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Polarization in heavy quark decays [Elektronische Ressource] / Kadeer Alimujiang

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Polarization in heavy quark decaysDissertationzur Erlangung des GradesDoktor der Naturwissenschaften“”am Fachbereich Physikder Johannes Gutenberg-Universit¨atin MainzKadeer Alimujianggeboren im Uigurischen Autonomen Gebiet Xinjiang, V. R. ChinaMainz, 2006Date of the oral examination: 30.11.200666AbstractIn this thesis I concentrate on the angular correlations in top quark decays and theirnext–to–leading order (NLO) QCD corrections. I also discuss the leading–order (LO)angular correlations in unpolarized and polarized hyperon decays.In the first part of the thesis I calculate the angular correlation between the top quarkspin and the momentum of decay products in the rest frame decay of a polarized topquark into a charged Higgs boson and a bottom quark in Two-Higgs-Doublet-Models:+t(↑)→ b+H . The decay rate in this process is split into an angular independent part(unpolarized) and an angular dependent part (polar correlation). I provide closed formformulae for theO(α ) radiative corrections to the unpolarized and the polar correlationsfunctions for m = 0 and m = 0. The results for the unpolarized rate agree with theb bexisting results in the literature. The results for the polarized correlations are new. Ifound that, for certain values of tanβ, theO(α ) radiative corrections to the unpolarized,spolarized rates, and the asymmetry parameter can become quite large.

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Polarization in heavy quark decays
Dissertation
zur Erlangung des Grades
Doktor der Naturwissenschaften“

am Fachbereich Physik
der Johannes Gutenberg-Universit¨at
in Mainz
Kadeer Alimujiang
geboren im Uigurischen Autonomen Gebiet Xinjiang, V. R. China
Mainz, 2006Date of the oral examination: 30.11.2006Abstract
In this thesis I concentrate on the angular correlations in top quark decays and their
next–to–leading order (NLO) QCD corrections. I also discuss the leading–order (LO)
angular correlations in unpolarized and polarized hyperon decays.
In the first part of the thesis I calculate the angular correlation between the top quark
spin and the momentum of decay products in the rest frame decay of a polarized top
quark into a charged Higgs boson and a bottom quark in Two-Higgs-Doublet-Models:
+t(↑)→ b+H . The decay rate in this process is split into an angular independent part
(unpolarized) and an angular dependent part (polar correlation). I provide closed form
formulae for theO(α ) radiative corrections to the unpolarized and the polar correlations
functions for m = 0 and m = 0. The results for the unpolarized rate agree with theb b
existing results in the literature. The results for the polarized correlations are new. I
found that, for certain values of tanβ, theO(α ) radiative corrections to the unpolarized,s
polarized rates, and the asymmetry parameter can become quite large.
InthesecondpartIconcentrateonthesemileptonicrestframedecayofapolarizedtop
+quark into a bottom quark and a lepton pair: t(↑)→X +` +ν . I analyze the angularb `
correlations between the top quark spin and the momenta of the decay products in two
different helicity coordinate systems: system 1a with the z–axis along the charged lepton
momentum, and system 3a with the z–axis along the neutrino momentum. The decay
rate then splits into an angular independent part (unpolarized), a polar angle dependent
part (polar correlation)and an azimuthal angle dependent part (azimuthalcorrelation). I
presentclosedformexpressionsfortheO(α )radiativecorrectionstotheunpolarizedparts
and the polar and azimuthalcorrelationsin system 1a and 3a form = 0 andm =0. Forb b
theunpolarizedpartandthepolarcorrelationIagreewithexistingresults. Myresultsfor
the azimuthal correlations are new. In system 1a I found that the azimuthal correlation
vanishes in the leading order as a consequence of the (V −A) nature of the Standard
Model current. The O(α ) radiative corrections to the azimuthal correlation in systems
1a are very small (around 0.24% relative to the unpolarized LO rate). In system 3a the
azimuthal correlation does not vanish at LO. The O(α ) radiative corrections decreasess
the LO azimuthal asymmetry by around 1%.
In the last part I turn to the angular distribution in semileptonic hyperon decays.
Using the helicity method I derive complete formulas for the leading order joint angular
decay distributions occurring in semileptonic hyperon decays including lepton mass and
polarizationeffects. Comparedtothetraditionalcovariantcalculationthehelicitymethod
allowsonetoorganizethecalculationoftheangulardecaydistributionsinaverycompact
and efficient way. This is demonstrated by the specific example of the polarized hyperon
0 + − − − − + 0decayΞ (↑)→ Σ +l +ν¯ (l =e ,μ )followedbythenonleptonicdecayΣ →p+π ,l
which is described by a five–fold angular decay distribution.
66Contents
1 Introduction 1
+2 Polarization Effects in t(↑)→b+H 7
2.1 The Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Virtual one–loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Vertex corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Quark self–energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3 Renormalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Renormalized virtual one–loop corrections . . . . . . . . . . . . . . 27
2.3 Real gluon emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 The amplitude squared for the real emissions . . . . . . . . . . . . . 31
2.3.2 The phase space integration . . . . . . . . . . . . . . . . . . . . . . 34
2.3.3 Integration of the soft gluon factor . . . . . . . . . . . . . . . . . . 42
2.3.4 Full real emission contributions . . . . . . . . . . . . . . . . . . . . 49
2.4 TotalO(α ) results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50s
+3 Angular correlations for t(↑)→b+` +ν in system 1a 59`
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 The Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Virtual one–loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.1 Vertex corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2 Renormalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.3 Renormalized virtual one–loop correction . . . . . . . . . . . . . . . 80
3.4 Real gluon emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4.1 The amplitude squared for the real emissions . . . . . . . . . . . . . 84
3.4.2 Phase space integration. . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5 TotalO(α ) results for system 1a . . . . . . . . . . . . . . . . . . . . . . . 94s
3.6 Azimuthal correlation atO(α ) . . . . . . . . . . . . . . . . . . . . . . . . 101s
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
+4 Angular correlations for t(↑)→b+` +ν in system 3a 109`
4.1 The Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 Virtual one–loop correction . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3 Real gluon emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4 TotalO(α ) results for system 3a . . . . . . . . . . . . . . . . . . . . . . . 120sii CONTENTS
4.5 Azimuthal correlation atO(α ) . . . . . . . . . . . . . . . . . . . . . . . . 126s
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 Helicity Analysis of Semileptonic Hyperon Decays 139
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2 The helicity amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3 Unpolarized decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.4 The rate ratio Γ(e)/Γ(μ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.5 Single spin polarization effects . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.5.1 Polarization of the daughter baryon . . . . . . . . . . . . . . . . . . 152
5.5.2 Polarization of the lepton . . . . . . . . . . . . . . . . . . . . . . . 153
5.5.3 Decay of a polarized parent baryon . . . . . . . . . . . . . . . . . . 154
5.6 Joint angular decay distribution . . . . . . . . . . . . . . . . . . . . . . . . 157
5.7 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Appendix 161
A Notations 161
A.1 Dirac matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.2 Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.3 Gell–Mann matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.4 The CKM matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.5 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.5.1 Outer lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.5.2 Propagators in the Feynman gauge . . . . . . . . . . . . . . . . . . 166
A.5.3 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
B Calculation of the loop integrals 168
B.1 Feynman parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B.2 Scalar loop integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
C Polylogarithm 178
D Basic integrals for the phase space integrations 181
D.1 Two–body phase space integration R (P;p ,k) . . . . . . . . . . . . . . . . 1812 b
+D.2 Basic z–integrals for t(↑)→H +b . . . . . . . . . . . . . . . . . . . . . . 183
+D.3 Basic z–integrals for t(↑)→b+` +ν . . . . . . . . . . . . . . . . . . . . 186`
D.4 Coefficient functions for system 1a. . . . . . . . . . . . . . . . . . . . . . . 188
D.5 Coefficient functions for system 3a. . . . . . . . . . . . . . . . . . . . . . . 191
D.6 Basic integrals for the azimuthal calculation . . . . . . . . . . . . . . . . . 194
E Some technical notes on the semileptonic hyperon decays 196
E.1 T–odd contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
E.2 Full five-fold angular decay distribution . . . . . . . . . . . . . . . . . . . 197
Bibliography 201List of Figures
1.1 Top quark pair production at the Tevatron . . . . . . . . . . . . . . . . . . 2
1.2 Single top quark production at the Tevatron . . . . . . . . . . . . . . . . . 3
2.1 Definition of the polar angle θ . . . . . . . . . . . . . . . . . . . . . . . . 8P
2.2 LO rates as functions of m /m in model 1 . . . . . . . . . . . . . . . . . 13H t
2.3 Asymmetry parameter as a function of m /m in model 1 . . . . . . . . . 13H t
2.4 LO rates as functions of m /m in model 2 . . . . . . . . . . . . . . . . . 14H t
2.5 Asymmetry parameter as a function of m /m in model 2 . . . . . . . . . 14H t
2.6 The tanβ dependence of asymmetry parameter in model 2 . . . . . . . . . 15
+2.7 Feynman diagrams for the virtual one–loop corrections in t→H +b. . . 15
2.8 The quark self–energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.9 Contribution of 1-particle irreducible diagram to a fermion propagator. . . 24
2.10 Contribution of 1-particle irreducible diagram to a quark propagator. . . . 26
+2.11 Feynman graphs for the real gluon emission in t(↑)→H +b . . . . . . . 31
2.12 Three–body phase space as sequence of two–body phase spaces . . . . . . . 34
2.13 The P–rest frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.14 NLO QCD corrections to the rates in model 1 . . . . . . . . . . . . . . . . 54
2.15 NLO QCD corrections to the asymmetry parameter in model 1 . . . . . . . 54
2.16 NLO QCD corrections to the rates in model 2 . . . . . . . . . . . . . . . . 55
2.17 NLO QCD corrections to the asymmetry parameter in model 2 . . . . . . . 55
2.18 NLO QCD corrections to the tanβ dependence of α in model 2 . . . . . . 56H
+3.1 The helicity systems 1a, 2’a and 3a in t(↑)→b+` +ν . . . . . . . . . . 60`
+3.2 Feynman diagrams for the decay t(↑)→b+` +ν at LO . . . . . . . . . 61`
2 +3.3 LO y –spectrum in t(↑)→b+` +ν . . . . . . . . . . . . . . . . . . . . . 67`
3.4 LO charged lepton spectra with and without NW–approximation. . . . . . 69
3.5 Feynman diagrams for the virtual one–loop corrections in t→b+`+ν . . 71`
+3.6 Feynman diagrams for the real emissions in t→b+` +ν . . . . . . . . . 84`
3.7 Unpolarized LO and NLO charged lepton spectra in system 1a. . . . . . . 106
3.8 Polarized LO and NLO charged lepton spectra in system 1a. . . . . . . . . 106
3.9 Azimuthal charged lepton spectra in system 1a. . . . . . . . . . . . . . . . 106
+4.1 The helicity system 3a for t(↑)→b+` +ν . . . . . . . . . . . . . . . . . 109`
4.2 QCD NLO corrections to the azimuthal neutrino spectrum . . . . . . . . . 126
4.3 Unpolarized LO and NLO neutrino spectra in system 3a. . . . . . . . . . . 134
4.4 Polarized LO and NLO neutrino spectra in system 3a.. . . . . . . . . . . . 135iv LISTOFFIGURES
4.5 Azimuthal LO and NLO neutrino spectra in system 3a . . . . . . . . . . . 135
25.1 q –dependence of the six independent helicity amplitudes.. . . . . . . . . . 143
+ −5.2 Phase spaces for the electron and muon modes in Ξ→ Σ +` +ν¯. . . . 151`
− −5.3 Polarization of the μ in the (μ ,ν¯ ) c.m. system. . . . . . . . . . . . . . . 154μ
5.4 Definition of the polar angles θ, θ and χ . . . . . . . . . . . . . . . . . . . 155P
5.5 Definition of the polar angles θ , θ and φ . . . . . . . . . . . . . . . . . . 156B B B
5.6 Definition of the polar angles θ, θ and χ in the joint angular decay distri-B
0 0 + 0 −butionofanunpolarizedΞ inthecascadedecayΞ → Σ (→p+π )+` +ν¯.158`
B.1 The contour of the k –integration . . . . . . . . . . . . . . . . . . . . . . . 1690
E.1 Definition of the three polar angles θ, θ and θ in the semileptonic decayB P
0 + −of a polarized Ξ into Σ + ` + ν¯ followed by the nonleptonic decay`
+ 0Σ →p+π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
E.2 Definition of the three azimuthal angles φ , φ and χ in the semileptonic` B
0decay of a polarized Ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Chapter 1
Introduction
The top quark, the most massive fundamental particle in the Standard Model (SM), was
/discovered in 1995 by the CDF and DO collaborations at the Tevatron collider at Fermi-
/lab [1, 2]. The top quark mass was measuredat CDF and DO to bem =174.2±3.3GeVt
(polemass)[3]. Theexistenceofthethetopquarkwaspredictedastheelectroweakisospin
partner of the b–quark (see Table 1.1), which was discovered in 1977 [4]. Although the
top quark discovery was anticipated the large mass of the top quark was a big surprise.
The top quark has a mass slightly less than the mass of the gold atom, approximately
twice that of W and Z bosons, the carriers of the electroweak force, and thirty–five times
thatofthenextmostmassivefermion, theb–quark. TheSMneitherpredictsnorexplains
the observed mass hierarchy.
At the top mass scale the strong coupling constant is small, α (m )∼ 0.1. Therefores t
QCD effects involving the top quark are well behaved in the perturbative sense, i.e. the
top quark decays provide an ideal tool for studying perturbative QCD.
The interplay between the large top mass and its spin is of crucial importance in
studying the SM. The decay width of the top quark is dominated by the two–body chan-
+nel t → b + W because the top quark has a mass above the Wb threshold, and the
Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix element V is close to unity withtb
other channels heavily suppressed. The top quark decay width increases with the top
quark mass. The top quark decay width Γ = 1.39GeV for the pole mass of 174 GeVt
was calculated for this channel in the SM to second order in QCD [5] and to first or-
der in EW [6] corrections. Consequently the SM result of the top quark lifetime is
−24τ ≈ 0.5× 10 s. This is much shorter than the typical time for the formation oft
−24QCD bound states τ ≈ 1/Λ ≈ 3×10 s, i.e. the top quark decays long before itQCD QCD
can hadronize [7]. Therefore top decays provide a very clean source of information about
the structureof the SM. In particularthe time scale is much shorterthan the typicaltime
requiredfortheQCDinteractionstorandomizeitsspin. Thereforethepolarizationofthe
top quark is preserved in the decay and can be studied through the angular correlations
betweenthedirectionofthetopquarkspinandthemomentaofthedecayproducts. Mea-
suring the top quark polarization and comparing it to the theoretical predictions would
present a clean test of the SM.
Until recently the precision of the measurements of the top quark have been limited
because of the relatively small top production cross section leading to a small number of2 1. Introduction
fermions bosons
st nd rd1 generation 2 generation 3 generation γ³ ´ ³ ´ ³ ´
ν ν νe μ τ ± 0leptons , e , μ , τ W ,ZR R Re μ τL L L³ ´ ³ ´ ³ ´
u c t
quarks , u ,d , c ,s , t ,b gR R R R R Rd s bL L L
Table 1.1: The Standard Model of the elementary particles. The matter is made up of the
fermions, the quarks and lepton which are divided into three families or generations. The left–
handed fermions form isospin doublets while right–handed fermions are isospin singlets. The
interaction between fermions are mediated by gauge bosons.
events. Atpresenttheworld’sonlysourceoftopquarksistheFermilabTevatroncollider.
AttheTevatronthetopquarksareproducedmainlyinpairsbypp¯collisionatacenter–of–√
massenergyof s =1.96TeV.AttheTevatronenergytopquarkpairsareproducedfrom
¯ ¯quark annihilation (qq¯→tt) 85% of the time and from gluon annihilation (gg→tt) 15%
of the time (see Fig. 1.1). Single top quark production occur via electroweak production
0 ∗ 0 ∗ 0¯ ¯mechanism qq¯ →W →bt or qg→qW →qbt with roughly a factor 3 suppression [9].
g t
q t g t
g g
g t
q t g t
( 85% ) ( 15% )
¯ ¯Figure 1.1: Leading order Feynman diagrams for top quark pair production via qq¯→tt or gg→tt
at the Tevatron. The total production cross section is ca. 7pb.
The top quark pair production cross section depends strongly on the top quark mass.
The theoretical predictions for the top quark pair production cross section at QCD NLO√
are around 6.7pb [13, 14] (at m = 175GeV, s = 1.96TeV). These predictions aret
1 /in good agreement with the measurements at CDF of 7.32±0.85pb [15] and at DO
+1.9of 7.1 pb [16]. With such a pair production cross section, in order to produce large−1.7
number of top quarks for precision studies, the collider must have a very high luminosity.
¯In the Run I period (1992-1996) there were around 1200 tt pairs in the interaction
/region where the detectors DO and CDF are situated. In the Run II period (2001 till
now) this number increased dramatically (see Table 1.2). The future colliders are also
powerful top quark factories. The Large Hadron Collider (LHC) at CERN is planned to
start in 2007, initial measurements in 2008, and the first precision measurements perhaps
−1in 2009 with integrated luminosity between 1 and 10 fb . The planned International
Linear Collider, the ILC [8] will hopefully run from 2015. Highly polarized top quarks
1Preliminary results, not yet submitted for publication as of September 2006.