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A QCD-parton calculation of associated Higgs boson production in hadron-hadron collision Zakarauskas, Pierre 1984-06-13

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A QCD-PARTON CALCULATION OF ASSOCIATED HIGGS BOSON PRODUCTION IN HADRON-HADRON COLLISION by PIERRE ZAKARAUSKAS B.Sc.,Universite Du Quebec A Chicoutimi, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department Of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1984 © Pierre Zakarauskas, 1984 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date i i Abstract This thesis contains a study of the reaction proton+proton or proton-antiproton into a Higgs boson and a pair of heavy quarks, in the region of high energy and high momentum transfer. The Higgs boson mass is treated as a free parameter. Numerical results are obtained through a Monte Carlo integration. Several differential cross sections relevant to experiment are given. Table of Contents Abstract ii List of Tables v List of Figures v Acknowledgement vChapter I INTRODUCTION 1 Chapter II LOCAL GAUGE TRANSFORMATIONS 8 Chapter III HIGGS MECHANISM 14 Chapter IV THE GLASHOW-WEINBERG-SALAM MODEL 20 Chapter V PARTON MODEL AND HADRON-HADRON COLLISION 33 Chapter VI HIGGS BOSON PHENOMENOLOGY 44 Chapter VII CALCULATION OF ASSOCIATED PRODUCTION OF HIGGS BOSON AND HEAVY FLAVOR IN PROTON-ANTIPROTON COLLIDERS 64 Chapter VIII RESULTS 77 Chapter IX DISCUSSION AND CONCLUSION 93 BIBLIOGRAPHY . ... 8 APPENDIX A - FEYNMAN DIAGRAMS AND QCD RULES 101 APPENDIX B - COLOR SUMMATION CALCULATION 107 APPENDIX C - THE MONTE-CARLO INTEGRATION ROUTINE 110 APPENDIX D - CALCULATION OF THE TRACE 122 APPENDIX E - PRINTOUT OF THE AMPLITUDE SQUARED OF THE PROCESS 127 APPENDIX F - QUARK AND GLUON DISTRIBUTION PARAMETRIZATIONS 1 44 APPENDIX G - HADRON-HADRON COLLIDERS 146 List of Tables Number of charm quarks, nc, and number of charged leptons, nA, in the final state particles of reaction (VII.1) after weak decays of the hadronsF, F and the Higgs boson 81 Fixed target cross section for reaction (VII.1) 82 Very high energy cross sections 83 V List of Figures + - + -1. Feynman diagram for W W —>W W , without scalar contribution 6 2. Scalar contribution to the process W+W~—>W + W 6 3. Potential III.3 for the case |> < 0 15 4. Vertices of the self-interacting scalar meson 18 5. Vector-scalar vertices 19 6. Feynman diagram for electron-muon scattering 34 7. Feynman diagram for electron-proton scattering 35 8. V(<P) for different values of fx 48 9. Branching ratio of the H° in function of mH 51 10. Feynman diagram for Z°—> Y+ H decay 54 11. Z° —> H° + ICH decay diagram 512. Feynman diagram for e+e~—> H° + Z° 55 o 13. Feynman diagram for gg —> H 57 14. Cross sections for pp—> H° + X 9 o 15. Feynman diagram g +c —> c + H 60 o 16. Feynman diagrams for the process qq —> H qq 61 17. Cross sections for compton-like process, for mH = 1 0 GeV/c2 62 18. Total cross section for processes 2), 5) and 6) for mH = 410 GeV/c1 • 3 19. Feynman diagrams for the background to the process hadron+hadron —> H° + anything 65 20. Feynman diagrams for qq —> F + F + H° 67 21. Feynman diagrams for gg —> F + F +H° 68 22. Total cross section for the process (VII.3) as a function of Vs, for pp collision, with m4 = 10 GeV/c*. 84 v i 23. Total cross section for the process (VII.4) as a function of ^s, with = 10 GeV/c* 85 24. Total cross section in pp from the sum of subreaction (VIII. 3) and (VIII.4), as a function of i/s, with mH = 10 GeV/cz 86 25. Total cross section for the process (VII.1) as a function of the mass of the heavy quark produced with the Higgs boson 7 26. Total cross section for the process (VII.1) as a function of the mass of the Higgs boson 88 27. Differential cross section dc/ /dEH for four different sets of the parameters m^ , m^ , and \fs 89 28. Differential cross section do' /dE« for four different set of the parameters m^, mh , and Vs" 90 29. Differential cross section da /dh± for four different sets of the parameters mh , mK and *Js^ 91 30. Differential cross section dc/dki for four different sets of the parameters mH , mK, and ifs 92 31. Order of particle generation in the Monte-Carlo method in particle physics 111 vii Acknowledgement I would like to thank my research supervisor, Dr. John Ng, for his abundant help and guidance during the course of this work. His enthusiasm for research has been a continuous source of motivation for myself. I also want to thank my wife, Louise, for her continuous moral support. She always made sure I got enough of all the good things life has to offer apart physics. I gratefuly acknowledge financial assistance from the Natural Sciences and Engineering Research Council. 1 I . INTRODUCTION Within the last decade the world has witnessed a total revolution in the understanding of particle physics. Until then, the weak interactions (WI) were described by a phenomenological, non-renormalizable Fermi interaction of four fields at a point. The WI are the weakest, after gravitation, of the basic known forces of nature. They are responsible for the beta decay of the neutron, for example, and other relatively slow processes in nuclear and particle physics. On the other hand, the newly developed quark model of that time could account for the previously mind boggling hundreds of "elementary" particles produced in strong interactions (hereafter SI), or their decay products. What was still badly needed however, was a theory of SI itself. The four known forces acting on matter in the universe, -gravitation, electromagnetism (EM), WI and SI, -did not seem to have much in common. Then, at the end of the 60's, EM and WI were "unified" within the framework of a gauge model, the Glashow-Weinberg-Salam model (GWSM)1 . A few years later, it was the turn of SI to be described by a gauge theory -quantum chromodynamics (QCD)2 . Now, most models of particle interaction are based on the gauge idea. Among them, are the 1 For historical accounts and references, see Nobel lectures of (Glashow, 1980), (Weinberg S. 1980), (Salam 1980) 2 For a review of QCD, see (Reya, 1981) 2 grand unified theories (GUTS) 1 whose goal is to unify EM, WI and SI into a single interaction with a non-abelian gauge group. Its main prediction is the instability of the proton, which is being intensively tested in many laboratories. Other gauge models are: technicolor* , supergravity3 , and several alternatives to the GWSM . What makes the concept of gauge invariance attractive is its inherent elegance. Its main feature is the following. You start with a symmetry you know to be valid, (or hypothesize to be valid), in general in a world where the matter fields are spin 0 bosons or spin 1/2 fermions. You require this symmetry to be conserved locally, i.e. at any point in space-time. To do so, you must introduce a new boson field, which will mediate some new interaction between the matter particles, in such a way that the symmetry remains non-violated. Hence, you have "deduced" a force from the symmetry requirement. It has been known for quite some time that EM can be "deduced" this way from the phase invariance in quantum mechanics (Fock, 1927), (Weyl, 1929). It was seen as merely an elegant way of linking EM and QM. Yang and Mills (1954) broadened the class of symmetries that can be "localized" this way, to include non-abelian symmetries. A non-abelian symmetry can be compared to a rotation in space -the order in which you apply the transformations is important. In this analogy, an abelian theory would be a rotation in some For a review of GUTS and their phenomenology, see (Langacker, 1981 ) 2 see for example (Susskind, 1979) 3 For a review of supergravity, see (van Niewenhuizen, 1981) * For example (Georgi and Glashow, 1974), (Pati and Salam, 1973) 3 plane. The forces generated by a non-abelian symmetry are much more complicated than those generated by an abelian one, mainly because the particles or fields responsible for carrying the interactions are "charged" themselves. But the Yang-Mills theory did not attract much attention for a while, because the boson particles you must introduce to carry the interactions must be massless, giving rise to long-range forces we do not observe. The forces generated by non-abelian symmetries did not seem to correspond to any of the known forces. One had to solve the problem of giving a mass to the vector bosons if one wants the theory to describe WI which are short range. The solution to this problem had to wait till 1964, when Higgs (1964) invented the spontaneous symmetry breaking (SSB) scheme. At the price of introducing a elementary scalar field, the vacuum would be made to be non-trivial. Real particles propagating through such a vacuum would interact with it, giving them effectively a mass, in much the same way as the apparent mass the of electron may be greatly affected when it travels through a lattice or a plasma. A few years later, Weinberg, Salam and Glashow came up independently with a model for the weak and electromagnetic interactions, using a non-abelian symmetry based on isospin, represented by an SU(2) group, and an abelian symmetry U(1). The SU(2) group has three generators, which implies three bosons mediating the interactions; the U(1) group has one. The symmetry is broken by introducing an interacting doublet of 4 complex scalars, endowed with a negative mass-squared. Of the four degrees of freedom brought in by the scalars, three are used to give mass to three of the four bosons. The fourth degree of freedom appears as a physical elementary field, with o a real mass. It is called the Higgs boson, symbolically H . The GWS model accounted well for what was known at the time of the WI, but it predicted a new component to the weak force; a neutral one. An example of it would be the reaction 1> q —> v q in which a neutrino, a particle which interacts only through WI, interacts with a quark and remains a neutrino. To see this experimentally, one would send a neutrino beam on a target, and wait to detect a deposition of energy and momentum, with no lepton produced. (The charged WI would produce a charged lepton in the final state). These neutral current interactions have extensively been measured and studied from their discovery in 1973 till now. Since then, the existence of the neutral boson has been comfirmed by its spectacular discovery in proton-antiproton collisions, at the collision beam facilities at CERN (Conseil European de Recherche Nucleaire), in the 1983 summer (Arnison et al. 1983). Its discovery had to wait so long because no particle accelerator in the world could reach the center of mass energy necessary to its production, since its mass was predicted to be 91 GeV/c . The next very important task facing the experimentalists is to look for the Higgs boson. The discovery of the Higgs boson would be a badly needed confirmation that the mechanism which endows the gauge bosons with masses is the spontaneous symmetry 5 breaking mechanism. This is a corner stone of the GWS model, and indeed, of nearly all unification theories based on the gauge principle. The main obstacle to its discovery, if it exists, is that unlike the intermediate vector boson W+, W and Z°, its mass and decay products are free parameters of the theory. These factors make its production, and especially its identification, very difficult. Several production mechanisms have already been suggested. Those pertinent to hadron-hadron collisions generally lack a clear signature. However, if the H° is too massive, its production will not yet be possible in the cleaner electron-positron collider rings. For the e+e~ colliders that are planned now the highest energy of 200 GeV would be reached by LEP II at CERN. On the other hand, a hadron collider of cm. energy 5 to 40 TeV (1 TeV = 1000 GeV) is being planned. One more argument may be given in favor of the existence of an elementary scalar, independently of the spontaneous symmetry breaking scheme. It concerns the high-energy behavior of the theory (Halzen and Martin, 1984). The predicted cross-section for any process must not diverge, i.e. the probability of occurence of this process must remain less than one. If one calculates the cross-section for the elastic scattering of a pair of charged W , from the three diagrams of Figure 1. 6 -V - + -Figure 1 - Feynman diagram for W W —>W W , without scalar contribution one finds that their sum diverges as s/M^ as s—>©°, (where the square of the total energy is denoted by s). A simple solution is to introduce a scalar particle to cancel this divergence, through the diagram of Figure 2. Figure 2 - Scalar contribution to the process W W —>W W The coupling of the h particle must be proportional to the W mass to cancel the divergences of the other diagrams. Therefore, if we had not introduced the Higgs boson to give mass to the gauge bosons, a la SSB mechanism, we would have 7 been forced to invent it to cancel out divergences in other processes! This thesis is divided into two parts. The first one covers the background material pertinent to Higgs mechanism and phenomenology, and includes the first six chapters. Chapter II gives a general treatment of gauge theories. The third chapter introduces the phenomena of spontaneous symmetry breaking and the important Higgs mechanism. The Glashow-Weinberg-Salam model is developed in chapter IV. Chapter V brings in the hadron contribution. There is presented the extremely useful, yet simple parton model. Using it, one may use perturbative QCD and derive useful predictions for experiments. We get to the core of the subject in chapter VI with the known phenomenology of the "standard" Higgs boson. This is where is rooted any analysis of Higgs boson production. The whole work relies heavily on it. The second part of the thesis includes chapters seven through nine. The starting point of the calculations is described in chapter VII, and the results are to be found in chapter VIII. The details of the calculations, in particular the matrix element squared, and the Monte-Carlo integration routine developed, have been confined to appendices. I summarize the work and suggest possible routes of extensions in chapter IX. 8 II. LOCAL GAUGE TRANSFORMATIONS Because local gauge invariance is at the heart of today's attempts to unify and/or explain fundamental interactions in physics, we will start with a brief account of this important subject. GENERAL CASE; FERMIONS: We start with the Lagrangian for free fermions. We demand that Jf. be locally invariant under transformations of itee a simple Lie group G, and Y transforms as a certain representation of G. The generators of G have representation matrices Ta which satisfy [ T*,TJ - i Cube Tc (II.2) where the Cabc are tne totally antisymmetric structure constants. If the fermion fields, under infinitesimal transformations, transform as (II.3) it is easy to check that the free Lagrangian d^ree is not 9 invariant under this transformation. The derivative introduces which spoils the invariance of $irte . The local property of the symmetry is expressed by the x-dependence in 9. To make invariant, one introduces the covariant derivative Dh; where a set of new 4-vector "gauge" fields have been introduced. Now, if one demands that the covariant derivative has the same transformation property as ^ itself, i.e. (II.5) then one must introduce vector gauge fields which transform under infinitesimal transformations as; In this expression, the second term is the transformation law for the adjoint multiplet under G. This implies that the gauge fields A^ carry the non-abelian quantum numbers, i.e. they are 10 "charged". We need now to introduce in the lagrangian a kinetic term for the vector gauge fields. In analogy with the abelian case (QED), a possible antisymmetric second rank tensor for the fermion field is; (II.7) which leads us to define h fi c (II.8) a Under infinitesimal transformation, transforms as a multiplet under G; >a _ r a i r c (II.9) Pu^ ~ f~+ Cahc 9 F The combination F F^,^ is then invariant under G. Notice that; 1: A mass term for the gauge field would not be invariant (unless the gauge field was invariant under G). 2: The kinetic energy term for the gauge field implies triple and quadruple vertices, since b A C (11.10) 11 The G-invariant Lagragian is finally; ABELIAN CASE; U(1) SYMMETRY: The U(l) case is simply QED. There is only one generator; therefore the structure constant is 0 and the gauge field tensor is; FMJ, = ()hA» - XAh) (II-12) The complete Lagrangian is then just the usual QED Lagrangian, with the field A^ being readily related to the vector potential of electromagnetism. SU(2) CASE; This is the Yang Mills case where SU(2) is usually taken to be an isospin symmetry, relating to "internal" isospin quantum numbers. Equations (II.2) to (11.11 ) hold with the identification Ca be ~ ^ a- b c and the two-dimensional representation matrices can be chosen to be the usual Pauli matrices , i = l,2,3. SU(3) CASE; The local SU(3) symmetry has found an application 12 in the attempt to develop a fundamental theory of strong interactions (Reya, 1981). The fermions are quarks, the eight gauge particles are called gluons, and the internal quantum number on which the symmetry is based is called color. QCD is an exactly locally invariant theory, i.e. the Lagrangian (11.11) applies without any modification. There are eight generators of the SU(3) group, usually labelled "X,-, i=1,...,8. The most important properties of QCD are asymptotic freedom and confinement: Its effective coupling constant, at a given momentum transfer squared Q , is given by: _L - _L | (II.M) in the leading approximation, and where sn^ is the number of quark flavors. As long as 16, o^(Q) grows smaller at large 2 Q . This is called asymptotic freedom, and is a most useful feature of QCD, as it permits perturbative treatment of many "hard" scattering processes. In fact, QCD is the only candidate theory which explains this behavior of the SI coupling constant, corresponding to the phenomenon of "scaling" in experiments (see chapter V). If the coupling constant grows smaller at large Q and correspondingly short distances, the opposite is also true. Lower energy transfer interactions correspond to larger distances and large couplings, which leads to the notion of quark confinement. Quark confinement means that quarks are 13 forever confined within hadrons and cannot appear isolated. Confinement has not been derived from QCD yet, but the behavior of the QCD coupling constant makes it qualitatively plausible. Asymptotic freedom and confinement are the most important reason QCD is now considered the complete theory of strong interactions. 14 III . HIGGS MECHANISM The Higgs mechanism can cause the spontaneous symmetry breaking of some locally invariant Lagrangians (Higgs, 1964). But before to studying this case, one has to see the effect of the spontaneous breaking of a globally invariant lagrangian. SPONTANEOUSLY BROKEN SYMMETRY: Let us consider the case of two real scalar fields and ; The effective potential is chosen for illustration to be; (in. 1 > which is invariant under rotation U; (III.2) (III.3) and one can distinguish two cases: -case 1: p. > 0. The minimum of V occurs at ^ = ^ = 0 and this will give simply a degenerate doublet of mass -case 2: p. < 0. The minimum occurs at .1 (III.4) 15 and there is a continuum of degenerate states at the minimum. The potential for this case is represented in Fig. 3 below. Figure 3 - Potential III.3 for the case ^< 0 One can always define coordinates so that the physical vacuum is at in the classical field theory, that is, in the quantum field theory; <°|<M°> ~~^r Ol<f>a/0> =° (III.5) To do perturbation theory around the classical minimum, one has to expand in powers of cf> = (h - V instead of Cp . <p of course, is still expanded around the value zero. (Ill.6) 16 The important feature here resides in the mass terms. The field 11 has acquired a (mass) - -2 jA > 0, while the ^ particle is massless. This is an example of the Goldstone theorem, which states that if a theory has an exact continuous symmetry of the Lagrangian which is not shared by the vacuum, a massless particle must occur. HIGGS MECHANISM: In the case of a locally invariant gauge theory, there is no massless Goldstone boson when the symmetry is spontaneously broken. The would-be Goldstone boson combines with the massless gauge boson to give a massive vector boson. This is the Higgs mechanism. To illustrate that point, let us consider the simple case of the Abelian. gauge theory with Lagrangian (III.7) (III.8) where The Lagrangian (III.7) describes a charged scalar interacting with itself, and with a gauge field A^. If ji/ < 0, it 17 describes scalar QED. The Lagrangian is invariant under the local transformat ions (III.9) 2 When > 0, <p develops again a vacuum expectation value. It is <o|'<t>|o>^ ; v' i (in.io> Let us use polar variables to parametrize cp , and expand about a specific vacuum point. This is done to show more clearly the physical content of the theory. The new set of coordinates is <f>00 = ^ iv + 7fxH e (in.ID Consider now the gauge transformation (III.9) with x)/V. The transformed fields are: and A*A; = A*- -^M 18 (III. 13) This is refered to as the U-gauge in the literature (Abers and Lee, 1973). If one substitutes these new expressions for the fields into the Lagrangian (III.7), and expands, one gets: Kinetic t-- i&im) - wf; \ } 3 a 5 terins •TniSS in tetacti'oti tet-ms Self - inteticton of *l (III . 14) The scalar meson has acquired a mass 3 AV - ^ = 2 ju. and self-interactions represented by the vertices of Fig. 4. \ )• 7 Figure 4 - Vertices of the self-interacting scalar meson 19 P The interactions between the vector gauge boson and the scalar meson will give rise to the vertices of figure 5. Figure 5 - Vector-scalar vertices The gauge boson has also acquired a mass, which is the aim of this mechanism. We can henceforth build gauge theories giving rise to short-range interactions. Moreover, the theory, altough having its symmetry explicitely broken, is still renormalizable. This was demonstrated by t'Hofft (1971). The Higgs mechanism finds its best applications in the GWS model, which will be described in the next chapter. 20 IV. THE GLASHOW-WEINBERG-SALAM MODEL The GWS model is usually introduced first with one doublet and one singlet of fermions only. The other known fermions can easily be introduced thereafter. This is the path I will follow. BASIC LAGRANGIAN: One wants to identify the massive vector bosons arising in the Higgs mechanism with the intermediate vector bosons (IVB) carrying the WI. In the phenomenologically successful IVB theory, the lagrangian for weak interactions is given by: r jUvA k.c.) where J, = 1/ )/A) -L )e is the leptonic charged current in its A e ,\ a so-called V-A form, and h.c. stands for "hermitian conjugate". On the other hand, the lagrangian for the electromagnetic interactions is given by <Lc. = C lL A-X (IV'2) where Jp/ = e / e is the electromagnetic current. Then, to unify EM with WI, one needs at least 3 gauge — o t •*. bosons, W , W , Z , to couple with the currents , and Je/ec. The simplest group with three such generators is SU(2). However, if and C are to form a doublet under SU(2), as 21 suggested by the form of the current J/pr , Q.elec cannot be a generator of the group, because the electric charges of the doublet do not add up to zero, whereas all 2X2 SU(2) representations matrices are traceless. One is then led to introduce a fourth gauge boson Z . The smallest group is now SU(2) ®U(1). Assuming that only V-A ± interactions occur for W , one takes the generators of the a. SU(2) groups to be the isospin operators T , whose 2-D representation may be taken to be the Pauli matrices divided by two. (IV.3) L = (^e/^ ^s then a doublet under SU(2) and R = eR is a singlet, as wanted- The subscript R or L means that only the right-handed or left-handed component of the lepton wave function is selected. One does it by multiplying the spinor by the projection operator (1 - ^s) to obtain the left-handed component, or (1 + Vs) for the right-handed component, i.e. 1.. •= (W (IV.4) The generator of U(1) is chosen such that the electric charge is a linear combination of the U(1) generator and the generator T3 of SU(2). One can choose 22 (IV.5) as generator of the CJ(1) group. The basic Lagrangian L of the GWS model may be split into 4 parts as follow; J = J * J +J J (IV.6) The gauge part of the Lagrangian is; with oc fc •a m) (IV-8> GAUGE Fi'fiO TfyJafi The leptonic part of the Lagrangian is; titrmon ~ L L I % 8 ~ \ }Z Aw) L tRaKr^+^'6jK ,IV-9) 23 where g = coupling constant associated with SU(2) g' = coupling constant associated with U(1) Notice one cannot have a bare mass term of the form e_e. , which is forbidden by SU(2) invariance. Also, the terms e^eR and ei eL vanish, because they contain the products of orthogonal operators (1 - Y$ ) and (1 + Ys ). a. To give a mass to the gauge bosons A^ and the electron, let us introduce a doublet of complex Higgs scalars (IV.10) They have Y = 1 to satisfy (IV.5), and transform as a doublet under SU(2). The scalar part of the lagrangian is then (IV.11) The most general renormalizable Higgs potential V(<£) is (Flores and Sher, 1982); V(9) = M^V + Ktfvf <IV-12> 24 Also, one is free to add a coupling between the scalar doublet and the leptons, of the form; It has been introduced by Yukawa (1935) to explain the nuclear binding force between nucleons, through the exchange of bosons. There cannot be terms of the form LtL because, being the product of three SU(2) doublets, they cannot form an SU(2) invariant. One needs to include both a singlet and a doublet, hence the form of (IV.13). One now must spontaneously break the SU(2) ®U(1) symmetry. SPONTANEOUS SYMMETRY BREAKING (SSB): Assume once more that p. < 0. The two minima are at|<?/ = v/ 2 with v = One now requires the neutral scalar field to develop a vacuum expectation value (VEV). One must let the VEV of the charged scalar vanish, in order not to have a charged vacuum. This leads to 1 YL/KAWA (IV.13) The coupling of the form is known as the Yukawa coupling. (IV.14) and again one expresses the scalar fields in polar coordinates 25 to bring out the physical meaning of the theory. Using the unitary transformation; lu$) = e (IV-15) the scalar fields read in the new coordinates; (IV.16) The four real components of $ are now distributed in 3 components for and one for the scalar °l. The symmetry breaking scheme that Weinberg and Salam adopted breaks both SU(2) and U(1)y, but preserves U(l)e/fc. One can check this using the condition for a generator 21 to leave the vacuum invariant; where ($) is the 2X2 matrix representation of the operator o - 0 (IV.17) For the generators of SU(2) X U(1), we find . (IV.18) 26 But CX^X = i(z3H)<4\ ~-o <iv.,9) The photon, and only the photon will then remain massless. Transforming now to the U-gauge (IV.15): 4>-4'-- Utf)<|> = U)/2 L -* 1' = U i \ eR:^R,' 8^r Bfj. (iv. 20) r A;- T°A';= ^n[T/\;-iu"(f)^w«)]u'(f) and A^ still transforms according to (II.6). 'In terms of the new fields, the Lagrangians become; ^•P J. r 'a r- I ol GAUGE I pi> The mass of the electron arises from the Yukawa term; 27 (IV.23) and is seen to be me = vGe/*{2. The coupling of the remaining Higgs boson t to the electron is i/a ?y~ (iv.24) which will be of primary importance to produce and detect it. The scalar field will give rise to gauge boson masses via the term with Let us isolate the vector mass terms, which are those terms quadratic in vector fields, into a Lagrangian J1 , subset of 28 If »e define ^ r ^ ^ + M ^ , 3 with a ,o >a\ at (IV.28) The Lagrangian ^ becomes X* -" All wIm** tMi?*?" <IV-29) (IV.30) The massless vector gauge boson can then be identified with the photon. For convenience, one usually introduces an angle dw (Weinberg angle), which relates the coupling g of the SU(2)^ to the coupling g' of the U(1)V group. Explicitly 29 J hn 6w (iv.3D So that 4f*V 1 /COS 6. <!»•"> The interaction between the leptons and the gauge fields can now be read off from the lepton lagrangian in (IV.22). Re-expressing $\Kftoh in function of the new fields W~, and Ar, one gets, using (IV.32); Lu,- JL* i^(\(i^^/eriu\tj\) 3 COS 01* a? >n' ^ ^ y - * (IV.33) Equating the coupling between the photon A^ and the electron to the electromagnetic coupling e gives the relation: S sin Qw = €. (IV.34) The IVB coupling is consistent with low energy phenomenology provided we identify 30 3 - _! - (IV.35) I/a ml ^3 The only missing piece is the mass of the Higgs boson which can be worked out from <$M\t* • This is equal to: z (IV.36) that is, completely undetermined in this model. However, upper and lower limits have been derived, which we will consider in chapter VI. ADDITION OF QUARKS: Let us introduce the first two families of quarks. The incorporation of the third family follows the same line of argument. According to Cabbibo's picture of WI (Cabibbo, 1963), the hadronic charged currents are represented by the weak-isospin doublets L (J'l > U- (1\ ' S'"/WS (IV.37) > . ci - J cos 6c + S  st'n 6<l where 5' -'J Sin Bz + S cos 0c and &c is refered to as the (Cabbibo) mixing angle. The Lagrangian terms corresponding to (IV.22) and (IV.23) are 31 (IV.38) q = u, d, s, c and (IV.39) respectively. The Lagrangian piece (IV.38) gives rise to exactly the same kind of results as for the lepton case. If we now perform SSB by replacing by its expectation value (IV.14) we obtain a serie of mass terms equivalent to (IV.23). (IV.39) We must now chose the Yukawa couplings Gj ,...,G6 so that u,d,s and c are mass eigenstates; Gi -" ^ 1/2 G 3 1 %s 1/3 AT (IV.40) Gt, - ^c ifa /at 6s- : " G2 t*f> Be Gc - f G2 cot^n Bc 32 The generalization to three generations introduces two more quark mixing angles, but the rest of the procedure stays essentially the same. 33 V. PARTON MODEL AND HADRON-HADRON COLLISION In order to calculate the production rates in hadron-hadron and lepton-hadron collisions, some simplifying hypotheses are needed about the structure of hadrons. Such a set of hypotheses, well supported by experience, forms the parton model1 . Let us describe its sources, links with QCD and applications to hadron-hadron collision. When high energy electrons or neutrinos are scattered from nucleons their angular distributions look as if they were scattering from hard, pointlike constituents inside the nucleons. It is a repetition, at higher energies, of Rutherford's experiment. These pointlike constituents of nucleons have been given the name "partons", and the partons interacting electromagnetically or weakly with the leptons have been identified (theoretically) as quarks. Another class of nucleon constituents, mediating the QCD force between the quarks, are "gluons". The gluons do not interact through WI, and are electrically neutral. They are spin-1 bosons, and carry the "color" charge which gives rise to strong interactions. Therefore, gluons interact with gluons, making quantitative predictions of QCD very difficult. Still, the method of lattice gauge theory2 managed to produce an acceptable hadron spectrum. The lattice gauge theory is a non-perturbative way of getting predictions from a theory with a large coupling constant, like QCD at low momerttum transfer. 1 see for example (Close, 1979) 2 for a review see (Drouffe and Itzykson, 1978) 34 Also, when experiments reach very high energy, the phenomenon of scaling occurs, and one may use perturbative QCD to calculate production rates. SCALING: The parton model picture stems from a property of lepton-proton scattering, called scaling. Here are the foundations of it. When one calculates the amplitude for lepton-lepton scattering, say —> e , one gets from the amplitude corresponding to the diagram of Fig. 6 Figure 6 - Feynman diagram for electron-muon scattering the cross-section: Q CM 6 x U SVn'fi aw i 1 (v.D where E : energy of the incoming electron in the lab frame E': energy of the scattered electron in the lab frame V : energy transfer to the muon in the lab frame z Q : minus one times the momentum transfer squared 35 0 : scattering angle of the electron in the lab frame M : rest mass of the muon On the other hand, if one wants to calculate inclusive electron-proton scattering, one is forced to introduce structure functions in the hadronic rate tensors W . The hadronic tensor W is the piece which must be introduced in the spin-summed amplitude squared at the location of the photon-proton vertex in evaluating the rate corresponding to the Feynman diagram of Fig. 7 ) Figure 7 - Feynman diagram for electron-proton scattering. (See appendix B for introduction to Feynman diagrams). It serves to parametrize our ignorance of the form of the current at the proton end of the photon propagator. The most general form for the proton structure function is 36 p being the proton 4-momentum. The W are in general 2 pis functions of V and Q . Gauge invariance q W =0 gives us some relations between the coefficients, and one is left with only two independent structure functions; (V.3) assuming parity-conserving interactions. Contracting with the leptonic rate tensor L and expressing the cross-section in the laboratory frame where the initial proton is at rest yields; JV = faXei Jo/I w.(v.q')+3 wsv,( (v.4) One can now compare expressions (V.1) and (V.4) and deduce that if the virtual photon scatters off a pointlike Dirac particle, the structure functions reduce to: the equations being written this way to form dimensionless 2 ratio CO= 2MV/Q only. 37 For scattering from the proton, in general one expects a 2 dependence of these functions on 2/ and Q separately. But, at high momentum transfer the phenomena of scaling occurs, i.e. is observed to hold empirically (Bjorken, 1969). The energy and momentum-transfer dependence of the process behaves exactly as if the electrons were scattering off hard, pointlike constituents inside the protons, i.e. like (V.5). This is why it was said earlier that it is a repetition, at higher energies, of the Rutherford scattering experiment. Impulse approximation: The parton model contains implicitly the equivalent of the impulse approximation in nuclear physics. The basic assumptions are the following: 1. During the time of interaction one can neglect interactions between the partons. 2. Final state interactions can be ignored. That is to say the parton is quasi-free in the proton, and can be considered free at very high energy. The effect of confinement acts much later, when the scattered parton has moved a distance of the same order as the size of the proton. In terms of Feynman diagrams, this means that we consider 3 for fixed Co and Q >, 1 GeV; (V.6) 38 subprocesses with different initial or final states as being non-interfering. One does not have to worry about the other "spectator" partons to the lowest order. These assumptions are extremely useful for calculations of processes in hadron-hadron scattering. Another element that one needs is the parton momentum distribution, which will be discussed below. Electromagnetic spin 1/2 structure function: Now we will express (V.1) and (V.4) using the Mandelstam invariants s, t This set of variables makes explicit the Lorentz invariance of any quantity expressed in terms of it. The relation (V.1) becomes and u: t - Cp-p' 7/ - ( P-9 ) (V.1) (V.8) It will be useful later to know that 5rtc (V.9) If one wants to compare (V.8) to inelastic electron-proton scattering cross-section (V.4), one uses the parton model, 39 where it is hypothesised that inelastic, electron-proton scattering comes from the sum of incoherent elastic scattering of electrons on the partons in the target. If these partons have spin 1/2 and couple to the photon the same way the ju~ couples to the photon, then one can easily obtain an expression for the cross-section. Going into a reference frame where the proton has infinite momentum, one effectively "freezes" the slow interactions. One defines (V.10) The relation (V.8) can be written in terms of the momentum of a parton with the substitutions s—>xs u—>xu t—>t dV^J fa* I /SJ+U4N!X 8(f + X(5*U)) (V.11) Jt du I t2 2 \ S* The proton being supposedly made of several partons, denoted by the index i, one has to sum over these, and integrate over the probability function f(x) for a parton i to have a momentum fraction between x and x+dx, to get the cross-section for electron-proton scattering: itiu/er* t2 2 £z J L s+u  (v- 12) 40 where the relation (V.9) has been used in rewriting the delta function. We are now ready to compare with the general expression for e-p scattering, eq. (V.4) which we recast into the form (V.13) Comparing the coefficients in (V.12) and (V.13), one gets (V.14) This is the Callan-Gross formula (Callan and Gross, 1969) for spin 1/2 parton model. Identifying the spin 1/2 partons with the quarks, and denoting f^ by the symbol q., one puts for the proton and the neutron (V.15) if7(*)^ iUv+J*)-* H^k^^ + s*)*-Sum rules and momentum parton distributions: The fundamental relations the quarks distributions u(x) and d(x) must obey come 41 from the isospin properties and zero net strangeness of the proton and neutron: 0 r f.'dx LSOO - SrioJ 1 0 (V.16a) Or 5 - I Jx luu) - u a)J J = Jo J* E «kx) - Jc*). (V.16b) A similar relation for electrically neutral partons (gluons) comes from momentum conservation i (V.17) where £ is the fraction of momentum carried by the gluons. Then the gluon momentum distribution G(x) must obey L ix x Gcx) - £ (V.18) The value of £ turned out in experiments to be about 42 £-0.5 (Smith, 1974). That means that half of the proton momentum is carried by the gluons. The quark momentum distributions can also be deduced from experiments after inverting equations (V.1.5) e/v &P 1 (V.19) Jcx) - il (x) - F 2 (x)J A IS A few particular parametrizations are given in appendix F. The most useful clues come from the l>- hadron and e-hadron deep inelastic scattering. Hadron-hadron scattering; The calculation of a process in QCD for hadron-hadron scattering proceeds according to the following scheme. a) Calculate the subprocess in a perturbative way, using QCD rules, and other models, such as the Weinberg-Salam model. This gives a sub-cross-section C , (x•,u,s,t) where x. is the J jab ' ' fraction of the momenta u, s and t carried by the incoming parton i. b) Convolute &iu6 with the parton distributions f (x) of the incoming hadrons, which reads; 43 Let us now come back to the subject of Higgs boson, to examine its properties in more detail. This will enable us to calculate the sub-cross-sections related to Higgs production in hadron collisions. 44 VI . HIGGS BOSON PHENOMENOLOGY Here, we are coming to the heart of the subject -the Higgs boson properties. The knowledge of these is essential to the development of a strategy in the Higgs boson "hunt". Its mass and couplings to matter are needed to predict the production mechanisms and detection modes. Mass of the Higgs boson: There exist several arguments giving rise to upper and lower mass bounds on the Higgs boson. Only one, a lower bound, is derived from solid experimental facts. All other ones depend on theoretical expectations. One upper bound on the mass of the Higgs boson comes from the unitary restriction in the elatic scattering W+W —>W^W (Lee et al., 1977). The subscript L denotes a longitudinally polarized particle. As we have seen in chapter I, the process violates unitarity, and this is removed by the scalar contribution. The scattering amplitude for this process is, after cancellation of the divergences T (VI.1) The amplitude T has the usual partial-wave expansion T ~ l& ir Ltytl) t Please) (VI.2) Partial wave unitarity requires jt.| < 1. Here, j = 0, and 45 fal < Hidi ~ l.S TeV/c* <VI-3) Of A more refined calculation yields m M < 1 TeV/c (Lee, Quigg and Thacker, 1977). If mH lies above this limit, perturbation expansion breaks down and higher order terms are as important as the lowest order one. It can be shown (Veltman, 1977) that as mH increases, the parameter of the scalar potential (IV.11) increases, and when /)>>1, perturbative theory is meaningless. This happens at around mH= 1 TeV/c'. There is nothing wrong in itself for the perturbative expansion technique to break down. A perturbative theory is merely a desirable condition. It has been shown that a nonperturbative Higgs sector would show very little effect in current phenomenology (Appelquist and Bernard, 1980). The Higgs sector itself could give rise to some new physics, for example with bound states of elementary Higgs bosons. Another upper bound on the Higgs comes from a study of the h triviality of the scalar ?\^ interaction (Callaway, 1983). There is evidence that the theory is a trivial theory, i.e. the interaction screens itself and is equivalent to a free field theory. The ^ interaction coupled with fermions and/or vector bosons might not be trivial however. This could happen only within certain limits, one of which being a bound on the o H mass. For the standard model: 46 d^A <: \2J fm.^^O &V/c* (vi.4) Lower bounds on the Higgs mass have been proposed from several sources. The decay K —>7T 1 1 gives a lower bound of about 325 MeV/cjfor the mass of the Higgs boson (Willey and Yu, 1982). If the Higgs boson was any lighter, it would appear as a resonance peak in the invariance mass of the lepton pair. Other lower bounds come from studies of the radiatively corrected Higgs potential. However, all conclusions derived from the studies of the Higgs potential are suspect1 . The technique of the scalar potential is developed in the following way. We introduced in (IV.11) the classical scalar potential V(<£)= fJ. I$>j t^/^/ which is the most general renormalizable scalar field expression. The "effective" potential for the quantum field can be written in terms of the classical field . ^c (Jona-Lasinio, 1964), (Coleman and Weinberg ,1973). According to quantum field theory, one can calculate effective potentials at the n order level, corresponding to n-loop graphs. For example, the effective potential corresponding to the zeroth order correction (tree graph) is the classical potential (IV.10). The first order quantum correction includes the sum over all one-loop graphs of the theory, etc. At the first order, including scalar + vector loops, the effective potential is (Jackiw, 1974) 1 Ng, private communication 47 y i i i where B = 3[2g + (g + g' ) 3/1 024 77" and is the minimum of the potential. With this potential, one finds u. -- 4H ol<J>c = fX^3BJcr' <VI-6) The effective potential for different values of the parameters is plotted in figure 8, extracted from (Flores and Sher, 1982). 3 Notice that one can have ^ negative, (non-tachyonic scalar mass) and still achieve spontaneous symmetry breaking. The a elegant hypothesis jA - 0, due to Coleman and Weinberg (1973), gives a calculable value mc^ = 10.4 GeV/c . In this hypothesis, no mass scale is introduced at the level of the bare Lagrangian. Also if ja < 4Bcr, the spontaneously broken vacuum is not stable. It could "tunnel through" a lower lying vacuum at <£= 0. The probability of tunnelling increases as m^ decreases, and this brings various limits on m , depending on what one assumes about the conditions prevailing at the beginnning of the universe. Figure 8 - V(<£) for different values of 49 If one assumes that the universe has somehow been brought in the asymmetric-state vacuum after its birth, one requires the lifetime of this state to be more than the age of the universe. This leads to a lower bound of mH > 260 MeV/c*. If one instead takes the position that the universe was in the symmetric state just after its birth, and underwent a phase transition to the spontaneously symmetry-broken vacuum, then this later must lie below the ^=0 point. The lower limit on mH becomes 7 GeV/cZ (Weinberg, 1976). It was also pointed out by Linde (1976) that the lifetime of this transition must be substantially smaller than the age of the universe, and he got a limit m > 0.99 m„ . The inclusion of fermion loops makes drop by 6 Mev(mf/15 GeV/c*) . Therefore, for m^< 30 GeV/c2, the fermion contribution is negligible. If the top quark mass (or any other heavy quark mass) is larger than 30 GeV/c*, m,^ will drop significantly. If m^> 100 GeV/c2, mCK, becomes negative, and more care in the Coleman-Weinberg mechanism is needed to derive meaningful results. Couplings of the Higgs boson: The coupling (IV.24) was derived in the model with one fermion doublet and singlet. We saw in (IV.39) that many Yukawa coupling terms , each with its constant G- must be introduced. Adjusting the G4- to reproduce the fermion mass spectrum, one gets for the coupling of the Higgs boson to fermions: (VI.7) 50 which means that the probability of a reaction producing a Higgs boson will be proportional to the square of the mass of the fermion it is coupled to. It also means that the Higgs will decay almost exclusively into the heaviest particle kinematically allowed. The coupling of the Higgs to photons and gluons is made only through loop diagrams. The coupling to + o W" and Z bosons, however, is; -2 c Nil (Gf 4z) <VI-e) and will be dominant when the available energy allows it. Decay of the Higgs; The decay rates for the Higgs boson into leptons are given by (Sudaresan and Watson, 1972); fVz rr \ 1 mH I For quarks, simply multiply by three, because of the color degree of freedom. A plot of the branching ratios of the H° is given in fig. 9 extracted from (Ellis, Gaillard and Nanopoulos, 1976). Figure 9 - Branching ratio'of the H in function of from (Ellis, Gaillard and Nanopoulos, 1976) 52 The decay rate of the Higgs boson into vector bosons is fOA VV0= Gfkl /m„ Cbxia(3^-Vx + V) (vi.10) 2 2 where x = 4 mv/mH. This rate becomes rapidly very large as mH increases. The consequence is that, because of their width, Higgs of mass greater than 700-800 GeV/c may never be observed (Ali, 1981). Higgs signature: Once one has produced the Higgs boson, how does one know about it? One characteristic of the Higgs boson is its strong tendency to decay into the heaviest particle kinematically allowed. So, even before analysing in detail the spin and angular distributions of its reaction products, one might suspect Higgs bosons had been produced if the final state contained an anomalously large fraction of heavy particles. If the Higgs mass lies above the b-quark threshold, but below the top quark one, then it would decay predominantly into b-quark pairs. If its mass is below twice the W mass, but above t-quark threshold, it would decay mostly into t-quark pairs, which would then decay into b-quarks. Thus, the observation of events with one or two jets of invariant mass m^, containing at least two bottom quarks would be the signature for a Higgs of mass 2mfc< m^ < 2m^. Moreover, each production mechanism will produce something different along with the Higgs, and can be used to discriminate it from the background. 53 Production of the Higgs boson; Here will be presented the principal mechanisms that have been proposed up to now, in the search for the Higgs boson. The strategy is to produce particles that have very large couplings to the H , and look o for the signal of a H which could be produced with it, radiated from it, or decay from it, dependent on the process. The first three processes are more pertinent to e*~e machines, and the last two are appropriate for hadron-hadron collisions. 1) Decay of the Z°: a) The Z° can decay into a Higgs and photon, through a fermion loop or a W" loop (Cahn et al., 1978) represented by the diagram of figure 10. 54 %/ 0 Figure 10 - Feynman diagram for Z —> j+ H decay. with a ratio ml (VI.11) 0 - + s. However, the background for this process, Z -> 1 1 / is so large that the process 1a) would be buried in it (Barbiellini et al., 1979) b) Z decay along the channel Z —> H + 1 1 (Bjorken, 1976) represented by the diagram of figure 11, Figure 11 - Z —> H + XX decay diagram. 55 where Z denotes a virtual Z . The branching ratio for this decay channel is: which is observable for a high-luminosity Z factory. A Z factory is an e*e~ collider where the center-of-mass collision energy can be tuned to the Z mass, allowing a very large Z production rate. The process peaks at large dimuon mass. The angular distribution and dilepton mass distribution may be used to distinguish Higgs bosons and other scalar particles, elementary or not (Kalyniak et al., 1984). The main drawback to the process 1b) is that it works only if the mass of the Higgs is less than about 60 GeV/cz. To circumvent the problem one needs to produce a Higgs together 0 . o with a Z , from a virtual Z . O 4. _ O o 2) Bremsstrahlung from a virtual Z : e e —>H + Z illustrated by the diagram of figure 12. 4. _ , O o Figure 12 - Feynman diagram for e e —> H + Z . 56 The production rate peaks atjs =m? + 2mH (Glashow et al., 1978) and the total rate is encouraging for e+e~ machines, predicting -35 2 2 a cross-section of 4 X 10 cm for a Higgs of 10 GeV/c for a center-of-mass energy of 104 GeV. The problem is that one has to wait for the LEP II project to be completed. For the pp colliders, the production rates corresponding to this process are below the minimum acceptable (Ellis et al., 1976). 3) Decay of quarkonia: The form of the Higgs coupling to fermions makes it worthwhile to investigate heavy quarkonia decay. For the upsilon particle radiative decay has been calculated using a non-relativistic quark model (Wilczek, 1977). However, if the Coleman-Weinberg estimate is correct, this decay is not accessible. One of the best ways to produce a relatively light Higgs is through the decay of the (still unobserved) toponium state: -r 0 (VI.13) an upper limit branching ratio (VI.14) which has a huge branching ratio for mT < m : 57 r(JT~" n y) y (vi.15) PC JT — HA* In the case m^ > m^, the Jf will decay mostly through weak decay, thus depleting the branching ratio for J^. —> H + Y . Also, all of the above processes share the same characteristic that their rates are insignificant in a hadron-hadron collider, because of the impossibility of "sitting on" a resonance, as with e*e~ colliders. The following processes may lead to more sizeable cross-section at hadron-hadron colliders. 4) Gluon-gluon fusion: The fusion of two gluons into a single Higgs boson through a quark loop as in the Feynman diagram of figure 13 allows the use of the important gluon component of .the hadrons (Georgi et al., 1978). Figure 13 - Feynman diagram for gg —> H . It also exibits the interesting feature of counting all possible quark loops, even for quarks that are so heavy they 58 would not be produced in the laboratory. This comes from the particular form of the Higgs coupling to fermions, which being proportional to the fermion mass, cancel out a fermion mass term in the denominator of the phase space integration to yield a cross-section which is not sensitive to the quark mass, but proportional to the square of the number of heavy quarks. This process has been described as a "heavy quark counter" because of this feature. For Higgs bosons of masses less than 2MW, the background is several orders of magnitude larger than the signal (Keung, 1981), and there is no hope to discriminate them. The background process is the creation of a fermion pair through quark-antiquark annihilation. The cross-section for _ o o pp—> H + X through gg—> H and the estimated Drell-Yan background are given in Fig. 14, from (Keung, 1981) 59 z y/s (GeV) Figure 14 - Cross sections for pp—> H + X through the process gg—> H° (solid curve) and background 2 (dotted curves) for mM = 10 GeV/c from (Keung, 1981) 60 However, if the mass of the Higgs is such that it can decay into a pair of vector bosons, the signal may become more important than the background (Cahn and Dawson, 1984). More calculations are needed. 5) Compton-like processes: This is the compton scattering of gluons from heavy sea quarks, illustrated in fig. 15 The signal and background are estimated in (Barger and al., 1982) and reproduced in fig. 17. The authors claim the final state would produce a dramatic signature. The final state would be the same as the one described and calculated later in this thesis. However, for the reaction 5) the rate is very low, of order 1 picobarn or less, because of the very small c-quark content of the proton and antiproton. 6) Vector-boson fusion: Cahn and Dawson (1984) have proposed another mechanism as part of a study of very massive Higgs boson production. It makes use of the large vector boson coupling to the Higgs boson, according to the process illustrated in Fig. 16 61 Figure 16 - Feynman diagrams for the process qq —> H qq The total cross section for m^ = 5 Myy for this process, together with processes 2) and 5) at SSC energies, are shown in Fig. 18, extracted from (Cahn and Dawson, 1984). After having examined the principal channels suggested up to now for Higgs production, we are now in position to introduce the new mechanism on which this work is based. 62 100 pp or pp — HX L (from gc — He) T 1 1—I I I I 1 1 1—i—i r i T JQ 10b-0.1 10 diffrpctive charm Figure 17 - Cross sections fo'r compton-like process, for mH • 10 GeV/cz. The solid curve represents the signal, the dotted curve is the background for the Higgs decaying into a tau pair. Figure 18 - Total cross section for processes 2), 5) and 6) for mu = 410 GeV/c \ 64 VII. CALCULATION OF ASSOCIATED PRODUCTION OF HIGGS BOSON AND  HEAVY FLAVOR IN PROTON-ANTIPROTON COLLIDERS In the last chapter, we surveyed several mechanisms through which the Higgs boson could be produced in present or planned accelerators. It was suggested that H° bremsstrahlung from Z° bosons in e*e~ annihilation at the Z° resonance will o provide the cleanest signal, if the H mass, m^, is less than o the Z mass. In fact the luminosities of currently planned machines such as LEP and SLC will restrict the detectability to z m^ < 50 GeV/c . It becomes important to know what are the possibilities of producing and detecting the H in proton-antiproton colliders, such as the CERN SPS collider, the FNAL Tevatron or even the SSC (see appendix G for properties of these colliders). These machines will be capable of taking the o z search for the H up to the mass range of m «- 1 TeV/c , far beyond the range reached by e4e~ colliders available in the foreseeable future. This is why estimates of the production cross-sections of the H • in hadron-hadron colliders are now very important for the planning and the designing of colliders experiments. All of the following will thus be concerned with proton-antiproton or proton-proton collision only. We saw in the last chapter that the H production mechanism with the highest cross-section was the gluon-gluon fusion. If mH< 2 mw one expects to observe two back-to-back jets which are isotropic with respect to the beam direction, each containing at least one heavy flavored particle. Unfortunately, the process illustrated in Fig. 19 can also lead 65 to two heavy quarks and it is estimated to be an overwhelming background, Q Figure 19 - Feynman diagrams for the background to the process hadron+hadron —> H° + anything. One can attempt to suppress the background by considering H production in conjunction with heavy quarks. If one has enough incoming energy a possibility is f? irons (VII.1) where F(F) denotes a hadron which contains a heavy quark such as the b- or t-quark. One expects that the remaining hadrons (VII.1) do not contain heavy quarks, as indicated in SPS collider data. Sequential weak decays will then lead to up to twenty c-quarks or 4 b-quarks and 4 c-quarks, or 4 c-quarks plus 8 charged leptons in the final state for the case of t-quarks. Table I gives the number of c-quarks and charged leptons obtainable after F(F) and H° decays. Other intermediate combinations of c-quarks and charged leptons are 66 all possible. One ends the chain of sequential decays at the c-quarks in anticipation that the tagging of charm or beauty hadrons may become a possibility with rapidly developing vertex detectors (Stone, 1983) Within the framework of QCD parton model (see appendix A for QCD rules) the production of M° that will result in accompanying heavy quark final states can proceed via at least three mechanisms: 1) gluon-heavy quark scattering (Barger et al., 1982) h+ f.-lf.-) - fj(fj) + H" WI.2) 2) Higgs bremsstrahlung from heavy quarks in light quarks (q(-) annihilation ?« f; + £ + H° (VII.3) 3) Higgs bremsstrahlung from heavy quarks in gluon-gluon fusion a a L I U° (VI 1.4) where the subscripts denote colour indices of the gluons and quarks and f(f) is a heavy quark (antiquark) such as the b- or t-quark. The Feynman diagrams depicting processes 2) and 3) 67 are given in Figs. 20 and 21. Mechanism 1) has been introduced in the last chapter. Figure 20 - Feynman diagrams for qq —> f + f + H . Figure 21 - Feynman diagrams for gg —> / + ? +H 69 It makes essential use of the heavy quark (antiquark) content in the sea component of the hadron wave function. The actual size of this component is not very well known but can be estimated from hadronic charm production data (Field and Feynman, 1977). In general the probability of finding a heavy quark in the proton (antiproton) is expected to be very small: less than a per cent or so. However, it is a two-body final state and hence less suppressed by phase space. Reaction (VII.2) gives a rate which is of order Ol^y where ®s is the colour fine-structure constant and y denotes the Yukawa coupling of the Higgs boson to the quarks and is given by y = 2 mfGF . On the other hand reactions (VII.3) and (VII.4) are both ,z 2 , of the order CXS y ; hence are down by a factor ws compared to the previous mechanism. They also have three-body final state phase space suppression. These are compensated by the large probability of finding light quarks or antiquarks and gluons in both the p and p. Hence, one would expect the mechanisms (VII.2) to (VII.4) to give comparable rates of H° production in pp annihilations. In this thesis the results of a complete calculation of the processes (VII.1) using both reactions (VII.3) and (VII.4) as the fundamental subprocesses are given. The parton picture has been assumed to convolute the initial quark and gluon distributions over the fundamental subprocesses. The calculations for both mechanisms are presented in this chapter. In chapter VIII the results of the numerical calculations are 70 given. Distributions in energy of the Higgs boson and the behaviour of the production rate as a function of the center-of-mass energy, and other essential kinematical variables are also given. Chapter IX contains discussions of the results and their experimental implications are given. Calculation of the subreaction amplitude: As the subreactions (VII.3) and (VII.5) result in the same final state, they are indistinguishable at the macroscopic level. Since they have different initial states at the parton level they add incoherently. Together they give us a first-order QCD estimate for the semi-inclusive process (VII.1). In the final state, the meson F and its charge conjugate contain at least a heavy quark of flavour c,b or t. We will concentrate on the six-quarks model. The case where the ff forms a resonance such as toponium (T) results in p t p -* H°4 Jt /,ajrohS (VII.5) which has been estimated to be small1 . This is due to the smallness of the wave functions at the origin for this process. We will now discuss the two mechanisms (VII.3) and (VII.4) separately. 1 Ng and Zakarauskas, unpublished 71 Quark-antiquark annihilation mechanism:'" The u- and d-type quarks are mainly responsible for this process, since they are the dominant quark components of the proton wave function. o Also due to the small coupling that the H has with u- and d-quarks, H° bremsstrahlungs off the initial quarks can be neglected. To lowest order in 0(i one needs only to calculate the diagrams depicted in Fig. 20. Within the framework of perturbative QCD model, the cross section for the reaction (VII.1) is given by first calculating the elementary subprocesses (VII.3), then convoluting with the quark and antiquark distributions in the proton and antiproton. The amplitude for (VII.4) is (see fig. 20 for kinematics) + per imutsi ion s C ic «* k ") where i,j,k,l = 1-3 (quark color indices) a,b = 1-8 (gluon color indices) l\,l> = 1-4 (Lorentz indices) T* =^<Lare the SU(3) matrices, introduced by Gell-Mann (see appendix I). The cross section C^f-» jf°r this elementary process is given by 72 (VII.1) 36 V? ir where s = Q ? (QJ^+QL) • For the value of c(i we used the running coupling constant (11.14) with n the number of quarks flavors equal to 6. The value A - 0.2 GeV has been chosen for this QCD parameter. The matrix element element squared is given by 31 2 A ~* 1 + ^ 2,2 ^Q^^K) AT 2 (VII.8) The spin and color degrees of freedom have been summed over. Also the Feynman gauge is used for the gluon propagator. The contribution to the cross section of (VII.1) stemming from VII.3 is then: (VII.9) with s = x(x2s and x( and being, respectively, the fractions 73 of momenta carried by the quark and antiquark in their parent hadrons. For numerical calculations we have used two different parametrizations of the quark distribution functions, to get an estimate of the uncertainty introduced by the quark distributions. They are written explicitly in equations (F.1a) to (F.2b). The differences in the results in using one or the other parametrization were no more than a few per cent. The contributions from the sea quarks in the proton or antiproton have also been omitted, since their importance is of the few percent level. The integrations are performed using a Monte-Carlo method described in Appendix C. Gluon-gluon fusion mechanism: This mechanism takes advantage of the large gluon component in both proton and antiproton wave functions as well as the large coupling of Higgs boson to heavy quarks in order to compensate for phase space and &s suppression discussed before. As a result it also carries with it the not so well-measured gluon distribution functions, thus leading to uncertainties in the estimates of the production cross sections. We will further discuss these points later and also exhibit quantitatively these uncertainties. The calculation proceeds by evaluating the Feynman diagrams shown in fig. 21. The amplitudes are given by (k*-* k) (VII.10) 74 for the diagram of fig. 21a, M ^-AT* T«;UXK)/,v;.(*) (VII.11) for the diagram of fig. 21b, and 2 '/2 '/«< for the diagram of fig. 21c, with A = g^m^G^ 2 Here, c, and are the polarization 4-vectors of the incoming gluons. The SU(3) structure constants are given by the f4(,c . To evaluate the square of the amplitude given by 2 (VII.13) the traces are obtained by using the symbol manipulation program REDUCE. The REDUCE program written for this is given in appendix D. The gauge invariance of the result has been 75 checked by making the substitution <f,—> g( or c\—> gt. The initial gluon polarization and colour states are then averaged, and the final state spins and colour factors are summed over. The resulting output for the amplitude squared is given at length in appendix E. The total cross section (/(s,mH ,ny ) for the subprocess is o obtained by integrating over the phase space for the H , f and f. Using the parton model assumptions one convolutes over the gluon distributions via (VII.14) to obtain the total production rate. The lower limits of the x( and x. z integrals are given by the kinematical requirements of z x x s > (mt + 2m,) . (VII.15) The condition of eq. (VII.15) requires that the events generated in the Monte-Carlo calculations satisfy the kinematics for heavy particle production. The production cross section depends on the gluon distributions. From general CPT arguments it is expected that G (x) has the same form as G (x); thus any uncertainty in the gluon distributions will be doubled in the cross section O'(s). We will study this below. 76 To this end, two specific parametrizations representing extreme cases (F.3 and F.4) are chosen. The differences in the cross sections coming from the use of one or the other of the gluon momentum distributions gives an estimate of the approximate size of the uncertainty in the results due to an incomplete knowledge of this distribution. In addition to restricting the generated events to be physical ones, one has to take into account that QCD perturbative calculations have strict validity only in the high energy deep inelastic region. One should therefore avoid the region of phase space where the gluons or quarks become soft and thereby invalidate the use of the parton model. Hence the integrations have been restricted to take place in the region of high momentum transfer. The event generating routine requests that all scalar products between the 4-momenta of the incoming and outgoing particles be larger than 3 GeV . More stringent cuts may be imposed to reproduce experimental configurations. Several differential cross-sections have also been generated by the Monte-Carlo integration routine. These may be extremely useful in selecting experimental cutoffs and hence reducing the background. The differential- cross sections calculated are those relative to the Higgs boson and heavy quark's kinetic energies and transverse momenta. 77 VIII. RESULTS In this chapter are presented the results of the Monte-Carlo calculation of the two processes described last chapter. The free parameters of the theory are m^, the mass of the top quark, and m^. The variables on which the total cross-section depends are the center-of-mass energy of the proton-antiproton pair, and the lower cutoff on relative transverse momenta of the produced particles. Fig. 22 to 24 display the production cross section versus the cm. energy of the pp for = 10 GeV/c for two values of mR, corresponding to the b-quark with m^, = 4.5 GeV/c* and a 35 GeV/c* t-quark. The cross sections from quark-antiquark annihilation and gluon-gluon fusion are shown separately in fig. 22 and 23, and they are added in fig. 24. The quark-antiquark channel reaches a peak in the picobarn range around Vs = 2 TeV. However, it dominates over the gluon fusion mechanism at lower energies in the range < 60 GeV. At these relatively low energies one is required to use partons with large x in order to produce the final state particles. The gluon momentum distribution is steeply peaked toward small x as opposed to the quark distributions. This can be uniderstood by noticing that the gluons are rediated from the quarks, and therefore must show a radiative spectrum. As a result there are fewer gluons at large x. On the other hand, the gluon fusion is totally dominating at high energies, where small x still makes the incoming parton very energetic. There are two types of curves in all of the figs. 22 to 78 30. The dotted lines represent the results of the calculations done using the scale-violating gluon momentum distribution given by (F.3). The continuous lines are results using the scaling distribution (F.4). The difference between the two is indicative of the effects of scaling versus scale violating gluon distributions. The intersection points are reflections of the particular values of x where the two parametrizations of G(x) cross each other. Explicitly, for the case of pp collider at the Tevatron, 2 the production of 10 GeV/c Higgs boson in conjunction with a t-quark pair of mass 35 Gev/c* is well over 100 pb. Interestingly, the production in conjunction with two b-quarks has the same cross section, in spite of the fact that the Yukawa coupling is proportional to the quark mass. This suppression is here overcome by kinematics and quark dynamics. The kinematic reason is that the subenergies of the two gluons must be such that > 80 GeV for the t-quark case and and this is hindered by the rapidly falling gluon distribution functions. Then the dynamical enhancement occurs for the cross section via the propagator effect which favours smaller quark masses. This results in the crossing over of the production cross section at Vs = 2 TeV. A detailed examination of 0^- as a function of m^ is given 2 in Fig. 25 for m^ = 10 GeV/c . The upper curves and points correspond to the expected production rate at FNAL, the lower ones to SPS collider. Here, there is a rise in the cross section which reaches a peak at mf = for Vs = 540 GeV and 79 irij =20 GeV/c" for Vs = 2 TeV. In table II are given the total cross sections relevant for lower energies (Vs = 45 GeV) for different values of m^ and mK. This will be of relevance for a 1 TeV p scattering on fixed target where one can probe much smaller cross sections than possible with colliders, due to the higher luminosity. The production cross section is also a very sensitive function of mh and this is depicted in fig.26. Here we have chosen the reference value of m = 4.5 GeV/c . The behaviour seen as m^ varies is mainly due to the propagator effect of the heavy quark. From equations (VII.7) and (VII.10) to (VII.12) we see that four of six denominators in the amplitude have their minimum values near mH when either h-k or h-ic is small. o This corresponds to collinear H bremsstrahlung from the heavy quark (or antiquark). We also calculated the cross-section for cm. energies of 10, 20 and 40 TeV, for a wide range of m^, up to 1 TeV/c2. These energies are relevant to the planned SSC (Super Supraconductor Collider), a pp or pp collider which would be built in U.S. before 1995. The results are reproduced in table III, for two values of the cutoff on 4-momenta scalar 2 Z products, 3 GeV and 100 GeV . The former value is the QCD cutoff, introduced last chapter, guaranteeing applicability of perturbative QCD. The 100 GeV2 value may be more relevant to experimental cutoffs, especially at the SSC. Coming back to present day energies, in figs. 27 to 30 are plotted several differential cross sections, four different 80 values of" m^ , mK and /s. The distributions in energies of the H° and the heavy quark are compared in figs. 27 and 28. In general, the H° has an average energy higher than the heavy o quarks, the mean value being 40-50 GeV for the H , and 10-15 GeV for the fermions, in the case of a 10 Gev/c* Higgs produced with a pair of b-quarks at the Fermilab collider. The transverse momentum of H° is shown in fig. 29. It is seen that these distributions are peaked at hx = 15 GeV at Fermilab and hi = 5 GeV at CERN, and the peak increases for heavier H°. Similarly, the transverse momenta of the heavy quarks produced are given in fig. 30. They have the same features as hi with the peak located at kj. = 5 GeV for CERN and kj. =15 GeV for Fermilab, which is still a high value. The numerical calculations needed to produce these curves have been performed on a VAX-780, and necessitated approximatively 60 hours of CPU time. 81 Table 1 - Number of charm quarks, nc, and number of charged leptons, n^, in the final state particles of reaction (VII.1) after weak decays of the hadronsF, F and the Higgs boson. The first column denotes the heavy quark flavour contained in F. The second, third and fourth column entries give the values of nc if the heavy quark decays non-leptonically and the values of (nc ,njj ) if they decay semileptonically. The headings of columns give mass ranges of H °. F 2mc < mH < 2mi 2m b < mH < 2m^. mH > 2m^ c 4 6 12 (4,2) (4,4) b 6 8 14 (4,2) (4,4) (4,6) t 12 1 4 20 (4,4) (4,6) (4,8) 82 Table 2 - Fixed target cross section for reaction (VII.1). Cross section, in picobarns, for the production of a Higgs boson of mass m^ and a pair of charm quarks (mc = 1.5 GeV/c2), or bottom quarks (mk = 5 GeV/c*), in proton-antiproton collision, with centre-of-mass energy (/s~ = 45 GeV, corresponding to a fixed target experiment using a 1 TeV antiproton beam. m \ m H (GeV/c7) 10 5 2 1 0.5 1 .5 1 .3 X -H 10 9.0 X -3 10 0.4 3.5 20.0 5 2.0 X -3 10 1.0 X -2 10 0.18 0.65 1 .5 83 Table 3 - Very high energy cross sections Total cross section, in Picobarns, for the process pp or pp—> H° + F + F, with mt = 35 GeV/cZ, for different values of /s and mH, and two values of the scalar product cutoff. The gluon distribution used is the scale violating one (F.4). cutof f (GeV ) s (TeV) m (GeV/c ) / / ° 50 100 250 500 1000 2 159 1 . 1 7.2X10 Z ~3 1.3X10 -s 2X1 0 3 10 s 10 3 8X1 0 800 13 0.18 -3 3X10 20 s 4X1 0 V 4X1 0 3 10 1 000 8.5 0.2 40 5X1 06 7 10* 5 2X1 0 2X10V 410 5.4 2 18 0.6 -2. 7X1 0 -3 10 -5 2X1 0 100 10 1000 380 100 8.4 0.17 -3 3X1 0 20 3000 1000 600 1 40 4.5 0.18 40 5500 3700 2500 1000 350 1 .7 84 I02 10s I04 Vs GeV Figure 22 - Total cross section for the process (VII.3) as a function of Vs, for pp collision. 85 86 Ss (GeV) Figure 24 - Total cross section in pp from the sum of subreaction (VIII.3)and (VIII.4), as a function of v's, with mH - 10 GeV/c*. 87 It J 10 100 c b Mx (GeV/c2) Figure 25 - Total cross section for the process (VII.1) as a function of the mass of the heavy quark produced with the Higgs boson, for m = 10 Gev/c . Discrete points refer to the value of the cross section at the masses of the c- and b-quarks. The continuum portion, starting at mK = 20 GeV/c* corresponds to the t-quark contribution. The dotted line has been added to guide the eye. 88 Figure 26 - Total cross section for the process (VII.1) as a function of the mass of the Higgs boson. Figure 27 - Differential cross section d^/dEn° for four different sets of the parameters mH, m^, and 90 Figure 28 - Differential cross section dcr/dE* for four different set of the parameters mH, mM and Vi. Figure 29 - Differential cross section dC /dhi for four different sets of the parameters mH, mK and Vs. Figure 30 - Differential cross section do- /dkj for four different sets of the parameters mH, mK, and ys. 93 IX. DISCUSSION AND CONCLUSION This thesis presents the QCD-parton model calculation of the production cross section of a Higgs boson plus a heavy quark pair in proton-antiproton or proton-proton collisions. We employed QCD and the Weinberg-Salam model of electroweak interactions, where the SU(2) ®U(1) group is broken by a doublet of scalar fields. After the spontaneous symmetry breaking occurs, one is left with one real scalar particle, called the Higgs boson. Now that the W~ and Z bosons have been found at CERN, the Higgs boson is the only particle predicted by the Weinberg-Salam model yet to be discovered. Thus, it is the whole concept of spontaneous symmetry breaking which would be confirmed in the event of a positive identification of a Higgs particle. This mechanism gives masses to particles in the popular gauge theories, hence the great importance of getting some experimental evidence supporting or invalidating it. The total production rate for the reaction (VII.1) has been calculated for center-of-mass energies ranging from 45 GeV to 40 TeV, as well as for a wide range of the Higgs mass, from 1 GeV/c to 1 TeV/c . The dependence of the cross section on m^, h± , kJf E/, and E^ has been calculated, and should be useful to place experimental cuts or discriminate against the background. At this point we compare the results of our calculations of reaction (VII.l) with that of the estimate using the bremsstrahlung technique of Ellis et al (1976). There the 94 production cross section is given by (IX.1) in the rest system of the heavy quark. The differential cross section dG\j is that of pair production of heavy quarks without the Higgs boson, i.e., _ _ (IX.2) p + p _^ F+F + y If we take this cross section to scale like mj* , then we see that CTH will be governed by the charm-quark pair production. Using CT ~ 10 cm , one obtains cr ~ 10 cm for m, =10 GeV/c c " hj and Vs = 540 GeV.This is about two orders of magnitude larger than our calculation. The phenomenological estimate given above includes all possible mechanisms for the physical process to occur,including (VII.2), (VII.3) as well as (VII.4). Our calculations are only good to first order in QCD. Furthermore, eq. (IX.1) gives an overestimate since it does not take into account the transverse momentum of the heavy quark and other kinematic-suppression factors. One can expect the real value of the croos section to lie somewhere in between the two calculations. The rates for high cm. energy and high mH, even with the 2 100 GeV cutoff, remain quite impressive compared to the three other production rates, calculated by Cahn and Dawson(1984), 95 and reproduced in Fig. 17. A comparison is difficult, because it is not clear what cutoff(s), if any, has been used by Cahn and Dawson. A analysis of these processes and their background is under way1 . Any production rate larger than 1 pb. is large enough for the corresponding process to be observable in present or planed collider rings (see appendix G). Then one must deduce from it all the cuts needed to detect the signal and discriminate it against the background; certain cuts may depend on the detectors used, like minimum transverse momenta or energies. The background rates must also be calculated and compared to the signal rate. If the former is larger than the total signal rate, there is still a chance that the signal and background have markedly different angular distributions, energy spectrum or some other kinematic variable dependence. A careful analysis of the signal and its background in the different decay channels of the Higgs boson is needed. This work could also be extended in assuming a different Higgs sector in the symmetry breaking mechanism, leading to several Higgs bosons, of both charged and neutral types. The coupling constants in these alternate models are very nearly free however, in constrast to the minimal scalar field case of the Weinberg-Salam model. This would introduce one or more new free parameters to the calculations. The main sources of uncertainty on the calculations presented in this thesis are brought by the gluon distribution 1 Ng, Bates and Zakarauskas 96 and the behavior of the amplitude squared in the low momentum transfer region. The uncertainties relative to the gluon distribution have been explicitely calculated in most cases. Only the SSC region calculations (Table III) have been covered using only one gluon distribution (F.4), because the scaling one, (F.3), is no longer appropriate at these energies. The low momentum transfer region has been completely avoided by using the 3 GeV3 cutoff on scalar products. All events within this region, which correspond often to larger sub-cross-sections, have been discarded by the Monte-Carlo phase space generator. But because these events have generally low p^ or a small opening angle between two of the final state components, they would also be discarded in real experiments. What has been done in this thesis is the complete first-order calculation of the process (VII.1). It points out the importance of this process in the search for the Higgs boson. Second order QCD corrections are expected to be of order ds times the calculated rate, or less. The strong coupling constant ols is approximatively 0.2 in the kinematic region considered. Of course, a more specific calculation of the signal and bakcground rates, including all kinematic cuts, geometry of detectors, decay and hadronization of reaction products, would have to be done before an experiment looking for the H° proceeds on a particular set of accelerator and detector. It must be pointed out that other models of electroweak interactions and grand unified theories all include at least 97 one scalar boson which corresponds to the Higgs boson in" the Weinberg-Salam model. Thus, the calculation exposed in this thesis is relevant to all these theories. 98 BIBLIOGRAPHY Abers,E.S., Lee,B.W. (1973). Phys. 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(1982). Phys. Rev. D26, 3287 Yang,C.N. and Mills,R.L. (1954). Phys. Rev. 96, 191 101 APPENDIX A - FEYNMAN DIAGRAMS AND QCD RULES In this appendix are collected the set of rules and conventions used throughout this work, as well as an introduction to Feynman diagrams and cross-section calculations. Appendix C. gives more details about cross-section integration. The conventions used in the calculations relative to Dirac matrices and cross-section calculations have been adopted from Bjorken and Drell (1964). The scalar product of two 4-vectors, p and q, is defined as: The gamma matrices must be introduced when solving the problem of describing the motion of spin 1/2 particles (fermions). The equation one must solve is the Dirac equation (for the non-interacting case) (A.1 ) where g is the metric (A.2) 0 (A.3) The f are spinors, which are usually represented as 4-component 1 02 vectors. The / are the so-called gamma matrices, obeying the following anticommutation relations {f.f}-- vYyV- if i I being the 4X4 unit matrix. According to quantum field theory, interactions between particles described by a Lagrangian give rise to scattering amplitudes which may be represented by an expansion in terms of Feynman diagrams, examples of which are given in figs. (VII.1) to (VII.5). To each Lagrangian corresponds a unique set of Feynman diagrams at each order of the expansion parameters g 's, the g 's being constants measuring the strength of an interaction. If the interaction is very weak, the first order expansion will yield a good approximation. If not, one has to calculate the next order, etc. The Feynman diagrams are constructed in the following way. From each Lagrangian may be derived a set of vertices and propagators. The vertices are nodes in a diagram; the propagators are lines in between the nodes. To each vertex is associated a strength gt-. An expansion at the n~" order in g^. is the set of all possible diagrams in which the product of the g(- is g{- or less. One looks for all the ways to combine available propagators and vertices from a given Lagrangian, with the same initial state A and final state B. The sum of these constitutes the A -—> B process n order expansion. 103 The Feynman diagrams really are elegant mnemonic devices, allowing one to represent visually any given process, and find all the allowable ways to combine vertices and propagators into a calculable expression. To each vertex or propagator corresponds either a scalar, a g or a gamma matrix times a scalar. To external lines are assigned either spinors, for fermions, or polarization 4-vectors for spin-1 bosons. All the terms are then combined to form the transition amplitude of the process. The matrix element squared expresses the probability for the transition to happen. One then sums over allowed configurations in phase space, multiplies by the appropriate kinematic factors to get the total cross-section c/, or some differential cross section relative to any desired variables x( , dO'/dxJ dx2.. .dx,, . The integration over phase space using the Monte-Carlo integration method is explained in appendix C. The term "on-mass shell" refers to real particles, as opposed to virtual ones; for the former, total energy, momentum and mass obey the relativistic relation (A.5) 1 04 The rules associated with vertices and propagators in QCD are (Politzer, 1974); Propagators gluon K2 J (A.6) fermion i vertices (A.8) 9 (A.9) T(-- as defined below i,j,k,l = 1-3 (quark color) a,b = 1-8 (gluon color) Polarization sum (A.10) 105 Color sum The qqV vertex involves the factor Ta= ^Att where the are the SU(3) matrices. The T<x' s obey the commutation rules [ja , T J ~ 1 iaU Tc (A. 1 1) {L,Llr H.ilm^lu L (A-12) where fabc are antisymmetric and the are symmmetric under interchange of any two indices. I(3j is the 3X3 unit matrix. Some identities that will be used in.appendix B involving the matrices Ta and symbols fatc are: Tr (TaTb) = i&i (A*13) Tr (TaTt la Tc ) ~ " /2 C§ U (A.15) 0 ' (A.16) 106 face/ fhei ~ 5 5d (A.17) 107 APPENDIX B - COLOR SUMMATION CALCULATION Having introduced the QCD rules in appendix A, we are now in a position to calculate the color factors for different terms of the amplitudes. For the amplitude (VI.6), using the relations (A.13) to (A.17), the color factor of the matrix element squared is Because we are averaging over initial color, we must also divide by a factor 9 for quarks. This is the total number of different color states for two incoming quarks. Hence, the factor 2/9 is obtained after averaging over quark colors. In the process (VII.4), there are two gluons in the initial state. Therefore, to average over color, we must divide by an overall factor of 64, which is the number of different color combinations for two incoming gluons. For the calculation of the color factors, the different terms arising from the squaring of the amplitude (VII.10) to (VII.12) are divided into five classes. For the squares of M, to Mfc, and the cross terms in between M, to M3, or Mv to M6, the color factor is: (B.1) (B.Z) 108 use eq. (A.12) = Tr(TVrrVif.icUrr'r) use eqs. (A.14) and(A.15) r " ^3 Cut ~ i fabc i ( SAkc * i iaic) use eq. (A.16) we get with eq. (A.17) >k/3 For the cross terms between one of M , M2 or M5 on one hand, and M5, Mg or M? on the other, the color factor is: Tr(TaTVT4)=-ii<$.i -~Ah For Mo and M* squared: filc ru r« • = f f urn (B.4) For cross terms between one of My, M5, or M6 on one hand, and M^ or M^ on the other, the color factor is: (B.5) ^ ate And finaly, for cross terms in between My, M^ or M^ on one hand, and M^ or Mg on the other: 109 , (B.6) These color factors are substituted when summing the different terms of the amplitude squared. They are essential to get the correct gauge invariance. V (Jake "* i Ld) - "L7^ &*:-6 1 10 APPENDIX C - THE MONTE-CARLO INTEGRATION ROUTINE In this appendix is given a brief outline of the Monte-Carlo method, followed by a listing of the program used in this work to integrate over phase space and parton momenta. For a complete description of the Monte-Carlo method in particle physics, see (Byckling and Kajantie, 1973). What we want to do is to integrate the amplitude squared over phase space to get the sub cross section & , and then over x2 and x? to get the total cross section. The Monte-Carlo method consists of generating (simulating) events, and then calculate the probability of it to happen, through the amplitude squared. The average of these for a large number of events converges toward the total cross section faster than standard integration method when the dimensionality D of the integral is large. In our case D = 7. To get differential cross sections is as easy. Suppose you want to calculate the derivative of the cross section relative to some angle 6. You create a vector V of dimension 100 or so, define 9 from the 4-vectors generated by the simulation, and calculate for each event in which position in V the event falls. That is, if for one event the angled is in between 0 and 1.8 , you add the probability corresponding to it into the first element of V. The probability acts here as a weight to the event. The cross section of a process Q +Q --> k + k + h where k, k and h are the heavy quark, antiquark and Higgs boson 4-momenta respectively is: 111 2 JaK J3I< A M^K (inlEt WE, (CD where F = 2s is the flux and |M| is the amplitude squared. The delta function insures that the conservation of energy momentum is respected. What the Monte-Carlo process does is generate 4-momenta of real particles in the center of mass of decaying or virtual particles, and then boost them back into the laboratory c.m. The simulation happens as if the reaction was taking place as in the diagram of figure 31. K A Figure 31 - Order of particle generation in the Monte-Carlo method in particle physics. Therefore, in our case, it is as if the incoming quark and antiquark were producing a fictitious particle G, which would decay into a real heavy quark k and a fictitious particle X. The 4-momentum of k in the G center of mass is stored, as well as a boost matrix to pass from the G c.m to the laboratory c.m. We then go in the X c.m. and let it decay into a heavy antiquark and a real Higgs boson. Their 4-momenta in the X c.m. are calculated, then boosted back to the G c.m. All 1 12 three particles' 4-momenta are finally boosted back to the laboratory cm. where they are used to calculate the cross section. To this end, all possible scalar products between the 4-momenta are formed and substituted into the amplitude squared. The result is then stored and the whole process is repeated N times. To cast the integral (C.l) into a form compatible with the Monte-Carlo technique , one integrates it over the & function 2 and introduces a fictitious mass Mx = (k + h) , corresponding to a virtual particle with 4-momentum X = k +h. One must also make another change of variables to spherical coordinates of the k in the G cm. and the h in the X cm. What is left is an integral over this Mx variable and the angular coordinates of the two real particles, as in & r ± JM]_2 J_ JM* JUxo. d/l*„ ill ill (C.2) F 2SF (irrf I 2 •± _ -» where |k| is the k momentum in the X center of mass, and |k| is the k momentum in the G c.m. To produce only physical events, the X particle is restricted to have a mass of at least mH + mK, but at most fs - m . The Monte-Carlo program generates X's masses and real particles angular distributions randomly, using a random number generator called GGUBFS out of the IMSL library. GGUBFS acts on an argument SEED, which is changed every time it is used. The Jacobian for the variable change from Mx and the XL's to the random numbers generated 1 1 3 between 0 and 1 is (4TT) [E - mu - 2m.,]. The Monte-Carlo routine has been tested on the phase space (|M|Z 5 1), where any mistake in the program would show up as a suspect momentum distribution, or an asymmetry between particles. The total phase space integration has been compared to analytical calculation and found to agree within 1 %. 1 14 0001 REAL'S P. XI. VI. THETA1. PHI1,SEED.V2,XI. X2, CI, 02. C1P 0002 REAL*B G1M. C2P. C2M, CUTOFF, CUT2, VOLUME, ST 0003 REAL*8 DCADRE, F, A, B, AERR. RERR. ERROR. SC 0004 REAL'S BK4.4), E2, MH, MK, MH2. MK2, MX, MX2, S, PI 0005 REAL'S K(4). AK(4), EX2, PX2.B2<4,4).XI2, THETA2. PHI2.H(4> 0006 REAL'B K2<4). AK2<4),H2(4>,AK3<4>.H3(4).SUPX 0007 REAL'S C0STHETA1,C0STHETA2,WI,FLUX, C.CONSTANTE OOOB REAL'B DI, D2. D3, D4, D5, D6, BUMW 0009 REAL'8 HSP1. HSP2. C1SH, 02SH,C1SP1.01SP2.C2SP1, C2SP2. P1SP2 0010 REAL'B HC, LAMBDA1, LAMBDA2. W, 8UMW2, INTECRALE 0011 REAL'B EH(100).EK(100).EAK(100>.ETH(100>.ETK<100>.ETAK(100) 0012 REAL'B XFE(100), RAP(100), Y, XF 0013 REAL'B ENERGY,TRANSH, TRANSK, TRANSAK, BINR 0014 REAL'S Rl(4), R2(4), RT1,RT2, UPS1,UPS2,PHIT1,PHIT2, PQ 0015 REAL'8 DISTRIBUTION, DENSITY, STOT 0016 REAL'S MAI.MA2.MA3.MA4.MAS. 0017 C Ml1A, M12A, M13A, M14A,M15A,M16A.M17A,M1BA.M22A, M23A. M24A, M25A, 0018  M26A. M27A.M2SA.M33A.M34A.M35A.M36A.M37A.M38A. M44A. M45A, M46A. 0019 C M47A,M48A.M55A,M56A,M57A,M5BA,M66A.M67A,M68A, M77A, M78A, M88A 0020 INTEOER START,DATA, IER 0021 EXTERNAL F 0022 COMMON C,STOT,ST 0023 COMMON EH, EK, EAK, ETH, ETK, ETAK 0024 C 0025 C RECEIVE PARAMETERS 0026 C— 0027 C GIVE THE CHOICE DF THE FORM OF ENTRY 0028 C 0029 16 WRITE(6,19) 0030 19 FORMAT< ' DO YOU WANT TO START A NEW CALCULATION (TYPE 0) OR 0031 C CONTINUE A PREVIOUS ONE (TYPE 1)') 0032 READ(5.18) DATA 0033 18 FORMAT (II) 0034 IF (DATA . NE. O .AND. DATA . NE. 1) WRITE<6,17), STOP 0035 17 FORMAT(' YOU MUST ENTER 0 OR 1') 0036 C 0037 C READ ENTRIES THROUGH TERMINAL 0038 C 0039 WRITE(6,20) 0040 20 FORMAT(' ENTER BEPERATELY CUTOFF,P. MH AND MK') 0041 READ(5.10) CUTOFF,P, MH, MK 0042 WRITE(6,29) 0043 29 FORMATC ENTER STEEPNESS OF OLUON DISTRIBUTION') 0044 READ< 5. 10)8T 0045 WRITE (6,30) 0046 30 FORMAT(' ENTER NUMBER OF EVENTS DESIRED') 0047 READ (5,12)N 0048 IF (DATA . EQ. 0) THEN 0049 START - 1 0050 BUMW «= O. 0051 6EED - 12345. O 0052 END IF 0053 C 0054 C READ ENTRIES THROUGH FILES 20 AND 21 0055 C 0056 IF (DATA . EQ. 1) THEN 0057 WRITE (6.11) 1 1 5 0058 11 FORMAT<' ENTER IJ, SEED AND SUMW'> 0059 READ(5.12) START 0060 12 FORMAT (17) 0061 READ(5.9)SEED. SUMW 0062 9 F0RMAT(D1B. 10) 0063 10 FORMAT(Fl5. 8) 0064 DO 15 1*1. 100 0065 15 READ<27.96)EH(I).EK(I).ETH(I),ETK(I),RAP(I), XFE(I) 0066 END IF 0067 C 0068 C INITIALIZE VARIABLES 1 0069 C 0070 WRITE(20. 98) 0071 98 FORMAT ( ' P MK MH CUTOFF ST ') 0072 URITE(20.99) P.MK.MH,CUTOFF. ST 0073 99 FORMATUD12. 5, D8. 3) 0074 WRITE(20. 97) 0075 97 FORMAT(' IJ SEED 0076 C BUMW X-SECTI ON ') 0077 MH2 = MH • MH 0078 MK2 - MK * MK 0079 C - MH «• 2. * MK 0080 8T0T - 4. * P#P 0081 RT1-0. 0082 RT2=00083 C • 0084 C ESTABLISH 4-VECT0R OF QUARKS IN LAB SYSTEM 1 0085 C 0086 100 DO 1000 IJ-START,N 0087 150 XI - OOUBFS(SEED) 0088 C1M=1. 0089 CIP-EXP(-ST) 0090 XI * -LOC(Q1M+XHMC1P-01M> )/ST 0091 Gl - 8T» EXP(-ST«X1> 0092 X2 - OCUBFS(8EED) 0093 CUT2 - C*C/(ST0T*X1) 0094 C2M=EXP(-ST»CUT2) 0095 C2P - C1P 0096 X2«= -L0C(C2M+X2*(C2P-C2M) >/ST 0097 C2 « ST*EXP(-ST*X2) 0098 UPS1 • XI - (RT2*«2.>/(X2 # 2 «P> 0099 UPS2 - X2 - (RT1**2. )/(Xl * 2. *P) 0100 IF(UP81 LE. 0.) 00 TO 150 0101 IF(UPS2 LE. O. ) 00 TO 150 0102 C 0103 C PICK-UP ANOLE FOR TRANS. MOMENTUM OF PARTONS 1 0104 C 0105 PI • 3. 141592654 0106 PHIT1 - 2. • PI * OCUBFS(SEED) 0107 PHIT2 - 2. • PI * OOUBFS(SEED) 0108 C 0109 C CALCULATE PARTONS 4-MOMENTA IN LAB SYSTEM 1 0110 C 0111 Rl(l> - UPS1*P • RT1*«2. /<4. »UPS1»P) 0112 RK2) - RT1 * COS(PHITl) 0113 RK3) - RT1 » SIN(PHITl) 0114 Rl(4) - UPS1*P - RT1*»2. /(4. »UPB1*P) 0115 R2(l> - UPS2*P • RT2««2. /(4. «UPS2»PJ 0116 R2(2> - RT2 * C0S(PHIT2) 0117 R2(3> - RT2 * 8IN(PHIT2> 0118 R2(4) - -UPS2*P + RT2**2. /(4.*UPS2*P> 0119 C 0120 C CALCULATE VELOCITY VI AND RAPIDITY XI OF CM IN LAB FRAME 1 0121 C ; 0122 PG - SQRT((Rl(2>+R2(2>)**2. •MR2(3)+R1(3))**2. +(R2(4>+Rl(4))*«2. ) 0123 VI - PQ / (Rl(l) + R2(l)) 0124 XI - L0C((1. + Vl)/(1. - Vl))/2. 0125 C 0126 C E2 IS ENERGY IN CM 1 0127 C ;  0128 S - X1«X2*ST0T 0129 E2 - SQRT(S) 0130 C 0131 C CHECK IF ENOUGH ENERGY IS AVAILABLE FOR REACTION TO OCCUR 1 0132 C 0133 IF(S . LE. (MH + 2»MK)»*2. )00 TO 150 0134 C 0135 C PICK-UP ANGLES OF ROTATION 1 0136 C 0137 PI * 3. 141592654 0138 C0STHETA1 • 2.« GGUBFS(SEED > -1. 0139 THETA1 - ACOS<COSTHETA1) 0140 PHU - 2. * PI • GGUBFS(SEED> 0141 C 0142 C COMPOSE ROTATION MATRIX 1 0143 C 0144 CALL BOOST(BI. XI. THETA1. PHI 1> 0145 C 0146 C PICK UP INVARIANT MASS OF THE DFF MASS-SHELL QUARK 1 0147 C 0148 SUPX- E2 - HK 0149 MX - SUPX - OGUBFB(SEED>« (SUPX - (MH+HK)) 0150 MX2»MX»MX 0151 C 0152 C CALCULATE 4-MOMENTA OF X AND K QUARKS IN QUARKS CM 1 0153 C 0154 K2(l) - (8 • MK2 - MX2)/ (2. *E2) 0155 K2<2> - O. 0156 K2(3) - 0. 0157 K2(4) - 8QRT(K2(1> *K2(1> - MK2) 0158 EX2 - E2 - K2(l> 0159 PX2 » -K2(4) 0160 C —* 0161 C COMPOSE BOOST MATRIX BETWEEN QUARKS CM AND X CM 1 0162 C 0163 V2 • PX2/EX2 0164 XI2« L0G((1. • V2>/(1. - V2))/2. 0165 C0STHETA2 - 2. * GGUBFS(SEED)-1. 0166 THETA2 - ACOS(CDSTHETA2) 0167 PHI2 -2 • PI* OGUBFS(SEED) 0168 CALL BOOST(B2, XI2, THETA2. PHI2) 0169 C 0170 C CALCULATE 4-MOMENTA OF HIGGS AND ANTI-QUARK IN X CM 1 0171 C 1 1 7 0172 H3<1) - (MX2+ MH2 - HK2)/(2. »MX) 0173 H3<2) -0. 0174 H3<3) - 0. 0175 H3<4> - SORT<H3(1) * H3(l) - MH2) 0176 AK3U) • MX - H3(l) 0177 AK3(2) - 0. 0178 AK3(3) > 0. 0179 AK3(4) - -H3<4) 0180 C 0181 C TRANSFORM THE 4-MOMENTA BACK INTO FRAME 2 1 0182 C 0183 CALL MULT<B2.H3» H2> 0184 CALL MULT(B2, AK3, AK2) 0185 C 0186 C TRANSFORM THE 4-MOMENTA BACK INTO LAB FRAME 1 0187 C 0188 CALL MULT(B1. H2, H> 0189 CALL MULT(B1,AK2.AK) 0190 CALL MULT(B1. K2. K) 0191 C 0192 C CALCULATE THE SCALAR PRODUCTS 1 0193 C 0194 CALL SCALP(AK, Rl, C2SP2) 0195 CALL SCALP(AK, R2, 01SP2) 0196 CALL SCALP(K. Rl. C2SP1) 0197 CALL SCALP(K. R2. ©1SP1) 0198 CALL SCALP(Rl, H. 02SH) 0199 CALL SCALP(R2. H,C1SH) 0200 CALL SCALP(K. AK. P1SP2) 0201 CALL SCALP(K, H. HSP1> 0202 CALL 8CALP(AK, H. H8P2) 0203 C 0204 C PUT CONDITIONS ON VALIDITY OF CALCULATION 1 0205 C IN PERTURBATIVE QCD 1 0206 C 0207 IF (01SP1 .LT. CUTOFF) 00 TO 150 0208 F (G1SP2 . LE. CUTOFF) 00 TO 150 0209 IF (02SP1 .LE. CUTOFF) 00 TO ISO 0210 F (C2SP2 . LE. CUTOFF) 00 TO 150 0211 IF (P18P2 . LE. CUTOFF) 00 TO 150 0212 F (C1SH LE. CUTOFF) 00 TO 150 0213 IF (02SH LE CUTOFF) 00 TO 150 0214 F (HSP2 . LE. CUTOFF) 00 TO 150 0215 IF (HSPl . LE. CUTOFF) 00 TO 150 0216 C 0217 C CALCULATE THE DENOMINATORS OF AMPLITUDE 1 0218 C 0219 DI - MH2 • 2. *HSP2 0220 2 - MH2 + 2. *HSP1 0221 D3 - -2 •018P1 0222 4 - -2.•02SP2 0223 DS - -2. •01SP2 0224 6 - -2. •02SP1 0225 CALL. AMPL(S. DI. 02. D3. D4. D5. D6. 0226 C MK2.MH2.HSPl.HSP2. 01BH. 02SH.018P1, 0227  01SP2. 02SP1. 02SP2. PISP2. HC.MAI.HA2. MA3. MA4. MAS. M11A. M12A. M13A. 0228 C M14A.M15A.M16A.M17A.niBA.M22A.M23A.M24A.M25A.M26A.M27A.M28A, 1 18 0229 C M33A. H34A. M35A, M36A.M37A,MSBA,M44A. M4SA. M46A. M47A, M48A. M55A. 0230 C M56A. M57A. H58A. M66A, M67A.M6BA.M77A,H7BA.M88A) 0231 C 0232 C CALCULATE THE SCALE VIOLATING GLUON DISTRIBUTION 1 0233 C 0234 SC - LOG(25. *S> 0235 BC - SC / LOO(125.0) 0236 SC * LOG (SO 0237 CALL DIST<XI.X2<SC.DISTRIBUTION) 0238 C . — 0239 C CALCULATE PHASE SPACE DENSITY AND ELEMENT OF INTEORALE 1 0240 C 0241 LAMBDA1 - SORT((8 - MK2 - HX2>**2-4. *MX2»MK2) 0242 LAMBDA2 - 8QRT((MX2 - MH2 - MK2>»*2 - 4.*MK2*MH2) 0243 VOLUME « (G1P-C1M)»(G2P-G2M> 0244 DENSITY « VOLUME/(Gl*02) 0245 CONSTANTE - 4. 488D+04 • MK2/(LOO(25. *S))*«2 0246 FLUX - 2. *S 0247 Ul - ((4. •PI)*«2«(E2 - 2«MK - MH)/(32. »S»MX)) • LAMBDA1 *LAMBDA2 0248 W - WI *HG * DISTRIBUTION • CONSTANTE • DENSITY /<(2.«PI>#»5»FLUX) 0249 SUMW - SUMW + W 0250 88 FORMAT( ' IJ=M10) 0251 C 0252 C WRITE OUT designed EVENTS 1 0253 C 0254 IF (IJ . LE. 10) THEN 0255 WRITE(17. 89) 0256 89 FORMAT( ' IJ H(I) AMI) K(D') 0257 DO 90 I • 1.4 0258 WRITE(17,91> IJ. H(I). AMI), MI) 0259 91 FORMAT<19. 3D20. 8) 0260 90 CONTINUE 0261 WRITE(17.92) 0262 92 FORMAT(' ') 0263 WRITE(17. 93) 0264 93 FORMAT(' IJ HG WI 0265 C E2 DISTRIBUTION  XI 0266  X2'> 0267 WRITE(17,94) IJ, HO,WI, E2. DISTRIBUTION, W. XI, X2 0268 94 FORMAT(17, 7D15. 5) 0269 WRITE(17,92) 0270 END IF 0271 0272 IF (XI .LE. -.01) THEN 0273 WRITE(17,89) 0274 DO 85 I - 1,4 0275 WRITE(17,91> IJ. H(I), AMI), MI) 0276 85 CONTINUE 0277 WRITE(17. 92) 0278 WRITEU7.930279 WRITE(17.94) IJ, HC.W1. E2. DISTRIBUTION, W. XI, X2 0280 WRITE(17.92) 0281 END IF 0282 0283 IF (W . LE. 0. > THEN 0284 WRITEU7.93) 0285 WRITEU7. 94) IJ, HG, WI, E2. DISTRIBUTION. W. XI. X2 1 19 0386 END IF 0287 0288 IF <W . LE. 0) THEN 0289 WRITE(6,8) 0290 WRITE(17. 7)IU. HC 0291 WRITE(17. 9)MA1,MA2. MAS. MA4. MAS, Ml 1A. M12A. M13A. M14A, M15A, M16A, M17A, 0292 C M1BA.M22A.M23A,M24A,M2SA,M26A,M27A.M28A,M33A. M34A. M3SA. M36A. M37A, 0293  M38A. M44A. M45A, M46A.M47A. M4BA,MSSA. M56A.MS7A. M58A, M66A, M67A. M68A, 0294 C M77A,M7BA. M8BA 0295 B FORMAT<' THERE 16 A NEGATIVE CROSS-SECTION') 0296 7 FORMAT ( UO. D20. 8) 0297 END IF 0298 C 0299 C BIN THE ENEROY AND TRANSVERSE ENERGY OF H.K.AK 1 0300 C •• 0301 ENERGY «( 2*P - MH - 2»MK>/3. 0302 TRANSH • SORT(H(2)*H(2) • H(3)»H(3)> 0303 TRANSK - SORT(K(2)*K(2) + K(3)»K(3)> 0304 CALL BIN(H(1), EH. 0.. ENERGY. U> 0305 CALL BIN(Kd), EK. 0. , ENERGY. W> 0306 CALL BIN(TRANSH. ETH. 0. , ENERGY, U) 0307 CALL BIN(TRANSK, ETK, 0. . ENERGY, W> 0308 C 0309 C BIN THE RAPIDITY Y AND FEYNMAN SCALING VARIABLE XF OF HIGGS 1 0310 C 0311 Y - ABS(0.5*L00((H(1)+H(4))/(H(1)-H(4)))) 0312 XF - ABS(H(4)/P) 0313 CALL BIN(Y,RAP,0. , 4. . W) 0314 CALL BIN(XF.XFE,O. , 1. .W) 0315 C 0316 C WRITE DOWN ANSWER EVERY 10O0 EVENTS. IN CASE 8YSTEM 1 0317 C BREAKS DOWN 1 0318 C 0319 RAT - MOD(IJ,1000) 0320 IF (RAT . EQ. 0) THEN 0321 WRITE(20,95)IJ,SEED.BUMW,SUMW/IJ 0322 OPEN (UNIT « 27) XFORT-I-DEFSTAUNK, Default STATUS- 'UNKNOWN' used in OPEN »tate»ent CPEN (UNIT - 27)3 in module OLUONtMAIN at line 322 0323 DO 112 1-1. 100 0324 WRITE(27,96)EH(I).EK(I).ETH(I).ETK(I).RAP(I>, XFE(I> 0325 95 FORMAT (17, D25. 10, D20. 10, D15. 6) 0326 96 FORMAT(6D15. 8) 0327 112 CONTINUE 032B CLOSE (UNIT - 27) 0329 END IF 0330 1000 CONTINUE 0331 0332 C CALCULATE THE INTEORALE 0333 0334 INTEGRALE- SUMW/N 0335 WRITE(17.43)N 0336 43 FORMAT( ' N «• ', 17) 0337 WRITE(17,42)CUT0FF 033B 42 FORMAT(' CT0FF='D15. 6) 0339 WRITE(17»44)P 120 0340 44 FORMAT( ' p - ', D15. 6> 0341 WRITE(17, 45)MH 0342 45 FORMAT< ' MH - ',015.6) 0343 WR1TE<17,46)MK 0344 46 FORMAT( ' MK • '.D15. 6) 0345 WRITE(17,55)1NTEGRALE 0346 55 FORMAT( ' X—SECTION - '.D15.6, ' PICOBARN') 0347 BINR » 0. 0348 DO 111 I- 1,100 0349 WRITE<18,56) BINR*ENERCY/100. . EH(I)/N, EK(I)/N 0350 WRITE(19,56) BINR*ENER0Y/100., ETH(I)/N,ETK<I)/N 0351 WRITE(22,57) BINR»4./100. ,RAP(I)/N 0352 WRITE(23,57) BINR/100. ,XFE(I)/N 0353 57 FORMAT(2D15. 4) 0354 BINR BINR + 1 0355 56 FORMAT (3D 15. 4) 0356 Ul CONTINUE 0357 END 0001 0002 SUBROUTINE DIST(XI.X2,SC.DISTRIBUTION) 0003 C CO04 C CALCULATES THE SCALE VIOLATING GLUON DISTRIBUTION FROM 1 0005 C PARAMETERS XI, X2 AND THE SCALE PARAMETER SC 1 0006 C 0007 REAL*8 XI.X2.SC.DISTRIBUTION, El, E2 0008 El - -0. 93*SC + 0. 36*SC»*2 CDC? 2 - 2. 9 + l.B3»SC 0010 DISTRIBUTION « X1«»E1 * X2**E1 0011 DISTRIBUTION - DISTRIBUTION • (1-X1)«*E2 • d-X2)*»E2 0012 DISTRIBUTION - DISTRIBUTION * (2.01 - 2. 73*SC • 1. 29*SC«»2)*«2 0013 DISTRIBUTION - DISTRIBUTION / (X1«X2> 0014 RETURN 0015 END 0001 0002 SUBROUTINE B0OST(B.XI,THETA,PHI) 0003 REAL «B B(4,4). XI. THETA, PHI 0004 Bd. 1)- COSH(XI) 0005 B(1.2)= -BINH(XI) • BIN(THETA) 0006 B(1.3> = O 0007 Bd. 4)» BINH(XI )»COS(THETA) OOOB B(2. 1)* 0. 0009 B(2.2>« COS(PHI)•COS(THETA) 0010 B(2.3)« -SIN(PHI) 0011 B(2. 4>- SIN(THETA>*C0S(PHI) 0012 B(3.1)- 0. 0013 B(3. 2)« BIN(PHI)*COS(THETA) 0014 B(3.3>« COS(PHI) 0015 B(3. 4)- BIN(THETA)*SIN(PHI) 0016 B(4. 1>- SINH(XI) 0017 B(4,2)- -COSH(XI)*SIN(THETA> 0018 B(4. 3)- O. 0019 B(4.4)- COSH(XI)*COS(THETA) 0020 RETURN 0021 END 121 0001 0002 SUBROUTINE BIN(F.AR.INF.SUP. W) 0003 C 0004 C CLASSES F INTO ONE OF 100 BINS BETWEEN INF AND SUP AND PUT 1 0005 C IT INTO ARRAY AR 1 0006 C 0007 REAL*8 F. AR<100), INF. SUP,POS 0008 COMMON EH.EK.EAK,ETH. ETK, ETAK 0009 POS - INT<100. # (F — INF)/<SUP - INF)) • 1. 0010 IF (POS . OT. 100. ) POS - 100. 0011 AR(POS) - AR(POS) + W 0012 RETURN 0013 END 0001 0002 SUBROUTINE SCALP(VI,V2, B) 0003 C 0004 C TAKE THE SCALAR PRODUCT OF THE TWO 4-VECTORS VI 1 0005 C AND V2 AND PUT THE RESULT INTO 8 1 0006 C 0007 REAL*B VI(4), V2(4).B 0008 8 - Vl(l)*V2(l) - V1(2)*V2(2> - V1(3)»V2(3> - V1(4)»V2(4> 0009 RETURN 0010 END OOOl 0002 SUBROUTINE MULT(B, VI,V2> 0003 C 0004 C CALCULATES THE PRODUCT BETWEEN THE MATRIX B AND 1 0005 C VECTOR Ml AND PUTS RESULT INTO V2 1 0006 C 0007 REAL«8 B(4.4), Vl(4>, V2(4). PH 0008 DO 300 1-1,4 0009 PH-O. 0010 DO 301 J»l. 4 0011 301 PH - BU.J) • VKJ) • PH 0012 V2(I) - PH 0013 300 CONTINUE 0014 RETURN 0015 END 1 22 APPENDIX D - CALCULATION OF THE TRACE Trace calculations come in evaluating Feynman diagrams involving fermions. Standard methods for calculating the traces are given in Bjorken and Drell (1964). When the number or length of traces to evaluate become too large to manage, one may now use one of a few number of programs designed to this end. One of them is REDUCE (see UBC REDUCE), which was used in a program for evaluating the amplitude squared of the process (VII.4). In this appendix is given a listing of this REDUCE program. The input is included in the program, and consists of the numerators of the amplitude (VI1.10) to (VII.12). The output is a FORTRAN code, in term of the scalar products of the outgoing particles 4-momenta. 123 1 »SICN0N OUCH TOM PAGES=80 PROUTE=PHYS 2 •**» 3 »SQURCE »REDUCE 4 OFF ECHO; 5 X 6 1 1 7 1 THIS PROGRAM CALCULATES THE AMPLITUDE 1 6 1 SQUARED FOR THE PROCESS 1 9 1 CLUON+CLUON —> QUARK •*• ANT I-QUARK • HIGGS 1 10 111 112 113 1 DEFINE VECTORS AND MASSES OF COMPONENTS 1 14 1 ; 15 MASS 01=0. 02=0. P1»MK. P2=MK, H-MH; 16 MSHELL Gl,G2,PI.P2, Hi 17 VECTOR E1.E2; IB LET G1.E1=0. G2. E2=0; 19 LET Gl. G2=S/2i 20 OPERATOR V2» U2> GM> GMH, 21 X ' 22 1 GIVE THE RULES FOR SUMMATION 1 23 1 OVER POLARIZATION OF GLUONS24 1 ; 25 LET El. El • -2; 26 FOR ALL P LET El.P * El.P = -P.P + 2 *( P.C1*Q C2+P. C2»Q. Gl)/S; 27 FOR ALL R, Q LET El.R • El.Q = -R.Q • 2 * (R. C1*Q. C2 + R.G2+Q.Gl>/S; 2B LET E2. E2 = -2; 29 FOR ALL P LET E2. P * E2. P = -P. P • 2 *( P.G1*Q G2+P. 02*0 Gl)/S; 30 FDR ALL P. Q LET E2. P * E2 Q = -P. Q + 2 » (P.G1*Q. 02 + P C2+Q. Cl)/S, 31 XOFF MCD; 32 FACTOR S, P1.G1, P1.C2. P2. Gl. P2. 02, PI. P2, H. Gl, H C2. H. PI. H P2; 33 ^ 34 1 DEFINE NEW OPERATORS TO SIMPLIFY THE 1 35 1 TYPING OF THE AMPLITUDE 1 36 1 ; 37 FOR ALL T. U LET 0M(T-MJ> = C(L. T)*C(L, U) • MK; 38 FOR ALL H LET OMH(H) = C(L. H) 2«MK; 39 LET V2 = C(L,PI) - MK; 40 LET U2 = C<L,P2) + MK; 41 LET P1C1 • 0M<G1 - Pl)i 42 LET P2C1 - 0M<P2 - 01)i 43 LET P2G2 = 0M(P2 - G2>; 44 LET P1G2 = CM<C2 - Pl)» 45 LET 0E2 = G(L,E2)i 46 LET 0E1 - G(L,E1)< 47 LET VERTEX - 2*C1. E2*0E1 + El.E2«(G(L.G2)-G(L. Gl)) 48 -2*02. E1»GE2; 49 LET MH*«2 - MH2; 50 LET MK*«2 » MK2; 51 OFF NAT; 66 X 67 1 WRITE THE AMPLITUDE COMPONENTS 1 68 I AND THE SUM OVER U AND V SPINORS69 1- . 70 LET Ml • U2 * CMH(H) * GE2 • P1C1 * 0E1 • V2i 71 LET M2 • U2 * 0E2 * P202 • P1G1 • ©El • V2i 72 LET M3 « U2 * CE2 * P2C2 * CE1 • CMH(-H) * V2; 124 73 LET M4 - U2 » 0E1 * P2G1 » GE2 • ©MH(-H) » V2; 74 LET M5 = U2 • GE1 » P2G1 » P1G2 * GE2 * V2; 75 LET M6 « U2 » OMH(H) * OE1 * P1G2 • GE2 «V2; 76 LET M7 - U2 * VERTEX « GMH(-H) • V2; 77 LET MS - U2 * GMH(H) * VERTEX * V2; 78 X 79 1 WRITE THE COMPLEX CONJUGATE OF THE AMPLITUDE 1 80 1 ; 81 LET MIR * GE1 • P1G1 * GE2 • GMH(H); 82 LET M2R - GE1 • P1G1 * P2C2 • GE2; 83 LET M3R - OMH(-H) * 0E1 • P2C2 • CE2; 84 LET M4R « CMH(-H) • GE2 * P2G1 • CE1; 85 LET M5R • CE2 » P102 * P2G1 * 0E1; 86 LET M6R - GE2 • P1G2 • 0E1 « GMH(H); B7 LET M7R - CMH(-H) • VERTEX; 88 LET M8R - VERTEX * OMH(H); 89 X 90 1 WRITE THE SQUARE OF THE AMPLITUDE 1 91 1 (WITHOUT THE DENOMINATORS)92 1 ; 93 OFF NAT; 94 OFF ECHO; 166 XWRITE "M23H - ", M2*M3R; 167 XWRITE "M33H = ", M3*M3R; 168 XWRITE "M34H = ". M3*M4R; 169 XWRITE "M35H «= ". M3*M5R; 170 XWRITE "M36H = ". M3*M6R; 171 XWRITE "M37H = ". M3»M7R; 172 XWRITE "M3BH - M3*MBR; 173 XWRITE "M44H - ", M4#M4R; 174 XWRITE "M45H • M. M4*M5R; 175 XWRITE "M46H - ". M4»M6R; 176 XWRITE "M47H • ". M4»M7R; 177 XWRITE "M48H • ", M4»MBR; 17B XWRITE -M55H - ", M5»M5R; 179 XWRITE "M56H - M5*M6R; 180 XWRITE "M57H - '*. M5»M7R; 181 XWRITE "M58H - ". M5*M8R; 182 XWRITE "M66H • ". M6»M6R; 183 XWRITE "M67H « ". M6»M7R; 184 XWRITE "M6BH - "» M6*MBR; 185 XWRITE "M77H = ". M7«M7R; 186 XWRITE "M8BH • ". MB»MBR; 187 XSHUT ZGLU0NRES23; 188 XON NAT; 189 X 190 1 REWRITE THE AMPLITUDE IN FILE CLU0NRES2 1 191 I IN A FORM READABLE BY FORTRAN 1 192 1 193 XON FORT, 194 XOUT GLU0NRES2; 218 XWRITE "LET M46H »,,,M46H; 219 XWRITE "LET M47H «",M47H; 220 XWRITE "LET M48H «".M48H; 221 XWRITE "LET M55H «"»M55H; 222 XWRITE "LET M56H -M,M56H; 223 XWRITE "LET M57H -".M57H; 224 XWRITE "LET M58H M58H; 125 225 XWRITE "LET M66H -". M66H; 226 "/.WRITE "LET M67H »",M67H; 227 XWRITE "LET M68H -".M68H; 228 XWRITE "LET M77H "",M77H; 229 XWRITE "LET M78H *".M78H; 230 XWRITE "LET MB8H M88H; 231 XSHUT CLU0NRES2; 232 XON ECHO; 233 x 234 1 DEFINE DENOMINATORS 1 235 1 AND DENOMINATORS SQUARED236 1 ; 237 LET DI - S - 2*P1. (G1+G2); 238 LET D2 = S - 2*P2. (G1+C2); 239 LET D3 • -2*01. PI; 240 LET D4 - -2*P2. 02; 241 LET D5 = -2*P2. CI; 242 LET D6 = -2*P1. C2; 243 LET D12 = Dl*Dli 244 LET D22 = D2*D2; 245 LET D32 • D3»D3; 246 LET D42 - D4*D4; 247 LET D52 » D5»D5; 248 LET D62 = D6*D6; 249 7.IN GLU0NRESULT2; 250 LET H «• Gl + G2 - PI - P2, 251 LET P1.P2 = (Gl + 02). (PI + P2) - (S + 2*MK*»2 - MH»*2)/2; 252 LET MH#*2 - MH2; 253 LET MK**2 = MK2; 254 %0UT GLU0NRESULT2; 255 -/.WRITE "Mil « "; 256 V.M11H; 327 7. 328 1 PUT ALL COMPONENTS OF AMPLITUDE SQUARED 1 329 1 OVER SAME DENOMINATOR 1 330 1 ; 331 LET MUA = Mil * D22 * D42 • D52 * D62 » S2; 332 LET M12A « M12 * DI * D22 * D4 * DS2 * 062 • S2; 333 LET M13A = M13 * DI * D2 * D3 * D4 * D52 • D62 • S2; 334 LET M14A = M14 * DI * D2 * D3 * D42 » D5 * D62 • S2; 335 LET M15A = Ml5 * DI * D22 * D3 * D42 * D5 * D6 * S2; 336 LET M16A = M16 • D22 » D3 * D42 • DS2 * D6 * S2; 337 LET M17A - M17 * DI * D2 * D3 * D42 » D52 * D62 * S; 338 LET M18A = M18 • D22 * 03 * D42 * D52 • D62 • S; 339 LET M22A = M22 • D12 • D22 » D52 * D62 • S2i 340 LET M23A = M23 • D12 • D2 • D3 » DS2 • D62 • S2; 341 LET M24A = M24 • D12 * D2 * D3 * D4 • D5 * D62 * S2; 342 LET M25A - M25 * D12 * D22 » D3 * D4 * D5 * D6 * S2; 343 LET M26A » M26 * DI * D22 » D3 * D4 * DS2 • D6 • S2; 344 LET M27A = M27 * D12 « D2 * D3 * D4 * D52 * D62 *S; 345 LET M28A = M28 * DI • D22 » D3 * D4 • D52 • D62 *S; 346 LET M33A = M33 » D12 * D32 * D52 * D62 * S2; 347 LET M34A = M34 » D12 • D32 » D4 * D5 » D62 • S2; 348 LET M35A - M35 * D12 » D2 * D32 * D4 * D5 * D6 * S2; 349 LET M36A - M36 * DI * D2 * D32 * D4 • D52 * D6 * S2; 350 LET M37A = M37 » D12 • D32 * D4 * D52 * D62 »S; 351 LET M3BA » M38 * DI * D2 * D32 * D4 » D52 * 062 «S; 352 LET M44A = M44 » D12 * D32 » D42 » D62 * S2; 126 353 LET M45A - M45 • D12 * D2 » D32 • D42 » D6 « S2; 354 LET M46A - M46 • 01 * 02 • D32 • D42 * D5 » D6 » S2; 355 LET M47A - M47 * D12 »D32 • 042 • D5 • D62 *S; 356 LET M48A = M48 * DI » 02 • D32 • D42 * D5 • D62 *Si 357 LET M55A - M55 • D12 * D22 » D32 • D42 • S2; 358 LET M56A - M56 * DI • D22 • D32 • D42 • D5 • S2; 359 LET M57A - M57 « D12 # D2 » D32 • D42 * D5 » D6 »S; 360 LET M5BA = M58 * DI * D22 * D32 • D42 * D5 * D6 *S; 361 LET M66A - M66 * D22 • D32 • D42 • D52 • S2; 362 LET M67A - 1167 » DI • D2 « D32 * D42 • D52 • D6 *S; 363 LET M6BA - M68 * D22 • D32 • D42 * D52 • D6 *S; 364 LET M77A - M77 • D12 * D32 » D42 » D52 » D62; 365 LET M78A - M78 • DI * D2 * D32 • D42 • D52 • 062. 366 LET M88A = M88 * D22 * D32 * D42 • D52 • D62. 367 LET MB8A • M88 • D22 « D32 • D42 • DS2 * D62; 368 X 369 1 REGROUP THE TERMS ACCORDING TO 1 370 1 COLOR FACTOR 1 371 1 ; 372 LET MAI - MilA + M22A + M33A + M44A + M55A + M66A + 373 2»(M12A + M13A • M23A •»• M45A • M46A * M56A), 374 LET MA2 - 2*(M14A + M15A + M16A + M24A • M25A + M26A 375 + M34A + M35A + M36A); 376 LET MA3 « M77A •»• M88A + 2*M78A; 377 LET MA4 - 2*(M17A + M18A + M27A + M28A • M37A + M38A); 378 LET MA5 - 2»(M47A • M48A + M57A + M58A •»• M67A + M68A); 379 XOUT CLU0NRESULT3J 380 XWRITE "MAI MAI; 381 XWRITE "MA2 «".MA2; 382 XWRITE "MA3 •".MA3; 383 XWRITE "MA4 •". MA4; 384 XWRITE "MA5 «",MA5; 385 XWRITE "M77A »".M77A; 386 XWRITE "Mil -".Mils 387 XWRITE "D12 D12; 388 X16*MAl/3 -2*MA2/3 • 12*MA3 • 6«MA4 - 6«MA5; 389 XSHUT CLU0NRESULT3; 390 MTS; 391 SIG 127 APPENDIX E - PRINTOUT OF THE AMPLITUDE SQUARED OF THE PROCESS Here is given the amplitude squared of the process (VII.4), called by the subroutine AMPL of the routine GLUON, whose listing appears in appendix D. The denominators inserted in lines 852 to 887 come from the propagators in the amplitude (V .10) to (VII.12). The color factors in line 896 have been calculated in appendix C. The variables in the numerator are defined as follow: S = s MK2 = m^ MH2 = G1SP1 = j'fl HSP1 H- pi etc. 128 0001 SUBROUTINE AMPL<S, DI. D2. D3. D4. 05. 06. 0002 C MH2. NH2. HSPl. HSP2. 01SH. 02SH. 01SP1. 0003  01SP2,C2SP1.02SP2. P1SP2. HG> 0004 C 0005 C CALCULATES THE AMPLITUDE OF THE PROCESS 1 0006 C 0007 0008 REAL«8 S, MK2, MH2. HSPl. HSP2.01SH.G2SH. C1SP1. 01SP2, C2SP1. C2SP2. P1SP2 0009 REAL*8 Ml 1. M12, M13.M14. M15.M16.M17.M18.M22. M23. M24. M25. M26, M27, M28 0010 REAL*8 M33.M34,M35.M36.M37,M3B.M44.M45.M46. M47. M48. MSS. M56. M57, M58 0011 REAL*8 M66.M67,M68.M77.M7B.MBS.DI.D2. D3. D4. D5. D6. HG 0012 REAL*8 Ml1A.M12A.M13A,M14A.Ml5A,M16A, M17A. M18A 0013 REAL»8 M22A,M23A,M24A.M2SA.M26A,M27A. M28A 0014 REAL»B M33A,M34A. M35A,M36A.M37A,M38A 0015 REAL*8 M44A, M45A. M46A, M47A, M4BA 0016 REAL*8 M55A, M56A, M57A. M5BA 0017 REAL*8 M66A,M67A.M6BA 0018 REAL*8 M77A.M7BA.MBBA 0019 REAL*B MAI.MA2, MA3. MA4. MA5 0020 0021 Mil «<-32. *S**2*MK2**3 0022 C -8. *S**2*MK2**2*MH2+32. *S*»2*MK2**2*(-HSP2+C1SH+ 0023  GlSP2)-8. *S**2*MK2*MH2*C1SP2+16.*S**2*MK2*(HSP2*01SH+2. *G1SH* 0024 C G1SP1+2. *GlSPl*01SP2>-8. *S**2*MH2*01SP1*C1SP2+16. *S**2*HSP2*Q1SH 0025 C *01SPl+64. *S*MK2**2*<-ClSH*C2SPl-©2SH*ClSPl+2. *C1SP1*02SP1-61SP1 0026  *02SP2-C1SP2»C2SP1> + 16.*S*MK2*MH2*C2. »©1SP1*Q2SP1+01SP1*02SP2+ 0027 C 61SP2»C2SP1)+32. *S*MK2*(-HSP2*01SH*C2SPl-HSP2*02SH»ClSPl+4. *HSP2 0028  *QlSPl*02SPl-4. *GlSH*01SPl*Q2SPl-4. *01SP1*01SP2*02SP1)+32. *S*MH2 0029 C »GlSPl*GlSP2*G2SPl-64. *S*HSP2*C1SH*01SP1*02SP1+256. *MK2*C1SP1* 0030 C 02SP1»<G1SH#G2SP1+Q2SH»G1SPH-G1SP1*C2SP2+G1SP2*02SP1)-<64. *MH2* 0031  G1SP1«G2SP1)*(01SP1*C2SP2+G1SP2*C2SP1> + 128. *HSP2*01SP1*C2SP1*< 0032 C C1SH*C2SP1+C2SH*G1SP1>)/S**2 0033 M12 »8. *S**3*MK2**2+4.*S**3*HK2*(HSP2+2. »G1SP1) 0034 C +4. *S**3*HSP2*C1SP1+ 0035  16. *S**2*P1SP2*MK2**2+16. *S**2*P1SP2*MK2*(HSP2+01SP1> + 16. *S**2* 0036 C P1SP2*HSP2*G1SP1-16.»S**2*MK2**3+B. *S**2*MK2**2»<-HSP2-HSP1+Q1SH 0037  -C2SH-2. *C2SPl-2. *Q2SP2)+B. «S**2*MK2*(-HSP2*01SP2-HSP2*e2SPl-0038 C HSPl*01SPl+01SH*ClSPl+01SH*Q2SP2-02SH*©lSP2-4. *01SP1*C2SP1>+B. *S 0039  **2*01SPl*(-HSP2*01SP2-2. *HSP2*©2SP1+»1SH*02SP2-02SH*©1SP2> + 16. * 0040 C S*PlSP2*MK2»<-GlSH»G2SP2-G2SH«QlSP2-4. •018P1»02SP1> + 16. »S*P1SP2* 0041  ClSPl»<-4. *HBP2*62SP1-Q1SH*Q2SP2-G2SH*01SP2)+16. *S*MK2*«2*<C1SH* 0042 C Q2SP2+Q2SH*QlSP2+4.*01SPl*G2SPl-2. *01SPl»02SP2-2. *C1SP2*G2SP1+4. 0043  *C1SP2*G2SP2> 0044 M12=M12+16. *S*MK2*<2. *HSP2*01SP1*02BP1-H8P2*G1SP1*02SP2-0045 C HSP2*GlSP2*02SPl+2.»HSPl*01SPl»02SPl+2. •HSP1»01SP2»G2SP2-G1SH* 0046  GlSPl*G2SPl-01SH*01SP2*02SP2-C18H*G2SPl*G2SP2+02SH*ClSPl**2+2. • 0047 C 02SH*C1SP1*©2SP1+02SH*C1SP2**2+C2SH*Q1SP2*02SP1-2. *01SP1**2* 0048  ©2SP2-2. *©lSPl*C19P2*C2SPl+4. »01SPl*G2SPl**2+4. *©1SP1»02SP1* 0049 C Q2SP2>+16.»S*GlSPl*(-2.*HSP2*01SPl*62SP2+2. *HSP2*G2SP1**2+2. * 0050  HSPl*ClSP2*G2SP2-01SH*01SP2*G2SP2-2. «G1SH*C2SP1*C2SP2+G2SH*C1SP2 0051 C **2+2. *02SH*01SP2*Q2SP1)+64.•P1SP2*Q1BP1*Q2SP1*<01SH*©2SP2+G2SH* 0052  019P2>+64. *MK2*01SP1*02SP1*<-01SH*C2SP2-02SH»GlSP2+2. *eiSPl* 0053 C 02SP2+2. *01SP2*02SPl-4. *01SP2*02SP2>+32. »01SP1*(2. *H8P2*01SP1* 0054  ©2SPl*Q2SP2+2. »HSP2*GlBP2*G2SPl**2-4. *HSP1*01SP2*02SP1*02SP2+ 0055 C 01SH*GlSPl*G2SP2**2+GlSH*GlSP2*G2SPl*02SP2+2. »G1SH*G2SP1**2* 0056  02SP2-G2SH*01SPl»018P2*G2SP2-G2SH*01SP2**2*02SPl-2. *02SH*C1SP2* 0057 C ©2SP1**2) 1 29 0038 M12«t112/S»»2 0039 W13 —2. *S**3*P1SP2*MH2 0060 C +8. «S**3»MK2»»2*4. «S»*3*MK2*(HSP2+HSPn+4. *S*# 0061 C 3»HSP2»HSP1 + 16. *S**2*PlSP2«MK2*«2+4. •S«»2*P1SP2*HK2*MH2+16.*S»«2 0062 C *PlSP2*MK2*<HBP2+HSPl>+4.»S**2*PlSP2*l1H2*tClSP2+02SPl >+8. »S**2* 0063 C P1SP2*<2. *HSP2»HSP1+01SH*C2SH>-16. •S»«2*MK2»«3+4. *S**2»MK2**2» 0064 C MH2- (16. #B»*2*HK2»*2) * (HSP2+HSP 1+01 SP I+G2SP2 > +4. *6«*2*HK2* (-2. • 0065 C HSP2»«2-2. •HSP2*HSPl-HSP2*01SH+HSP2*02SH-2. •HSP2»01SP1-2. «HSP2* 0066 C C1SP2-2. *HSPl*«2+HSPl*GlSH-HSPl*02SH-2. *HSPl»G2SPl-2. *HSP1*C2SP2 0067 C -2. *01SH#02SPl+2. *QlSH*02SP2+2. •Q2SH»ClSPl-2. »02SH*018P2>+4. •S** 0068 C 2»MH2*<ClSPl*G2SP2*018P2*02SPl>-<8. •S**2)*(HSP2*HBP1*01SP2>HSP2* 0069  HSPl*G2SPl+HSP2*GlSH«02SPl+HSPl*02SH*QlSP2>+32. •S*P1SP2*MK2*< 0070 C C1SP1«C2SP2+C1SP2*C2SP1>-<8. *S*P1SP2*HH2)*(C1SP1*G2SP2+01SP2* 0071  G2SP1) 0072 tl 13=M13-(16. *S*P1SP2)»< HSP2«01SH*G2SP1+HSP2*C2SH*G1SP1•HSP1»G1SH» 0073 C C2SP2+HBP1»C2SH*C1SP2+Q1SH#C2SH#C1SP2+G1SH#C2SH*02BP1>*32. »S*MK2 0074  **2*<C1SH»02SP1+Q1SH*02SP2+G2SH*01SP1+©28H#01SP2*2. •G1SP1*02SP1-0075 C GlSPl*C2SP2-QlSP2*02SPl+2. *C1SP2*G2SP2>+8. *S»MK2*MH2*<-2. *C1SP1* 0076  02SPl-01SPl*02SP2-01SP2*02SPl-2. *GlSP2*02SP2>+8. *S*HK2*<2. »HSP2* 0077 C 01SH*Q2SPl+2. »HSP2*GlSH#G2SP2+2.•HSP2*G2SH«01SPl+2. *HSP2»G2SH* 0078  01SP2+B. •HSP2*01SPl«G2SPl-4.*HSP2»ClSPl#G2SP2-4. »HBP2*G1SP2« 0079 C 02SP1+2. •HSPl»01BH*G2SPH-2. »HSP1*C1SH*02SP2*2. »HSP1«Q2SH*G1SP1 + 0080  2. *HSPl*Q2SH*GlSP2-4. *HSPl*GlSPl»G2SP2-4. »HSP1*01SP2*C2SP1+B. • 0081 C HBP 1*C1SP2«G2SP2-G1SH**2*G2SP1+01SH**2*02SP2+01SH*Q2SH*Q1SP1 + 0082  GlSH*C2SH*QlSP2+GlSH*02SH»Q2SPl+QlSH«C2SH*Q2_P2+2. •Q1BH*018P1« 0083 C G2SP2+2. *01SH*ClSP2*C2SP2+2. »G1SH«Q2SP1**2+2. *C18H*02SP1*C2SP2+ 0084  C2SH»*2»GlSPl-C2SH**2*GlSP2+2.*G2SH»QlSPl*ClSP2+2. *Q2SH*G1SP1* 0085 C 02SP1+2. •02SH*GlSPl*02SP2+2. *02SH*G1SP2**2+B. •01SP1»C1SP2»G2SP2+ 0086  8. *G1SP1*Q2SP1*G2SP2> 0087 M13-M13-(B. *S*MH2)«(G1SP1*01SP2*G2SP2+C1SP1«G2SP1* 0088 C 02SP2+C1SP2*«2*G2SP1+01SP2*G2SP1**2>+16. »S*<2. *HSP2**2*01SP1* 0089  G2SP1-HSP2*HSP1*G1SP1*G2SP2-HSP2*HSP1*G1SP2»G2SP1+HSP2*G1SH* 0090 C ClSP2*C2SPl+HSP2*ClSH*C2SPl**2+2.*HSP1**2*C1SP2*C2SP2+HSP1*G2SH* 0091  01SP2«*2+HSPl*02SH*01SP2*G2SPl)+32.*P1SP2*<C1SH**2*G2SP1*02SP2+ 0092 CGISH*C2SH*01SP1*02SP2+01SH*G2SH*C1SP2*G2SP1*C2SH**2*C1SP1»C1SP2) 0093 C +32. »MK2*(-G1SH**2*G2SP1*C2SP2-01SH*C2SH*C1SP1*G2SP2-G1SH*C2SH* 0094  ClSP2*G2SPl-4. *QlSH»ClSPl*C2SPl*C2SP2-4. *C1SH*C1SP2*Q2SP1*C2SP2-0095 C G2SH**2»01SPl*ClSP2-4. *C2SH*GlSPl*GlSP2*G2SPl-4. *C2SH*G1SP1* 0096  GlSP2«02SP2-2. *C19P1»»2#C2SP2**2-12. #019Pl*ClSP2*02SPl*C2SP2-2. • 0097 C G1SP2»*2*C2SP1»»2>+16. *MH2»<GlSPl**2*G2SP2**2+6. *G1SP1*01SP2* 009B  C2SPl*C2SP2+GlSP2*«2*C2SPl*«2>-64.*<HSP2«C1SH»C1SP1*C2SP1*C2SP2+ 0099 C HSP2*G2SH*G1SP1*C1SP2*02SP1+HSP1*01SH*G1SP2*C2SP1*02SP2+HSP1• 0100  G2SH*G1SP1*01SP2*02SP2> 0101 M13«M13/S**2 0102 M14 -8. *S**2*P1BP2**2*MH2 0103 C +32. *S**2*P1SP2*MK2**2+16. *S**2*P1SP2*MK2«< 0104  HSP2+HSP1-2.»GlSH-01SPl-01SP2>+8.*S**2»PlSP2*01SH**2+4. *S**2*MK2 0105 C *MH2*(01SPl+ClSP2)+8. *B**2*MK2*(-HBP2*G1SH+HSP2*C1SP1-HSP2*G1SP2 0106  -HSP1*Q1SH-HSP1*01SP1+HSP1*G1SP2)+B. *S**2*MH2*0lBPl*QlSP2-<8. *S 0107 C **2*GlSH)*<HSP2#GlSPl+HSPl*GlSP2>-32.»S*PlSP2»*2*ClSH*02SH+32. »S 0108  *P1SP2»MK2»(01SH*02SH+01SH*02SP1+01SH*02SP2+02SH*01SP1+02SH» 0109 C C1SP2+2. #01SPl»Q2SP2+2. •01SP2*02SPl>-<32. *S*P1SP2*MH2)*(01SP1* 0110  G2SP2+01SP2*G2SP1)+16. *8*P1SP2*(HSP2*G19H*G2SP1+HSP2*C2SH*01SP1 + 0111 C HSP1*01SH*C2SP2+HSP1*C2SH*01SP2-01SH**2*02SP1-G1SH**2»02SP2)-( 0112  64. *S*MK2**2)*(01SP1*02SP2+G1SP2*02SP1) 0113 M14-M14+16. *S*MK2*(-2. *HSP2*G1SP1 0114 C *02SP2-2. *HSP2*GlSP2*02SPl-2.•HSPl*01SPl»C2SP2-2. •HSP1»C1SP2* 1 30 0115 C 02SPl+QlSH**2*02SPl+01SH»«2»02SP2+GlSH«©lSPl»©2SPl+3.*01SH#01SP1 0116  •C2SP2+3. *G1SH*G1SP2*G2SP1+C1SH*C1SP2»G2SP2+C2SH*01SP1»»2-2. • 0117 C G2SH»ClSPl»GlSP2+02SH*GlSP2*»2+2. *QlSPl»»2*02SP2+2. »Q1SP1»G1SP2» OUS  C2SP1+2. *QlSPl*01SP2*02BP2+2. *G1SP2*«2*02SP 1)-(16. *9*MH2*01SPi» 0119 C 01SP2)*<G2SP1+02SP2)+16. *S*Q1SH»<HSP2»01SP1»G2SP1+HSP2«01SP1» 0120  02SP2+HSPl«01SP2»02SPl+HSPl«GlSP2*G2SP2)+64. *P1SP2*01SH*Q2SH*< 0121 C Q1SP1*C2SP2+G1SP2»Q2SP1>+64. *MK2»<-G1SH»G2SH*G1SP1*02SP2-G1SH» 0122  G2SH»C1SP2*G2SP1-G1SH*G1SP1*G2SP1»02SP2-01SH«01SP1*02SP2**2-01SH 0123 C *01SP2*02SP1*#2-C1SH«Q1SP 2«02SP1*G2SP2-G2SH*01SP1*«2«G2SP2-G2SH* 0124  ClSPl*GlSP2*02SPl-02SH«GlSPl*ClSP2*G2SP2-C2SH«G18P2««2«G2SPl-2. « 0125 C 01SPl**2»G2SP2*«2-4. •GlSPl»GlSP2»G2SPl*02SP2-2. *01SP2»»2»02SP1** 0126  2)+32 *MH2*<ClSPl««2»C2SP2«*2+2. •C1SP1«C1SP2*C2SPI*C2SP2+G1SP2** 0127 C 2*G2SPl««2)-32.*(HSP2*G1SH«01SP1*G26P1*02€P2+HSP2«01SH«G1SP2* 0128  G2SP1##2+HSP2*02SH#01SP1»»2»G2SP2+HSP2»G2SH»GISP1*01SP2*02SP1+ 0129 C HSP1»01SH»01SP1*C2SP2»»2+HSP1*01SH*01SP2*G2SP1*G2SP2+HSP1»C2SH» 0130  G1SP1#01SP2»C2SP2+HSP1*02SH»C1SP2»«2*G2SP1> 0131 M14«M14/B*»2 0132 M15 —8. »S»*3#P1SP2*MK2 0133 C +4. »S*«3»PlSP2*QlSH-4. *S»«3«MK2*HSPl-4. »S»*3* 0134  HSP1*C1SP2-16. •S»»2«PlSP2«*2«MK2+8. •S**2*P1SP2**2*G1SH+16 *S**2* 0135 C PlSP2*MK2**2+8. •S**2»P1SP2*MK2»(HSP2-HSP1-G1SH+G2SH+01SP1-G1SP2+ 0136  2. •02SP1+2. *02SP2)+8. *8»»2*P1SP2*<HBP2*02SP1-HSP1»Q1SP2-01SH# 0137 C 01SPl-ClSH»02SPl-01SH*02SP2>-<8.•S**2*MK2**2>*(G18H+G1SP1+G1SP2> 0138  +4. •S*»2*nK2#(-HSP2*01SPl+HSP2»ClSP2+HSPl*GlSPl+HSPl*GlSP2+2. » 0139 C GlSH»C2SPl+2. *02SH#C18Pl+4.«GlSPl»02SP2+4. •01SP2*02SP1)+8. »S**2* 0140  (HSP1*C1SP1*C1SP2+HSP1*C1SP2*G2SP1+HSP1*01SP2*C2SP2-01SH»C1SP1* 0141 C C2SP2+G2SH*01SPl*ClSP2>-<16.«S«P1SP2*«2)«<G1SH«02SP1+G2SH*01SP1> 0142  +16. •S«PlSP2*MK2»<GlSH*C2SPl+G2SH*QlSPl-4. #G1SP1*C2SP1+4. •01SP1* 0143 C G2SP2+4. »G1SP2»C2SP1> + 16. »S»PlSP2»C-2. •HSP2*G1SP1*G2SP1+HSP1* 0144  G1SP1»G2SP2+HSP1»C1SP2«C2SP1+G1SH*C1SP1«02SP1-01SH«01SP1«G2SP2-0145 C GlSH*G1SP2»G2SP1+G1SH*G2SP1*G2SP2+G2SH«0ISP 1«G1SP2-02SH«G1SP2* 0146  G2SP1) 0147 M15-M15-<32. *S«MK2««2)«<G1SP1»G2SP2+G1SP2*G28P1)+B. »S»MK2«<-2. • 0148 C HSP2»GlSPl*G2SP2-2. *HSP2«GlSP2»G2SPl+2. •HSP1*G1SPl»02SP2+2. *HSP1 0149  #GlSP2*G2SPl+3. *GlSH*GlSPl*G2SPl+3. •G1SH*G1SP1»G28P2+G1SH«G1SP2* 0150 C G2SPl+01SH*GlSP2*G2SP2-G2SH«GlSPl»«2-2 *G2SH*GlSPl«GlSP2-2. *G2SH 0151  •01SPl»G2SP2-G2SH*01SP2*«2-2. *G2SH*GlSP2*G2SPl+4. *G1SP1*«2*G2SP1 0152 C -2. *GlSPl**2*G2SP2+2. •01SPl»01SP2*Q2SPl+2. •GlSPl*GlSP2*G2SP2-4. • 0153  GlSPl*G2SPl»G2SP2-4. »GlSPl*G2SP2**2+2. »G1SP2*«2»G2SP1-4. «G1SP2* 0154 C 02SPl»*2-4. •01SP2«02SP1*02SP2> + 16. •8«<HSP2*01SP1»*2»02SP1-HSP2« 0155  GISP1*C1SP2»C2SP1-HSP2*C1SP1*C2SP1»C2SP2-HSP2*01SP2*G2SP1*»2-0156 C HSP1»G1SP1»01SP2*C2SP1+HSP1»G1SP2»»2*02SP1+01SH»G1SP1•»2*02SP2+ 0157 CGISH*G1SP1»G2SP1»G2SP2+01SH*C1SP1•G2SP2**2-G2SH»G1SP1**2«G1SP2-0158 C G2SH*GlSPl»01SP2*G2SPl-02SH*ClSPl*01SP2*G2SP2>+32. *P1SP2«(G1SH* 0159  C1SP1»G2SP1*02SP2+G1SH»01SP2*02SP1**2+Q2SH*C1SPH»«2*©2SP2+Q2SH* 0160 C 01SP1*G18P2*02SP1> 0161 Ml5-M15+32.*MK2*(-G1SH*01SP1*G2SP1*02SP2-01SH*G1SP2* 0162 C 02SPl*»2-02SH«ClSPl»«2*G2SP2-G2SH*GlSPl»01SP2*02SPl+4. *01SP1«»2» 0163  02SPl»C2SP2-2.«01SPl*»2*02SP2«»2+4. •01SPl*018P2*028Pl**2-4. • 0164 C ClSPl«ClSP2»C2SPl*C2SP2-2. •C1SP2»»2»G2SP1»«2)*32. »<2 •HSP2»C1SP1 0165  ••2»G2SPl*02SP2+2. »HSP2»01SP1«01SP2*02SP1»«2-HSP1»G1SP1*»2#02SP2 0166 C »«2-2. •HSPl«OlSPl*01SP2»028Pl*02SP2-HSPl*G18P2»»2*02SPl»»2-2. * 0167  01SH»01SP1»*2*02SP1•02SP2-01SH»C1SP1»02SP1#02SP2«»2-G1SH«G1SP2« 0168 C 02SPl*«2»G2SP2+2. •G2SH*G1SP1»*2*G1SP2»02SP1+G2SH*G1SP1«01SP2* 0169  G2SP1»G2SP2+G2SH*G18P2»«2*02SP1*»2) 0170 M15-M15/S»»2 0171 M16 —8. «S»«3*PISP2»MK2 131 0172 C +2. *S»*3»P1SP2*MH2-B. •S#»3»t1K2»HSPl-4. *S»#3» 0173  HSP2«HSP1-M6. »S**2«PlSP2«MK2*«2-4. *S»«2»P1SP2*MK2»MH2+16. «S»»2« 0174 C P1SP2«MK2»<C1SP1+02SP1>-<4.«S«»2*P1SP2»MH2)»<01SP1«-02SP1 >-16. «S 0175 C •*2*MK2**3-4. •S»»2«MK2«*2«MH2+16. •S*«2»MK2«-»2« <-H8P2+HSP1)-M3. *S 0176 C »*2*MK2*(HSP2*H8Pl+2. *HSPl»01SPl+2. »HSPl*02SPl+2. •018H*C2SPl+2. • 0177  C2SH*ClSPl+2. •C1BPl»C2SP2+2. *G1SP2»G2SP1)-(4. •S»*2*HH2)»<01SP1* 0178 C 02SP2+01SP2*02SP1)+8. •S««2»HSP2»<HSP1«C1SP1+HSP1*C2SP1+01SH* 0179 C G2SPl+G2SH*ClSPl)-64. *S*P1SP2#MK2*G1SP1*C2SP1+16. •S»P1SP2«MH2« 0180  GlSPl*G2SPl+64. •S*MK2*«2*(-Q1SH»G2SP1-Q2SH*G1SP1+01SP1»G2SP1-0181 C 01SP1•C2SP2-01SP2*G2SP1) +16. *5*MK2*MH2»(G1SP1*G2SP1•©1SP1«C2SP2+ 0182  C1SP2*G2SP1) 0183 M16-M16+32. *S*MK2«(-HSP2*ClSH*G2SPl-HSP2*G2SH*01SPl+2. »HSP2 0184 C »C1SP1«G2SP1-2. *HSP1*G1SP1«G2SP1-C1SH*C1SP1*G2SP1-01SH*C2SP1*»2-0185 C G2SH*C1SP1••2-G2SH*G1SP1*G2SP1-01SP1••2*G2SP2-G1SP1»01SP2»G2SP1-0186  01SP1*C2SP1*C2SP2-G1SP2*C2SP1*»2)+B. «S#MH2«(C1SP1»«2»C2SP2+G1SP1 0187 C *G1SP2»G2SP1+Q1SP1«Q2SP1*Q2SP2+G1SP2»G2SP1*»2>+16. *S*HSP2»<-2. « 0188  HSP1*G1SP1*G2SP1-01SH*G1SP1*G2SP1-01SH»C2SP1**2-G2SH*G1SP1»»2-0189 C G2SH«G1SP1*G2SP1>+256.•HK2»C1SP1«G2SPHMG1SH«G2SP1+G2SH»C1SP1+ 0190 C 01SPl*C2SP2+01SP2»G2SPl)-(64. *HH2«G1SP1»62SP1)»(G1SP1»02SP2+ 0191 C Q1SP2»G2SP1)+12B. *HSP2»01SP1«02SP1«<G1SH»C2SP1+02SH«C1SP1) 0192 M16»M16/S«»2 0193 M17 -4. #S**2»P1SP2»MK2 0194 C -S««2»PlSP2*f1H2+4. *S»«2*MK2#*2+S*«2*MK2«MH2+4. • 0195  S»*2»MK2»<HSP2+HSP1>+2.•S»»2*HSP2*HSP1+8. *S*P1SP2»MK2*(G1SH-C2SH 0196 C +C1SP1-02SP1>+2. »S«P1SP2*MH2»<-G1SP1+02SP1>+4. *S*P1SP2»Q1SH»<-0197  GlSH+G2SH)+8. »S«MK2»«2«(01SP2-G2SP2)+2. *S»MK2«MH2*(-G1SP2+C2SP2) 0198 C +4. «S«MK2»(HSP1»G1SH-HSP1*02SH+2.«HSP1»G1SP1-2. «HSP1»02SP1+01SH 0199  ••2-ClSH*G2SH+2. »ClSH#01SPl+2.•ClSH*01SP2-2. »01SH»02SPl-2. «G2SH* 0200 C G1SP2+4. *0ISP 1*01SP2-2. *01SP1*G2SP2-2.»G1SP2*G2SP1)+2. •S*MH2«(-0201  2. »G1SP1»01SP2+G1SP1«G2SP2+Q1SP2*G2SP1)+4. »S«(HSP2*HSP1*C1SP1-0202 C HSP2»HSP1"028?1+HSP2»G1SH*01SP1-HSP2»C1SH»G2SP1+HSP1*01SH«01SP2-0203  HSP1»G2SH*C1SP2>+B. »P1SP2«(G1SH**2*G2SP1+G1SH*G2SH*G1SP1-G1SH* 0204 C G2SH*G2SP1-02SH*«2*G1SP1> 0205 M17-M17+B. »MR2#<-C1SH»«2»G2SP1-01SH*02SH*C1SP1 0206 C +01SH«G2SH»C2SPl-2. •GlSH»GlSPl»G2SPl-2. •GlSH*GlSPl*G2SP2-2. «G1SH 0207  •GlSP2*G2SPl+2. *G1SH*G2SP1**2+G2SH**2*G1SP1-2. »C2SH*GlSPl««2+2. • 0208 C G2SH*01SPl«C2SPl+2. »G2SH*GlSPl*C2SP2+2. *C2SH*ClSP2«G2SPl-2. » 0209  01SPl»*2#02SP2-6. •GlSPl*01SP2*C2SPl+6. «ClSPl»C2SPl*02SP2+2. * 0210 C QlSP2*02SPl»«2>+4. »MH2*(G1SPl«*2*G2SP2+3. •GlSPl*GlSP2*G2SPl-3. • 0211  01SP1»02SP1*02SP2-01SP2»G2SP1**2)+B. *(-HSP2»01SH*01SPl»02SPl + 0212 C HSP2»C1SH«G2SP1*«2-HSP2»G2SH»G1SP1»*2+HSP2*G2SH*01SP1«02SP1-HSP1 0213  »01SH»G1SP1»02SP2-HSP1»C1SH»G1SP2*G2SP1+HSP1•C2SH»G1SP1«C2SP 2+ 0214 C HSP1*G2SH*G1SP2»G2SP1) 0215 M17-M17/S 0216 M1B -(4.»S**2*P1SP2»MK2 0217 C -S#«2«PlSP2«MH2+4. #S«*2«MK2»«2+S»»2«MK2«MH2*4. • 021B  S»«2*MK2»(HSP2+H8P1)+2. •S*»2*HSP2»HSP1+B. •S«P1SP2*«K2*(01BP1-0219 C G2SP1)+2. •S»PlSP2*MH2»(-GlSPl+G2SPl>+8.•S»MK2*»2»CC1SH-C2SH+ 0220  01SP2-G2SP2)+2. *S*MK2*MH2#<-01SP2+02SP2)+4. *S»MK2«<HSP2»C1SH-0221 C HSP2»G2SH+2. *HSPl«GlSPl-2. •HSPl*G2SPl+4. •GlSH»01SPl-2. «G1SH» 0222  02SP1-2. «02SH*01SPl+4. •01SPl«01SP2-2. •01SPl*C2SP2-2. •01BP2*G2SP1 0223 C >+2. »S»MH2»(-2. »G1SP1*C1SP2+G1SP1»G29P2+01SP2»02SP1)+4. •S*HSP2*< 0224  HSPl»ClSPl-HSPl*C2SPl+2. •01SH»01BP1-01SH*Q2SP1-02SH#C1SP1>+16. # 0225 C MK2»(-3. #01SH«01SPl*G2SPl+01SH*G2SPl**2-02SH»GlSPl«»2+3. *G2SH» 0226  ClSPl«G2SPl-GlSPl»*2»02SP2-3.»01SPl#01SP2»02SPl«-3. •01SP1402SP1* 0227 C 02SP2+GlSP2»C2SPl»«2)+4. •MH2»(01SP1»»2»G2SP2*3. »01SP1«G1SP2» 0228  G2SP1-3. «019P1«G2SP1«G2SP2-01SP2*G28P1**2)-H3. •HSP2»(-3. •C1SH* 1 32 0229 C G1SP1»C2SP1+C1SH*G2SP1**2-C2SH*G1BP 1**2+3. *G2BH*Q1BP1*02SP1>>/S 0230 M22 «<B. *S**3»HK2**2 0231 C +8. *S*«3*MK2*<ClSPl+Q2SP2>+8. *S**3*Q1SP1*C2SP2+16. 0232 C «S»*2*P1SP2*MK2**2+16.*S**2*P1SP2*MK2* < 01SP1+02SP2) +16. *S**2* 0233  P1SP2*01SP1*02SP2-16.*S**2*I1K2**3-<16. *S**2*MK2**2)*<01SP2+02SP1 0234 C > + 16. *S**2*MK2*(-QlSPl*01SP2-2.*01SPl»C2SPl-2. *C1BP2*C2SP2-C2SP1 0235  *02SP2)-<32. *S**2*G1SP1*02SP2)*<G1SP2+028P1)-<64. *S*P1SP2*MK2>*< 0236 C 01SPl*Q2SPl+01SP2*C2SP2)-<64. *S*P1SP2*C1SP1*C2SP2)«<C1SP2+G2SP1) 0237 C +64. *S*MK2**2*<01SPl*02SPl+QlSP2*G2SP2)+32. *S*MK2*<-0ISP 1**2* 0238  G2SP2+GlSPl*GlSP2*G2SPl+2. *GlSPl*G2SPl**2-01SPl*02SP2**2+2. * 0239 C 01SP2**2»G2SP2+01SP2*02SPl*G2SP2>+32. *S*CISP1*02SP2*<-Q1SP1» 0240  02SP2+2. *ClSP2**2+3. »C1SP2*C2SP1+2. »C2SP1**2)+256. *P1SP2*C1SP1* 0241 C GlSP2*Q2SPl»Q2SP2-256. *HK2»C1SP1»01SP2»C2SP1»G2SP2+128. *G1SP1* 0242  G2SP2*< 01SP1*G1SP2*02SP2+01SP1»02SP1*02SP2-01SP2**2*C2SP1-01SP2* 0243 C G2SP1*«2))/8**2 0244 M23 -8.»S**3*MK2**2 0245 C +4. »S**3*f1K2*(HSPl+2. *02SP2)+4. *S**3*HSP1*Q2SP2+ 0246  16. «S**2*P1SP2*MK2««2+16.*S**2*P1SP2*MK2*< HSP1+02SP2> +16. *S**2» 0247 C P1SP2*HSP1*Q2SP2-16.*S**2*MK2**3+B. *S**2»MK2**2*<-HSP2-HSP1-01SH 0248  +02SH-2. »GISP1-2. *01SP2)+B. *S**2*MK2*<-HSP2*02SP2-HSP1*C1SP2-0249 C HSPl*Q2SPl-01SH*C2SPl+02SH«QlSPl+Q2SH*C2SP2-4. *©1SP2*Q2SP2)+B. »S 0250 C »»2*G2SP2*(-2. *HSP1»01SP2-HSP1*C2SP1-G1SH»02SP1+02SH»G1SP1> + 16. * 0251  S*PlSP2*MK2*<-01SH*Q2SPl-02SH»018Pl-4. »01SP2»G2SP2)+ 16. »S»P1SP2» 0252 C 02SP2*(-4. »HSP1»01SP2-G1SH»02SP1-02SH*G18P1) + 16. *S*MK2**2*<01SH* 0253  02SPl+02SH»01SPl+4. *016Pl*02SPl-2. •01SPl*G2SP2-2. »©lSP2*C2SPl+4. 0254 C *©1SP2*C2SP2)+16. *S»MK2*<2. *HSP2*01SP1*02SP1+2. *HSP2*C1SP2*©2SP2 0255  -HSPl»QlSPl»Q2SP2-HSPl*01SP2*©2SPl+2. »HSP1»G1SP2»G2SP2+G1SH* 0256 C 01SP2*02SPl+2. »C1SH*C1SP2*C2SP2+C1SH*C2SP1**2+C1BH*C2SP2»*2-G2SH 0257 C *01SPl*GlSP2-G2SH*GlSPl*G2SPl-G2SH*ClSP2*G2SP2+4. *01SP1*C1SP2* 0258  G2SP2-2. *QlSPl*C2SP2**2+4. *ClSP2*»2*C2SP2-2. *C1SP2*C2SP1»G2SP2)+ 0259 C 16.»S*G2SP2»<2.»HSP2»C1SP1»C2SP1-2.*HSPl*GlSPl*G2SP2+2. *HSP1* 0260  ClSP2**2+2. *G1SH*C1SP2*C2SP1+G1SH*C2SP1**2-2. *G2SH*G1SP1*G1SP2-0261 C G2SH*G1SP1*G2SP1) 0262 M23-M23+64. *P1SP2*01SP2*02SP2*(01SH*G2SP1+02SH*C1BP1> + 0263 C 64. *HK2*01SP2*G28P2*(-GlSH*G2SPl-G2SH*018Pl-4. *G18Pl»G28Pl+2. * 0264  GlSPl*G2SP2+2. *G1SP2»G2SP1)+32.*G2SP2*(-4. *HSP2*G1SP1*G1SP2* 0265 C 02SP1+2. *HSPl»G18Pl*01SP2*G2SP2+2. *HSP1*01SP2**2*G2SP1-G1SH* 0266  GlSPl*02SPl*G2SP2-2. *01SH*G1SP2**2*G2SP1-G1SH*G1SP2*G2SP1**2+ 0267 C G2SH*GlSPl**2*02SP2+2. *02SH*G1SP1*G1SP2**2+028H*01SP1*G1SP2* 0268  G2SP1) 0269 M23»M23/B**2 0270 M24 —B.*S*«3*P1SP2*HK2 0271 C +4. *S**3*PlSP2*01SH-4. *S**3*MK2*HSP2-4. *S**3* 0272  H8P2«G1SP1-16. *S**2«P1SP2**2*MK2+B. *S**2*P1SP2**2*018H+16. *S**2« 0273 C PlSP2»MK2**2+8. *S**2*P1SP2*MK2*(-HSP2+HSP1-G1SH+C2SH-01SP1+01SP2 0274  +2. *02SPl+2.*G2SP2)+8. *S**2*P1SP2*(-HSP2*C1SP1+HSP1*G2SP2-G1SH* 0275 C 01SP2-01SH*02SP1-01SH*02SP2)-(B.»S**2*MK2»*2>*(01SH+G1SP1+C1SP2> 0276  +4. *S**2*MK2*(HSP2*GlSPl+HSP2*ClSP2+HSPl*01SPl-HSPl*GlSP2+2. * 0277 C GlSH»G2SP2+2. *G2SH»01SP2+4. *01SPl*02SP2+4. *G1SP2*02SP1)+8. *S**2* 0278  (HSP2*G1SP1*G1SP2+HSP2*G1SP1*02SP1+HSP2*01SP1*G2SP2-G1SH*G1SP2* 0279 C G2SP1+02SH*01SP1*01SP2>-(16.*S*P19P2**2)*(018H»G2SP2+02SH*G1SP2> 0280  +16. »S*P18P2*HK2*(018H*02SP2+G2SH*GlSP2+4. *01SPl»02SP2+4. *G1SP2* 0281 C G2SP1-4. *G1SP2*02SP2) + 16. *S*P1SP2*(HBP2*018P1*02SP2+HSP2*G1SP2* 0282  02SP1-2. *HSP1*G1SP2*G2SP2-G1SH*01SP1*G2SP2-G1SH*018P2»G2SP1+G1SH 0283 C «G1SP2*02SP2+01SH*02SP1*G2SP2+02SH*G1SP1*01SP2-02SH*01SP1*G2SP2 > 0284 C -<32. *S*HK2**2)*(G1SP1*02SP2+01SP2*02SP1> 0285 M24-M24+B. *S*t1K2*(2. *HSP2* 1 33 0286 C QlSPl»02SP2+2. *HSP2#01SP2*02SPl-2. •HSPl»01SPl»02SP2-2. *HSP1» 0287 C 01SP2*02SP1+015H*01SP1*02SP1+01SH»C1SP1*G2SP2+3. *C1SH*C1SP2* 0288  02SP1+3. •01SH*01SP2«02SP2-C2SH»GlSPl»»2-2. •C2SH»01SPl»ClSP2-2. • 0289 C 02SH»QlSPl»C2SP2-02SH»ClSP2»«2-2.»02SH»Q1SP2*C2SP1+2. *01SP1**2* 0290  02SP2+2. *0tSPl»0lSP2»02SPH-2.•0lSPl»QlSP2*C2SP2-4. *018P1«G2SP1* 0291 C 02SP2-4. »01SPl*C2SP2»»2-2. •GlSP2**2»G2SPl+4. •GlSP2*»2*C2SP2-4. * 0292  QlSP2»C2SPl»*2-4. »01SP2»G2SP1»02SP2) + 16. •S*(HSP2*G1SP1»*2*02SP2-0293 C HSP2»C1SP1*C1SP2*C2SP2-HSP1*0ISP1*01SP2»C2SP2-HSP1*C1SP1*02SP2*» 0294 C 2+HSP1#01SP2»*2*C2SP2-HSP1*C1SP2«C2SP1•G2SP2+C1SH*C1SP2*«2*C2SP1 0295  +01SH*C1SP2*C2SP1••2+01SH»G1SP2»02SP1•02SP2-02SH*01SP1»Q1SP2««2-0296 C ©2SH«01SPl»ClSP2*G2SPl-G2SH»01SPl»QlSP2»02SP2>+32. •P18P2«(G1SH« 0297  01SP1«G2SP2««2+G1SH«G1SP2*C2SP1•02SP2+C2SH«G1SP1*01SP2*G2SP2+ 0298 C G2SH»CIBP2**2»C2SP1> 0299 M24-M24+32. «MK2«C-01SH*01SP1»02SP2*»2-01SH*016P2* 0300 C 02SPl*G2SP2~G2SH*GlSPl*GlSP2*G2SP2-G2SH*GlSP2*»2*G2SPl-2. •01SP1 0301  ••2«G2SP2»«2-4. •GlSPl»01SP2»02SPl»02SP2+4. •01SP1»01SP2»02SP2»»2-0302 C 2. #01SP2*»2«C2SPl»«2+4. *ClSP2»*2*G2SPl»G2SP2>+32. •(-HSP2 0303 C *Q1SP1**2 0304  «02SP2»«2-2. *HSP2«Q1SP1*01SP2*02SP1*G2SP2-HSP2*01SP2**2*C2SP1**2 0305 C +2. *HSP1*01SP1*G1SP2«C2SP2*«2+2. *HSP1»G1SP2*»2»02SP1•C2SP2-01SH» 0306  GlSPl*G2SPl»G2SP2»*2-2. •C1SH*G1SP2»*2*G2SP1*G2SP2-01SH*G1SP2* 0307 C G2SPl**2«02SP2+02SH*G18Pl«*2*G2SP2»*2+2. •02SH*01SP1*01SP2*«2* 0308  02SP2+02SH*G1SP1*G1SP2*02SP1«G2SP2) 0309 t124=M24/S*»2 0310 M25 -2. *S»«4*P18P2+8. »S»*3*P1SP2»»2 0311 C -B. *S*»3»PlSP2»MK2-(4. *S*»3*P1SP2) 0312  •<01SP1+G1SP2+G2SP1+02SP2>-B.»S»«3*MK2«»2-<4. •S»*3)»(G1SP1*G2SP2 0313 C +01SP2*02SP1)+16. »S»»2«P1SP2»»3-16. •S**2*P1SP2*«2«MK2-(B. »S*»2» 0314  P1SP2*«2)*<01SP1+G1SP2+G2SP1+G2SP2)+B. •S«»2«P1SP2«MK2»(G1SP1+ 0315 C C1SP2+C2SP1+G2SP2)+B.•S»«2*P1SP2*(2.*GlSPl*GlSP2+GlSPl*G2SPl-3. • 0316  GlSPl*G2SP2-3.•GlSP2*G2SPl+01SP2*02SP2+2. »C2SPl»02SP2>+8. »S*»2» 0317 C MK2«(3. »ClSPl*C2SPl+2. •GlSPl»02SP2+2. •GlSP2*G2SPl+3. »G1SP2»C2SP2 0318  )+8. »S««2»<C1SP1«*2»02SP2+C1SP1«01SP2»C2SP1+C1SP1*G1SP2»02SP2+ 0319 C G1SP1*G2SP1•G2SP2+C1SP1»C2SP2»«2+C1SP2«»2*02SP1+C1SP2*G2SP1**2+ 0320  01SP2»02SPl*02SP2>-<64. *S*P1SP2*»2>*(C1SP1»G2SP2+C1SP2*G2SP1)• 0321 C 64. •S»PlSP2*MK2»<ClSPl»C2SP2+ClSP2*C2SPl>+32. #S»P1SP2*(GISP 1**2* 0322  G2SP2+G1SP1*02SP2**2+G1SP2«*2*G2SP1+G1SP2«G2SP1**2>-<16. •S»MK2>* 0323 C < 01SP1##2«02SP2+G1SP1*C1SP2«G2SP1+G1SP1»C1SP2*C2SP2+C1SP1*02SP1• 0324  G2SP2+G1SP1«G2SP2**2+G1SP2»»2»C2SP1+G1SP2»G2SP1••2+01SP2*C2SP1• 0325 C G2SP2) 0326 M23-M23+16. »S» <-2. #G1SP1*»2»C1SP2*G2SP2-G1SP1•*2»C2SP1*C2SP2+ 0327 C QlSPl»»2«C2SP2«»2-2.•G1SP1*G1SP2**2*G2SP1-G1SP1*G1SP2»G2SP1**2-0328  2. •01SPl*GlSP2*G2SPl*G2SP2-GlSPl*G18P2»G2SP2««2-2. •01SP1»C2SP1» 0329 C 02SP2»«2+GlSP2»»2»G2SPl»»2-GlSP2»*2»G2SPl»G2SP2-2. •C1SP2»G2SP1»» 0330  2*C2SP2>+64. #PlSP2»<01SPl«*2«02SP2»*2+2. «C1SP1»C1SP2*C2SP1»C2SP2 0331 C +01SP2»»2»C2SPl*»2>+64.•MK2#<-01SPl«*2«©2SP2«»2-2. »©1SPI«01SP2* 0332  C2SPl«02SP2-ClSP2»»2«Q2SPl»«2>+32. »<-GlSPl«»3«C2SP2««2+GlSPl««2« 0333 CGISP2»C2SP2**2+G1SP1»*2»G2SP1•C2SP2»»2-C1SP1•»2»G2SP2»«3+G1SP1• 0334 CGISP2««2*G2SP1••2-G18P2*«3*G2SP1••2-01SP2**2*G2SP1••3+01SP2»«2» 0335 C C2SP1»»2»02SP2> 0336 M23-M25/B«»2 0337 M26 —8. *S**3*P1SP2«MR2 0338 C +4. »S*»3»PlSP2»02SH-4. •S«»3»MK2»HSPl-4. *S*«3« 0339  HSP1«C2SP2-16. *S»*2«P1SP2«*2»MK2+B. •S»»2»P1SP2»»2»C2SH+16. »S»»2* 0340 C P1SP2»MK2*«2+B. *S»»2»P18P2»t1K2»(HSP2-HSPl+01SH-G2SH+2. •01SP1+2. • 0341  G1SP2+C2SP1-02SP2)+B. »6»»2»P1SP2«<HSP2*G1SP1-HSP1»02SP2-©2SH« 0342 C G1SP1-G2SH*G1SP2-G2SH*G2SP1)-(B.*S»«2*MK2»»2)»(02SH+C2SP1+G2SP2) 1 34 0343 C +4. »S*#2«MK2#<-HSP2#02SPl+HSP2#02SP2+HSPl#02SPl+HSPl#G2SP2+2. » 0344 C 01SH#02SPl+2. #02SH#01SP1+4. #GlSPl#C2SP2+4. •01SP2*02SP1>+8. #S#*2* 0345  <HSP1#01SP1#02SP2+HSP1#Q1SP2#G2SP2+HSP1#02SP1#02SP2+C1SH»C2SP1# 0346 C 02SP2-02SH#C1BP2*G2SP1)-(16.»S»P15P2*#2)•(01SH»02SP1+02SH*01SP1) 0347  +16. #S#PlSP2#MK2#<01SH#G2SPl+02SH#01SPl-4. #01SPl#C2SPl+4. #01SP1* 0348 C 02SP2+4. *01SP2#C2SP1) + 16.*S#PlSP2#<-2.#HSP2#01SP1*02SP1+HSP1» 0349  01SP1#Q2SP2+HSP1#QISP2#C2SP1-01SH*G1SP1*Q2SP2+01SH*©2SP1»02SP2+ 0350 C 02SH#C1SP1*C1SP2+C2SH«01SP1«02SP1-02SH#01SP1*G2SP2-02SH*01BP2* 0351  G2SP1) 0352 M26-M26-(32. •S#MK2**2>#(G1SP1#02SP2+G1SP2#G2SP1>+a •S»MK2«(-2. » 0353 C HSP2#ClSPl«02SP2-2. #HSP2«ClSP2#C2SPl+2. #HSPl*01SPl#Q2SP2+2. #HSP1 0354  *01SP2*02SPl-2. #GlSH#01SPl#G2SP2-2.#G1SH#C1SP2#G2SP1-01SH#G2SP1 0355 C «*2-2. #ClSH#C2SPl»02SP2-ClSH»02SP2##2+3. #Q2SH*C1SP1»C2SP1+C2SH# 0356 C 01SPl#C2SP2+3. #G2SH*01SP2»02SPl+C2SH#ClSP2*C2SP2-4. #C1SP1#»2* 0357  G2SP2-4. #GlSPl#ClSP2#G2SPl-4.#ClSPl#GlSP2*G2SP2+4. #01SP1#02SP1*» 0358 C 2+2 •GlSPl#02SPl»G2SP2+2.#01SPl#02SP2##2-4. #01SP2*#2#G2SPl-2. « 0359 C ClSP2»G2SPl##2+2. •C1SP2»G2SP1#C2SP2)+16. #S#<-HSP2*G1SP1##2«G2SP2 0360 C -HSP2«G1SP1#01SP2«G2SP1+HSP2*01SP1«02SP1##2-HSP2»G1SP1»G2SP1» 0361  G2SP2-HSP1#01SP1#C2SP1*G2SP2+HSP1#01SP1»C2SP2#»2-G1SH»01SP1• 0362 C 02SP1#02SP2-01SH*©1SP2»02SP1#G2SP2-G1SH#G2SP1##2*C2SP2+Q2SH» 0363  ©lSPl*QlSP2#©2SPl+C2SH*01SP2##2»02SPl+C2SH#ClSP2«02SPl##2>+32. * 0364 C P1SP2# < 01SH»01BP 1#02SP1#02SP2+01SH#01SP2*G28P1**2+02SH*G1SP1##2» 0365 C G2SP 2+G2SH*G1SP1*01SP 2»02SP1) 0366 M26-M26+32. #MK2# <-01SH*G1SP1#©2SP1#G2SP2-0367 C 01SH»01SP2»02SP1#«2-02SH#C1SP1•#2#G2SP2-C2SH«C1SP1»01SP2*C2SP1+ 036B  4. *GlSPl«#2#G2SPl#G2SP2-2. #ClSPl##2#G2SP2#*2+4. #01SP1#01SP2 0369 C *G2SP1 0370 C ##2-4. •01SPl#ClSP2#02SPl*02SP2-2. #GlSP2##2*02SPl#«2>+32. «(2. • 0371  HSP2#01SPl##2#C2SPl#G2SP2+2.»HSP2#C1SP1#G1SP2#G2SP1**2-HSP1# 0372 C GlSPl##2#G2SP2»#2-2.#HSP1#01SP1*G1SP2#G2SP1*©2SP2-HSPI#01SP2##2* 0373  e2SPl##2+eiSH»GlSPl»*2#G2SP2#*2+GlSH#GlSPl#GlSP2*G2SPl#G2SP2+2. » 0374 C 01SH«G1SP1*G2SP1••2#Q2SP2-02SH#C1SP1##2#01SP2#©2SP2-C2SH*C1SP1• 0375  ClSP2*#2#G2SPl-2. »G2SH#G1SP1#01SP2*C2SP1*#2) 0376 M26=M26/S##2 0377 M27 =4. *S*#2*P1SP2*MK2 0378 C -2. »S*#2#PlSP2#ClSH+4. #S#*2#MK2**2+2. #S##2#MK2# 0379  <HSP2+HSP1+01SH+2. »01SPl+2. #G2SP2)+2. #S##2#(HSP2*G1SPI+HSP1# 0380 C G2SP2>+4. #S#PlSP2##2#<-01SH+02SH>+4. #S#P15P2«MK2»<ClSH-02SH+2. * 0381  G1SP1-2. *01SP2-2. #G2SPl+2. *G2SP2)+4. #S#P1SP2#<HSP2*01SP1-HSP2* 0382 C Q2SP1-HSP1#G1SP2+HSP1#02SP2+01SH*01SP2+©1SH*G2SP1>+4. #S*MK2#(-0383 C 01SH*C2SPl-GlSH#C2SP2-02SH#ClSP2+©2SH#©2SP2-2. »©lSPl#C2SPl-2. * 0384  GlSPl#G2SP2-2. #GlSP2»C2SPl-2.#C1SP2#G2SP2)+4.»S*(-HSP2»G1SP1* 0385 C GlSP2-HSP2 #01SP1*02SP1-HSP1#01SP2*G2SP2-HSP1#©2SP1•C2SP2+C1SH* 0386  G1SP2#G2SP1-01SH#G2SP1#C2SP2-C2SH#G1SP1*G1SP2+G2SH#G1SP1#G2SP2)+ 0387 C 8. •PlSP2*<01SH#ClSPl#©2SP2+eiSH#eiSP2#©2SPl-02SH#ClSPl#02SP2 0388  -C2SH 0389 C •01SP2*02SP1) 0390 M27-M27+8. #MK2#<-©1SH#©18P1#02SP2-01SH#©1SP2*©2SP1+©2SH# 0391 C G1SP1#C2SP2+C2SH#G1SP2#C2SP1-2.»G1SP1##2#©2SP2-2. •©1SP1•©1SP2* 0392  G2SP1+2. #©1SPl#GlSP2#G2SP2+2.#G1SP1#Q2SPl#C2SP2-2. #018P1#©2SP2*» 0393 C 2+2. #ClSP2#«2#G2SPl+2. #01SP2#©2SPl##2-2. •01SP2*G2SP1#©2SP2)+B. •( 0394  -HSP2*01SP1••2#©2SP2-HSP2*01SP1•©!SP2*02SP1+HSP2»G1SP1•02SP1• 0395 C ©2SP2+HSP2*01SP2#02SP1*»2+HSP1*©1SP1#©1SP2«©2SP2-HSPI«01SP1« 0396  G2SP2*«2+HSP1#G1SP2»»2»G2SP1-HSP1*01SP2»G2SP1•G2SP2-01SH*0ISP2»* 0397 C 2*02SP1-C1SH#01SP2*02SPI##2+01SH#G1SP2#02SP1#02SP2+C1SH#G2SP1#*2 0398  «02SP2+G2SH#01SP1#01SP2##2+02SH*01SP1*G1SP2#G2SP1-02SH#G1SP1« 0399 C G1SP2#G2SP2-02SH#G1SP1#G2SP1*G2SP2) 1 35 0400 M27-M27/S 0401 M2B «4. #S*«2*P1SP2«HK2 0402 C -2. »S*«2»PlSP2«C2SH+4. •S««2»MK2«2+2. *S«*2«MK2* 0403 C (HSP2+HSP1+02SH+2. *01SPl+2. «C2SP2)+2. •S**2*(HSP2*01SP1+HSP1* 0404  G2SP2>+4. »S»PlSP2«»2#<01SH-G2SH)+4. »S*P1SP2*MK2»(-01SH+G2SH+2. • 0403 C Q1BP1-2. *ClSP2-2. »02SPl+2. »C2SP2)+4. #S»P1SP2»(HSP2*C1SP1-HSP2» 0406 C 02SP1-HSP1*C1SP2+HSP1•G2SP2+G2SH«01SP2+G2SH«G2SP1)+4. •S«MK2»( 0407  01SH»01SP1-C1SH*C2SP1-02SH*61SP1-C2SH#C1SP2-2. *01SP1*02SP1-2. * 0408 C 01SPl»02SP2-2. •01SP2»G2SPl-2. »Q1SP2»Q2SP2>+4. *S»<-HSP2»G1SP1» 0409 C 01SP2-HSP2*C18P1*C2SP1-HSP1*G1SP2*C2SP2-HSP1*C2SP1*C2SP2+C1SH* 0410 C01SP1*C2SP2-01SH»G2SP1#02SP2-02SH«01SP1•©1SP2+G2SH*G1SP 2«02SP1> + 0411 C B. *PlSP2*<-ClSH*01SPl»G2SP2-ClSH*019P2»02SPl+G2SH»01SPHfG2SP2+ 0412  G2SH*C1SP2»Q2SP1> 0413 M28-M28+8. *MK2#(G1SH*01SP1*Q2SP2+01SH*01SP2»02SP1-C2SH 0414 C *ClSPl»C2SP2-C2SH»GlSP2»02SPl-2. »ClSPl**2»G2SP2-2. *C1SP1»C1SP2» 0415  C2SP1+2. »01SPl«ClSP2«G2SP2+2. •GlSPl*G2SPl*G2SP2-2. »01SP1*G2SP2*« 0416 C 2+2. •GlSP2**2»G2SPl+2. *01SP2»G2SPl*»2-2. •G1SP2*02SP1»G2SP2)+8. •< 0417  -HSP2»GISP1*»2»G2SP2-HSP2*01SP1»G1SP2«G2SP1+HSP2*C1SP1»G2SP1* 0418 C C2SP2+HSP2«Q1SP2*02SP1*»2+HSP1*01SP1»G1SP2*C2SP2-HSP1»C1SP1• 0419  G2SP2««2+HSP1•©1SP2*»2»C2SP1-HSP1•©1SP2*G2SP1«G2SP2-C1SH*G1SP1* 0420 C 01SP2*02SP2-01SH*01SP1*02SP1#C2SP2+01SH#G1SP2«G2SP1*C2SP2+G1SH* 0421  028P1•*2*02SP2+02SH»Q1SP1»01SP2»»2+02SH*01SP1*G1SP2»02SP1-G2SH* 0422 C 01SP2**2*02SP1-G2SH*©1SP2*028P1**2> 0423 M28-M2B/S 0424 M33 -(-32. «S»*2*MK2««3 0425 C -8. «S*»2*MK2**2«MH2+32. *S»*2*MK2*«2*(-HSP1+©2SH+ 0426  ©2SP1)-8. *S»«2*MK2«MH2*C2SP1 + 16. *S«*2*MK2«(HSP1*©2SH+2. *C2SH* 0427 C C2SP2+2. •G2SPl«GSSP2)-8. •S**2«MH2*G2SP1*C2SP2+16. »S*»2*HSP1*G2SH 0428 C *G2SP2+64.»S*MK2**2»(-C1SH»C2SP2-C2SH«C1SP2-C1SP1»©2SP2-C1SP2* 0429  G2SP1+2. *G1SP2*G2SP2> + 16. *S«MK2*MH2*(G1SP1«G2SP2+G1SP2*G2SP1+2. • 0430 C GlSP2«C2SP2)+32. *8«MK2*<-HSPl«GlSH*C2SP2-HSPl*C2SH»01BP2+4. «HSP1 0431  »GlSP2»G2SP2-4. •G2SH»GlSP2*G2SP2-4.•01SP2«G2SPl*G2SP2>+32. »S«MH2 0432 C •GlSP2»G2SPl«G2SP2-64. •S*H5Pl*G2SH*GlSP2*C2SP2+236. *MK2»C1SP2« 0433  02SP2»(01SH*G2SP2+02SH«G1SP2+01SP1«G2SP2+G1SP2#028P1)-(64. •MH2* 0434 C G1SP2*G2SP2)*<01SP1*G2SP2+G1SP2*Q2SP1>+128. *HSP1*01SP2«02SP2»< 0433  C1SH*G2SP2+G2SH*G1SP2))/S»»2 0436 M34 —8. «9*«3*P1SP2*MK2 0437 C +2. •S*«3#PlSP2*HH2-8.•S#»3»MK2*HSP2-4. «S»«3» 0438  HSP2«HSP1 + 16. •S«*2«PlSP2*HK2*«2-4. #S*»2*P1SP2«MK2#MH2+16. «S*«2* 0439 C P18P2*MK2»<ClSP2+C2SP2>-(4. *S««2*P1SP2*MH2)»<G1SP2+C2SP2>-16. «S 0440  »«2»MK2*«3-4. «S«»2«MK2«*2*MH2+16. *B»#2*MK2«»2*<HSP2-HSP1)+8. *S»* 0441 C 2*MK2»CHSP2»HSPl+2.»HSP2»01SP2+2.»HSP2»G2SP2+2. *01SH#G2SP2+2.• 0442  C2SH*ClSP2+2. *019Pl»C2SP2+2. *G1SP2*C2SP1> 0443 M34-M34-(4. «S»»2*MH2)•(G1SP1• 0444 C Q2SP2+G1SP2»C2SP1>*8. *S#*2*HSP1*(HSP2*G1SP2+HSP2*02SP2+©1SH« 0443  C2SP2+02SH*0lSP2>-64. •8»P1SP2*MK2*C1SP2*G2SP2+16. •S«P1SP2»MH2* 0446 C 01SP2«G2SP2+64. •S*HK2#»2»<-01SH»Q2SP2-G2SH*C1SP2-G1SP1«C2SP2-0447  01SP2*Q2SP1+01SP2»G2SP2>+16. »S»MK2*MH2»<G1SP1»C29P2+G1SP2*G2SP1+ 0448 C G18P2«02SP2>+32. «S*HK2*(-2. *HSP2*01SP2*028P2-H8P1»G1SH«02SP2-0449  HSPl»C2SH«ClSP2+2. *HSP1*C1SP2»G2SP2-C1SH»C1SP2*C2SP2-C1SH«C2SP2 0450 C •*2-G2SH*01SP2**2-Q2SH«G1SP 2*C2SP2-Q1SP1*G1SP2*G2SP2-G1SP1*G2SP2 0451  •«2-01SP2«*2*02SPl-01SP2*02SPl»02SP2)+8. »S»MH2»<01SP1*01SP2» 0452 C 02SP2+C1SP1*©2SP2«*2+01SP2**2«G2SP1+C1SP2*Q2SP1*G2SP2> + 16. *S» 0453  HSPHM-2. »HSP2#C1SP2*02SP2-C1SH*01SP2»02SP2-C1SH*C2SP2**2-G2SH* 0454 C GlSP2»*2-C2SH»GlSP2»02SP2)+256. »MK2*C1SP2»02SP2»(01SH»G2SP2+02SH 0455  •C1SP2+G1SP1*G2SP2+G1SP2*C2SP1)-(64. •MH2»C1SP2*02SP2)»(C1SP1« 0456 C 02SP2+01SP2«02SP1)+128.•HSP1*G1SP2*02SP2*(01SH*G2SP2+02SH*01SP2) 136 0457 M34-M34 /S«*2 0458 M35 —8. »S*»3»P1SP2*MK2 0459 C +4. *S*»3*PlSP2«C2SH-4. »S*#3*MK2»HSP2-4. *B*«3» 0460  HSP2*02SP1-16. *S»*2»P1SP2»*2*MK2+B. »S»*2»P1SP2»*2*026H+16. #S*»2» 0461 C P1SP2*MK2«*2+B. •S»»2*PlSP2*MK2«<-HSP2+HSPl+GlSH-G2SH+2. »ClSPl+2. 0462 C *01SP2-C2SP1+C2SP2>+S. •S»«2«P1SP2*<-HSP2*G2SP1+HSP1«G1SP2-G2SH* 0463 C 01SPl-C2SH»GlSP2-G2SH»02SP2>-<8. *S*#2*MK2»*2>*<©2SH+02SP1+C2SP2) 0464  +4. *S»#2*MK2«(HSP2»02SPl+HSP2«Q2SP2+HSPl»Q2SPl-HSPl*G2SP2+2. * 0465 C ©lSH*Q2SP2+2. *C2SH*01SP2+4. •01SPl*©2SP2+4. *©1SP2»©2SP1)+B. *S**2* 0466  (HSP2*01SP1*02SP1+HSP2»Q1SP2«©2SP1+HSP2«C2SP1•C2SP2+01SH#02SP1* 0467 C ©2SP2-C2SH*©1SP1*C2SP2)-<16.•S»P1SP2**2)*(C1SH*C2SP2+C2SH#C1SP2) 0468 C +16. *S*PlSP2»MK2*C01SH*©2SP2+©2SH*ClSP2+4. »©lSPl*©2SP2+4. »01SP2* 0469  G2SP1-4. »01SP2#C2SP2) + 16. *S#P1SP2»<HSP2»©1SP1»02SP2+HSP2»©1SP2* 0470 C G2SP1-2. *HSP1*G1SP2*G2SP2-01SH»G1SP2#02SP1+G1SH*C2SP1*©2SP2+C2SH 0471  »C1SP1»01SP2-G2SH*G1SP1•C2SP2-G2SH»01SP2*G2SP1+02SH»©1SP2«G2SP2) 0472 C -(32.•S»MK2»»2)»(GISP1»G28P2+G1SP2»G2SP1) 0473 M35-M35+B. •S»MK2*<2.*HSP2» 0474 C 01SPl*02SP2+2. *HSP2»01SP2«G2SPl-2. #HSPl»GlSPl*02SP2-2. »HSP1* 0475  GlSP2»02SPl-2. »01SH*01SPl»02SP2-2.»C1SH*C1SP2»02SP1-01SH«C2SP1«* 0476 C 2-2. »01SH»02SPl*02SP2-01SH»02SP2«»2+02SH*01SPl»02SPl+3. «©2SH» 0477 C ©lSPl«©2SP2+©2SH*©lSP2*©2SPl+3.*©2SH»©lSP2*©2SP2-4. »G1SP1»«2« 0478 C 02SP2-4. •01SPl*ClSP2»62SPl-4.•01SPl«©lSP2»©2SP2+2. *G1SP1*G2SP1* 0479  G2SP2-2. »C1SPl*C2SP2»*2-4. *01SP2*«2»©2SP1+2. *©lSP2*C2SPl««2+2. * 0480 C GlSP2*Q2SPl»Q2SP2+4.*G1SP2»G2SP2»»2>+16.»S»(HSP2»C1SP2*G2SP1*»2-04B1  HSP2*01SP2*02SP1*02SP2-HSP1*6ISP 1*61SP2*C2SP2-HSP1»01SP2»*2» 0482 C C2SP1-HSP1•©1SP2*C2SP1*02SP2+HSP1*G1SP2*C2SP2**2-C1SH*C1SP1• 0483 C G2SP1»G2SP2-G1SH*C1SP2*G2SP1•C2SP2-C1SH*G2SP1•C2SP2»*2+G2SH* 0484  GlSPl»»2*C2SP2+G2SH#GlSPl»ClSP2»C2SP2+C2SH#GlSPl»C2SP2*»2>+32. • 04B5 C P1SP2« < C1SH»©1SP1*©2SP2**2+©1SH*C1SP2»©2SP1»©2SP2+C2SH»G1SP1• 0486  G1SP2*G2SP2+G2SH*G1SP2*«2*C2SP1) 0487 M35=M35+32. »MK2« (-C1SH«G1SP1#G2SP2»»2-0488 C C1SH«C1SP2»C2SP1*02SP2-C2SH«C1SP1#C1SP2*02SP2-©2SH*C1SP2*«2* 0489  C2SP1-2. »01SPl*«2»©2SP2**2-4.•01SPl*01SP2»©2SPl»C2SP2+4. »01SP1* 0490 C GlSP2*C2SP2**2-2. •GlSP2«*2«G2SPl*«2+4. *01SP2**2*G2SP1*02SP2)+32. 0491  #(-HSP2*01SPl«*2*02SP2**2-2.•HSP2*01SP1»01SP2*02SP1*02SP2-HSP2« 0492 C ©lSP2*«2*02SPl»«2+2. *HSPl*ClSPl*GlSP2*C2SP2»*2+2. •HSP1«C1SP2**2* 0493  G2SP1•C2SP2+01SH*C1SP1*01SP2*G2SP1*02SP2+G1SH*G1SP2**2*G2SP1**2+ 0494 C 2. *G1SH«G1SP2*02SP1*G2SP2*«2-C2SH«G1SP1**2*C1SP2*02SP2-02SH 0495  *©1SP1 0496 C »01SP2*«2»02SPl-2. •02SH*G1SP1*G1SP2»02SP2*«2) 0497 M35-M35/S**2 0498 M36 -8. »S*»2*P1SP2*«2»MH2 0499 C +32. »S**2»P1SP2«MK2*«2+16.»S«*2»P1SP2»HK2*( 0500  HSP2+HSP1-2. »G2SH-02SP1-02SP2)+8. »S*»2»PlSP2»02SH»»2+4. »S»*2»MK2 0501 C •MH2*<G2SP1+G2SP2)+B. »S*»2»MK2«<-HSP2»C2SH+HSP2*C2SPl-HSP2»C2SP2 0502  -HSP1*C2SH-HSP1*02SP1+HSP1«©2SP2)+B. •S*«2*MH2«G2SPl*G2SP2-<8. «S 0503 C **2#G2SH)•(HSP2*G2SP1+HSP1»02SP2)-32. »S*P1SP2»*2»01SH*02SH+32. *S 0504  »P1SP2*MK2»(G1SH*G2SH+G1SH*G2SP1+G1SH*G2SP2+G2SH»G1SP1+G2SH* 0505 C 01SP2+2. *01SPl*02SP2+2. *C1SP2*02SP1)-(32. #S*P1SP2»MH2)«(G18P1* 0506  G2SP2+01SP2*G2SP1) + 16. •S*P1SP2*(HSP2*G1SH*G2SP1+HSP2*G2SH*G1SP1 + 0507 C HSP1»C1SH«C2SP2+HSP1«C2SH*C1SP2-G2SH«*2*G1SP1-C2SH»«2»01SP2 >-( 0508  64. •S*HK2*«2)»(01SP1»02SP2+G1SP2«02SP1> + 16. »S*MK2»(-2. *HSP2 0509 C «G1SP1 0510  «C2SP2-2. *HSP2»GlSP2*02SPl-2.#HSPl»ClSPl*G2SP2-2. *HSP1*G1SP2« 0511 C G2SPl+01SH*02SPl**2-2. »01SH*02SP1»02SP2+018H*02SP2«*2+02SH««2« 0512  GlSPl+02SH*«2*01SP2+G2SH*01SPl*02SPl+3. »02SH»GlSPl»02SP2+3. »02SH 0513 C •GlSP2*G2SPl+G2SH*GlSP2»G2SP2+2.*GlSPl*02SPl»02SP2+2. »01SP1* 1 37 0314 C C2SP2«#2+2. *Q1SP2»G2SPl*«2+2.•C1SP2»C2SP1*G2SP2> 0313 M36-M36-(16. *S*MH2« 0316 C Q2SP1«02SP2)*(01SP1+01SP2)+16.•S*G2SH*(HSP2*G1SP1*02SP1+HSP2* 0517 C QlSP2*C2SPl+HSPl«ClSPl*02SP2+H8Pl#GlSP2»02SP2)+64. •P1SP2«01SH* 0518  02SH*(01SP1»Q2SP2+01SP2«02SP1)+64. »MK2*(-01SH*02SH*01SP1»02SP2-0519 C 01SH*C2SH*C1SP2»C2SP1-C1SH*01SP1»C2SP1•C2SP2-C1SH*01SP1»C2SP2**2 0520  -01SH*C1SP2*G2SP1*»2-G1SH»C1SP2*C2SP1«G2SP2-G2SH*C1SP1**2*G2SP2-0521 C 02SH«01SP1»G1SP2*02SP1-C2SH«01SPH>C1SP2*C2SP2-02SH#C19P2««2* 0522  02SP1-2. »01SPl*»2»02SP2**2-4.*01SPl»ClSP2»C2SPl*C2SP2-2. *C1SP2»» 0523 C 2»02SPl»»2)+32. »MH2*(01SPl*»2»02SP2*#2+2. *G1SP1»G1SP2»G2SP1* 0524  02SP2+G1SP2«*2*G2SP1*«2)-32.•(HSP2*01SH*G1SP1*02SP1«02SP2+HSP2» 0525 CGISH*G1SP2*G2SP1*»2+HSP2*02SH»GlSPl*«2«C2SP2+HSP2*G2SH*Q1SP1• 0526 C 01SP2»C2SP1+HSP1*01SH*G1SP1»C2SP2*»2+HSP1*C1SH»G1SP2»G2SP1*C2SP2 0 527  +HSP1»C2SH«C1SP1»C1SP2*C2SP2+HSP1*C2SH#01SP2*»2»02SP1) 0528 M36-H36/S**2 0529 M37 «(4. *S**2*P1SP2«MK2-S»«2»P1SP2«MH2 0530 C +4. »S*«2*MK2**2+S#*2«MK2*MH2+4. * 0531 C S*»2»MK2»(HSP2+HSP1>+2.•S*»2»HSP2»HSP1+8. *S»P1SP2»MK2»(-01SP2+ 0532  02SP2)+2. *S*PlSP2*MH2*(018P2-02SP2>+8. •S*MK2*«2*(-01SH+02SH-0533 C 01SP1+G2SP1)+2. *S«MK2»MH2«(01SP1-G2SP1)+4. •S*HK2*(-2. *HSP2*G1SP2 0534  +2. •HSP2*02SP2-HSPl*018H+HSPl»02SH-2. •01SH*02SP2-2. •02SH*61SP2+ 0535 C 4. *02SH*02SP2-2.*0lBPl»028P2-2.«©lSP2»Q2SPl+4. »02SP1*02SP2) 0536  +2. *S* 0537 C MH2»(QlBPl«02SP2+01SP2»02SPl-2.#02SPl»G2SP2)+4. »S*HSP1»(-HSP2* 0538  01SP2+HSP2«G2SP2-GlSH»02SP2-02SH»GlSP2+2. •02SH*G2SP2) + 16. »MK2»( 0539 C 3. •GlSH*01SP2»G2SP2-01SH«G2SP2*«2+G2SH*01SP2««2-3. «02SH»01SP2« 0540  G2SP2+3. •01SPl»ClSP2*Q2SP2-GlSPl*02SP2»*2+01SP2»»2»©2SPl-3. • 0541 C GlSP2*02SPl*G2SP2)+4. *MH2*(-3.*0ISP 1*01SP2«02SP2+0ISP1*02SP2**2-0542  01SP2»*2«02SPl+3. •01SP2*02SPl»02SP2)+8.•H8P1»(3. »G1SH#01SP2« 0543 C G2SP2-QlSH«02SP2**2+G2SH*GlSF2«*2-3.*02SH«01SP2»02SP2)>/S 0544 1138 -4.•S**2»P1SP2*MK2-S*«2*P1SP2*MH2 0545 C +4. #S**2«HK2**2+S*«2*MK2»l1H2+4. # 0546  S««2«MK2«(HSP2+HSPl>+2. «S*»2«HSP2«HSPl+8. »S»P1SP2«MK2#(-01SH+ 0547 C G2SH-GlSP2+G2SP2)+2.•S*PlSP2»MH2»(01SP2-C2SP2)+4. •S*P1SP2*G2SH*( 0548  01SH-02SH)+8. »S»MK2»*2»(-01SP1+02SP1>+2. *S»MK2*MH2*(01SP1-02SP1) 0549 C +4. •S*MK2«(-HSP2*ClSH+HSP2*G2SH-2.»HSP2»C18P2+2. •HSP2*G2SP2-G1SH 0550  »G2SH-2. #01SH*C2SPl+02SH«*2-2.»G2SH«GlSP2+2 •02SH»C2SP1+2. »02SH« 0551 C 02SP2-2. #01SPl»02SP2-2. •GlSP2*G2SPl+4. *G2SP1»C2SP2)+2. •S»MH2»( 0552  ©lSPl*02SP2+ClSP2»C2SPl-2. *G2SPl»02SP2>+4. •S*(-HSP2*HSP1»G1SP2+' 0553 C HSP2«HSP1»02SP2-HSP2»018H*G2SP1+HSP2«02SH*G2SP1-HSP1»G2SH«01SP2+ 0554  HSP1*C2SH»G2SP2>+B.•P1SP2«(-G1SH**2*G2SP2-G1SH*G2SH*G1SP2+01SH« 0555 C ©2SH»G2SP2+©2SH»*2*©1SP2> 0556 M3B-M38+B. «MK2*(01SH«*2*02SP2+Q1SH*C2SH*C18P2-0557 C 01SH*G2SH*02SP2+2. •01SH»01SPl*02SP2+2.•01SH*01SP2»02SPl+2. *01SH* 0558  01SP2*02SP2-2. •01SH*02SP2**2-©2SH«*2»©lSP2-2. *©2SH»©1SP1#©2SP2+ 0559 C 2. •G2SH*GlSP2*«2-2. •02SH*01SP2*02SPl-2. •02S++»01BP2«02SP2 0560  +6. •G1SP1 0561 C »01SP2»C2SP2-2. *ClSPl»©2SP2««2+2.•01SP2»«2*C2SPl-6. *©1SP2*02SP1# 0562  02SP2>+4. »MH2*<-3. »01SP1*©1SP2»02SP2+©1SP1*G2SP2»*2-©1SP2»*2« 0563 C 02SP1 +3. •© 1SP2»C2SP1*G2SP2 > +8.•(HSP2*C1SH*C1SP1»©2SP2+HSP2»©1SH* 0564  01 SP2«02SP 1 -HSP2*02SH*01 SP 1 •02SP2-H8P2*02SH«C 1 SP2»02SP 1+HSP1 • 0565 C C1SH»C1SP2»G2SP2-HSP1«Q1SH»02SP2**2+HSP1»G2SH«G1SP2»»2-HSP1«G2SH 0566  «01SP2«G2SP2) 0567 M3B-M3B/S 0568 M44 -(-32. »S*»2»MK2*»3 0569 C -8. •S*«2#MK2*«2*MH2+32. »S**2»MK2**2»(-HSP1+01SH+ 0570  01SPD-8. *S»«2«MK2»MH2*0ISP 1 + 16. *S*»2*MK2»(HSPl«01BH+2. *©1SH« 138 0571 C Q1BP2+2. •01SP1»01SP2)-B. *S»*2»MH2»Q1SP1*C1SP2+16. »S««2«HSP1»C1SH 0572  »01SP2+64. •S«MK2««2*(-Q1SH*G2SP2-Q2SH»C1SP2-01SP1»G2SP2-G1SP2» 0573 C 02SP1+2. •G1SP2»02SP2)+16. •S*riK2*MH2*<GlSPl*02SP2+01SP2*G2SPl+2. • 0574  Q18P2*02SP2>+32. •S»MK2«(-HSPl»01SH*G2SP2-HSPl*02SH*QlSP2+4. »HSP1 0575 C *01SP2*02SP2-4. •GlSH»GlBP2»C2SP2-4. »ClSPl*01BP2*G2SP2>+32. •S#MH2 0576 C •01SPl*01SP2«02SP2-64. •S»HSPl»G18H»01SP2*Q2SP2+256. *HK2*01SP2« 0577 C 02SP2«<GlSH*C2SP2+G2SH»GlBP2+01SPl»02SP2+QlSP2*Q2SPl>-<64. *MH2* 0578  01SP2»Q2SP2>*<Q1SP1»Q2SP2+01SP2*02SPI>+128. •HSP1*G1SP2»G2SP2»< 0579 C 01SH*C2SP2+G2SH#01SP2>)/S»*2 0580 M4S -8. *S«*3*MK2»«2 0581 C >4. »S«*3*HK2«(H8P1+2. #018P2)+4. •B*«3«HSP1»C1SP2+ 0582 C 16. •S»*2«P1SP2»MK2»*2+16. *S*«2»P1SP2«MK2*<HSP1+Q1SP2>+16. *S*»2* 0583  P1SP2*HSP1»01SP2-16.•S«»2*MK2»*3+B. •S**2*MK2*«2«(-HSP2-HSP1+01SH 0584 C -02SH-2. •C2SP1-2. #G2SP2)+B. *S#*2*MK2»<-HSP2«C1SP2-HSP1»G1SP1-0585 C HSPl»C2SP2+GlSH*ClSP2+01SH*G2SPl-02SH»GlBPl-4. »ClSP2*02SP2>+8. #8 0586 C *»2«01SP2«<-HSPl*01SPl-2. •HSP1»02SP2+G1SH«C2SP1-Q2SH«G1SP1>+16. * 0587 C S»PlSP2«MK2»<-01BH«G2SPl-C2SH*ClSPl-4. *C1SP2*C2SP2)+16. •S*P1SP2* 0588  01SP2*(-4. *HSP1»G2SP2-01SH*Q2SP1-Q2SH*01SP1>+16. »S»MK2**2»(01SH* 0589 C C2SPl+02SH»01SPl+4. «G18Pl#02SPl-2. »01SPl*02SP2-2. *C1BP2»Q2SP 1+4. 0590  »C1SP2*C2SP2) 0591 M45-M45+16. *S«MK2*<2. *HSP2*01SP1*02SP1+2. *HSP2»C1SP2*C2SP2 0592 C —HSP1*G1SP1»02SP2-HSP1*01SP 2*02SP1+2. *HSP1»G1SP2«02SP2-01SH* 0593  G1SP1»02SP1-GlSH»01SP2»C2SP2-C1SH*C2SP1»C2SP2+G2SH»G1SP1••2+02SH 0594 C •01SPl«G2SP2+G2SH*GlSP2**2+2.•02SH*ClSP2»02SP2-2. »018P1*C1SP2» 0595  02SP2-2. •GlSP2«*2«G2SPl+4.*GlSP2*G2SPl*G2SP2+4. •G1SP2*G2SP2**2> + 0596 C 16. •S*01SP2*(2. »HSP2*01SPl*C2SPl-2. •HSPl*GlSP2«G2SPl+2. *HSP1» 0597  02SP2»#2-01SH*01SPl»02SPl-2.»01SH»G2SPl»02SP2+02SH»GlSPl»»2+2. • 0598 C C2SH*01BPl»C2SP2)+64. »P18P2*G1SP2*G2SP2*(01BH*G2SP1+02SH*G1SP1) + 0599  64. *MK2«GlSP2»C2SP2»<-GlSH»02SPl-02SH»C16Pl-4. »GlSPl»02SPl+2. • 0600 C ClSPl«G2SP2+2. •C1SP2«C2SP1)+32.»C18P2»<-4. #HSP2«G1SP1«G2SP1» 0601  G2SP2+2. »HSPl»GlSPl»02SP2**2+2.»HSP1»C1SP2*02SP1»C2SP2+01SH» 0602 C 01SPl«G2SPl»G2SP2+01SH*GlSP2»G2SPl**2+2. *G1SH«G2SP1*G2SP2*«2-0603  02SH»01SPl**2«G2SP2-G2SH*GlSPl*GlSP2«G2SPl-2. •G2SH*01SP1*G2SP2«» 0604 C 2 > 0605 M45«M45/S««2 0606 M46 —2. •S«*3»P1SP2#MH2 0607 C +8. *S»*3*MK2»«2+4. *S*«3»MK2*(HSP2+HSP1>+4. »S»* 060B  3*HSP2*HSP1 + 16. *S««2«P18P2*MK2**2+4. *S«»2»P1SP2»MK2*MH2+16. «S*»2 0609 C *PlSP2»MK2»<HSP2+HSPl)+4.*S»»2*PlSP2*MH2»(GlSPl+G2SP2)+8. »S»»2« 0610  P1SP2#(2. •HSP2»HSP1+01SH*G2SH)-16. »S*»2«MK2**3+4. *S*«2*MK2»«2» 0611 C MH2-C16. •S«»2*MK2**2>*(HSP2+HSP1+G1SP2+02SP1>+4. *S««2*MK2»(-2. • 0612  HSP2**2-2. »HSP2»HSPl+HSP2*ClSH-HSP2»C2SH-2. *HSP2»C2SP1-2. »HSP2* 0613 C 02SP2-2. •HSPl«*2-HSPl*GlSH+HSPl«02SH-2. •HSPKG1SP1-2. •HSP 1*0ISP2 0614  +2. •GlSH*G2SPl-2. •GlSH»G2SP2-2. *G2SH*GlSPl+2. «02SH*GlSP2>+4. *S*« 0615 C 2*MH2«<QlSPl»G2SP2+01SP2»C2SPl>-<8. »S»»2)»<HSP2»HSP1»G1SP1+HSP2» 0616  HSPl»02BP2+HSP2*02SH*ClSPl+HSPl*018H*02SP2)+32. •S»P1BP2»MK2»< 0617 C GlSPl*02SP2+ClSP2*C2SPl)-<8.•S»P1SP2»MH2)«(G1SP1»C2SP2+01SP2« 0618  C2SP1)-(16. *S*P1SP2>*<HSP2*G1SH*C2SP1+HSP2*C2SH*G1SP1+HSP1»C1SH* 0619 C 02SP2+HSPl*G2SH*G18P2+01BH»02SH*GlSPl+01SH*G2SH*028P2>+32. »S»MK2 0620  **2*< G1SH*02SP1+01SH»02SP2+02SH»01SP1+02SH*G1SP2+2. *01SP1»02SP1-0621 C QlSPl»G2SP2-QlSP2«02SPl+2. *01SP2*02SP2) 0622 M46-M46+8.*S*MK2»MH2*<-2. *01SP1• 0623 C G2SPl-01SPl*G2SP2-01SP2*G2SPl-2. *01SP2«02SP2)+B. •S»HK2*<2. *HSP2* 0624  GlSH*C2SPl+2.*HSP2*G18H»G2SP2+2.*HSP2»02SH*ClSPl+2. *HSP2»02SH« 0625 C 01SP2+8. *HSP2*GlSPl*02SPl-4.*HSP2*G1SPl*02SP2-4. «HSP2*G1SP2« 0626  G2SP1+2. »HSPl»01SH»C2SPl+2.•HSPl*GlSH*G2SP2+2. •HSP1*02SH*01SP1+ 0627 C 2. «HSPl»C2SH*01SP2-4. «H8Pl*GlSPl*G2SP2-4. *HSPl*ClSP2»C2SPl+8. * 139 0628 C HSP1*C1SP2*C2SP2+01SH**2*Q2SP1-C1SH**2*C2SP2+C1SH*C2SH*01 SP 1 + 0629 C 01SH*C2SH*ClSP2+ClSH»Q2SH»02SPl+ClSH*C2SH*C2SP2+2. *01SH*C1SP1* 0630 C 02SP1+2. *QlSH*QlSP2*02SPl+2.*01SH*G2SPl*C2SP2+2. *G1SH*C2SP2**2-0631 C 02SH**2*ClSPl+Q2SH**2«ClSP2+2.»C2SH*C1SP1**2+2. *02SH*C1SP1*G1SP2 0632 C +2. *C2SH*01SP2*G2SP1+2. »C2SH*C1SP2*C2SP2+8. *C1SP1*G1SP2*G2SP1+8. 0633  *01SP2*02SPl»C2SP2)-(8. *S*MH2)*(01SP1**2*C2SP2+01SP1*01BP2*C2SP1 0634 C +018P1*02SP2**2+01SP2*02SP1*G2SP2>+16. *S*(2. *HSP2**2*G1SP1*G2SP1 0635  -HSP2*HSP1*01SP1*02SP2-HSP2*HSP1*01SP2*02SP1+HSP2*C2SH*G1SP1**2+ 0636 C HSP2*02SH*QlSPl*Q2SP2+2. *HSP1**2*G1SP2*02SP2+HSP1*01SH*01SP1* 0637 C C2SP2+HSP1*C1SH*C2SP2**2> 0638 H46-M46+32. *P1SP2*<G1SH**2*Q2SP1*02SP2+C18H* 0639 C 02SH*01SPl*C2SP2+01BH*G2SH*ClSP2*C2SPl+C2SH**2*ClSPl*01SP2>+32. * 0640  MK2*(-01SH**2*02SP1*02SP2-01SH*02SH*G1SP1«G2SP2-01SH*G2SH*G1SP2* 0641 C 029P1-4. *G1SH*01SP1*02SP1*02BP2-4. *C1SH*G1SP2*02SP1*G2SP2-02SH*« 0642  2*01SPl*01SP2-4. *02SH*01SP1*G1SP2»G2SP1-4. *02SH*G1SP1*G1SP2* 0643 C 02SP2-2. »01SP1**2*02SP2**2-12.*01SP1*01SP2*02SPl*G2SP2-2. *G1SP2 0644 C **2*02SP1»*2>+16. *HH2*(QlSPl**2*C2SP2**2+6. *01SP1*01SP2*G2SP1* 0645  02SP2+GlSP2**2*02SPl**2)-64.*(HSP2*01SH*01SP1*02SP1*©2SP2+HSP2* 0646 C 02SH»C1SP1*01SP2*02SP1+HSP1*01SH»01SP2*02SP1*G2SP2+HSP1*G2SH* 0647  01SP1*01SP2*G2SP2) 0648 M46-M46/S**2 0649 M47 -<-4. *S»«2*P1SP2»HK2+S**2*P1SP2*MH2 0650 C -4.*S**2»MK2**2-S**2*MK2*MH2-( 0651 C 4. *S**2*MK2)*(HSP2+HSP1)-2.*S**2#HSP2*HSP1+8. *S*P1SP2*MK2*( 0652  -01SP2 0653 C +02SP2)+2. »S*PlSP2»MH2*<01SP2-02SP2)+8. *S*«K2**2»<-G1SH+02SH-0654  01SP1+02SPD+2. *S*MK2*MH2*(G1SP1-02SP1 >+4. *S*MK2»(-2. *HSP2*G1SP2 0655 C +2. *HSP2*G2SP2-HSPl*01SH+HSPl*G2SH-4. *GlSH*GlSP2+2. *G1SH*G2SP2+ 0656  2. *02SH*01SP2-4. *01SPl*GlSP2+2.*G1SPl*G2SP2+2. *01SP2*02SP1) 0657 C +2. *S* 0658  MH2*<2. *01SP1*01SP2-01SP1*02SP2-01SP2*G28P1)+4. *S*HSP1*(-HSP2* 0659 C 01SP2+HSP2*G2SP2-2. *G1SH*G1SP2+G1SH*G2SP2+G2SH*G1SP2> + 16. *MK2*( 0660  3. *GlSH*GlSP2*G2SP2-GlSH*G2SP2**2+G2SH*GlSP2**2-3. *C2SH*G1SP2* 0661 C G2SP2+3. «01SPl*01SP2*02SP2-GlSPl*G2SP2**2+01SP2**2*G2SPl-3. * 0662 CGISP2*02SP1*02SP2)+4. *HH2*(-3.*01SP1*C1SP2*02SP2+01SP1*02SP2**2-0663 C GlSP2**2*02SPl+3. *01SP2*02SPl»G2SP2)+8. *HSP1*(3. *G1SH*G1SP2* 0664  G2SP2-01SH*G2SP2»*2+C2SH*01SP2**2-3. *G2SH*G1SP2*02SP2))/S 0665 M48 —4. *S**2*P1SP2*MK2 0666 C +S**2*PlSP2*MH2-4. *S**2*MK2**2-S**2*MK2*MH2-< 0667  4. *S**2*MK2)*(HSP2+HSP1)-2. *8**2*HSP2*HSP1+8. *S*P1SP2*MK2*( 0668 C -01SH+ 0669  G2SH-ClSP2+G2SP2>+2.•S*PlSP2*MH2*(GlSP2-02SP2)+4. *S*P1SP2*G1SH*( 0670 C 01SH-02SH)+8. *S*MK2**2*(-01SP1+02SP1)+2.*S*MK2*MH2*(01SP1-G2SP1) 0671  +4. *S*MK2*(-HSP2*01SH+HSP2*02SH-2.*HSP2*GlSP2+2. »HSP2*C2SP2-G1SH 0672 C **2+GlSH*02SH-2. *01SH*C1SP1-2.*GlSH*GlSP2+2. *GlSH*028P2+2. *G2SH* 0673  01SP1-4. *01SPl»GlSP2+2.*01SPl*C2SP2+2.»G1SP2*G2SP1)+2. *S»MH2*<2. 0674 C *GlSPl*01SP2-01SPl*02SP2-01SP2*02SPl)+4. *S*(-HSP2*HSP1*01SP2+ 0675  HSP2*HSP1*02SP2-HSP2*01SH*G1SP1+HSP2*G2SH*G1SP1-HSP1*01SH*G1SP2+ 0676 C HSP1*01SH*C2SP2) 0677 M48-M48+B. *P 1SP2* < -C1 SH**2*02SP2-01 SH*C2SH*C 1SP2+018H* 0678 C 02SH»C2SP2+G2SH**2*01SP2)+8.«MK2*(01SH»*2*G2SP2+G1SH»G2SH*C1SP2-0679  GlSH*G2SH»02SP2+2. *G18H*GlSPl»02SP2+2.*G1SH»C18P2*02SP1+2. »01SH* 0680 C 01SP2*02SP2-2. *01SH*028P2**2-02SH««2»GlSP2-2. •02SH*G1SP1«G2SP2+ 0681  2. *C2SH*CISP2**2-2. *C2SH*ClSP2»C2SPl-2. *G2SH*G1SP2*G2SP2 0682 C+6. *01SP1 0683 C *01SP2*02SP2-2. »01SPl*02SP2*»2+2. »01SP2**2*G2SPl-6. *G1SP2*02SP1» 0684 C G2SP2)+4. »MH2*(-3. *01SP1*01SP2*02SP2+G1SP1*02SP2*«2-G1SP2**2* 140 0665 C 02SP1+3. »01SP2«02SPl»02SP2>+8. *<HSP2»C1SH»01SP1#02SP2+HSP2»C1SH» 0686 C 01SP2*G2SP1-HSP2*C2SH»C1SP1*C2SP2-HSP2»C2SH»C1SP2*G2SP1+HSP1* 0687  01 SH»C 1 SP2»G2SP2-HSP 1 *C 1 SH*G2SP2»»2+HSP 1 #C2SH*G 1 SP2**2-HSP 1 »G2SH 0688 C •01SP2*Q2SP2> 0689 M48-M48/S 0690 M5S "(8. »S**3*MK2*»2 0691 C +8. #S**3»MK2*<C1SP2+C2SP1)+B.•S*«3*C1SP2*C2BP1+16. 0692 C *S««2*P1SP2«MK2*«2+16. •S»*2»P1SP2*MK2#<01SP2+G2SP1 > + 16. «S*»2« 0693  P18P2*C1SP2»C2SP1-16. »S»»2*MK2»»3-(16. »S»»2*MK2*»2)•(01SP1+C2SP2 0694 C > + 16. •S*«2*t1K2*(-01SPl*01SP2-2.•01SPl»02SPl-2. •C1SP2*02SP2-G2SP1 0695  •G2SP2)-(32. •S»*2*G1SP2*C2SP1)»<ClSPl+C2SP2>-<64. •S*PlSP2«liK2)«< 0696 C 01SPl*02SPl+01SP2*C2SP2>-<64. *S»P1SP2*C1SP2*G2SP1 >»<01SP1+G2SP2> 0697 C +64. *S«MK2»*2«<GlSPl»G2SPl+01SP2*C2SP2>+32. »S*MK2»C2. *G1SP1»«2» 0698  G2SP1+01SP1*01SP2»G2SP2+01SP1*G2SP1»G2SP2-01SP2#*2*G2SPI-01SP2» 0699 C C2SPl**2+2. *GlSP2*G2SP2**2)+32. *S*G1SP2*G2SP1*<2. *01SPl*«2+3. • 0700 C 01SPl*G2SP2-GlSP2*G2SPl+2.*G2SP2**2>+256.•P1SP2»018P1#01SP2* 0701  G2SP1*02SP2-2S6. *MK2*01SP1*G1SP2«G2SP1»02SP2+12B. •G1SP2*G2SP1«<-0702 C C1SP1»»2»02SP2+01SP1»G1SP2»02SP1-01SP1•02SP2«»2+G1SP2*G2SP1• 0703  G2SP2)>/S**2 0704 M56 -8. #S»*3*MK2»»2 0705 C +4. *S»«3*MK2# < HSP2+2. »02SP1)+4.»S»*3»HSP2*02SP1 + 0706  16. »S««2»P1SP2»MK2««2+16 •S««2«P1SP2*MK2«<H8P2+028P1>+16. •S**2* 0707 C P1SP2*HSP2*02SP1-16. •S««2»MK2«»3+8. •S**2*MK2»«2»(-HSP2-HSP1-C1SH 0708 C +02SH-2. *01SPl-2: »01SP2)+8.«S»»2»MK2»<-HSP2*G1SP1-HSP2*G2SP2-0709 C HSPl»G28Pl-01BH«C2SP2+Q2SH«GlSP2+02SH*02SPl-4. •C1SP1*C2SP1)+B. *S 0710 C *«2«G2SPl*(-2. #HSP2*01SP1-HSP2*G2SP2-G1SH*02SP2+02SH«G1SP2) + 16. • 0711  S*PlSP2*MK2»<-01SH*02SP2-G2SH»01SP2-4. *018P1»02SP1) + 16. *S»P1SP2* 0712 C 02SPl#<-4. »HSP2«01SP1-01SH»G2SP2-02SH»C1SP2>+16. »S»MK2*»2*(C1SH* 0713  02SP2+C2SH»01SP2+4. •01BPl*02SPl-2. •GlSPl»02SP2-2. •01SP2»C2SPl+4. 0714 C •G1SP2*G2SP2) 0715 M56-M56+16. «S«MK2«(2.*HSP2»01SP1«G2SP1-HSP2*G1SP1*G2SP2-0716 C HSP2*ClSP2*G2SPl+2. «HSPl*01SPl*G2SPl+2. »HSPl»ClSP2*02SP2+2. •CISH 0717  »C1SP1*G2SP1+01SH»G1SP1»C2SP2+G1SH*G2SP1*#2+G1SH»G2SP2«»2-G2SH* 0718 C 01SPl»ClSP2-C2SH*ClBPl»C2SPl-C2SH#ClSP2*C2SP2+4. •01SP1«»2*C2SP1 + 0719  4. •GlSPl«01SP2»Q2SPl-2. •01SPl«02SPl«02SP2-2. •G1SP2*G2SP1«»2> 0720 C +16. • 0721  S«C2SP1•(2. »HSP2»C1SP1»»2-2. •HSP2»C1SP2*G2SP1+2. »HSP1*C1SP2* 0722 C G2SP2+2. «01SH*01SPl*02SP2+ClSH*G2SP2*»2-2. •02SH»01SP1»Q1SP2-02SH 0723  *GlSP2»C2SP2)+64. »PlSP2»GlSPl»02SPl»<018H»C2SP2+C2SH»GlSP2>+64.# 0724 C MK2»GlSPl*C2SPl«(-01SH*C2SP2-G2SH»ClSP2+2. *01SPl*G2SP2+2. *G1SP2» 0725  G2SP1-4. *01SP2*C2SP2)+32. »C2SP1»<2. *HSP2*01SPl#*2*02SP2+2. *HSP2» 0726 C 01SPl»GlSP2*G2SPl-4.»HSP1*G1SPl*GlSP2*G2SP2-2. »G1SH*G1SP1**2* 0727  G2SP2-GlSH»0lSPl*G2SP2»*2-01SH*GlSP2*G2SPl*G2SP2+2. •G2SH*G1SP1•* 0728 C 2»G1SP2+Q2SH*G1SP1«01SP2*G2SP2+02SH«G1SP2**2*02SP1> 0729 M56«M56/S**2 0730 M57 —4. «S»«2*P1SP2*MK2 0731 C +2. «S*«2#PlSP2»G29H-4. »S«*2«MK2»»2+2. »S**2»MK2 0732  »(-HSP2-HSPl-G2SH-2.*01SP2-2.»02SPl)-(2. •8»»2>»<HSP2»02SP1+HSP1» 0733 C GlSP2)+4. •S«PlSP2*»2»(-01SH+G2SH>+4. •S*PlSP2»MK2*(GlSH-G2SH+2. * 0734  01SP1-2. »01SP2-2. *G2SPl+2. »G2SP2)+4. •S»P1SP2«(HSP2»01SP1-HSP2* 0735 C G2SP1-HSP1*01SP2+HSP1«02SP2-02SH*G1SP1-02SH»62SP2)+4. »S«MK2» <-0736  01SH«01SP2+ClSH»C2SP2+02SH»01SPl+02SH»GlSP2+2. »01SPl*G2SPl+2. » 0737 C ClSPl»02SP2+2. *01SP2»02SPl+2. *01SP2»02SP2) 0738 M57-M57+4. *S*(HSP2*G1SP1* 0739 C 02SP1+HSP 2*G2SP1«G2SP2+HSP1«01SP1*01SP 2+HSP1»01SP2*02SP2-G1SH* 0740 CGISP2*C2SP1+G1SH#02SP1*G2SP2+G2SH«G1SP1*01SP2-G2SH*G1SP1»C2SP2)+ 0741 C 8. •P1SP2»(©1SH*01SP1»02SP2+01SH»C1SP2*C2SP1-C2SH»G1SP1»G2SP2 141 0742 C -02SH 0743 C •Q1SP2*Q2SP1>+8. *MK2*<-G1SH»G1SP1*Q2SP2-Q1SH*01SP2*02SP1+G2SH* 0744 C 01SPl»02SP2+C2SH*ClSP2»02SPl-2.»C1SPl**2*02SP2-2. *C1SP1*C1SP2« 0745  02SP1+2. «01SPl*QlSP2*02SP2+2.*01SPl«02SPl*02SP2-2. *01SP1*02SP2«» 0746 C 2+2. »01SP2**2*02SPl+2. »01SP2*C2SPl#*2-2. #01SP2*02SPl*02SP2>+8. *( 0747 C -HSP2»01SPl»«2»02SP2-HSP2»GiSPl*ClSP2»G2SPl+HSP2*01SPl*G2SPl« 0748 C 02SP2+HSP2»C1SP2»C2SP1*#2+HSP1 »01 SP 1 «Q 1SP2*02SP2-H8P1*G1SP1* 0749 C 02SP2««2+HSP1#01SP2»#2«02SP1-HSP1*©1SP2*G2SP1*G2SP2+G1SH*G1SP1» 0750 C 01SP2*C2SP1-01SH*CIBP1»C2SP1*C2SP2+G1SH»G1SP2*G2SP1*C2SP2-G1SH* 0751  G2SP1»C2SP2»*2-02SH»G1SP1*«2»C1SP2+02SH»C1SP1••2«02SP2-C2SH# 0752 C 01SP1*C1SP2«G2SP2+02SH»C1SP1*G2SP2»«2> 0753 M57-M57/S 0754 H58 —4. *B«*2*P1BP2»MK2 0755 C +2. «S»#2#PlSP2*01SH-4. »S*»2*MK2««2+2. »S*«2«MK2 0756  •<-HSP2-HSPl-01SH-2. »01SP2-2.*02SPl)-(2. •S«-»2>*<HSP2»C2SP1+HSP1* 0757 C 01SP2>+4. *S*PlSP2«»2»(01SH-02SH)+4. •S»PlSP2«t1K2»(-01SH+02SH+2. • 0758 C G1SP1-2. *GlSP2-2. #C2SPl+2.*©2SP2>+4. #S*P1SP2»<HSP2»C1SP1-HSP2* 0759 C 02SPl-HSPl«01SP2+HSPl»02SP2-01SH«01SPl-01SH»©2SP2>+4. «S»MK2»( 0760  01SH*02SPl+018H*C2SP2+©2SH*01SPl-02SH*02SPl+2. »0ISP 1*02BP1+2. • 0761 C 01SPl»G2SP2+2. •01SP2»G2SPl+2 *018P2*G2SP2)+4. *S»<HSP2*01SPI» 0762 C 02SP1+HSP2»02SP1•02SP2+HSP1*0ISP 1*01SP2+HSP1•©1SP2*G2SP2-G1SH* 0763  01SP1*02SP2+01SH«C2SP1*02SP2+G2SH*C1SP1*C1SP2-02SH*01SP2*G2SP1>+ 0764 C 8. *P1SP2«(-01SH*01SP1»G2SP2-G1SH*C1SP2»C28P1*02SH»Q1SP1*G2SP2+ 0765  02SH»01SP2«G2SP1> 0766 M58-M58+8. *riK2«<01SH*GlSPl*G2SP2+01SH*GlBP2#G2SPl-02SH 0767 C »GlSPl#02SP2-02SH»01SP2»G2SPl-2.*ClSPl«»2»G2SP2-2. »G1SP1*G1SP2» 0768 C G2SP1+2. *GlSPl*01SP2*G2SP2+2.•01SPl*G2SPl«G2SP2-2. «G1SP1*G2SP2*« 0769  2+2. »ClSP2»»2*G2SPl+2 •GlSP2*G2SPl**2-2. *GlSP2«02SPl*02SP2)+8. •( 0770 C -HSP2»01SP1•»2*©2SP2-HSP2»01SP1»G1SP2*G2SP1+HSP2»G1SP1»C2SP1• 0771  02SP2+HSP2*G1SP2»G2SP1••2+HSP1«01SP1*G1SP2*G2SP2-HSP1*01SP1• 0772 C G2SP2««2+HSP1*01SP2**2*G2SP1-HSP1»C1SP2«G2SP1*02SP2+G1SH*G1SP1•* 0773  2*G2SP2-G1SH«G1SP1*G2SP1•G2SP2+01SH*G1SP1*G2SP2**2-G1SH*G2SP1* 0774 C G2SP2**2-02SH*C1SP1••2»G1SP2+G2SH»C1SP1»01SP2»G2SP 1-G2SH*G1SP1• 0775  01SP2*G2SP2+G2SH»G18P2*G2SP1»02SP2) 0776 M5B-M58/S 0777 M66 -<-32. »S»»2»MK2**3 0778 C -8. •S*»2«MK2«»2#MH2+32.•S»»2*MK2**2« <-HSP2+G2SH+ 0779  C2SP2)-8. »S«*2»MK2«MH2»02SP2+16. »S«»2»HK2»(HSP2»02SH+2. •G2SH* 0780 C G2SP1+2. »C2SPl»G2SP2>-8.*S»»2»MH2»C2SP1»C2SP2+16. »S»»2*HSP2»G2SH 0781  *02SPl+64. •S»MK2**2*(-01SH*G2SPl-02SH*GlSPl+2. *0ISP1*02SP1-G1SP1 0782 C •C2SP2-G1SP2»G2SP1>+16. *S*MK2*MH2*<2.•G1SP1«G2SP1+018P1*02SP2+ 0783  G1SP2»C2SP1)+32. #S*MK2*<-HSP2*GlSH*02SPl-HSP2*G2SH*01SPl+4. #H8P2 0784 C »01SPl»02SPl-4. »C2SH*01SPl»Q2SPl-4.»01SPl»C2SPl»Q2SP2>+32. »S*MH2 0785  *01SPl*G2SPl«G2SP2-64. •S»HSP2»G2SH»G1SP1»02SP1+256. •MK2*01SP1« 0786 C 02SPl«(01SH«G2SPl+G2SH»GlSPl+01SPl»G2SP2+GlSP2»02SPl)-t64. »MH2* 0787  G18P1*G28P1)*(01SP1#02SP2+G1SP2»G25P1) + 12B. •HSP2*01SP1«G2SP1«( 0788 C C1BH»C2SP1+02SH»01SP1))/S»«2 0789 (167 --4. •S»»2*P1SP2»WK2+S««2»P1SPZ*MH2 0790 C -4. •S**2«MK2##2-B»*2»MK2#MH2-( 0791  4. •S**2«flK2)*(HSP2+HSPl)-2. *S»«2«HSP2»HSPl+8. •S»P1SP2»MK2*<G1SH-0792 C 02SH+G1SP1-02SP1)+2. •S»P1SP2»MH2*<-Q1SP1+02SP1>+4. •S»P1SP2*C2SH» 0793  <-ClSH+G2SH>+8. •S«MK2««2«<ClSP2-02SP2>+2. *S»HK2*HH2«<-01SP2+ 0794 C 02SP2>+4. »S»MK2*(HSPl*01SH-HSPl*02SH+2. »H6Pl»QlSPl-2. •HSP1*02SP1 0795  +01SH»02SH+2. •018H»02SP2-G2SH**2+2. »02SH*ClSPl-2. •02SH*02SPl-2. * 0796 C 028H«G2SP2+2.#01SPl»02SP2+2.«01SP2»02SPl-4. •02SPl*02SP2>+2. *S« 0797 C MH2«<-01SPl*02SP2-01SP2«02SPl+2. «02SP1*02SP2> 079B M67-M67+4. *S»(HSP2«HSP1• 142 0799 C 01SP1-HSP2»HSP1*02SP1+HSP2«C2SH*C1SP1-HSP2»Q2SH«C2SP1+HSP1*C1SH* 0800 C C2SP2-HSPl*02SH«G2SP2>+8.*P1SP2«<Q16H*»2»G2SP1+G1SH»C2SH»G1SP1-0801 C 01SH*G2SH»C2SP1-02SH»#2»01SP1>+B. »MK2*(-C1SH«*2»02SP1-C1SH«G2SH* 0802 C 01BPl+QlSH#02SH»G2SPl-2. *01SH«01SPl«G2SPl-2. •01SH*01SPl*02SP2-2. 0803 C «01SH»01SP2*C2SPl+2. »GlSH»02SPl*»2+Q2SH«#2«ClSPl-2. «02SH*01SP1*« 0804 C 2+2. »02SH»ClSPl*G2SPl+2.*G2SH»ClSPl»G2SP2+2. •02SH»ClSP2»C2SPl-2. 0805 C *01SPl#»2*02SP2-6. «01SPl*01SP2*G2SPl+6. *GlSPl»Q2SPl*G28P2+2. • 0806 C 01SP2»C2SPl*«2)+4.*MH2*(01SPl«»2«02SP2+3. *01SPl*01SP2«G2SPl-3. • 0807 C 01SP1*G2SP1»C2SP2-01SP2*C2SP1**2)+B. «<-HSP2*01SH*GlSPl«C2SPl+ 0808  HSP2»01SH*G2SP1»»2-HSP2»G2SH»G1SP1##2+HSP2*G2SH#Q1SP1»G2SP1-HSP1 0809 C *G1SH*01SP1*C2SP2-HSP1*G1SH«G1SP2»02SP1+HSP1*02SH*01SP1*G2SP2+ 0810  HSP1*G2SH*G1SP2*G2SP1) 0811 M67-M67/S 0812 M68 -(-4. «S»»2*P1SP2»HK2 0813 C +S«»2»PlSP2»MH2-4. •S»«2»MK2««2-S**2*MK2*MH2-< 0814  4. •S»#2«MK2>»<HSP2+HSP1) 0815 C -2. «S»«2»HSP2»HSPl+8.»S*P1SP2»MK2»CG1SP1-0816  G2SP1)+2. *S»P1SP2«MH2»<-G1SP1+C2SP1>+8. »S«MK2**2»<G1SH-C2SH+ 0817 C 01SP2-Q2SP2>+2. •S»MK2»MH2*<-QlSP2+Q2SP2>+4. «S*MK2«<HSP2*G1SH-0818  HSP2»G2SH+2. *HSPl*01SPl-2.*HSPl»G2SPl+2. *01SH»C2SPl+2. «C2SH* 0819 C 01SP1-4. •02SH»C2SPl+2. »G1SPl*G2SP2+2. *01SP2»02SP1-4. *G2SP1»C2SP2 0820 C )+2. «S»MH2*<-QlSPl«02SP2-01SP2»G2SPl+2. »Q2SPl*C2SP2>+4. #S*HSP2«< 0821  HSP1*C1SP1-HSP1»G2SP1+G1SH*G2SP1+G2SH*G1SP1-2. #G2SH*G2SP1 > +16. « 0822 C MK2*<-3. *01SH*G1SP1»02SP1+C1SH»G2SP1**2-G2SH*G1SP1*«2+3. »G2SH« 0823  GlSPl»G2SPl-01SPl**2*G2SP2-3.•ClSPl*01SP2»C2SPl+3. »G1SP1*G2SP1» 0824 C G2SP2+QlSP2*02SPl*«2>+4. »l1H2*CClSPl»«2*C2SP2+3. #G1SP1*C1SP2* 0825  G2SP1-3. •GlSPl«02SPl*G2SP2-GlSP2*G2SPl**2)+8. *HSP2«<-3. »G1SH» 0826 C 01SPl*C2SPl+GlSH»G2SPl**2-C2SH*GlSPl**2+3. •02SH»G1SP1*G2SP1)>/S 0827 M77 »8. *S»P1SP2»MK2 0828 C -2. *S»PlSP2«MH2+8. •S*MK2**2+2.»S«MK2*MH2+8. *S»MK2«( 0829  HSP2+HSP1)+4.*S»HSP2«HSP1+16.*MK2»(C1SH*01SP2-G1SH»C2SP2-G2SH» 0830 C O1SP2+G2SH»G2SP2+G1SP1*01SP2-G1SP1«G2SP2-C1SP2*G2SP1+G2SP1»C2SP2 0831  )+4. •MH2»(-G15Pl»GlSP2+GlSPl»C2SP2+GlSP2»G2SPl-C2SPl»G2SP2>+8. • 0832 C HSP1*(01SH*01SP2-G1SH*02SP2-G2SH*G18P2+G2SH*02SP2> 0833 M7B -8. *S*P1SP2*MK2 0834 C -2. *S*P1SP2«MH2+B.*S«MK2*»2+2.»S«MK2«MH2+B. «S*MK2«< 0835  HSP2+HSP1)+4. *S*HSP2»HSPl+4. »P1SP2«<-GlSH»«2+2. »C1SH»G2SH-C2SH*« 0836 C 2)+4. •MK2«(01SH*«2-2. *GlSH*G2SH+2. *G1SH*G1SP1+2. »GlSH*GlSP2-2. • 0837  01SH*02SPl-2. »GlSH*G2SP2+G2SH**2-2. «G2SH*GlSPl-2. *G2SH«GlSP2+2. « 0838 C 02SH«G2SPl+2. «02SH*02SP2+4.*GlSPl«GlSP2-4. «01SPl*G2SP2-4. •C1SP2* 0839  G2SP1+4. *G2SP1«G2SP2)+4. «MH2»<-01SP1*G1SP2+G1SP1*G2SP2+G1SP2* 0840 C G2SPl-G2SPl»02SP2>+4.•<HSP2»G1SH»G1SP1-HSP2»C1SH*G2SP1-HSP2»C2SH 0841  *C1SP1+HSP2*C2SH*C2SP1+HSP1»G1SH«G1SP2-HSP1*G1SH«G2SP2-HSP1#G2SH 0842 C *G1SP2+HSP1*02SH*G2SP2) 0843 MSB -8. *S*P1SP2*I1K2 0B44 C -2. »S»P1SP2*MH2+B. #S»MK2»»2+2. *S*MK2*MH2+B. #B»MK2»< 0845  HSP2+HSP1)+4. «S*HSP2»HSP1 + 16.»MK2«(G1SH»G1SP1-G1SH*02SP1-G2SH* 0B46 C 019P1+G2SH*02SP1+G1SP1»C1SP2-G1SP1«G2SP2-G1SP2»G2SP1+02SP1»G2SP2 0847 C )+4. *MH2*(-018P1«G18P2+G1SP1*G2SP2+C1SP2*02SP1-G2SP1»02SP2)+B. • 0848 C HSP2*(01SH»01SP1-C1SH*02SP1-G2SH«G1SP1+02SH»02SP1> 0849 C 0850 C DO THE DIVISION BY THE PROPAGATORS 0851 C 0852 MUA - MU/(D1#D3*D1*D3) 0853 M12A - M12/<D1«D3*D3*D4> 0854 M13A • M13/(D1»D3«D2*D4) 0855 M14A - M14/<D1#D3»D2«D5) 143 0656 M15A - M13/<D1*D3*D5*D6> 0837 M16A - M16/(D1*D3*D1*D6) 0838 M17A - M17/<Dl*D3*D2*B> 0839 M1SA - M1B/<D1*D3*D1*6> 0B60 M22A - M22/<D3*D4*D3*D4) 0861 M23A - M23/<D3*D4*D2*D4> 0862 M24A - M24/<D3*D4*D2*D5) 0863 M25A - M25/<D3*D4*D5*D6> 0864 M26A - M26/<D3#D4*D1*D6> 0865 M27A - M27/<D3*D4*D2*S> 0866 M28A - M28/<D3*D4*D1*8> 0867 M33A - M33/(D2#D4*D2«D4) 0868 M34A - M34/<D2*D4*D2*D5> 0869 M33A « M35/<D2*D4*D3*D6) 0870 M36A - M36/<D2*D4*D1*D6> 0871 M37A « M37/(D2*D4*D2*S> 0872 M38A = M38/(D2»D4*D1*8> 0873 M44A • 1144/(D2*D5*D2*D5> 0874 M45A - M45/(D2*D5*D5*D6) 0875 M46A - M46/(D2*D5*D1»D6> 0876 H47A - M47/<D2»D5*D2*S) 0877 M48A - M48/(D2*D5*D1*8) 0878 M55A - M55/(D5*D6*D5*D6> 0879 MS6A - M56/(D5*D6«D1*D6) 0880 (157A - M57/(05*D6*D2*S> 0881 M58A - M58/(D5*D6*D1*S) 0882 M66A - M66/(D1*D6*D1*D6> 0883 M67A - M67/<D1*D6»D2*S> 0884 M68A - M68/<D1*D6*D1*S) 0885 M77A - M77/<D2»S*D2*S> 0886 M7BA - M7B/(D2»S*D1*S) 0887 M8BA - M88/<D1*S*D1*S) 08B8 0889 MAI - MilA • M22A + M33A + M44A + M5SA + M66A + 0890 C 2. *<M12A + M13A • M23A + M45A • M46A + M36A) 0891 MA2 - 2. »(M14A • M13A • M16A + M24A + M25A • M26A 0892 C + M34A • M35A • M36A) 0893 MA3 • M77A • M88A + 2. *M78A 0894 MA4 - 2. *(M17A + M1BA + M27A + M28A + M37A + M38A) 0895 MAS - 2. *<M47A • M48A + M57A • MS8A • M67A • M6BA) 0B96 HO - 16. *MAl/3. -2. *MA2/3. • 12. *MA3 + 6. *MA4 -6. *MA5 0897 RETURN 089B END 1 44 APPENDIX F - QUARK AND GLUON DISTRIBUTION PARAMETRIZATIONS One of the sources of uncertainty of rate predictions in pp or pp interactions is the shape of the parton distribution employed to convolve over parton momenta. The quark distribution can be directly measured from lepton-proton interactions (Field and Feynman, 1977), because they can interact through the electromagnetic force. The discrepancy between the possible quark distributions arises from experimental uncertainty, and the different parametrizations used to fit data. Gluons distributions on the other hand, are only indirectly probed in hadron-hadron interactions. The uncertainty on them is much larger, as is reflected in the differences in cross sections they give rise to (Fig. 22 and 23) . For quarks, we used the following distributions (Peierls et al., 1977) V? au) = z.n (i- *)3 (F-1a) (F.lb) and (Barger and Phillips, 1974) Vx woo = Q57Y0-xf + 0.%ICl-if + Oill (h(F'2a) vrd(^= o.ov 0-xl)3+ 0.20k G-xWaK/-**)7 (p.2b> 145 For gluons, we used the simple ansatz (Brodsky and Farrar, 1973) 7(GM = 3 (F.3) and the scale-violating distribution (Baier et al., 1980) ^ . -.13/0 + O.36X,1 1.7+/•«/) XGcx) = (2.01-l.Kp + /.2?yo (M) (F.4) with and Q = 5 GeV The choices for the gluon distributions are motivated by the fact that they represent two extreme expectations on the actual gluon distributions. 146 APPENDIX G - HADRON-HADRON COLLIDERS Here is a table of the three planned or existing hadron-hadron colliders, and the estimated values of their cm. energy, luminosity and corresponding event rates for a reference cross sectin of 1 Picobarn. Also listed is their starting year of operation. There are two values of luminosity listed for the SSC, as it is not decided as yet if it will be a proton-proton or proton-antiproton collider. Collider year cm. energy Luminosity rate/1 Pb. (TeV) cm s _ SPS 1980 0.54 10 1 event/115 day FERMILAB 1986 2 10 1 event/11.5day SSC PP PP 1995? 10 to 40 1 event/1 1.5day 1 event/17 min 


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