CP Violation in Production and Decay of Supersymmetric Particles [thesis]

142 Pages·2004·1.892 MB·English

by  O. Kittel

CP Violation
in Production and Decay
5 of Supersymmetric Particles
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2 Dissertation zurErlangung des

1 naturwissenschaftlichen Doktorgrades
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derBayerischen Julius-Maximilians-Universita¨t Wu¨rzburg
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vorgelegt von
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Wu¨rzburg 2004
Contents
Prologue 6
1 Introduction 8
1.1 Motivation: Symmetries and models . . . . . . . . . . . . . . . . . . . 8
1.2 CPviolating phasesand electricdipole moments . . . . . . . . . . . 9
1.3 Methods for analyzingCPviolating phases . . . . . . . . . . . . . . . 11
1.3.1 Neutralino andchargino polarizations . . . . . . . . . . . . . . 11
1.3.2 T-odd and CP-odd triple-product asymmetries . . . . . . . . . 11
1.3.3 Statistical error and significance . . . . . . . . . . . . . . . . . 13
1.4 Organization ofthe work . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 CP violation in production anddecayof neutralinos 16
2.1 T-odd asymmetries in neutralino production and decay into slep-
tons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.2 T-odd asymmetries . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.4 Summary ofSection 2.1 . . . . . . . . . . . . . . . . . . . . . . 27
CONTENTS 3
2.2 CP asymmetry in neutralino production and decay into polarized
taus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Tauspin-density matrix and cross section . . . . . . . . . . . 31
2.2.2 Transverse tau polarization and CPasymmetry . . . . . . . . 32
2.2.3 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.4 Summaryof Section 2.2 . . . . . . . . . . . . . . . . . . . . . . 36
2.3 T-odd observables in neutralino production and decay into a Z bo-
son . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.1 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.2 Z boson polarization . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.3 T-oddasymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.4 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.5 Summaryof Section 2.3 . . . . . . . . . . . . . . . . . . . . . . 51
3 CPviolation in production and decayof charginos 52
3.1 CPasymmetry in chargino production anddecayinto a sneutrino . 53
3.1.1 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.2 CPasymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.3 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.4 Summaryof Section 3.1 . . . . . . . . . . . . . . . . . . . . . . 58
3.2 CPviolation in chargino production anddecayinto a W boson . . . 59
3.2.1 Spin densitymatrix of the W boson . . . . . . . . . . . . . . . 61
3.2.2 W boson polarization . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.3 T-oddasymmetries . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.4 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.5 Summaryof Section 3.2 . . . . . . . . . . . . . . . . . . . . . . 69
4 CONTENTS
4 CP violation in sfermion decays 71
4.1 Sfermion decaywidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 T-odd asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3.1 Decaychain via τ˜ τ χ˜0 . . . . . . . . . . . . . . . . . . . . . 76
1 → 2
4.3.2 Decaychain via τ˜ τ χ˜0 . . . . . . . . . . . . . . . . . . . . . 77
1 → 3
4.4 Summary of Chapter4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Summaryandconclusions 83
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A Basics of the MinimalSupersymmetricStandardModel 87
A.1 MSSM Lagrangian in termsof superfields . . . . . . . . . . . . . . . . 88
A.1.1 Superfield content . . . . . . . . . . . . . . . . . . . . . . . . . 88
A.1.2 The supersymmetric Lagrangian . . . . . . . . . . . . . . . . . 89
A.1.3 The soft SUSYbreaking Lagrangian . . . . . . . . . . . . . . . 91
A.2 MSSM Lagrangian in componentfields . . . . . . . . . . . . . . . . . 91
A.3 Mass matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.3.1 Neutralino massmatrix . . . . . . . . . . . . . . . . . . . . . . 93
A.3.2 Chargino massmatrix . . . . . . . . . . . . . . . . . . . . . . . 94
A.3.3 Stau massmatrix . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A.3.4 First andsecond generation sfermion masses . . . . . . . . . 96
CONTENTS 5
B Kinematicsand phasespace 97
B.1 Spherical trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
B.2 Kinematicsof neutralino/chargino production and decay . . . . . . 99
B.2.1 Momenta and spin vectors of the production process . . . . . 99
B.2.2 Momenta and spin vectors of leptonicdecays . . . . . . . . . 99
B.2.3 Phasespace for leptonic decays . . . . . . . . . . . . . . . . . 100
B.2.4 Energydistributions ofthe decayleptons . . . . . . . . . . . . 101
B.2.5 Momenta and spin vectors of bosonic decays . . . . . . . . . 102
B.2.6 Phasespace for bosonic decays . . . . . . . . . . . . . . . . . . 103
B.3 Kinematicsof sfermion decays . . . . . . . . . . . . . . . . . . . . . . 103
B.3.1 Momenta and spin vectors . . . . . . . . . . . . . . . . . . . . 103
B.3.2 Phasespace for sfermion decays . . . . . . . . . . . . . . . . . 104
C Spin-densitymatricesfor neutralinoproduction anddecay 106
C.1 Neutralinoproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C.1.1 Neutralinopolarization independentquantities . . . . . . . . 108
C.1.2 Neutralinopolarization . . . . . . . . . . . . . . . . . . . . . . 109
C.2 Neutralinodecayinto sleptons . . . . . . . . . . . . . . . . . . . . . . 111
C.3 Neutralinodecayinto staus . . . . . . . . . . . . . . . . . . . . . . . . 112
C.4 Neutralinodecayinto the Z boson . . . . . . . . . . . . . . . . . . . . 112
D Spin-densitymatricesfor charginoproduction and decay 117
D.1 Charginoproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
D.1.1 Charginopolarization independentquantities . . . . . . . . . 119
D.1.2 Charginopolarization . . . . . . . . . . . . . . . . . . . . . . . 120
D.2 Charginodecayinto sneutrinos . . . . . . . . . . . . . . . . . . . . . . 122
D.3 Charginodecayinto the W boson . . . . . . . . . . . . . . . . . . . . 122
6 Prologue
E Neutralinoand Charginotwo-body decaywidths 126
E.1 Neutralino decaywidths . . . . . . . . . . . . . . . . . . . . . . . . . 126
E.2 Chargino decaywidths . . . . . . . . . . . . . . . . . . . . . . . . . . 129
F Spin formalismfor fermionsandbosons 132
F.1 Bouchiat-Michel formulae for spin 1 particles . . . . . . . . . . . . . 132
2
F.2 Spin formulae for spin 1particles . . . . . . . . . . . . . . . . . . . . . 133
G Definitions andconventions 135
Bibliography 137
Prologue 7
Prologue
Supersymmetry and the search for new particles
Elementaryparticle physics hasmadeenormous progress in the lastdecades. The
electroweak and strong interactions of the fundamental buildingblocks of matter,
the quarks and leptons, are now described by the so called Standard Model (SM)
of particle physics.
The development of the SM was possible due to intensive efforts and successful
achievements in experiment and theory. On the one side, theoreticians have pro-
vided the physical models and the mathematical techniques necessary to define
and calculate observables. On the other side, experiments with particle accelera-
tors and detectors have not only allowed to find new fundamental particles, but
also high precision measurementshave madeitpossible to test the models.
However, there aregeneralarguments which pointtowards theexistence ofathe-
ory beyond the SM. One of the most attractive candidates for such a more funda-
mental theory is Supersymmetry (SUSY). SUSY transformations change the spin
of a particle field, and thus bosonic and fermionic degrees of freedom get related
to each other. Therefore, new particles are predicted in SUSY models like in the
minimal supersymmetric extension of the SM. If SUSY is realized in nature, the
supersymmetric partnersof the SMparticles have to bediscovered.
The next future colliders, like the Large Hadron Collider (LHC) at CERN or a
planned International Linear Collider (ILC), are designed to find these particles.
Their properties will be measured with high precision in production and decay
processes. The underlying physical model can then be determined by a compari-
son ofexperimental and phenomenological studies.
Chapter 1
Introduction
1.1 Motivation: Symmetries and models
Symmetries in physics have always played an important role in understanding
thestructureoftheunderlyingtheories. Forinstance,theexistenceofconservation
lawscanbeexplainedbyspecificsymmetrytransformationsunderwhichatheory
is invariant. Energy, momentum and angular momentum are conserved in field
theories with continuous spacetime symmetries.
Highenergymodelsofelementaryparticleinteractionshavetobeinvariantunder
thetransformations ofthePoincare´ group, whicharerelativisticgeneralizationsof
the spacetime symmetries. The interactions of the particles, the strong and elec-
troweak forces, can be understood as a consequence of the so called gauge sym-
metries.
Butnot only such continuous symmetries are crucial. The discrete symmetries
Charge conjugation C:interchange ofparticles with antiparticles
•
Parity P: transformation of the spacecoordinates x x
• → −
Timereversal T: transformation of the parametertime t t
• → −
are also essential for the formulation of relativistic quantum field theories. In par-
ticular, any Poincare´ invariant local field theory has to be symmetric under the
combined transformation of C,Pand T,which iscalledCPTinvariance.
1.2CPviolatingphasesandelectricdipolemoments 9
The discovery in the fifties that weak interactions violate C and P maximally, was
noted with substantial belief that a particle theory apparently still conserves the
combined symmetry CP. However, a few years later, in 1964, CP violation was
confirmed in the K-meson system.
This lead to a powerful prediction in 1972. The implementation of CP violation
made it necessary to include a phase in the quark mixing matrix, the so called
Kobayashi-Maskawa Matrix [1]. Such a phase is only CP violating if the matrix is
atleastof dimension three, andthus athird generation ofquarksand leptonswas
required. Thispredictionofathirdfamilywasmadelongbeforethefinalmember
of the second family, the charm quark, wasfound.
Current experiments with B-mesons verify the existence of one CP phase in the
quark mixing matrix of the SM. However, one phase alone cannot explain the ob-
served baryon asymmetry of the universe, as shown in [2]. The fact that further
sources of CP violation are needed leads to the crucial prediction of CP violating
phasesin theories beyond the SM.
Apartfromthat,afurtherimportantsymmetryofphysicswasbornin1974: Super-
symmetry (SUSY), whichrelatesfermionicandbosonicdegreesoffreedom. SUSY
models, like the the Minimal Supersymmetric Standard Model (MSSM), are one
of the most attractive theories beyond the SM. They give a natural solution to the
hierarchy problem and provide neutralinos as dark matter candidates. Further-
more, SUSY allows for grand unifications and for theories, which might include
also gravity.
1.2 CP violating phases and electric dipole moments
The MSSM might have several complex parameters, which cause CP violating ef-
fects. In the neutralino and chargino sector these are the Higgsino mass param-
eter µ = µ eiϕµ and the U(1) gaugino mass parameter M1 = M1 eiϕM1 [3]. The
| | | |
SU(2) gaugino mass parameter M can be made real by redefining the fields. In
2
the sfermion sector of the MSSM, also the trilinear scalar coupling parameter A
f
of the sfermion f˜can be complex, Af = Af eiϕAf.
| |
TheCPviolating phasesareconstrained byelectricdipolemoments(EDMs)[4]of
electrone,neutronn,199Hgand205Tlatoms. Theirupperboundsare,respectively:
d < 4.3 10 27 ecm [5], (1.1)
e −
| | ×
10 Introduction
d < 6.3 10 26 ecm [6], (1.2)
n −
| | ×
d < 2.1 10 28 ecm [7], (1.3)
Hg −
| | ×
d < 1.3 10 24 ecm [8]. (1.4)
Tl −
| | ×
The CP phase in the quark sector of the SM give contributions to EDMs, which
generally arise at two loop level, and respect the bounds of the EDMs. In SUSY
models, however, neutralino and chargino contributions to the electron EDM can
occur at one loop level, see Fig. 1.1. For the neutron EDM in addition also gluino
exchange contributions are present due to a phase of the gluino mass parameter.
The phases of the SUSY parameters are thus constrained by the experimental up-
perlimitsof the EDMs. Inthe literature, three solutions are beingproposed [9]:
TheSUSY phasesare severelysuppressed [10,11].
•
SUSY particles of the first two generations are rather heavy, with masses of
•
the order ofa TeV[12].
There are strong cancellations between the different SUSY contributions to
•
the EDMs, allowing a SUSY particle spectrum of the order of a few 100 GeV
[13–15].
Due to such cancellations, for example, in the constrained MSSM [14], the phase
ϕ is not restricted but the phase of µ is still constrained with ϕ < 0.1π [14].
M1 | µ|
If lepton flavor violating terms are included [15], also the restriction∼on ϕ may
µ
disappear.
Therestrictions ontheSUSYphasesarethusverymodeldependent. Independent
methods for their measurements are desirable, in order to clarify the situation.
In order to determine the phases unambiguously, measurements of CP sensitive
observables are necessary. Such observables are non-zero only if CP is violated,
i.e. they are proportional tothe sineof the phases.
γ γ
χ˜ e˜
−j −1,2
e− ν˜e e− e− χ˜0j e−
Figure 1.1: SUSY contributions to the electron EDM.