Institut f¨ur Theoretische Physik Fakult¨at Mathematik und Naturwissenschaften Technische Universit¨at Dresden Implementation of τ-lepton Decays into the Event-Generator SHERPA Diplomarbeit zur Erlangung des akademischen Grades Diplom-Physiker vorgelegt von Thomas Laubrich geboren am 7. August 1980 in Karl-Marx-Stadt Dresden 2006 Ä Eingereicht am 31. M¨arz 2006 1. Gutachter: Prof. Dr. R¨udiger Schmidt 2. Gutachter: Dr. Frank Krauss Kurzfassung In der vorliegenden Diplomarbeit wird die Implementierung von τ-Lepton-Zerf¨allen in den Ereignis-Generator SHERPA beschrieben. Daf¨ur wird das Modul HADRONS++ entwickelt, wel- ches unstabile Leptonen und Hadronen in einer Kette zerfallen l¨asst. Die τ-Lepton-Zerf¨alle werden als erstes System vollst¨andig in das leicht-erweiterbare Modul eingebaut. Auf diese Weise wird zum einen dessen Infrastruktur getestet und zum anderen der Grundstein f¨ur die Implementierung von Hadronenzerf¨allen in SHERPA gelegt. Bei der Implementierung der Zerfallsketten spielen Spinkorrelationen eine große Rolle. Ein Algorithmus, der die korrekten Spinkorrelationen beim Zerfall ber¨ucksichtigt, wird im- plementiert und dargestellt. Um die Zerfallsamplituden in HADRONS++ zu berechnen, wird der Helizit¨atsformalismus verwendet. Die Grundlagen, auf denen dieser Formalismus basiert, werden gezeigt. Desweiteren wird die numerische Integration beschrieben, die eine wichtige Rolle f¨ur das Erstellen der Zerfallskinematik spielt. Die Parametrisierungen der einzelnen τ-Lepton-Zerfallskan¨ale werden detailliert beschrie- ben. Insbesondere wird die Chirale St¨orungstheorie und deren Extrapolation zu h¨oheren Energien betrachtet. Die Beschreibung vorkommender Resonanzen wird mittels des ph¨ano- menologischen K¨uhn-Santamar´ıa-Modells und einer effektiven Feldtheorie, der Resonanten Chiralen Theorie, vorgenommen. Beide Ans¨atze werden verglichen. Abstract In this diploma thesis the implementation of τ-lepton decays into the event-generator SHERPA is described. The module HADRONS++ which develops a decay chain for unstable leptons and hadrons is implemented. The τ-lepton decays are the first system being implemented into the easily-extendable module. On the one hand, the module’s infrastructure is tested in this way. On the other, the foundation for the implementation of hadron decays in SHERPA is laid. Spin correlation effects play a major role when implementing decay chains. An algorithm which takes account of the correct spin correlations during the decay is implemented and displayed. In order to compute the decay matrix elements in HADRONS++, the helicity formal- ism is used. The fundamentals on which this formalism is based are presented. Furthermore, the numerical integration algorithm is explained. It plays an important role for choosing the correct decay kinematics. The parameterisation of each τ-lepton decay channel is described in detail. Particularly, the Chiral Perturbation Theory and its extrapolation to higher energies are discussed. Oc- curring resonances are implemented according to the phenomenological K¨uhn-Santamar´ıa model and an effective field theory, the Resonance Chiral Theory. Both approaches are compared. Contents 1 Introduction....................................... ......................... 1 2 Chiral Perturbation Theory........................... ....................... 3 2.1 Motivation: Semileptonic τ-leptonDecays .................... 3 2.2 QCDandChiralSymmetry . .. .. .. .. .. .. .. .. 4 2.3 TheGoldstone-Theorem . 7 2.4 SymmetryRequirements. 8 2.5 Construction of an Effective Lagrangian . ....... 9 2.6 Extrapolation to Higher Energies . ...... 13 2.7 Breit-Wigner Parameterisation . ...... 13 2.8 ResonanceChiralTheory . 14 2.8.1 TheVielbeinField ............................. 14 2.8.2 Resonance Chiral Theory Lagrangians . 14 3 Implementation..................................... ........................ 17 3.1 HelicityAmplitudes .............................. 17 3.1.1 SpinorBasisandMassiveSpinors. 17 3.1.2 SpinorProducts .............................. 18 3.1.3 BasicBuildingBlocks . 19 3.2 NumericalIntegration . 20 3.2.1 Monte Carlo Integration . 21 3.2.2 ChoosingDecayKinematics . 22 3.3 SpinCorrelation ................................. 23 3.3.1 Production of Particles in the Hard Process . ...... 23 3.3.2 DecayofParticles ............................. 23 4 Parameterisation of Tau Decay Channels . ................... 27 4.1 LeptonicChannels ................................ 27 4.2 Semileptonic Channels: General Remarks . ....... 28 4.3 TheOne-PseudoscalarMode . 30 4.4 The Two-Pseudoscalar Mode with ∆S =0.................... 30 4.4.1 K¨uhn-Santamar´ıaModel. 31 4.4.2 ResonanceChiralTheory . 32 4.5 The Two-Pseudoscalar Mode with ∆S = 1................... 34 ± 4.5.1 K¨uhn-Santamar´ıaModel. 34 4.5.2 ResonanceChiralTheory . 35 4.6 TheThree-PseudoscalarMode . 37 4.6.1 K¨uhn-Santamar´ıaModel. 39 4.6.2 ResonanceChiralTheory . 42 4.7 TheFour-PionMode................................ 46 4.7.1 TheOne-ProngDecay ........................... 47 4.7.2 TheThree-ProngDecay . .. .. .. .. .. .. .. 48 4.8 TheRemainingModes ............................... 51 5 Results and Discussion............................... ....................... 53 5.1 LeptonicChannels ................................ 53 5.2 SemileptonicChannels . 54 5.2.1 ThePion/KaonChannel. .. .. .. .. .. .. .. 55 5.2.2 TheTwo-PionChannel .......................... 55 5.2.3 ThePion-KaonChannel. .. .. .. .. .. .. .. 57 5.2.4 TheThree-PionChannel . .. .. .. .. .. .. .. 59 5.2.5 TheKaon-Two-PionChannel . 61 5.2.6 ThePion-Two-KaonChannel . 63 5.2.7 TheFour-PionMode............................ 63 5.3 SpinCorrelations................................ 65 5.3.1 Intermediate Z-Boson ........................... 65 5.3.2 Intermediate W ±-Boson.......................... 65 6 Summary and Outlook.................................. .................... 69 A More on Chiral Perturbation Theory ...................... ................... 73 A.1 Axial and Vector Current at Order (p2) .................... 73 O A.1.1 The Left- and Right-Handed Currents . 73 A.1.2 TheOne-GoldstoneMode . 75 A.1.3 TheTwo-GoldstoneMode. 75 A.2 Effective Lagrangian at (p4)........................... 76 O B The Two-Body Decay................................... .................... 77 Bibliography......................................... ....................... 79 List of Tables 2.1 Definition of the Gell-Mann matrices. ...... 5 2.2 Values of the non-vanishing structure constants f abc............... 5 2.3 Notation for external gauge fields. ...... 10 2.4 Transformationproperties.. ...... 11 2.5 Notation of meson resonance fields in antisymmetric tensor notation. 14 3.1 , , -functions for different helicity combinations. ..... 19 X Y Z 3.2 Estimated error of various integration schemes . .......... 21 4.1 Resonances in the τ π π0ν , τ K K ν channel. 32 − → − τ − → − S,L τ 4.2 Resonances in the τ π K ν , τ K π0ν channel. 35 − → − S,L τ − → − τ 4.3 Parameters for the respective three-pseudoscalar channels............ 37 4.4 Notation of invariant masses in the three-pseudoscalar channel. 39 i 4.5 Parameters of the resonances in the three-pseudoscalar channels (FA, FS). 40 4.6 Three-meson channels with an anomaly term. ....... 41 4.7 Parameters of the resonances in the three-pseudoscalar channels (FV ). 42 4.8 Parameters of the resonances in the three-pseudoscalar channels (RχT). 45 4.9 Notation of invariant masses in τ 4πν mode. ................ 47 → τ 4.10 Parameters of the resonances in the four-pion channel (one-prong). 47 4.11 Contributions to the three-prong four-pion channel. .............. 48 4.12 Parameters of the resonances in the four-pion channel (three-prong). 49 6.1 List of all implemented τ-lepton decay channels. 71 List of Figures 2.1 Feynman diagram for the semi-leptonic channel. ......... 3 3.1 Decay algorithm for a particle with a spin density matrix............ 24 4.1 Feynman diagram for the leptonic channel. ....... 27 4.2 Feynman diagram for the semi-leptonic channel. ......... 28 4.3 Examples for strangeness-conserving and strangeness-changing channels. 29 4.4 Feynman diagram for the strangeness-conserving two-meson channel. 31 4.5 Feynman diagram for the loop contribution to the two-meson channel. 32 4.6 Feynman diagram for the three-meson channel. ........ 39 4.7 Feynman diagram for the parity-violating three-meson channel. 42 4.8 All contributions to the RχT form factors of the 3π-mode............ 44 5.1 Energy spectrum of the outgoing electron. ........ 53 5.2 The invariant mass distribution of the two-pion final state............ 56 5.3 The invariant mass distribution of the two-pion final state............ 56 5.4 The invariant mass distribution of the two-kaon final state. .......... 57 5.5 The branching ratio as a function of cd. ..................... 58 5.6 The invariant mass distribution of the pion-kaon final state. .......... 59 5.7 Invariant mass distribution of the three outgoing pions. ............ 60 5.8 Invariant mass distribution of the two outgoing pions. ............ 61 5.9 Invariant mass distribution of the three outgoing mesons. ........... 62 5.10 Invariant mass distribution of the two outgoing mesons. ............ 62 5.11 Invariant mass distributions of the 4π channel. ................. 64 5.12 Energy distribution of the outgoing pion. ......... 66 5.13 Invariant mass distribution of the outgoing two-pion final state. 66 5.14 Energy distribution of the outgoing pion. ......... 67 1 Introduction In 1975, the τ-lepton was discovered by M.L. Perl [1] when he and his colleagues