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Gauge Checks, Consistency of Approximation Schemes and Numerical Evaluation of Realistic Scattering Amplitudes

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Gauge Checks, Consistency of Approximation Schemes and Numerical Evaluation of Realistic Scattering Amplitudes Gauge checks, consistency of approximation schemes and numerical evaluation of realistic scattering amplitudes Vom Fachbereich Physik der Technischen Universit¨at Darmstadt zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Dipl.-Phys. Christian Schwinn aus Darmstadt Referent: Prof. Dr. P.Manakos Korreferent: Prof.Dr. N.Grewe Tag der Einreichung: 18.4.2003 Tag der Prufung:¨ 23.6.2003 Darmstadt 2003 D17 2 3 Abstract In this work we discuss both theoretical tools to verify gauge invariance in numerical calculations of cross sections and the consistency of approximation schemes used in realistic calculations. We determine a finite set of Ward Identities for 4 point scattering ampli- tudes that is sufficient to verify the correct implementation of Feynman rules of a spontaneously broken gauge theory in a model independent way. These identities have been implemented in the matrix element generator O’Mega and have been used to verify the implementation of the complete Standard Model in Rξ gauge. As a theoretical tool, we derive a new identity for vertex functions with several momentum contractions. The problem of the consistency of approximation schemes in tree level cal- culations is discussed in the last part of this work. We determine the gauge in- variance classes of spontaneously broken gauge theories, providing a new proof for the formalism of gauge and flavor flips. The schemes for finite width effects that have been implemented in O’Mega are reviewed. As a comparison with existing calculations, we study the con- − + − ¯ sistency of these schemes in the process e e → e ν¯eud. The violations of gauge invariance caused by the introduction of running coupling constants are analyzed. Zusammenfassung Diese Arbeit besch¨aftigt sich mit theoretischen Werkzeugen zur Uberpr¨ ufung¨ von Eichinvarianz in numerischen Berechnungen von Streuquerschnitten sowie mit der Konsistenz von N¨aherungsschemata, die in realistischen Rechnungen angewendet werden. Ein endlicher Satz von Ward Identit¨aten von 4 Punkt Streuamplituden wird bestimmt, der es erlaubt, die korrekte Implementierung der Feynmanregeln einer spontan gebrochenen Eichtheorie modellunabh¨angig zu verifizieren. Diese Identit¨aten wurden in den Matrixelementgenerator O’Mega implementiert und zur Uberpr¨ ufung¨ der Implementierung des vollst¨andigen Standardmodells in Rξ Eichung benutzt. Als theoretisches Hilfsmittel wird eine neue Identit¨at fur¨ Ver- texfunktionen mit mehreren Impulskontraktionen hergeleitet. Im letzten Teil der Arbeit wird das Problem der Konsistenz von N¨aherungen in tree-level Rechnungen diskutiert. Wir bestimmen die Eichinvarianzklassen in spontan gebrochenen Eichtheorien und geben einen neuen Beweis des Formalis- mus der Eich- und Flavorflips. Die in O’Mega implementierten Schemata zur Behandlung endlicher Zer- fallsbreiten werden vorgestellt. Zum Vergleich mit existierenden Rechnungen − + − ¯ untersuchen wir die Konsistenz dieser Schemata im Prozess e e → e ν¯eud. Verletzungen der Eichinvarianz durch die Einfuhrung¨ laufender Kopplungskon- stanten werden analysiert. 4 Die Philosophie steht in diesem großen Buch geschrieben, dem Universum, das unserem Blick st¨andig offenliegt. Aber das Buch ist nicht zu verstehen, wenn man nicht zuvor die Sprache erlernt und sich mit den Buchstaben vertraut gemacht hat, in denen es geschrieben ist. Es ist in der Sprache der Mathematik geschrieben, (. ), ohne die es dem Menschen unm¨oglich ist, ein einziges Wort davon zu verstehen; ohne diese irrt man in einem dunklen Labyrinth umher. Galileo Galiliei: zitiert nach: Albrecht F¨olsing: Galileo Galiliei Prozeß ohne Ende, Eine Biographie Contents 1 Introduction 1 2 Gauge invariance in numerical calculations: tool and challenge 5 2.1 Tree-level unitarity and gauge invariance . 6 2.2 Consequences of gauge invariance: Ward Identities . 7 2.2.1 Global Symmetries . 7 2.2.2 Quantum electrodynamics . 8 2.2.3 Yang-Mills-Theory . 9 2.2.4 Massive vector bosons . 10 2.3 Ward Identities as tool in numerical calculations . 11 2.3.1 Reconstruction of the Feynman rules? . 11 2.3.2 Numerical checks of Ward identities . 12 2.4 Gauge invariance classes . 13 2.4.1 QED . 13 2.4.2 Nonabelian gauge theories . 14 2.4.3 Forests, Groves and flips . 15 2.5 Gauge invariance and finite widths . 16 I Gauge invariance of tree level amplitudes 17 3 Slavnov Taylor Identities 19 3.1 STI for Green’s functions . 19 3.1.1 The general STI of Green’s functions . 20 3.1.2 STIs for amputated Green’s functions . 22 3.2 Zinn-Justin equation . 24 3.3 STI for physical vertices . 26 3.3.1 Graphical notation . 27 3.3.2 Tree level . 28 3.4 STI for vertices with several contractions . 30 3.4.1 3 point function . 31 3.4.2 4 point function . 32 4 Diagrammatical analysis of STIs 33 4.1 4 point functions . 34 4.1.1 Ward Identities . 34 4.1.2 Slavnov Taylor Identity . 36 4.1.3 Green’s function with 2 unphysical gauge bosons . 38 5 6 CONTENTS 4.2 Gauge parameter independence . 39 4.3 Gauge invariance classes . 42 4.3.1 Definition of gauge invariance classes . 42 4.3.2 Definition of gauge flips . 44 4.3.3 ff¯ → ffW¯ .......................... 44 4.3.4 General 5 point amplitudes . 46 4.3.5 N point diagrams . 47 II Reconstruction of Feynman rules 51 5 Lagrangian of a spontaneously broken gauge theory 53 5.1 Field content and symmetries . 53 5.1.1 Gauge fields . 53 5.1.2 Scalar fields . 54 5.1.3 Fermions . 55 5.1.4 Majorana Fermions . 55 5.1.5 Unbroken Symmetries . 56 5.2 Lagrangian . 56 5.3 Symmetry conditions and implications of SSB . 58 5.4 Input parameters and dependent parameters . 60 5.5 Example:Standard model . 62 5.6 Example: SUSY Yang-Mills . 63 6 Reconstruction of the Feynman rules from the Ward-Identities 65 6.1 Cubic Goldstone boson couplings . 66 6.1.1 Couplings of one Goldstone boson . 66 6.1.2 Couplings of 2 or 3 Goldstone bosons . 67 6.2 Gauge invariance of physical couplings . 68 6.2.1 Example: WWHH Ward identity . 68 6.2.2 Gauge boson and fermion couplings . 70 6.2.3 Symmetry of Higgs-Yukawa couplings . 70 6.3 Goldstone boson couplings . 71 6.3.1 Quartic Goldstone boson -gauge boson couplings . 71 6.3.2 Lie algebra structure of the triple Goldstone boson couplings 72 6.3.3 Global invariance of Goldstone boson Yukawa couplings . 73 6.3.4 Scalar potential . 73 6.4 Summary . 74 7 Gauge checks in O’Mega 75 7.1 Architecture of O’Mega . 75 7.2 Implementation of Ward identities . 77 7.3 Implementation of Slavnov-Taylor identities . 79 III Consistency of realisitic calculations: selection and resummation of diagrams 81 8 Forests and groves in spontaneously broken gauge theories 83 8.1 Definition of gauge flips . 83 CONTENTS 7 8.1.1 ff¯ → WW .......................... 84 8.1.2 Elementary flips in spontaneously broken gauge theories . 85 8.1.3 Flips for nonlinear realizations of the symmetry . 86 8.2 Structure of the groves . 87 8.2.1 Linear parametrization . 87 8.2.2 Nonlinear realizations . 88 8.2.3 Numerical checks . 91 9 Finite width effects 93 9.1 Dyson summation and violation of Ward Identities . 93 9.2 Simple schemes for finite width effects . 94 9.2.1 Step width . 95 9.2.2 U(1) restoring schemes . 95 9.2.3 SU(2) restoring schemes . 96 9.2.4 Loop schemes . 97 9.3 Numerical results for single W production . 98 9.3.1 Comparison of Matrix elements . 98 9.3.2 Results for cross sections . 101 10 Effective coupling constants 105 10.1 Running coupling constants and effective Weinberg angle . 105 10.2 Cubic vertices involving Goldstone bosons . 107 10.3 Gauge boson couplings . 110 10.3.1 Conditions from Ward Identities . 110 10.3.2 Modification of Feynman rules . 112 Summary and Outlook 115 IV Appendix 117 A BRS symmetry 119 A.1 BRS formalism . 119 A.2 Application to a spontaneously broken gauge theory . 120 A.3 Graphical notation . 122 B Nonlinear realizations of symmetries 125 B.1 General setup . 125 B.2 STIs for nonlinearly realized symmetries . 127 C More on STIs 131 C.1 STIs for amputated Green’s Functions . 131 C.2 STIs in unitarity gauge . 134 C.3 Ghost terms in the STI with 2 contractions . 136 C.4 Explicit form of STIs . 137 D Lagrangian and coupling constants 143 D.1 Parametrization of the general Lagrangian . 143 D.2 Relations among coupling constants . 145 8 CONTENTS E Explicit form of flips 149 E.1 Gauge flips . 149 E.2 Flavor and Higgs flips . 150 F Calculation of Ward Identities 153 F.1 Ward identities for 3 point functions . 153 F.2 WIs for 4 point function with one contraction . 157 F.3 4 point WIs with several contractions . 162 Bibliography 171 Chapter 1 Introduction The agreement between the theoretical predictions of the Standard Model (SM) of the electroweak interactions and experiment is established to an impressive degree [1]. The only missing ingredient is the Higgs boson that yet has to be discovered. However, there are compelling theoretical reasons for believing that the electroweak Standard Model is merely a low energy approximation to a more fundamental theory that should become visible at TeV scale energies. Thus indications on the underlying theory should be found by experiments at the LHC or at future linear colliders (see e.g. [2]). These future experiments pose the challenge to theorists to make predictions for processes with many particles in the final state. This is a general signature for processes with heavy, unstable particles in intermediate states that have to be considered to identify the nature of the new physics phenomena. Because the number of (tree-level) Feynman diagrams contributing to the scattering amplitude is growing rapidly with the number of external particles (it can be shown, that this growth is factorial in an unflavored φ3 theory [3]), it is necessary to perform completely automatized numerical calculations [5].
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