loop v a r i a b l e q must be r e g u l a r i z e d . This i s performed i n d=4-e dimensions using the method of dimensional r e g u l a r i z a t i o n , and y i e l d s 1 i [yk U+(l-x-y)h U][yk^+(l-x-y)h V] 9(1-x-y) c v 2 v = —4T W<* 16TT2M2 0 0 [x(l-x-y)-X] + tfr iV*7 { A + l n [ x - x ( i - x - y ) ] i ( c - 1 0 ) 105 where A = ^+ij)(l)+ln(4iru 2/m 2) and u Is the a r b i t r a r y mass scale introduced by the r e g u l a r i z a t i o n . The i n t e g r a l over y then gives c y v — [ (~ \ + I Q " M ^ X k J k i J + kjk*) ] ( C . l l ) 16TT 2M2 + i M — [ A + 1 - I Q + 2XI_ X ] + (terms - kjk^.kjkl*) pv When contracting indices on i t must be remembered that the metric i s i n 4-e dimensions so that g W =4-e. Thus P 16* 2 Next consider the i n t e g r a l c 2 p = 7T7 t A + X I-i 1 ( c * 1 2 ) c . = j _ i H l _ I (C.13) (2TT) 4* [q 2-m 2] [(q-h) 2- m2] Expanding with Feynman parameters gives c . - j J * i - r d x r d v n i ) 6 ( i - x - y ) ( 2 * ) * * t) [(q2- m2) x f( q2_ 2 q.h+M2- m2)y]2 (C14) Again the i n t e g r a l over the loop parameter must be regularized giving C = — — [ A - I n ] (C.15) 16*2 u S i m i l a r l y C - - / - ^ 1 = _ J _ A (c.16) (2TT )*• tq 2- m2][(q-k 1)2-m2] 16ir 2 106 APPENDIX D - PROPERTIES OF THE FUNCTION I(X) This appendix describes the function I(X) where 1(A) = /dx I ln[ X-x(l-x) X (D.l) Note that I(X) i s just I_^ as defined i n equation ( C . l ) . This function appears i n a l l of the matrix elements for spin-0 boson to two photon decay widths. Evaluating equation (D.l) gives The function XI(X) i s plotted i n figure 18. The r e a l part shows a very weak dependence on X f o r values of X » 1/4, and there i s a sharp peak at X=l/4. The imaginary part i s zero f o r X > 1/4, and peaks between 0 < X < 1/4. Both the r e a l and imaginary parts go to zero as X 0. P h y s i c a l l y X=l/4 corresponds to the threshold to produce a p a i r of r e a l rather 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 p a r t i c l e vertex. Hence one should see t h i s peaking behaviour whenever a threshold i s crossed. [In equation (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, and hence i n t h i s case the peak at X=l/4 i s suppressed.] Providing that one i s well below threshold, i . e . X » l / 4 , then equation (D.2) can be approximated by I(X) = + °(^~ 2) which i s useful In making simple comparisons. 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 loops i n equation (B.2.3) w i l l make small contributions below threshold. K X ) = - 2 [ s i n - l ( ^ r ) ] 2 J . 91 ? r 1+/1-4X1 r 1+/1-4X-I (D.2) Figure 18 - Plot of the Function XI(X) vs X The s o l i d (broken) curve shows the r e a l (imaginary) part. 108 APPENDIX E - FEYNMAN RULES FOR MINIMAL BROKEN SUPERSYMMETRY The e f f e c t i v e Lagrangian f o r broken supersymmetric standard models i s given i n equation (3.1.2)-(3.1.8) i n component f i e l d s . The component f i e l d content of the theory i s displayed i n table I I . In working with component f i e l d s , the usual Wess-Zumino gauge of supersymmetry [34] i s chosen. The gauge of the SU(2)xU(l) symmetry i s f i x e d to be the 't Hooft-Feynman gauge. The Fadeev-Popov (FP) ghost i s thus the same as that of the standard model [35]. Due to the mixing of the scalars a l l three f i e l d s H° (j=l,2,3) couple to the ghost f i e l d . As expected the pseudoscalars H^ and H^ do not couple to the FP ghosts. The relevant couplings f o r the c a l c u l a t i o n of the amplitudes of X° -»• "YY are given i n d i f f e r e n t sets below. The f i r s t set involves the scalar + couplings to fermions, sfermions, charged Higgs bosons H and t h e i r ± ± companion would-be Goldstone bosons, G , as well as the gauge bosons W and the FP ghosts. This i s displayed i n figure 19. The set of diagrams i n figure 16 i s gauge invariant i n the standard model and i n the two Higgs doublet model. Demanding that t h i s gauge invariance holds i n supersymmetry i s a reasonable condition, which greatly s i m p l i f i e d the ff^GH^ vertex (and consequently the H+H^O vertex). Figure 20 gives a l l the photon couplings. The mixed states of charged Hlggsinos and W-gauginos, x-̂  and x 2> have couplings to the Higgs scalars given i n figure 21. Sewing together the v e r t i c e s given below gives the f u l l set of Feynman diagrams displayed i n fig u r e 1 for H° to two photon decays. For the pseudoscalars the couplings are simpler. Only two are relevant; namely x^ a t l d x 2 a n < * fermion couplings are involved and these are represented i n figure 22. 109 Figure 19 - Feynman Rules f or Scalar H° Couplings H -imf V H H- -igMw+igM 2M, V M .+ V U . rij 2 2j " 2 i h V s G' 6" -IgM 2M w „--" f tn. ig^tv.U-.-v.u,.) ~ ' R / L 2C0S29, 2 2 ' ' " N l . . - 2 l m 2 v ' R / L w , , " f w H fl* t 2 M " V. U.j + V2 U2j v U +v U I Ij 2 2j 2 ' v U + 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 are used. Figure 20 - Feynman Rules f o r Photon Couplings 110 " ' i to i *L- o A / W W * ; : ~ lee.tp + p L v w w v < ^ ±iep P L / R Y 8 x + W* / w \ A ^ 7 . - , e [ { P,-P2 ) xV • ( P2" P 3 ) >' 5 »'X* ( p 3" P l , * ' 0 X ^ P | P 3 W X K i n 2 , e V ? ^ H. ( G 1 -MP-P^ W" * f9^ v V £ A A A < C ^ J " l e e f r / * Figure 21 - Feynman Rules f o r Chargino-Scalar H° Couplings < T 1 " 7 ^ r ( s - c + u . , + s + c _ u 2 i ) Hj I — e — < x 2 7r ( s + c - u . j + s - c ^ ) H. A 2 V ^ J 112 Figure 22 - Feynman Rules f o r Chargino-Pseudoscalar 11° Couplings H H X 2 g X s ^ ( V . S . C . + V . S . C J T ; H f f _T7 f Here k=4,5 and n f=+l (~1) for up (down) type fermions. Also n, =cosx (sinx) for k=4 (5). k 113 APPENDIX F ~ EQUIVALENT PHOTON APPROXIMATION The equivalent photon approximation (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 cross section for the electron-quark s c a t t e r i n g process i n f i g u r e 8b. It involves t r e a t i n g one of the exchange photons as a parton-like object, c a l c u l a t i n g the photon-quark cross section and then convoluting over a photon spectrum d i s t r i b u t i o n to obtain the electron-quark cross s e c t i o n . The photon-quark "subprocess" i s depicted i n f i g u r e 23. This leads to a matrix element given by _ " i g M = u ( q i ) [ i e e Y P ] u ( P l ) ( — E ^ - ) M a V ( - p 2 , - p 1 + q 1 ) £ C T ( p 2 ) ( F . l ) where M° V(k^,k 2) i s the Higgs to 2y decay width matrix element, and e a ( P 2 ) i s the p o l a r i z a t i o n vector of the Incoming photon. The usual Mandelstam variables w i l l be used. The structure of M C T V(k^,k 2) must be of the form k V M ^ . V = F ( g a V ( F . 2 . ) for a scalar Higgs, or k k M ( k l f k 2 ) - F [e - j ^ - ) (F.2b) for a pseudoscalar, as described e a r l i e r i n appendix B. 114 Figure 23 - Photon Quark Subprocess Substituting these into equation ( F . l ) leads to photon quark cross sections of the form a e 2 F 2 sCs-M 2) 2 2s (s-M 2) 2 a ( s c a l a r ) = — 3 — { ln[ — ] + — (s-M 2)ln[ — ] 4s 2 m 2 q M H 2 8 ( 8 - 4 ) — 2 - } (F.3a) and a e 2 F 2 s ( s - M 2 ) 2 a (pseudoscalar) = — ^ — J Xn[ 1 Y q 4s 2 m2qM^ 2s s(s- M 2 ) 2 3(s-M 2) 2 + — (s-Mplnt — ] - — } (F.3b) r e s p e c t i v e l y . These are then convoluted with the photon spectrum i n equation (4.1.4) to give the electron-quark s c a t t e r i n g cross sections. The re s u l t s are shown In equations (4.1.5) and (4.1.6). I d e n t i c a l steps are taken to obtain electron-electron cross sections with t h i s method, and for that matter any d i f f e r e n t i a l cross sections which may be needed. The main advantage i n using the EPA method i s that a l l i n t e g r a l s may be done a n a l y t i c a l l y . I t should be noted that f o r the electron-quark s c a t t e r i n g process, the electron's photon must be the one treated with EPA. This i s to avoid the subsequent complications which a r i s e when the quark i s convoluted over the usual parton d i s t r i b u t i o n s . APPENDIX G - MONTE CARLO INTEGRATION ROUTINE This appendix contains the FORTRAN code for numerically c a l c u l a t i n g a general 2 to N body sc a t t e r i n g cross section. I t i s a g e n e r a l i z a t i o n of a program f i r s t developed i n reference [36]. A l l of the i n t e g r a t i o n routines used were modified versions of t h i s program. The "amplitude" which must be supplied for the given process, r e f e r s to the matrix element squared. D i f f e r e n t i a l cross sections are e a s i l y obtained by using the BIN subroutine, which stores the desired v a r i a b l e with the appropriate weight f o r each event. The convergence of the routine was quite good except where noted i n chapter 4. A general d e s c r i p t i o n of the Monte-Carlo method can be found i n reference [37]. c _. C PROGRAM MONTE.FOR C C FILE 5 C FILE 6 C FILE 74 C FILE 75 C FILE 76 C FILE 77 INPUT FROM SOURCE OUTPUT TO SINK ANSWER EVERY 1000 PTS FOR DISPLAY STORE LATEST ANSWER FOR RESTART STORE MASSES FOR RESTART MISCELLANEOUS FOR DISPLAY c CC DECLARE VARIABLES c C II.Jd.KK ARE ABBREVIATIONS FOR OFTEN USED INTEGER EXPRESSIONS C NPTS IS THE NUMBER OF POINTS TO BE USED IN THE MONTE CARLO C N IS THE NUMBER OF PARTICLES IN THE FINAL STATE C SEED IS A PARAMETER NEEDED BY RANDOM NUMBER GENERATOR GGUBFS C P IS THE INITIAL PARTICLE MOMENTUM IN THE LAB CM FRAME C M(I) IS THE MASS OF PARTICLE I C M2(I) IS THE SQUARE OF M(I) C MSUM(I) IS THE SUM OF M(J) FOR J=I TO N C MX(I) IS THE MASS OF THE VIRTUAL PARTICLE ABOUT TO DECAY INTO C M(I) AND MX(I+1) C MX2(I) IS THE SQUARE OF MX(I) C THE MATRIX B(4,4,I) BOOSTS THE MX(I) CM FRAME ONE BACK C LAMBDA(I ) IS THE MAGNITUDE OF THE MOMENTUM OF PARTICLE I IN C THE MX(I) CM FRAME C STOT IS THE CM ENERGY SQUARED OF THE PROCESS IN LAB CM FRAME C X1.X2 ARE THE USUAL PARTON MOMENTUM FRACTIONS C S IS THE CM ENERGY SQUARED OF THE SUBPROCESS C V.XI ARE VELOCITY AND RAPIDITY OF ONE FRAME W.R.T. ANOTHER C THETA,PHI ARE THE USUAL ANGLES C K4V(4,I) IS THE MOMENTUM 4-VECT0R CF PARTICLE I C LK4V(4,I) IS K4V(4.I) AFTER BOOSTING TO LAB FRAME C DV1 IS A DUMMY 4-VECTOR USED FOR PROGRAMMING EASE C A IS THE SUBPROCESS AMPLITUDE SUPPLIED BY THE USER C W IS THE ELEMENT OF X-SECTION CALCULATED ON EACH LOOP PASS C SUMW IS THE SUM OF THE ELEMENTS W FOR ALL LOOP PASSES C UAC IS THE JACOBIAN FACTOR FROM THE INTEGRALS C INTEGRAL IS THE FINAL ANSWER c INTEGER 11,JJ,KK,NPTS.START,N,RAT REAL*8 M(9),M2(9),MSUM(9),MX(9),MX2(9),B(4,4,9),LAMBDA(8) REAL*8 SEED,P,DUMMY,STOT,X1,X2,S,V,XI.COSTHETA,THETA,PHI,PI REAL*8 K4V(4,9),LK4V(4,9),DV1(4) REAL*8 A.W,SUMW,JAC.FLUX,FACTOR,INTEGRAL COMMON SEED C CC PROGRAM SETUP c _ C NEW CALCULATION OR RESTART, NUMBER OF EVENTS c WRITE(6,12) 12 FORMATC NEW CALCULATION (TYPE 0) OR RESTART (TYPE 1) ?') READ(5,15)Jd 15 FORMAT(H) IF(JJ.NE.O.AND.JJ.NE.1) THEN WRITE(6,17) 17 FORMATC YOU MUST TYPE O OR 1') STOP 118 END IF WRITE(6,18) 18 FORMATC ENTER NUMBER OF EVENTS DESIRED (17)') READ(5,19)NPTS 19 FORMAT(17) c C NEW: READ ENTRIES THROUGH TERMINAL c IF(JJ.EQ.O) THEN START=1 SUMW=0.0 SEED=12345.0 WRITE(6.21 ) 21 FORMATC ENTER P (F15.8)') READ(5,22)P 22 F0RMAT(F15.8) WRITE(6,23) 23 FORMATC ENTER NUMBER (3-9) OF PARTICLES (11)') READ(5,15)N IF(N.LT.3.0R.N.GT.9) STOP DO 29 1 = 1 ,N WRITE(6.26)I 26 FORMAT(' ENTER M(',11,') (F15.8)') READ(5.22)M(I) M2(I)=M(I)**2 WRITE(76,32)M(I),M2(I) 29 CONTINUE END IF c C RESTART: READ ENTRIES FROM FILES c I'F(JJ.EO.I) THEN READ(75,31)START,N,SEED.SUMW,P,DUMMY 31 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) 32 F0RMAT(2D18.10) 33 CONTINUE END IF C - - - C INITIALIZE VARIABLES c DO 42 1=1,N MSUM(I)=0.0 DO 41 J=I,N 41 MSUM(I)=MSUM(I)+M(J) 42 CONTINUE KK=N-1 II=3*N-4 ST0T=4.0*P*P IF((2.0*P) .LE.MSUM(O) THEN WRITE(6,43) 43 FORMATC NOT ENOUGH ENERGY FOR REACTION') STOP END IF PI=3.141592654 MX(N)=M(N) MX2(N)=M2(N) 119 WRITE(6,48) 48 FORMAT(' BEGINNING MAIN MONTE CARLO LOOP',/) C -- CC BEGIN MAIN MONTE CARLO LOOP c . DO 999 IJ=START,NPTS c _ _ — C GENERATE XI,X2 AND CHECK IF ENOUGH ENERGY c - - - 50 X1 = 1 .0 X2=1.0 S=X1*X2*ST0T MX(1 )=SQRT(S) MX2(1)=S C IF(MX(1).LE.MSUM(1)) GO TO 50 C - C GENERATE VIRTUAL PARTICLE MASSES C - DO 65 I=2.KK MX(I) = (MX(I- 1)-MSUM(I- 1))*GGUBFS(SEED )+MSUM( I) MX2(I)=MX(I)**2 65 CONTINUE C - C 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 B00ST(B(1 , 1 . 1),XI,THETA,PHI) C C LET THE PARTICLES DECAY, GET THE BOOST MATRICES AND 4-VECTORS • C-- - - - DO 79 1=1,KK 79 CALL DECAY(MX(I),MX(I+1),M(I),K4V(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) C - - C BOOST 4-MOMENTA TO LAB FRAME c DO 88 1=1,N DO 81 K=1,4 81 DV1 (K)=K4V(K,I) DO 84 JK=1.I d=I-JK+1 IF(I.EO.N) d=I-JK IF(d.EO.O) GO TO 84 CALL MULT(B(1,1.d),DV1,LK4V(1.I)) DO 83 K=1,4 B3 DV1(K)=LK4V(K,I) 84 CONTINUE 88 CONTINUE CALL AMPLITUDE -- MUST BE LINKED TO, OR PART OF THE PROGRAM A=1 .0 C CALCULATE PHASE SPACE DENSITY AND ELEMENT OF INTEGRAL 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) 92 CONTINUE <JAC=(4 .0*PI )**KK DO 93 I=1,KK-1 93 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 C --- --- C WRITE OUT THE FIRST TEN EVENTS c IF(IJ.LE.10.0R.W.LE.0) THEN WRITE(77,101) 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 ) 102 FORMAT(17,1X,12,4D15.5) 103 CONTINUE WRITE(77,104) 104 FORMATC Id A W') WRITE(77,105)Id,A,W 105 F0RMAT(I7,2D12.4./) END IF C _ J C STORE ANSWER EVERY 1000 EVENTS FOR RESTART IF SYSTEM CRASHES c RAT=M0D(Id,1000) IF(RAT.EO.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 112 FORMATC FINISHED ',17,' POINTS') END IF c CC END MAIN MONTE CARLO LOOP C--- 999 CONTINUE c CC CALCULATE THE INTEGRAL AND OUTPUT c INTEGRAL=SUMW/FLOAT(NPTS) WRITE(77.113)NPTS,P 113 FORMATC NPTS= '.I7.4X.' P= '.D15.6) WRITE(77,114)INTEGRAL 114 FORMATC X-SECTION= '.D15.6) END C --- C MUST LINK TO RTNS.OBd FOR SUBROUTINES c . c RTNS.FOR CONTAINS MONTE-CARLO SUBROUTINES C c SUBROUTINE DECAY2(M1,M2.M3,BM2,BM3) FOR TWO CHAIN DECAY OF M1 INTO M2.M3 FIND BOOST MATRICES BACK REAL*8 M1,M2,M3.V2(4),V3(4),BM2(4,4),BM3(4,4) REAL*8 COSTHETA.THETA.PHI.V,SEED.PI,XI 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 ) IF(V.EO.-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 ) IF(V.EQ.-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 B00ST(B.XI,THETA,PHI) REAL *8 B ( 4 , 4 ) . XI. THETA, PHI B( 1 . 1 )•= DC0SH(XI ) 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 )= DSINH(XI) B(4,2)«= -DC0SH(XI )*DSIN(THETA) B(4,3)= 0. B(4,4)= DC0SH(XI)*DC0S(THETA) RETURN END SUBROUTINE BIN(F,AR,INF,SUP, W) CLASSES F INTO ONE OF 1C1 BINS EETWEEN INF AND SUP AND PUT 1 IT INTO ARRAY AR 1 c  REAL*8 F.AR(101),INF,SUP.W INTEGER POS POS = DINTOOO. * (F - INF)/(SUP - INF)) + 1 IF (POS .GT. 101) POS = 101 IF (POS .LT. 1) P0S=1 AR(POS) = AR(POS) + W RETURN END SUBROUTINE SCALP3(V1.V2,VSV) REAL*8 V 1 ( 4 ) , V 2 ( 4 ) , VSV VSV = V1(2)*V2(2) + V1(3)*V2(3) + V1(4)*V2(4) RETURN END SUBROUTINE MULT(B,V1,V2) c C CALCULATES THE PRODUCT BETWEEN THE MATRIX B AND 1 C VECTOR V1 AND PUTS RESULT INTO V2 1 c REAL*8 B ( 4 , 4 ) , V 1 ( 4 ) , V 2 ( 4 ) , PH DO 300 1=1,4 PH=0. DO 301 J=1,4 301 PH = B ( I , d ) * V1(d) + PH V 2 ( I ) = PH 300 CONTINUE RETURN END SUBROUTINE DECAY(M1,M2,M3,V3,M) C-- C FOR DECAY OF M1 INTO M2 AND M3, CALCULATE THE 4-VECT0R V3 C OF PARTICLE 3 IN M1 REST FRAME, THEN CALCULATE BOOST MATRIX C 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 V3(1) = (M1*M1+M3*M3-M2*M2)/(2.*M1) V3(2) = O. V3(3) = 0. V3(4) = DS0RT(V3(1) *V3(1) - M3*M3) PX2 = -V3(4) EX2 = M1 - V3(1) COMPOSE BOOST MATRIX BETWEEN QUARKS CM AND X CM V = PX2/EX2 I F ( V .NE. -1.) XI= DL0G((1. + V ) / ( 1 . - V ) ) / 2 . I F ( V .EO. -1.) RETURN COSTHETA =2.* GGUBFS(SEED)-1. THETA = DACOS(COSTHETA) PHI = 2. * P I * GGUBFS(SEED) CALL B00ST(M, XI, THETA, PHI) RETURN END SUBROUTINE SCALP(V1,V2.S) TAKE THE SCALAR PRODUCT OF THE TWO 4-VECT0RS V1 AND V2 AND PUT THE RESULT INTO S REAL*8 V1 ( 4 ) , V 2 ( 4 ) , S S = V1(1)*V2(1) - V1(2)*V2(2) - V1(3)*V2(3) - V1(4)*V2(4) RETURN END 123 APPENDIX H ~ GLOSSARY charging: This charged supe r p a r t i c l e i s a fermion. I t s mass eigenstate i s a c t u a l l y a mixture of the superpartners for the W-boson and the Higgs. c h i r a l symmetry: A symmetry which preserves the handedness of massless fermions. Technically t h i s means a Y 5 invariance. electroweak breaking scale; Energy scale at which the electromagnetic and weak forces are u n i f i e d . Experimentally established as being =246 GeV. Fadeev-Popov ghosts: Mathematical constructs known as ghosts are introduced i n addition to the physical f i e l d s to preserve gauge invariance. These are simply a convenient technical invention which allow us to express the complex mathematics of a ph y s i c a l process i n a simple graphical form. Goldstone boson: Massless p a r t i c l e which r e s u l t s whenever a continuous global symmetry i s spontaneously broken. Higgs mechanism: Process through which gauge bosons acquire mass, where a Goldstone boson becomes the l o n g i t u d i n a l component of the gauge boson. loops, loop diagram, loop p a r t i c l e : Physical processes can be represented g r a p h i c a l l y , with p a r t i c l e s represented 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 vertex. The diagrams for higher order processes w i l l involve loops formed by the p a r t i c l e s . one-loop approximation: Used with the graphical representation of a ph y s i c a l process, where only diagrams with one loop are included. renormalization: Procedure i n which divergent quantities that a r i s e from higher order corrections are absorbed into a r e d e f i n i t i o n of parameters, thereby making a perturbative expansion convergent. spontaneous symmetry breaking: The ground state of the system does not respect the same symmetry as the Lagrangian which i s used to describe i t . superparticles, superpartners: The supersymmetric partners to the usual standard model p a r t i c l e s , which d i f f e r by 1/2 integer unit of quantum spin. two photon fusion mechanism: Any process i n which the f i n a l state 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 to describe a method of producing Higgs bosons during 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 expectation value (VEV): The value which the scalar f i e l d acquires i n i t s ground state. would-be Goldstone boson: This r e f e r s to the degree of freedom which would normally be a Goldstone boson, but i s instead 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 scalars (or pseudoscalars) and fermions. In our case i t i s the Higgs boson-fermion i n t e r a c t i o n .

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