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Renormalization of Fermion Mixing

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Renormalization of Fermion Mixing Renormalization of Fermion Mixing Dissertation zur Erlangung des Grades ,^)oktor der Naturwissenschaften" am Fachbereich Physik der Johannes Gutenberg-Universitat in Mainz Roxana gchiopu geb. in Rimnicu Vilcea, Rumanien Mainz, den 11. Mai 2007 M'bZer (md ZoMg e^ougA, ZAe avocado w;H grow. Datum der miindlichen Pnifmig: 11. Mai 2007 Contents 1 Introduction and Motivation 1 2 Lagrange Density and Feynman Rules for Dirac and Majorana Fermions 5 2.1 Remarks on Conventions and Notations ................................................. 5 2.2 Lagrange Density Mass Term ................................................................. 6 2.2.1 Dirac Mass Term.......................................................................... 6 2.2.2 Majorana Mass Term.................................................................... 7 2.2.3 Dirac-Majorana Mass Term....................................................... 11 2.3 General Form for the Lagrange Density with Dirac and Majorana Fermions ...................................................................................................... 13 2.4 Feynman Rules......................................................................................... 15 2.4.1 Feynman Rules with Dirac Particles........................................... 16 2.4.2 Feynman Rules with Dirac and Majorana Particles................... 18 3 One-loop Fermion Self-Energies 23 3.1 General Expression for the Dirac Fermion Self-Energy .........................23 3.1.1 Diagrams with Vector Bosons ........................................................ 23 3.1.2 Diagrams with Scalar Bosons ........................................................ 28 3.1.3 Analysis of Divergent and Imaginary Parts....................................30 3.2 General Expressions for the Dirac and Majorana Fermion Self-Energies 31 3.2.1 Dirac Fermion Self-Energy .............................................................. 32 3.2.2 Majorana Fermion Self-Energy ........................................................ 34 3.3 Summary of Results for Fermion Self-Energies .....................................39 4 Divergences of Fermion Self-Energies 45 4.1 Analysis of the One-loop Fermion Propagator ..................................... 47 4.2 Imaginary Parts, Divergences and Gauge Dependence ............................ 55 5 Renormalization of the Free Fermionic Lagrangian 63 5.1 Renormalized Free Dirac Lagrangian ........................................................ 63 i CONTENTS 5.1.1 General Renormalized Free Lagrangian .....................................64 5.1.2 Hermitian Renormalized Free Lagrangian ..................................70 5.1.3 Re-diagonalized Maas Term Approach ........................................ 71 5.2 Renormalized Free Major ana Lagrangian .................................................. 80 5.2.1 General Renormalized Free Lagrangian .....................................80 5.2.2 Hermitian Renormalized Free Lagrangian ..................................85 5.2.3 Re-diagonalized Mass Term Approach ........................................86 5.3 Remarks on CPT Invariance ........................................................................88 6 Renormalization of the Interaction Lagrangian 93 6.1 Interaction Terms with Dirac Fermions ..................................................... 93 6.1.1 Interaction Terms with Vector Bosons ............................................94 6.1.2 Interaction Terms with Scalar Bosons ............................................98 6.2 Interaction Terms with Majorana Fermions ............................................. 100 6.3 Generic Fermion Interaction Terms..........................................................102 6.4 Feynman Rules Derived from the Renormalized Lagrange Density . 104 6.5 One-loop Corrections to Vertices................................................................107 7 Renormalization of the Quark Mixing Matrix 113 7.1 Corrections to the Quark-Antiquark-IV Vertex ....................................... 113 7.1.1 Corrections from the Renormalized Parameters ........................114 7.1.2 Corrections from the One-loop Diagrams....................................116 7.1.3 Discussion of the One-loop Renormalized Quark Mixing Matrix 119 7.2 Top Decay Rate ........................................................................................ 123 8 Neutrino Seesaw Mechanism and the Renormalization of the Mix­ ing Matrix 131 8.1 Seesaw Mass Term..................................................................................... 132 8.2 Lagrange Density for Seesaw Type I......................................................... 134 8.3 Lagrange Density for Seesaw Type II...................................................... 139 8.4 Model Restrictions and Field Renormalization Constants ..................... 145 8.5 Corrections to the Neutrino Mixing Matrix and Unitarity .....................149 8.6 Corrections to the Lepton-IV Vertex ...................................................... 150 8.7 Heavy Neutrino Decay Rate......................................................................156 9 Summary 159 A Dimensional Regularization 163 CONTENTS iii B One- and Two-point Integrals 165 B.l Definitions ....................................................................................................165 B. 2 Evaluation of the One- and Two-point Integrals ................................... 166 B.2.1 Complete Evaluation of Bo(p2; mi, mg).....................................168 B.2.2 Particular On-shell Cases for Eg, Ei and Their Derivatives . 171 B.2.3 Complete Evaluation of Eg and E^............................................. 172 C Matrix Manipulations for One-loop Calculations 175 C. l Matrix Inversion ........................................................................................175 C.2 Unitary Matrix ...........................................................................................176 C.3 Matrix Diagonalization .............................................................................. 177 Bibliography 183 Curriculum Vitae 184 IV CONTENTS Chapter 1 Introduction and Motivation The renormalization of the electroweak Standard Model has already a success­ ful history. In some of its sectors experimental measurements are sensitive to loop corrections and there, calculations with high precision are already mandatory. Sur­ prisingly, there are still open problems related to the renormalization of the quark sector starting with the one-loop corrections. The presence of the quark mixing matrix that needs to be renormalized and the presence of unstable particles raise difficulties in defining correct and complete renormalized parameters that preserve all symmetries of the Lagrangian. Going beyond the Standard Model, one has other models that allow for fermion mixing and there, a complete renormalization scheme is also required. For example, there are experimental evidences for neutrino oscillations and a model that describes the neutrino mixing has to be taken into account. One can also think of supersym ­ metric or other exotic particles. Mixing is present in the Standard Model for Dirac fermions (quarks), while beyond, it can involve also Majorana particles (possible candidates are neutrinos). The necessity of the renormalization for fermion mixing was already recognised in the late '70s [Sirlin]. In 1982, Aoki et al. [Aok82] were describing an on-shell renormalization scheme that led to a diagonal fermion propagator. This prescription was used in 1990 by Denner and Sack [Den90a] to calculate the one-loop counter terms of external legs and of the quark mixing matrix. It was the first attempt to provide an analytical result for a mixing matrix counter term. In the next decade, few articles followed, trying to use a similar prescription to calculate the counter term for quark or neutrino mixing matrices, for models with stable particles. In 1999, in [Gam99], a gauge parameter dependence problem was recognized in the renormalization prescription of [Den90a]. Starting from 2000 on, many authors tried to find a prescription that will have as result a renormalized fermion mixing matrix that is unitary, gauge parameter independent and that will lead to an UV-finite amplitude. These requirements were successfully accomplished for the corrections 1 2 1. Introduction and Motivation to the quark mixing matrix in [Den04], but treating ail quarks as stable particles. The difficult point remains the renormalization of the theory involving unstable particles. In [Esp02], it was shown that a correct on-shell renormalization scheme that accounts for unstable particles and that includes the absorptive contributions from the self-energies leads to wave function renormalization constants not related by hermiticity (i.e. the constant renormalizing the incoming fermion and the outgoing antifermion is different from the one that renormalizes the outgoing fermion and the incoming antifermion). The lack of hermiticity can destroy in this case the unitarity of the renormalized mixing matrix. The aim of this thesis is to provide a renormalization prescription for fermion mixing, in a general framework, a prescription that can be applied then to specific models. We base our calculations on a Lagrange density that describes generic in ­ teraction terms for Dirac and Majorana fermions, in models that allow for mixing. We also take into account the presence of unstable particles. The analysis of the renormalized Lagrangian is general, but the analytic formulas for the
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