PROPERTIES AND CALCULATIONS OF MULTI PARTON PROCESSES PROPERTIES AND CALCULATIONS OF MULTI PARTON PROCESSES PROPERTIES AND CALCULATIONS OF MULTI PARTON PROCESSES PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE RIJKSUNIVERSITEIT TE LEIDEN, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. J.J.M. BEENAKKER, HOOG- LERAAR IN DE FACULTEIT DER WISKUNDE EN NATUUR- WETENSCHAPPEN, VOLGENS BESLUIT VAN HET COLLE- GE VAN DEKANEN TE VERDEDIGEN OP DONDERDAG 5 OKTOBER 1989 TE KLOKKE 15.15 UUR DOOR WALTER THEO GIELE GEBOREN TE DEN HAAG IN 1957 Promotie-commissie: Promotor : Prof. Dr. F.A. Bcrends Referent : Dr. W.L.G.A.M. van Neerven Overige leden : Prof. Dr. R. de Bruyn Ouboter Prof. Dr. J.M.J. van Leeuwen Prof. Dr. C.J.N. van den Mcijdenbrrg This investigation is part of the research program of the Stichting voor Funda- menteel Onderzoek der Materie (FOM) which is financially supported by the Ne- derlandse Organisatie voor Wctenschappelijk Onderzor-k (NWO). Contents I Outline 1 II Introduction 5 1 The theoretical model 5 2 A historical review of the calculations 9 III The helicity method 15 1 Introduction 15 2 The Weyl-van der Waerden implementation 1G 2.1 The formalism 16 2.2 The momentum vectors 17 2.3 The spin-i particles 19 2.4 The spin-1 particles 20 IV The recursion relations 25 1 Introduction 25 2 The gluon recursion relation 26 3 The spinorial recursion relation 34 4 From currents to amplitudes and cross sections 36 4.1 Scattering of n gluons 36 4.2 The process producing qq and n gluons 38 4.3 The process producing qq, n gluons and a vector boson ... 38 V Solutions of the recursion relations 41 1 Introduction 41 2 QED examples for n photon emission 42 3 The gluon recursion relation 46 4 The quark current and its recursion relation 51 5 The amplitudes for specific helicity configurations 52 5.1 Scattering of n gluons 52 5.2 The process producing qq and n gluons 54 5.3 The process producing qq, n gluons and a vector boson ... 56 VI Multiple soft gluon radiation 61 1 Introduction 61 2 The gauge behaviour of the gluonic current 62 3 Single* soft gluon emission in an n-gluon process 65 4 Double soft gluon emission 67 5 Multiple soft gluon emission 71 6 Soft gluon emission in a process producing qq and n gluons ... 75 7 Soft gluon emission in a process producing qq, V and n gluons . 84 VII Exact expressions for multi gluon scattering 87 1 Introduction 87 2 The building blocks for multi gluon scattering 88 3 The 4- and 5-gIuon amplitudes 93 4 The 6-gluon amplitude 94 5 The 7-gluon amplitude ... 96 VIII Processes involving a vector boson end up to five partons 101 1 Introduction 101 2 Analytic results for the gluon and spinorial currents 104 3 The vector current for V —» qq + n gluons 109 4 The process V —* qqqq + n gluons 113 5 Squaring and crossing 116 6 The helicity currents 119 Samenvatting 125 Curriculum Vitae 127 List of Publications 128 Chapter I Outline The observation of jots at the PP collider in CERN by the UA1 and UA2 experiments [1] made it possible to interpret and organize the experimental data in a very clear and direct way. Jets are streams of hadrons moving more or less in the same direction. They can be understood as a result of large angle scattering of partons. These partons are the constituents of hadrons and are either qtiarks or gluons. Each outgoing parton which originates from such a large angle collision evolves into an observable jt-t through the mechanism of fragmentation. As long as one is interested in the global properties of jets, such as for example energy content or angular distributions, one can identify the outgoing parton with the jet. This viewpoint gives a very direct interpretation of the theoretical model in terms of experimental observables [2]. The model consists of two distinct parts. The first part describes the properties of the parton within the hadrons, such as momentum distributions of the partons. These distributions are the input parameters for the second part of the model. This describes the actual collision of two individual partons, each from one of the colliding hadrons. The interaction between these partons is described by Quantum Chromo Dynamics (QCD). The description of the partons within the hadron belongs to the realm of nonperturbative QCD, while the actual scattering process is described by perturbative QCD. As one can see the study of jets at the colliders is a direct test of the QCD description of the strong interactions. When one is also interested in the hadron contents of jets one adds to the above model another part, which describes the evolution of the outgoing partons into jets. Another important reason to study jets is given by the fact that any new physics, such as the detection of the top quark or the Higgs particle, is done by looking at the decay products of these heavy particles. The decay products are leptons and partons, which evolve into jets. The res»ilting experimental signal can be mimicked by other, more conventional processes which give exactly the same final state [3]. An example is the case where the top quark is heavier than the W mass. It is possible that in a PP collision a top quark pair is produced. These quarks do not evolve into jets because of their large mass, instead they decay via electroweak interactions into a final state with jets and/or leptons. Other processes, the background processes, can easily reproduce the same final state. So an important isstie is how to look for specific properties of the signal produced by the b quarks. For instance, one 1 considers the invariant mass distribution of the observed jets. One can then try to apply cuts on the signal which reduce the background in favour of the top quark signal [4]. Another important example is the search for the Higgs particle. Again the background processes threaten the ability to observe this particle, and all kinds of cuts must be applied to reduce this background with respect to that of the Higgs boson signal [5]. From the above it is clear that calculations of multi jet final states are crucial for the experiments at current and future colliders, especially with the increasing beam energies of the colliders. Any new and interesting physics is hidden in a background of multi jet final states, so it is important to be able to calculate the background processes. Unfortunately the calculations of such background processes are complicated. In this thesis we will study and develop methods to calculate these multi jet final states. In chap. 2 the parton model is described, the application of perturbative calculations is given and the shortcomings of the approach are discussed. Furthermore the difficulties in evaluating the processes are discussed following the historical development of the calculational methods. In chap. 3 we introduce our basic calculational tool, the Weyl-van der Waerden spinor calculus. This technique makes optimal use of the fact that in our calculations all partons are taken massless. This reduces a lot of algebra which is necessary in more conventional spinor calculus. The developed spinor calculus will be applied in chaps. 5, 6, 7 and 8, where the actual calculations of matrix elements are carried out. Chap. 4 introduces the recursion relations. These relations form a new technique, the basis for calculating multi parton matrix elements. In the next four chapters the recursive formulae will be applied to obtain relevant results. Firstly in chap. 5 the so called Parke-Taylor conjecture [6] and related conjectures are proven. The Parke-Taylor conjecture gives the matrix element for the scattering of two gluons to an arbitrary number of gluons for special helicity configurations of the gluons. The validity of this conjecture is important because approximate formulae for multi gluon processes are based on it [7]. The proof is extended to processes involving a quark pair with or without a vector boson. In chap. 6 we examine the soft gluon behaviour of multi parton processes. A number of factorization properties are proven for scattering amplitudes with an arbitrary number of gluons. These proofs make extensive use of the recursive techniques developed in the previous chapters. The subject of chaps. 7 and 8 is the explicit calculation of multi parton helicity amplitudes. Chap. 7 is concerned with n-gluon scattering, n < 8, while in chap. 8 the process involving a vector boson and up to 5 partons is calculated. References [1] UA2 contribution to the XXI international conference, Paris (1982); UA2 and UAl contributions to the Rome PP workshop (1983). [2] See for example: M. Jacob, Particle Physics, Cargese, 1987, eds. M. Levi, J.L. Basdevant, M. Jacob, D. Speiser, J. Weyers and R. Gastmans, and references therein. [3] See for a review: Z. Kunszt and W.J. Stirling, Proceedings of the Workshop on Physics at future Accelerators, CERN-87-07, Vol. II, 548. [4] H. Baer, V. Barger, H. Goldberg and R.J.N. Phillips, Phys. Rev. D37 (1988) 3152; R. Kleiss, A.D. Martin and W.J. Stirling, Z. Phys. C39 (1988) 393; S. Gupta and D.P. Roy, Z. Phys. C39 (1988) 417; Top Quark Signatures at the Tevatron Collider, F. Halzen, C.S. Kim and A.D. Martin, Durham University preprint, DTP/88/34 (1988); On Detecting New Heavy Quarks at Hadron Colliders, P. Agrawal and S.D. Ellis, University of Washington preprint, DOE-ER-40423-19-P8 (1988); Aspects of Top Quark Searches in TeV Hadron Collisions, J.L.