complementarity of symmetry tests at the energy and
TRANSCRIPT
Complementarity of Symmetry Tests at
the Energy and Intensity Frontiers
by
Tao Peng
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
(Physics)
at the
University of Wisconsin–Madison
2017
Date of final oral examination: 5/4/2017
The dissertation is approved by the following members of the Final Oral Committee:
Akif Baha Balantekin, Professor, Physics
Michael Ramsey-Musolf, Professor, Physics
Akikazu Hashimoto, Professor, Physics
Matthew Herndon, Professor, Physics
Wesley Smith, Professor, Physics
Pupa Gilbert, Professor, Geoscience, Chemistry, Materials Science, Physics
i
Complementarity of Symmetry Tests at
the Energy and Intensity Frontiers
Tao Peng
Under the supervision of Professors Michael Ramsey-Musolf and Akif Baha
Balantekin
At the University of Wisconsin–Madison
Abstract
We studied several symmetries and interactions beyond the Standard Model and
their phenomenology in both high energy colliders and low energy experiments. The
lepton number conservation is not a fundamental symmetry in Standard Model (SM).
The nature of the neutrino depends on whether or not lepton number is violated. Lep-
togenesis also requires lepton number violation (LNV). So we want to know whether
lepton number is a good symmetry or not, and we want to compare the sensitivity of
high energy collider and low energy neutrinoless double-β decay (0νββ) experiments.
To do this, We included the QCD running effects, the background analysis, and the
long-distance contributions to nuclear matrix elements. Our result shows that the
reach of future tonne-scale 0νββ decay experiments generally exceeds the reach of the
14 TeV LHC for a class of simplified models. For a range of heavy particle masses at
the TeV scale, the high luminosity 14 TeV LHC and tonne-scale 0νββ decay experi-
ments may provide complementary probles. The 100 TeV collider with a luminosity of
ii
30 ab−1 exceeds the reach of the tonne-scale 0νββ experiments for most of the range
of the heavy particle masses at the TeV scale.
We considered a non-Abelian kinetic mixing between the Standard Model gauge
bosons and a U(1)′ gauge group dark photon, with the existence of an SU(2)L scalar
triplet. The coupling constant between the dark photon and the SM gauge bosons ǫ
is determined by the triplet vacuum expectation value (vev), the scale of the effective
theory Λ, and the effective operator Wiloson coeffcient. The triplet vev is constrained
to <∼ 4 GeV. By taking the effective operator Wiloson coeffcient to be O(1) and Λ
> 1 TeV, we will have a small value of ǫ which is consistent with the experimetal
constraint. We outlined the possible LHC signatures and recasted the current ATLAS
dark photon experimental results into our non-Abelian mixing scenario.
We analyzed the QCD corrections to dark matter (DM) interactions with SM
quarks and gluons. Because we like to know the new physics at high scale and the
effect of the direct detection of DM at low scale, we studied the QCD running for a list
of dark matter effective operators. These corrections are important in precision DM
physics. Currently little is known about the short-distance physics of DM. We find
that the short-distance QCD corrections generate a finite matching correction when
integrating out the electroweak gauge bosons.
The high precision measurements of electroweak precision observables can pro-
vide crucial input in the search for supersymmetry (SUSY) and play an important
role in testing the universality of the SM charged current interaction. We studied the
SUSY corrections to such observables ∆CKM and ∆e/µ, with the experimental con-
straints on the parameter space. Their corrections are generally of order O(10−4).
Future experiments need to reach this precision to search for SUSY using these ob-
iii
servables.
iv
To my family
“It is a miracle that curiosity survives formal education.”
Albert Einstein
v
Acknowledgements
First and foremost, I would like to thank my advisors Professor Michael Ramsey-
Musolf and Professor Baha Balantekin for guiding me in the physics study and re-
search, and in the all the administrative paperwork. Professor Ramsey-Musolf taught
me courses in Particle Physics and Collider Physics Phenomenology, in which I learned
a lot of important theories and skills in theoretical physics and phenomenology. In
research, Professor Ramsey-Musolf also provided very good and interesting research
project topics. He guided me, worked in parallel with me, and offered much help in
research. Without his help, I could never be able to finish the research. Professor
Ramsey-Musolf also provided me very good opportunities to visit UMass and attend
workshops and conferences to communicate with and learn from colleagues.
I thank Professor Baha Balantekin very much for willing to be my advisor in
UW. He helped me a lot in my paperwork, which made it convenient for me to work
remotely with Professor Ramsey-Musolf.
In the research projects, I thank my collaborators Michael Ramsey-Musolf, Peter
Winslow, Grigory Ovanesyan, Wei Chao, Carlos Argüelles, Xiao-Gang He, and Haolin
Li. Without their work and help, I could not finish the projects efficiently. I especially
would like to thank Michael Ramsey-Musolf, Peter Winslow, Grigory Ovanesyan and
Wei Chao, because we did most of the research work in parallel but independently, so
that we can cross check and make sure that our results are all correct and reliable.
I am also grateful to all my graduation committee members: Professors Michael
vi
Ramsey-Musolf, Baha Balantekin, Akikazu Hashimoto, Matthew Herndon, Wesley
Smith. For my preliminary examination, I thank Professors Michael Ramsey-Musolf,
Baha Balantekin, Akikazu Hashimoto, and Sau Lan Wu for being on my committee
and for their feedbacks.
I would also like to thank Professor Akikazu Hashimoto for teaching me Ad-
vanced Quantum Mechanics for two semesters, and I would like to thank Professor
Ludwig Bruch for teaching me Theoretical Physics Dynamics.
I sincerely thank everyone in the High Energy Phenomenology and Theory group
in the University of Wisconsin-Madison for creating a friendly and nice environment.
I also thank everyone in my research group for their insights and ideas in group
meetings: Michael Ramsey-Musolf, Peter Winslow, Grigory Ovanesyan, Wei Chao,
Huaike Guo, Chien Yeah Seng, Haolin Li, Jiang-Hao Yu, Hiren Patel, Kaori Fuyuto,
Simon Shen, Satoru Inoue, Yong Du, Mario Pitschmann, Martin Gonzalez-Alonso,
and Sky Bauman.
Finally, I would like to thank my family for their support, understanding, and
encouragement.
vii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
1 Introduction 1
1.1 Particle physics standard model and beyond . . . . . . . . . . . . . . . 2
1.2 Lepton number and its conservation . . . . . . . . . . . . . . . . . . . . 3
1.3 Lepton number violation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Neutrinoless double beta decay . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Experimental tests and searches for LNV . . . . . . . . . . . . . . . . . 6
1.6 Future colliders at 100 TeV . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 The triplet scalar model and LHC studies . . . . . . . . . . . . . . . . 11
1.8 The motivation of non-Abelian kinetic mixing . . . . . . . . . . . . . . 13
1.9 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.10 Motivations for studying dark matter operator mixing and running . . 16
1.11 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.12 Charged current universality and the weak charge . . . . . . . . . . . . 19
2 Lepton number violation collider study at 14 TeV and comparison
with 0νββ decay 23
2.1 The models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
viii
2.2 The effect of LNV operator running . . . . . . . . . . . . . . . . . . . . 27
2.3 Constraint from neutrinoless double beta decay . . . . . . . . . . . . . 33
2.4 Collider studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Lepton number violation collider study at 100 TeV 52
3.1 LHC signal and backgrounds . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Cut analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 LHC results and comparison to 0νββ decay experiments . . . . . . . . 56
3.4 Comparison to machine learning results . . . . . . . . . . . . . . . . . . 60
4 LHC Signatures of Non-Abelian Kinetic Mixing 67
4.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Collider phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Triplet-like scalar decay branching ratios . . . . . . . . . . . . . . . . . 79
4.4 ATLAS recast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 More on the UV completion . . . . . . . . . . . . . . . . . . . . . . . . 85
5 QCD corrections for dark matter effective interactions 89
5.1 Purpose of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 The effective operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Loop corrections and the anomalous dimension matrix . . . . . . . . . 94
5.4 Phenomenological effects of QCD corrections . . . . . . . . . . . . . . . 98
5.5 Box graph corrections and factorizability . . . . . . . . . . . . . . . . . 103
5.5.1 Fermion dark matter . . . . . . . . . . . . . . . . . . . . . . . . 105
5.5.1.1 Dirac dark matter . . . . . . . . . . . . . . . . . . . . 107
ix
5.5.1.2 Majorana dark matter . . . . . . . . . . . . . . . . . . 113
5.5.1.3 Inelastic dark matter . . . . . . . . . . . . . . . . . . . 114
5.5.2 Scalar dark matter . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 SUSY radiative corrections 118
6.1 Parameter scans in MSSM . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Correction results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7 Summary 127
Bibliography 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
x
List of Tables
2.1 Cut-flow part 1/2. Designed for optimizing signal relative to back-
grounds. The backgrounds include diboson and charge-flip. For the cut
flow of jet-fake backgrounds, see Table ??. . . . . . . . . . . . . . . . . 45
2.2 Cut-flow part 2/2. Designed for optimizing signal relative to back-
grounds. The backgrounds include jet-fake. For the cut flow of diboson
and charge-flip backgrounds, see Table ??. . . . . . . . . . . . . . . . . 47
3.1 Cut-flow part 1/2 for 100 TeV. Designed for optimizing signal relative
to backgrounds. The backgrounds include diboson and charge-flip. For
the cut flow of jet-fake backgrounds, see Table ??. . . . . . . . . . . . . 56
3.2 Cut-flow part 2/2 for 100 TeV. Designed for optimizing signal relative
to backgrounds. The backgrounds include jet-fake. For the cut-flow of
diboson and charge-flip backgrounds, see Table ??. . . . . . . . . . . . 59
5.1 Operator basis and their corresponding dimensions. . . . . . . . . . . . 92
xi
List of Figures
1.1 Graphic illustration of 0νββ decay. . . . . . . . . . . . . . . . . . . . . 7
1.2 The light Majorana neutrino model that makes 0νββ decay . . . . . . . 7
1.3 The left-right symmetric model that makes 0νββ decay . . . . . . . . . 7
1.4 The R Parity Violation Supersymmetry model that makes 0νββ decay 8
2.1 The lepton number violating process extracted from neutrinoless double
beta decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 The lepton number violating process, with fictitious intermediate par-
ticles S+ and F 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Cross section of pp → e−e− + jets vs. mass of S+ and F 0, taking the
couplings C1 = C2 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 The one-loop correction to the dd→ uuee LNV process. There are two
more diagrams symmetric to the two diagrams on the lower part. . . . 29
2.5 The running of the Wilson coefficients. Assuming that at the TeV scale,
C1 = 1, C2 = C3 = C4 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 The process of the two pion-two electron operator in Eq. ??. . . . . . . 35
2.7 The charge-flip Z/γ∗ → e+e− process. . . . . . . . . . . . . . . . . . . . 40
2.8 The charge-flip tt process. The b’s are not tagged. . . . . . . . . . . . . 40
xii
2.9 The HT distribution for signal and backgrounds at 14 TeV. . . . . . . . 46
2.10 The ml1l2 distribution for signal and backgrounds at 14 TeV. . . . . . . 46
2.11 The MET distribution for signal and backgrounds at 14 TeV. . . . . . . 48
2.12 Significance of the e−e− + dijet signal as a function of integrated lumi-
nosity, assuming that the C1/Λ5 is consistent with the GERDA 0νββ
half-life limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.13 Current and future exclusion reach of 0νββ decay and LHC searches
for the TeV LNV interaction as a function of the coupling geff and mass
scale Λ. The coupling is defined as geff = C1/41 , where the C1 is the
coupling in Eq. ??. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.14 Current and future discovery reach of 0νββ decay and LHC searches
for the TeV LNV interaction as a function of the coupling geff and mass
scale Λ. The coupling is defined as geff = C1/41 , where the C1 is the
coupling in Eq. ??. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1 The HT distribution for signal and backgrounds at 100 TeV. . . . . . . 57
3.2 The ml1l2 distribution for signal and backgrounds at 100 TeV. . . . . . 57
3.3 The MET distribution for signal and backgrounds at 100 TeV. . . . . . 58
3.4 The leading lepton pT distribution for signal and backgrounds at 100
TeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
xiii
3.5 Current and future exclusion reach of 0νββ decay and 100 TeV LHC
searches for LNV interaction as function of the coupling geff and mass
scale Λ. The requirement for exclusion is S/√S + B ≥ 2. The cou-
pling is defined as geff = C1/41 , where the C1 is the coupling in Eq. ??.
The blue shaded areas are for the uncertainty of M0, whose borders
correspond to M0 = −1.0 and M0 = −1.99. . . . . . . . . . . . . . . . . 61
3.6 Current and future discovery reach of 0νββ decay and 100 TeV LHC
searches for LNV interaction as function of the coupling geff and mass
scale Λ. The requirement for discovery is S/√S + B ≥ 5. The cou-
pling is defined as geff = C1/41 , where the C1 is the coupling in Eq. ??.
The blue shaded areas are for the uncertainty of M0, whose borders
correspond to M0 = −1.0 and M0 = −1.99. . . . . . . . . . . . . . . . . 62
3.7 Comparison of the discovery reaches using cut-based analysis (left) and
machine learning analysis (right). The random forest method is used in
the machine learning analysis. The machine learning analysis was done
by Peter Winslow. In the right figure, the notation M means Λ and yeff
means geff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1 Feynman diagrams that may generate the non-abelian mixing effective
operator O(5)WX . The intermediate particles in the loops are (a) fermions,
(b) scalars, or (c) other sources from non-perturbative dynamics. . . . . 76
xiv
4.2 The Feynman diagrams for LHC production and the subsequent decay
of the particles in the non-Abelian mixing model with the triplet scalars.
Diagrams (a) and (b) are the scalar pair productions, followed by the
scalar decays mediated by the non-Abelian mixing operator O(5)WX . Di-
agrams (c) and (d) are production and decays of H and X. In all four
diagrams, the incoming vector bosons are all virtual. . . . . . . . . . . 76
4.3 LHC production cross sections for pp → V → φφ and pp → V → Xφ
at√s = 8 TeV, where φ = H+, H2. The mφ = 130 GeV, and mX = 0.4
GeV. For the processes with final states of a single charged scalar and
one neutral boson, we summed the cross sections for both charges, for
example: σ(H+H2) + σ(H−H2). . . . . . . . . . . . . . . . . . . . . . . 77
4.4 LHC production cross sections for pp → V → φφ and pp → V → Xφ
at√s = 8 TeV, where φ = H+, H2. The mφ = 300 GeV, and mX = 0.4
GeV. For the processes with final states of a single charged scalar and
one neutral boson, we summed the cross sections for both charges, for
example: σ(H+H2) + σ(H−H2). . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Branching ratios forH+ decays as a function of ǫ (upper horizontal axis)
and β/Λ (bottom horizontal axis), for the triplet vev vΣ = 1 GeV. The
dark photon mass is chosen as mX = 0.4 GeV. The top plot corresponds
to mH+ = 130 GeV, and the bottom plot corresponds to mH+ = 300
GeV. The solid black line is the branching ratio for H+ → W+X.
Branching ratios for other final states are as indicated by other colors. . 80
xv
4.6 Branching ratios for H+ decays as a function of ǫ (upper horizontal
axis) and β/Λ (bottom horizontal axis), for the triplet vev vΣ = 10−3
GeV. The dark photon mass is chosen as mX = 0.4 GeV. The top
plot corresponds to mH+ = 130 GeV, and the bottom plot corresponds
to mH+ = 300 GeV. The solid black line is the branching ratio for
H+ → W+X. Branching ratios for other final states are as indicated
by other colors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.7 Constraints on the cτ of X from the ATLAS exclusion. The ATLAS
exclusion in the (cτ , σ × BR) plane [99], where the region above the
parabola is excluded. The diagonal curves are the dependence of σ×BR
on cτ for different values of vΣ. This figure was made by our collaborator
G. Ovanesyan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.8 Constraints on the non-Abelian kinetic mixing model parameters, re-
casted from the ATLAS results in Ref. [99]. The curves give the exclu-
sion regions in the(vΣ, Λ/β) parameter plane for mX = 0.4 GeV (the
red region) and mX = 1.5 GeV (the yellow region). This figure was
made by our collaborator G. Ovanesyan. . . . . . . . . . . . . . . . . . 86
5.1 The diagrams for the one loop QCD corrections to the operators in
Table. ??. The grey dot represents insertion of the operators in Table. ??. 95
5.2 The Feynman rule for the χχgg vertex. . . . . . . . . . . . . . . . . . . 99
5.3 The dependence of the functions X(r), Y (r), S(r) on the new physics
scale Λ, with r = αs(µ)/αs(Λ), in the RG equation solutions in Eq. ??
and Eq. ??. Here we used the low energy scale µ = 1 GeV. . . . . . . . 99
xvi
5.4 Ratio of the NLO order to LO DM-nucleon cross sections from the
operators O3 and O4. There is no quark mass factor in operators O3
and O4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5 The QCD running effect on the relic abundance curve for the operators
O3 and O4. There is no quark mass factor in operators O3 and O4. . . . 104
5.6 Box graphs which describe the χ-quark interactions at the one loop level.106
5.7 Feynman rules for the Dirac DM and gauge boson interaction. . . . . . 108
6.1 Upper: ∆CKM vs M2, the parameter values are ml2= 120 GeV and µ =
M1 = 80 GeV. Lower: ∆CKM, the parameter values are ml2= 120 GeV
and M1 =M2 = 80 GeV. The resulting charginos are sufficiently heavy
as to obey the LEP limits. . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 ∆CKM vs M1, the parameter values are ml2= 120 GeV and µ = M2 =
80 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3 ∆CKM and ∆e/µ vs µ and M2, where µ = M2. The difference between
the two figures is that the upper figure has M1 = 500 GeV, while the
lower figure has M1 = 1 TeV. . . . . . . . . . . . . . . . . . . . . . . . 124
6.4 The ∆CKM and ∆e/µ scatter plots for parameters constrained by weak
charge and LHC results. The upper one shows only the constrained
plot. The lower one shows the comparison between the constrained
plot and the plot with completely random parameters. . . . . . . . . . 125
1
Chapter 1
Introduction
2
1.1 Particle physics standard model and beyond
The Standard Model (SM) is a great success in fundamental particles and in-
teractions. In this model, there are three types of particles: quarks, leptons, and
gauge bosons. The quarks include u, d, s, c, b, t quarks, the leptons include
e, νe, µ, νν , τ, ντ . The interactions include electromagnetic, weak, and strong interac-
tions, where the electomagnetic and weak interactions are unified in a SU(2)L⊗U(1)Y
electroweak theory, and the strong interaction is described by the SU(3)c quantum
chromodynamics (QCD) theory.
Although the Standard Model has huge and continuing success in experiments,
it is still often believed to be not sufficient to explain all phenomena and not a com-
plete theory of matter and interactions. For example, it does not incorporate general
relativity for the gravitation interaction. It does not have massive neutrinos and thus
does not explain the neutrino oscillation experiments. It also does not explain the dark
matter which is observed in cosmology. It has the hierachy problem, which is the large
difference between the magnitudes of weak force and gravity, and the large correction
to the Higgs mass. It also lacks naturalness, which means that many parameters differ
by many orders of magnitude. The baryon asymmetry, which means the imbalance in
baryonic and antibaryonic matter in the universe, is also a problem of SM.
To solve the above and other problems. Many theories or models beyond the
Standard Model have been proposed, such as the Grand Unified Theories, the seesaw
mechanisms for neutrino mass, string theory, extra dimensions, supersymmetry, and
other theories beyond the Standard Model.
3
In this paper, we study the test and search of lepton number violation, dark
matter, and supersymmetry.
1.2 Lepton number and its conservation
The lepton number in particle physics is defined as the number of leptons minus
the number of antileptons in a reaction process. Written in equation, it is Ltotal =
Ll −Ll, where Ll is the number of letpons, Ll is the number of antileptons, and Ltotal
is the total lepton number. Each lepton and antilepton has a lepton number value of
+1 and −1 respectively. Besides total lepton number of a system, each leptonic family
has its own lepton number, like Le, Lµ, and Lτ .
Lepton number conservation is a law which says that in particle reaction pro-
cesses, the total lepton number must remain the same. The lepton number for each
type of lepton must also remain the same. An example is β decay n → p + e− + νe,
in which the left hand side has lepton number 0, and the right hand side has lepton
number 0 + 1− 1 = 0. Another example is muon decay µ− → e− + νe + νµ, in which
the lepton number of each lepton type is conserved, respectively. Lepton number con-
servation is useful in determining whether a particle reaction process is possible to
happen.
1.3 Lepton number violation
While lepton number conservation is an important conservation law in the Stan-
dard Model, and is supported by many experiments, there are several reasons why we
care whether the lepton number is a good symmetry of nature.
First of all, the Standard Model requires lepton number conservation because
4
its Lagrangian is invariant under the global U(1)e ⊗ U(1)µ ⊗ U(1)τ rotations of the
lepton fields, assuming that there are no neutrino mass terms. This results in the
conservation of the total lepton number and the conservation of the lepton number of
each of the lepton types. However, this is just an “accidental" consequence of the fact
that there are no possible renormalizable Lagrangian terms that violate the lepton
numbers. In other words, there is no corresponding fundamental symmetry behind
the lepton number conservation. So in a new model beyond the Standard Model,
it is possible that the lepton number is violated. Moreover, lepton number is not
conserved in the SM at the level of quantum corrections. The B+L anaomaly implies
lepton number is not conserved.
The second reason is that there is a general obstacle to treating the lepton
number and baryon number as fundamental symmetries of nature, since they are
violated by non-perturbative electroweak effects. In Ref [1], it is shown that the
lepton number conservation law is violated by Bell-Jackiw anomalies, in models of
fermions coupled to gauge fields.
The neutrinoless double beta decay (0νββ decay), which will be explained in
the next subsection, is a typical lepton number violating process. If 0νββ decay is
observed, then the Schechter-Valle Theorem [2] indicates that neutrinos are Majorana
particles. This is important for neutrino property.
Another important factor is that leptogenesis requires lepton number violation.
In Ref [3], the following lepton number violating Lagrangian is presented:
L = LW.S. + N tR∂N
tR +MtN
tcRNR + h.c.+ hijN
iRl
jLφ
† + h.c., (1.1)
where N tR is a right-handed Majorana neutrino. In this model, the decays of NR:
5
NR → lL+ φ and NR → lL+φ have different day rates through the one-loop radiative
correction by a Higgs particle if CP is violated. This gives the net lepton number
production.
Finally, the lowest dimension non-renormalizable operator violates the lepton
number. For example, the dimension 5 operator
L5 =1
MLiLjHkH lǫikǫjl, (1.2)
which gives a neutrino mass
mν =λαβM
v2
2, (1.3)
violates lepton number.
1.4 Neutrinoless double beta decay
Beta decay n→ pe−νe is a process of radioactive decay in nuclear physics. In this
process, a neutron becomes a proton with the emission of an electron with some missing
energy. Double beta decay nn → ppe−e−νeνe is a process of radioactive decay in
nuclear physics. In this process, two neutrons are simultaneously transformed into two
protons inside a nucleus. Neutrinoless double beta decay (0νββ decay) nn→ ppe−e−
is a process like a double beta decay, but with no neutrinos in the final state. A
graphic illustration of 0νββ decay is as shown in Fig 1.1.
0νββ decay can happen in many models [63]. For example, the Light Majorana
neutrino model [6], in which the neutrino is a Majorana particle and at least one type
of neutrino has non-zero mass, the two neutrinos annihilate each other without going
out to the final state. This scenario is shown in Fig 1.2. Another model that makes
0νββ decay possible is the left-right symmetric model [7, 24, 9], in which a heavy
6
right-handed neutrino is involved. This scenario is shown in Fig 1.3. The R-Parity
Violating Supersymmetry (RPV SUSY) can also give rise to a 0νββ decay [10, 11, 12].
R-Parity is defined as PR = (−1)3(B−L)+2s, where B is bayron number, L is lepton
number, and s is spin. R-Parity was introduced as a new symmetry to eliminate the
possibility of B and L violating terms in the renormalizable superpotential in SUSY
models. So in the SUSY models which violates R-Parity (PRV SUSY), lepton number
violation is allowed. In particular, in RPV SUSY, two selectrons become a neutralino,
and emits two electrons without neutrinos. This gives the 0νββ decay. This scenario
is shown in Fig 1.4.
All those are possible models that can make 0νββ decay happen. There might
be more such models. Here in this paper we are trying to do a model independent
study, which applies to all these models for 0νββ decay.
1.5 Experimental tests and searches for LNV
There are several ways to test the lepton number violation. For examples: the
0νββ decay nn → ppe−e− mentioned before, which has a half-life longer than 1025
years [13, 14, 15, 16]. The conversion of muon type lepton to electron type lepton
µ− + (Z, A) → e+ + (Z − 2, A), with experimental branching ratio smaller than
10−12 [17]. The kaon decay K+ → µ+µ+π−, with experimental branching ratio smaller
than 3 × 10−9 [18]. Of all the potential lepton number violating processes, the 0νββ
decay is by far the most sensitive test of lepton number violation. This is why people
are most interested in 0νββ decay, and why the low energy experiments of the search
for lepton number violation are all about 0νββ decay.
There are several experimental searches for 0νββ decay. In 2001, the Heidelberg-
7
u
d
d
d
d
u
u
d
u
e−
e−
u
d
u
n
n
p
p
Fig. 1.1.— Graphic illustration of 0νββ decay.
d
d
u
e−
e−
u
W−
W−
νe
Fig. 1.2.— The light Majorana neutrino model that makes 0νββ decay
d
d
u
e−
e−
u
WR
WR
N
Fig. 1.3.— The left-right symmetric model that makes 0νββ decay
8
Moscow experiment gave a bound to the half-life of 76Ge, which is T1/2(76Ge) > 1.9×
1025 yr at 90% CL [13]. In 2006, a subset of the Heidelberg-Moscow experiment
announced a limit of T1/2(76Ge) = 2.23 × 1025 yr [14], but this result needs to be
confirmed. The EXO-200 experiments on 136Xe set a half-life limit of T1/2 > 1.1×1025
yr at 90% CL [15]. The GERDA Phase I experiment on 76Ge set the lower limit
T1/2(76Ge) > 2.1 × 1025 years at 90% CL, and the combination with the previous
experimental resutls about 76Ge sets T1/2(76Ge) > 3.0 × 1025 yr [16]. The latest
result of the KamLAND-Zen experiment on 136Xe gives the half-life T1/2 > 1.07×1026
yr at 90% CL [20]. The next generation of tonne scale experiments aim for a half-
life sensitiviy of ∼ 1027 years [19]. A comparison of several experiments, including
GERDA and EXO, are shown in Fig. 2 in Ref. [16].
The 0νββ decay experiments are on even-even nuclei, such as 76Ge and 136Xe,
because for the nuclei which has one atomic number higher have smaller binding
energy, preventing single beta decay. However, the nuclei which has two atomic number
higher have larger binding energy, making double beta decay allowed.
The above experiments are all classic searches at the intensity frontier. It is
d
d
u
e−
e−
u
e−
e−
χ0
Fig. 1.4.— The R Parity Violation Supersymmetry model that makes 0νββ decay
9
also possible to search for 0νββ decay at the energy frontier using LHC, to make
complementary studies.
One reason to do the energy frontier searches is that the 0νββ decay lifetime
measurement does not provide the means for determining the underlying mechanism.
To see this, let’s consider the case when 0νββ decay is generated by light massive
Majorana neutrino exchange [4]. We have the half life
1
T 0ν1/2
= G0ν |M0ν |2| 〈mββ〉 |2, (1.4)
where the mass 〈mββ〉 = |Σi|Uei|2mieiαi |, or else if the 0νββ decay is generated by
heavy particle exchange, assuming the dynamics is at scale Λ, then [24]
AH
AL
∼ M4W 〈k2〉
Λ5 〈mββ〉,⟨
k2⟩
∼ (100 MeV)2. (1.5)
For 〈mββ〉 ∼ 0.1 − 0.5 eV and Λ ∼ 1 TeV, we have AH/AL ∼ O(1). Therefore we
do not know whether it is heavy or light particle exchange. And it does not provide
the way to know the underlying mechanism. One the other hand, the underlying
mechanism can be better studied by going to the higher energy scale.
The best way to search for lepton number violation in high energy colliders is
to look for the same-sign dilepton signals, for example pp → e−e− + jets in which
the lepton number is violated by two. The ATLAS Collaboration [25] searched for
the same-sign dilepton signals at LHC for the Type II seesaw model, which is defined
as having an addition of Higgs triplet (H++, H+, H0) whose coupling to the (l, ν)L
doublets gives a small neutrino mass. Assuming pair production, couplings to left-
handed fermions, and a branching ratio of 100% for each final state, masses below
409 GeV, 398 GeV, and 375 GeV are excluded at 95% credence level for e±e±, µ±µ±,
10
and e±µ± final states, respectively. The ATLAS Collaboration also searched for the
same sign and opposite sign lepton pairs in LHC for supersymmetry models [26] pp→
gg + SS/OS lepton pairs, in which they set a limit of 550 GeV for gluino mass at
95% credence level. In 2015, ATLAS Collaboration did their latest search for the new
physics with same-sign dilepton signatures with an integrated luminosity of 20.3 fb−1 at
8 TeV. Exclusion limits are derived for a specific model of doubly charged Higgs boson
production [21]. CMS searched for same-sign dilepton and jets with an integrated
luminosity of 19.5 fb−1 at 8 TeV [22]. Constraints are set on several RPV SUSY models.
CMS also searched for same-sign leptons with two or more jets and missing transverse
momentum with an integrated luminosity of 2.3 fb−1 at 13 TeV [23]. Constraints
are set on various SUSY models, with gluinos and bottom squarks masses regions
excluded.
In Chapter. 2, we study the lepton number vilation signatures in LHC at an
energy of 14 TeV and comapre its discovery and exlcusion reach with the low energy
0νββ decay experiments.
1.6 Future colliders at 100 TeV
The second operational run of LHC is designed to be between 2015 and 2018. The
operating energy will be 13 TeV for 2015 to 2017. In 2016, it focused on increasing the
integrated luminosity and the collision rate. It achieved a luminosity of 1034cm−2s−1.
Other than LHC, people are thinking about building more powerful colliders in
the future. The FCC-pp is a proposed future energy-frontier hadron collider [27, 28],
which could make protons or heavy-ions collide at a center-of-mass energy of 100
TeV. It is designed to deliver an integrated luminosity of more than several 100fb−1
11
per year. An ultimate goal of an integrated luminosity of 30 ab−1 is proposed. The
SppC is also designed for a center-of-mass energy above 50 TeV and a luminosity of
1.2× 1035 cm−2s−1. These high energy and high luminosity can provide a much more
powerful way to search for the lepton number violation.
In Chapter 3, we will study searching for lepton number violation signals at a
100 TeV collider, wth high integrated luminosity of 3 − 30 ab−1, which is within the
design of the FCC-pp and SppC colliders.
1.7 The triplet scalar model and LHC studies
The next important symmetry we consider is the dark U(1)′ gauge symmetry
and its corresponding collider phenomenology. We look at the electroweak symmetry
breaking and the scalar sector of the Standard Model and see the possible extension
to it. Ref. [31] studied the predictions of a possible extension of the Standard Model
where the Higgs sector consists of a real triplet and an SU(2)L doublet. Because the
non-Abelian mixing model we studied in Chapter 4 is the extension of the model of
Ref. [31], we explain this model here.
Although the Standard Model has achieved great success, the scalar sector of
the theory which is responsible for the electroweak symmetry breaking (EWSB) has
to be confirmed experimentally, which is one of the primary goals of the LHC. In
Ref. [31], they considered the possibility that a light real triplet Σ = (Σ+,Σ0,Σ−)
that transforms as (1, 3, 0) under SU(3)C ⊗SU(2)L⊗U(1)Y is added to the Standard
Model scalar sector. This is a simple and natural extension to the SM scalar sector
because it only added a scalar triplet to the doublet. This model can also provide
a light charged scalar which can be a dark matter candidate. In this model, the
12
Lagrangian of the scalar sector is
Lscalar = (DµH)†(DµH) + Tr(DµΣ)†(DµΣ)− V (H,Σ) (1.6)
where H is the Standard Model Higgs doublet:
H =
(
φ+
φ0
)
, (1.7)
and Σ is the real triplet, which can be written as its components:
Σ =1
2
(
Σ0√2Σ+
√2Σ− −Σ0
)
(1.8)
A compact form of the most general renormalizable scalar potential is
V (H,Σ) = −µ2 H†H + λ0(
H†H)2 −1
2M2
ΣF +b44F 2 + a1 H
†ΣH+a22H†HF , (1.9)
where
F ≡(
Σ0)2
+ 2Σ+Σ−. (1.10)
Both H and Σ can be written in terms of their vacuum expectation value v0 and
x0 respectively as the following:
H =
(
φ+
(v0 + h0 + iξ0)/√2
)
(1.11)
and
Σ =1
2
(
x0 + σ0√2Σ+
√2Σ− −x0 − σ0
)
. (1.12)
The relations between the parameters can be obtained by minimizing the tree-level
potential.
After electroweak symmetry breaking, the mass term of the neutral scalars can
be written as:
V = ...+1
2
(
h0 σ0)
M20
(
h0
σ0
)
+ ..., (1.13)
13
where the mass matrix is
M20 =
(
2λ0v20 −a1v0/2 + a2v0x0
−a1v0/2 + a2v0x0 2b4x20 +
a1v204x0
)
, (1.14)
and similarly, we have the mass matrix for the charged scalars:
M2± =
(
a1x0 a1v0/2
a1v0/2a1v204x0
)
. (1.15)
The masses of the eigenstates of the neutral and charged scalars are given by:
(
H1
H2
)
=
(
cos θ0 sin θ0− sin θ0 cos θ0
)(
h0
σ0
)
, (1.16)
(
H±
G±
)
=
(
− sin θ± cos θ±cos θ± sin θ±
)(
φ±
Σ±
)
, (1.17)
where the θ0 and θ± are the mixing angles.
Ref. [31] find that in this model, the decay of the Standard Model like Higgs boson
into two photons can be different substantially from that of the Standard Model. If
the neutral tripletlike Higgs has a vanishing vev, the charged scalars can be long-lived,
which can have a distinctive single or double charged track plus MET signals at the
LHC. If the vev is non-vanishing, the γγ decays of the triplet-like neutral scalar can
have a large rate for the γγτν and γγbb states.
We then consider the scenario of the mixing between the SM SU(2)L and a dark
sector U(1)′ gauge group via non-Abelian kinetic mixing, with the presence of a scalar
SU(2)L triplet.
1.8 The motivation of non-Abelian kinetic mixing
The search for weakly coupled light vector bosons (dark photons) has been one of
the interests in recent years. This interest is because dark photons are possible cause
14
of the (g − 2) anomaly [37]. It is helpful to achieve the Sommerfeld enhancement for
dark matter annihilation, which is often needed in many DM scenarios to obtain the
right relic density. And by interacting with SM photons, dark photon can also allow
the existence of a dark sector.
The searches were in several different ways, like low energy and high energy
colliders, meson decays, and beam dump experiments [38, 39]. Previous theoretical
studies mostly considered that the interactions of the new vector bosons with the SM
fields are mediated by Abelian kinetic mixing between the SM hypercharge and the
dark U(1)’ gauge groups, or via the mass terms in the Lagrangian [40, 41, 42, 43, 44].
For both the Abelian and mass term mixing, the effects arise from the renormalizable
operators. The coupling between the vector boson and the SM fields is described by
a parameter ǫ which is constrained by experiments to be less than O(10−3). However,
this small value of ǫ is not natural and needs to be explained. In Chapter 4, we show
how the non-Abelian kinetic mixing between the U(1)′ and the SM SU(2)L gauge
groups can provide a natural explanation of the small ǫ. There we also discuss the
possibilities for future LHC tests of this scenario.
The idea of non-Abelian kinetic mixing is not original to us. Ref. [32] considered
a U(1)Y ⊗ SU(2)′ model, in which the SU(2)′ is a dark gauge group [46]. The dark
SU(2) gauge invariance requires an extra scalar triplet. However, for large values of
the dark triplet vev, a small ǫ value requires a small operator coefficient. Applications
in astrophysical anomalies and other constraints were studied in this model in a follow-
up work of them [33]. Ref [34] used this non-Abelian kinetic mixing to explain the 3.55
keV X-ray line. Ref. [35] considered the SU(2)L⊗U(1)′ kinetic mixing in a dimension
15
six operator
C
Λ2H†T aHW a
µνXµν , (1.18)
where H is the usual Standard Model Higgs doublet. The analysis gives the coupling
ǫ ∼ C(v/Λ)2. If this dimension six operator arises at one-loop with a mediator with
mass mϕ, one can show that Λ ∼ 4πmϕ. For Λ >∼ 10 TeV, which is mϕ>∼ 1 TeV,
the experimental constraints on ǫ can be satisfied for C ∼ O(1). The authors also
considered an explicit model, which has a scalar mediator ϕ ∼ (1, 3, 0, qD) and a dark
Higgs hD ∼ (1, 1, 0, qD) that generates the dark photon mass. They also analyzed the
collider signatures of the dark boson.
1.9 Dark matter
Dark matter is another important topic of physics beyond the Standard Model.
Dark matter is a type of matter that does not have electromagnetic interaction, which
means that it does not emit, absorb, or reflect electromagnetic waves, and is therefore
invisible via the electromagnetic spectrum. Dark matter has not been observed directly
due to the lack of electromagnetic interaction, but its existence can be inferred from
its interaction with visible matter via gravitational effects. Although not observed,
dark matter is very important. Dark matter constitutes more than 80% of the total
mass of the universe, and dark mass plus dark energy constitute more than 95% of
the energy density of the present universe. Thus, dark matter can influence the large-
scale structure of the universe, the galaxies, and can affect the cosmic microwave
background of the universe. It can also cause gravitational lensing.
In experiments, dark matter particles may be produced at high energy colliders
like LHC, and because of the lack of electromagnetic interaction, they will not be ob-
16
served by the detectors. But we can infer their existence by calculating the energy and
momentum carried by them, which is one part of the missing energy and momentum.
In theory, the current most popular hypothesis for dark matter is that it is
weakly interacting massive particles (WIMPs), which interact with other particles
through only gravitational force and weak force. The WIMPs can pass through ordi-
nary matter without being noticed because they are weakly interacting, but they are
massive because they take part in the gravitational interaction. The neutrino in the
Standard Model is an example of a WIMP particle. However, the neutrino mass is
too small, if any, to contribute to the large dark matter mass in the universe. Some
physics theories beyond the Standard Model can provide such dark matter particles
as the neutralinos in supersymmetry, particles in extra dimension theories, and axions
which were originally proposed to explain the neutron’s lack of electrical dipole mo-
ment (EDM), which needs explanation because the θQCD term is naturally non-zero in
QCD and it can induce an EDM for the neutron [45]. Experiments have not detected
these particles yet.
1.10 Motivations for studying dark matter operator mixing
and running
As shown in the above section, although we have evidence for the existence of
dark matter in the universe and the important role it plays in the universe, very
little is known about its features and its non-gravitational interactions. In the past
decade, several experiments have searched for dark matter interactions with atomic
nuclei. Some of these experiments claimed to have observed the signal [36, 47], while
17
others reported no signal [48, 49, 50, 51]. The purpose of the next generation of direct
detection experiments is to clarify and improve this situation.
Since little is known about the short-distance physics of the dark matter particle
interactions, it is useful to first study them in effective theory, which is in a model
independent way [52, 53, 54, 55, 56, 57]. For the low-energy DM interactions, the
heavy intermediate particles have been integrated out. So DM effective theory is
most appropriate for such interactions. In this way, we consider a set of operators
which can generate interactions between dark matter particles and Standard Model
particles. Basically, the effective theory has a set of non-renormalizable operators
which contains both the dark matter and Standard Model fields. Refs. [52, 53, 54, 55]
studied scenarios with spin-zero and spin-1/2 dark matter operators, and Refs. [56, 57]
studied spin one dark operators. One advantage of this approach is that we can use
the constraints from the LHC direct detection on dark matter in a model independent
way. For example, as shown in Refs. [58, 59]. But Refs. [60, 61, 62] pointed out
that this approach has limitations, and when the mediator is not heavy, more work is
needed to obtain the collider constraints on dark matter searches [62]. However, in the
case when the energy scale is low compared to the mediator mass, the effective theory
is still a useful method for studying the result of the direct detection experiments.
In this context of dark matter effective theory, the effects from beyond the leading
order can play an important role in some cases [56, 63, 64, 65, 66, 67, 68, 69, 70, 71,
72, 73, 74, 75]. The electroweak loop correction effects [64, 71] can generate mixing
between the spin-independent and spin-dependent dark matter operators for direct
detections. The meson-exchange currents which can be thought of as the long-range
QCD effects, were shown in Refs. [63, 66, 69] to play an important role in the theoretical
18
precision in the calculations of WIMP-DM cross sections. As shown in Ref. [69], in the
case of the isospin-violating dark matter model [76], these long-distance QCD effects
can lead to significantly different phenomenology [69]. Ref. [67] shows that the loop
effects can change the LHC monojet bounds on dark matter couplings by several orders
of magnitude. In Chapter. 5, we study the effect of the loop corrections and mixings
of the dark matter operators in a model independent way using effective theory.
1.11 Supersymmetry
Supersymmetry is one of the most well-motivated new physics beyond the Stan-
dard Model. It proposes a new symmetry which relates the bosons and fermions. In
this symmetry, each boson has a superpartner fermion and each fermion has a super-
partner boson. The spin of the superpartner differs from itself by a half-integer. For
example, the electron has a superpartner selectron with spin 0. If supersymmetry is a
perfect symmetry, each pair of superpartners should have the same mass. But since no
superpartners of the Standard Model particles have been observed in experiments, the
superpartners must have different mass from the SM particles. This difference can be
generated from a spontaneously broken symmetry. The simplest form of the sponta-
neously broken symmetry is the Minimal Supersymmetric Standard Model (MSSM),
which is the best candidate for supersymmetric theories. The benefit of supersymme-
try is that provides a potential solution to the hierarchy problem, and some natural
dark matter candidate, and a way for the gauge grand unification.
The searches for supersymmetric models have been ongoing for many years, in
both the measurement of low-energy observables, the dark matter density measure-
ment and collider experiments including the LHC. The first run of the LHC found
19
no direct evidence for supersymmetry, and thus many scenarios and parameter spaces
are constrained. The low energy precision measurement experiments can be a com-
plementary search for the LHC experiments.
1.12 Charged current universality and the weak charge
The universality of the charged current weak interaction is a feature of the Stan-
dard Model and it has been tested with high precision, thus, the experiments place
stringent constraints on the Beyond Standard Models with non-universality, which
includes the MSSM. So testing the universality is an important way to discover or
exclude MSSM models.
Two useful quantities to test the charged current universality are ∆CKM and
Re/µ [77]. The ∆CKM describes the deviation of the square sum of the first-row
Cabibbo-Kobayashi-Maskawa (CKM) matrix from unity:
∆CKM =(
|Vud|2 + |Vus|2 + |Vub|2)
− 1. (1.19)
The correction to the largest entry Vud can be related to the Fermi constant GβV
in the following way [77]:
GβV = GµVud
[
1 + ∆r(V )β −∆rµ
]
gV (0), (1.20)
where ∆r(V )β and ∆rµ are the corrections to the amplitudes of the β-decay and muon
decay respectively. Both of these corrections can be from the Standard Model and
Beyond Standard Model, like the MSSM.
In the difference (∆r(V )β −∆rµ), the SM W-boson propagator modifications cancel
due to universality, leaving only the non-universal corrections. The shift in the value
20
of ∆CKM due to beyond SM physics is then in the form
δ∆CKM = −2|Vud|2[
∆r(V )β −∆rµ
]
(BSM). (1.21)
The MSSM corrections to this difference, which are non-universal, were calculated at
one-loop in Ref [77] and studied numerically in Ref [78].
The experimental value of ∆CKM is currently [80]
∆CKM = −0.0001± 0.0006, (1.22)
and the largest theoretical numerical value from MSSM in Ref [78] can reach order
10−3 with some chosen parameter values.
The Re/µ is the ratio of pion decay branching ratios:
Re/µ =Γ [π+ → e+ν(γ)]
Γ [π+ → µ+ν(γ)]. (1.23)
The advantage of calculating Re/µ is that many hadronic uncertainties cancel
from this ratio. Recent work gives the deviation of SM prediction from experiments [81]
∆e/µ ≡ ∆Re/µ
Re/µ
≡Rexp
e/µ −RSMe/µ
RSMe/µ
= −0.0034± 0.0030± 0.0001, (1.24)
where the first error is experimental and the second is theoretical. The MSSM con-
tribution to this ∆e/µ was calculated in Ref [79] and numerically studied in Ref [78],
which shows that the numerical value can reach order 10−3.
The weak charge is another quantity to search for SUSY and new physics beyond
the SM [82, 83]. The parity-violating electron scattering experiments can provide
ways to measure it. The weak charge of the fermion is defined in the effective A× V
Lagrangian [83]:
Leff = − Gµ
2√2Qf
W eγµγ5efγµf , (1.25)
21
where Gµ above is the Fermi constant. At tree level in SM, the weak charges of
electrons and protons are QeW = 1 − 4 sin2 θW ≈ 0.1, and the one loop electroweak
corrections reduces the values to QeW = −0.0449 [82, 84] and Qp
W = 0.0716 [82]. This
significant suppression of their values in SM makes them more transparent to possible
effects of new physics beyond SM.
At tree level in SM, the fermion weak charge is
QfW = 2If − 4Qf sin
2 θW . (1.26)
With higher order corrections, the fermion weak charge can be written in the
following form [83]:
QfW = ρPV
[
2I3f − 4QfκPV sin2 θW]
+ λf (1.27)
By comparing with Eq. (1.26), we see that at tree level ρPV = κPV = 1 and λf = 0.
At one loop level we can write [83]:
ρPV = 1 + δρSM + δρSUSY (1.28)
κPV = 1 + δκSM + δκSUSY (1.29)
λf = λSMf + λSUSYf (1.30)
The variables ρPV and κPV can be written in terms of oblique parameters S, T
22
and U , which are defined as the following [85]
S =4s2c2
αM2Z
Re
{
ΠZZ(0)− ΠZZ(M2Z) +
c2 − s2
cs
[
ΠZγ(M2Z)− ΠZγ(0)
]
+ Πγγ(M2Z)
}New
,
T =1
αM2W
{
c2(
ΠZZ(0) +2s
cΠZγ(0)
)
− ΠWW (0)
}New
,
U =4s2
α
{
ΠWW (0)− ΠWW (M2W )
M2W
+ c2ΠZZ(M
2Z)− ΠZZ(0)
M2Z
+ 2csΠZγ(M
2Z)− ΠZγ(0)
M2Z
+ s2Πγγ(M
2Z)
M2Z
}New
, (1.31)
where "New" means that we only include new physics contributions to the self-energies.
Then the variables ρPV and κPV can be written as the following [83]:
δρSUSY = αT − δµV B (1.32)
δκSUSY =
(
c2
c2 − s2
)(
α
4s2c2S − αT + δµV B
)
+c
s
[ΠZγ(q2)
q2− ΠZγ(M
2Z)
M2Z
]SUSY
+( c2
c2 − s2
)[
−Πγγ(M2Z)
M2Z
+∆α
α
]SUSY
+ 4c2F eA(q
2)SUSY (1.33)
Any parameters combination leading to values of S, T and U lying outside the present
95% confidence limit will be ruled out.
23
Chapter 2
Lepton number violation collider study at 14
TeV and comparison with 0νββ decay
24
2.1 The models
In this chapter, we study the lepton number violation at 14 TeV LHC and the
low energy 0νββ decay experiments and compare their sensitivity. This discussion
is based on the work published in Ref [86]. The purpose of this section is to study
the lepton number violation at LHC, by searching for the same sign dilepton signals.
We can extract the effective LNV operators from the neutrinoless double beta decay,
which has the corresponding process shown in Fig. 2.1.
Here we used effective operators. The reason is that we try to be model inde-
pendent. It is also possible that new physics is too heavy to be produced on shell at
the LHC. Some new physics may be light enough to be produced on shell, but we still
would like to study the effect of new particle mass.
We have implemented the model in FeynRules, and generated events with Mad-
Graph and MadEvent for pp collisions at 14 TeV, carrying out showering, jet matching,
and hadronization with Pythia and detector simulation with PGS. We implemented
the model with the following simplest operator:
L = ǫij(QαLdRα)(Q
βiL dRβ)(L
jLL
cR) + h.c.+ LSM , (2.1)
but MadGraph was not able to generate the corresponding events, because MadGraph
could not generate events for lepton number violating operators with dimension higher
than 6. Therefore, we generated the events in an alternative way, using some heavy
fictitious intermediate particles S+ and F 0 [87], as shown in Fig. 2.2.
The particle S+ is a complex sclar field with electric charge +1, and F 0 is a
25
d
d
u
u
e−
e−
Fig. 2.1.— The lepton number violating process extracted from neutrinoless double
beta decay.
u
d
u
d
e− e−
S+ S+F 0
Fig. 2.2.— The lepton number violating process, with fictitious intermediate particles
S+ and F 0.
26
neutral Majorina fermion. Their masses were both initially set at 1 TeV. We later
varied their masses.
The Lagrangian which can lead to the process in Fig. 2.2 is
L = C1udS + C2eFS∗ + h.c.+ LSM , (2.2)
and written in Gauge invariant form, it is
Leff = C1QαLdRαD + C2ǫ
ijLiLFD
∗j, (2.3)
where D is the SU(2) doublet scalar field for the S+ particle, with hypercharge
1/2:
D =
(
S
S ′
)
(2.4)
and F is the SU(2) singlet fermion field for the F 0 particle, with hypercharge 0.
Gigen the Lagrangian in Eq. 2.2, we have the following formulae for the decay
widths of S+ and F 0, which have been verified numerically by MadGraph:
ΓS =mS
8π
(
Nc|C1|2 + θ(1− xF )|C2|2(1− xF )2)
, (2.5)
ΓF = θ(xF − 1)ΓF1 + ΓF2 , (2.6)
where
ΓF1 =Nc|C2|216π
mF
(
1− 1
xF
)2
, (2.7)
ΓF2 =Nc|C1|2|C2|2
256π3
mF
x2F
·[
2(
(1− xF )2 + xS(2xF − 3)
)
cot−1 (√s) + tan−1
(
xF−1√xS
)
√xS
+xF (4− 3xF ) + (xS + xF (4− xF )− 3) ln
(
1 + xS(1− xF )2 + xS
)
]
, (2.8)
xF ≡ m2F
m2S
, xS ≡ Γ2S
m2S
, (2.9)
27
and θ is the Heaviside step function:
θ(x) ≡{
0, x < 0,
1, x ≥ 0,(2.10)
and Nc ≡ 3 is the number of quark colors. We included the decay widths of S+ and
F 0 as parameters when generating events with MadGraph.
The use of particles S+ and F 0 not only enables us to generate events with
MadGraph, but what more importantly, it is a simplified model that can be matched
onto a more UV complete model, as shown in Fig. 2.2.
Fig. 2.3 shows the relation between σ(pp→ e−e−+jets) and the mass of S+ and
F 0, where the cross sections were obtained by running MadGraph. We can see that
the logarithm of the cross section is almost a linear function of mS+ and mF 0 , which
is expected because according to Fig. 2.2, we should have σ ∝ m−8S+m
−2F 0 . The curve
has a change in slope at 6 TeV, above which the particles S+ and F 0 can no longer be
integrated out.
For mS+ = mF 0 = 1 TeV, we have σ = 0.06913 pb. This means that for
an integrated luminosity of 300 fb−1, the number of events is roughly 20 thousand.
Therefore it is appropriate to generate 1 million events for the collider study.
2.2 The effect of LNV operator running
We want to constrain the coupling constant of the lepton number violating op-
erators using the low energy neutrinoless double beta decay experimental results, and
use the constrained coupling constant to do the collider studies. Because the neutrino-
less double beta decay and the collider experiments are at very different energy scales,
we would like to know the effect of the operator running with energy scale changes.
28
In particular, we would like to study the effect of the running of the lepton number
violating operators from new physics scales to EW scales, and then to hadronic scale.
To compute the running, we consider the following operators extracted from
neutrinoless double beta decay as the basis operators, which are the effective operator
from the operators in Eq. 2.2 when the particles S+ and F 0 are integrated out
O = ǫij(QLΓdR)(QiLΓdR)(L
jLL
cR), (2.11)
where
Γ = 1, σµν , ta, taσµν . (2.12)
In the above equation, σµν ≡ i2[γµ, γν ], and ta is the generator of the SU(3)c group.
We found that the following set of operators is a closed set of operators in the
mixing and running, and it also contains the operator (uLdR)(uLdR)(eLecR), which is
the simplest one, and the one we are interested in.
O1 = (uLdR)(uLdR)(eLecR), (2.13)
O2 = (uLσµνdR)(uLσµνdR)(eLe
cR), (2.14)
O3 = (uLtadR)(uLt
adR)(eLecR), (2.15)
O4 = (uLtaσµνdR)(uLt
aσµνdR)(eLecR). (2.16)
To study the mixing and running, we need to compute the one-loop correction
diagrams. The independent diagrams are shown in Fig. 2.4.
After calculations, we can show that when Γ = 1, the infinite terms of the
diagrams in Fig. 2.4 are
g2
16π2ǫ
1
((M)2)2−d/2
(
−O3 +1
4O4
)
, (2.17)
29
2 4 6 8 1010-15
10-12
10-9
10-6
0.001
1
mS+ =mF0 [TeV]
σ(p
p→
e-e-
jj)[p
b]
Fig. 2.3.— Cross section of pp → e−e− + jets vs. mass of S+ and F 0, taking the
couplings C1 = C2 = 1.
dR
dR
uL
uL
eL
eL
g
dR
dR
uL
uL
eL
eL
g
dR
dR
uL
uL
eL
eL
g
dR
dR
uL
uL
eL
eL
g
Fig. 2.4.— The one-loop correction to the dd → uuee LNV process. There are two
more diagrams symmetric to the two diagrams on the lower part.
30
where ǫ ≡ 2−d/2, and we worked in the modified minimal substraction renormalization
scheme.
Similarly, when Γ = σαβ, the infinite terms are
g2
16π2ǫ
1
((M)2)2−d/2(12O3 − 3O4) , (2.18)
When Γ = ta, the infinite terms are
g2
16π2ǫ
1
((M)2)2−d/2
(
−2
9O1 +
1
18O2 +
1
3O3 −
1
12O4
)
, (2.19)
when Γ = taσαβ, the infinite terms are
g2
16π2ǫ
1
((M)2)2−d/2
(
8
3O1 −
2
3O2 − 4O3 +O4
)
. (2.20)
Doing the same calculations for the upper-right graph in Fig. 2.4, which has the
same results as above.
For the lower-left graph in Fig. 2.4, the infinite terms are:
g2
16π2ǫ
1
((M)2)2−d/2
(
O3 +1
4O4
)
, if Γ = 1, (2.21)
g2
16π2ǫ
1
((M)2)2−d/2(12O3 + 3O4) , if Γ = σαβ, (2.22)
g2
16π2ǫ
1
((M)2)2−d/2
(
2
9O1 +
1
18O2 +
7
6O3 +
7
24O4
)
, if Γ = ta, (2.23)
g2
16π2ǫ
1
((M)2)2−d/2
(
24
9O1 +
2
3O2 + 14O3 +
7
2O4
)
, if Γ = taσαβ. (2.24)
31
For the lower-right graph in Fig. 2.4, the infinite terms are:
g2
16π2ǫ
1
((M)2)2−d/2
(
16
3O1
)
, if Γ = 1, (2.25)
0, if Γ = σαβ, (2.26)
g2
16π2ǫ
1
((M)2)2−d/2
(
−2
3O3
)
, if Γ = ta, (2.27)
0, if Γ = taσαβ. (2.28)
For other graphs, the results are the same as the above results due to symmetry.
Summing up all the graphs, we get the following corrections at the one-loop
level:
O1 → g2
16π2ǫ
(
32
3O1 +O4
)
, (2.29)
O2 → g2
16π2ǫ(48O3) , (2.30)
O3 → g2
16π2ǫ
(
2
9O2 +
5
3O3 +
5
12O4
)
, (2.31)
O4 → g2
16π2ǫ
(
32
3O1 + 20O3 + 9O4
)
, (2.32)
In order to calculate the anomalous dimension matrix, we define the renormal-
ization matrix Z as the following [88]:
OiR = (Z−1)ijOj
0 (2.33)
= (Z−1)ijZnq/2q Z
nl/2l Oj, (2.34)
where OR is the renormalized field, O is the unrenormalized field, and O0 is the
bare field. nq and nl are the number of quark fields and the number of lepton fields
respectively. The Zq and Zl are the wave function renormalization constants for the
32
quark and lepton fields, with values [89]:
Zq = 1 + δ2 = 1− g2
12π2ǫ, (2.35)
Zl = 1. (2.36)
The renormalized Lagrangian is
LReff =
∑
j
CjORj , (2.37)
and we require LReff be independent from the scale M :
Md
dMLR
eff = 0, (2.38)
and because the bare field O0 is M independent, the above equation means
Md
dMLR
eff =Md
dM
∑
j
CjORj =M
d
dM
∑
j
Cj(Z−1O0)j = 0, (2.39)
which is(
Md
dMC
)
Z−1O0 + C
(
Md
dMZ−1
)
O0 = 0 (2.40)
Md
dMCj +
∑
i
Ciγij = 0, (2.41)
where
γij =∑
k
(
Md
dMZ−1
ik
)
Zkj, (2.42)
is the anomalous dimension matrix.
Combining the above equations, we can obtain
γ =αs
2π
8 0 0 1
0 −8/3 48 0
0 2/9 −1 5/12
32/3 0 20 19/3
(2.43)
33
Eq. 2.41 is the Renormalization Equation. In matrix form, it is
Md
dMC + γTC = 0. (2.44)
The Wilson coefficients then evolve in the above way.
After considering the running of αs, we have solved the Renormalization Equa-
tion analytically. The analytical solution is too long to list here. The numerical
solution to the Renormalization Equation is shown in Fig. 2.5. Under this evolution,
we find, for example, that if only C1(M = Λ) is non-vanishing at the high scale, then
the magnitude of the Wilson coefficients Cj(M = 1 GeV) are: C1 = 0.203C1(Λ),
C2 = −0.007C1(Λ), C3 = 0.266C1(Λ), and C4 = −0.055C1(Λ).
2.3 Constraint from neutrinoless double beta decay
For the low energy neutrinoless double beta decay, we can write the effective
Lagrangian as
LeffLNV =
∑
j
Cj
Λ5Oj + h.c.. (2.45)
At the neutrinoless double beta decay scale, which is the GeV scale, it is no longer
appropriate to use the quark degrees of freedom. So we have to match the operators
Oj onto operators at hadronic degrees of freedom [90]. To do this, we followed Ref. [63]
to find the SU(2)L×SU(2)R chiral and parity transformation properties of the Oj. We
also did the Fierz transformation of O3,4 to quark bilinears which are all color singlets,
which gives an effective O1:
Ceff ≈ C1(1 GeV)− 5
12C3(1 GeV) = 0.092C1(Λ) (2.46)
where we omitted the contributions from C2,4(1 GeV) due to running from high to
low scale.
34
We can then write O1 in the notation of Ref. [63] as:
LeffLNV =
Ceff
2Λ5
(
O++2+ −O++
2−)
eLecR + h.c. , (2.47)
where ecR ≡ (eL)C and
Oab2± = qRτ
aqLqRτbqL ± qLτ
aqRqLτbqR (2.48)
with qTL,R = (u, d)L,R. Because O++2− has odd parity, and the 0νββ-decay process of
experimental interest is the 0+ → 0+ transition, we only keep the O++2+ part of (2.47).
At the hadronic level, the O++2+ eLe
cR operator can be matched onto the pion-
electron operator because they have the same SU(2)L×SU(2)R chiral transformation
and parity transformation properties:
Ceff
ΛO++
2+ eLecR + h.c.→ CeffΛ
2HF
2π
2Λ5π−π−eLe
cR + h.c. , (2.49)
where Fπ = 92.2 ± 0.2 MeV is the pion decay constant [91], and ΛH is the mass
scale related to the matrix element of the operator O++2+ . ΛH can be estimated as
ΛH = m2π/(mu +md) ≈ 2.74 GeV for mπ+ = 139 MeV and mu +md = 7 MeV [92].
The corresponding process of the two pion-two electron operator in Eq. 2.49 is
shown in Fig. 2.6.
Following Ref. [63], we can obtain the matrix element from the operator in
Eq. 2.49:
M2π0 =
1
12π
g2AG2FΛ
2H
Λue1γ
2γ0uTe2Oππ0 , (2.50)
where gA = 1.27 is the axial pion-nucleon coupling which is related to the coupling
gπNN by the Goldberger-Treiman relation.
The squared amplitude, after simplification, is then
∑
spin
|Mππ0 |2 = − 1
144π2
g4AG4FΛ
4H
R2Λ2ββ
|M0|2∑
spin
ue1γ2γ0uTe2 · uTe2γ2γ0ue1, (2.51)
35
200 400 600 800 10000.0
0.2
0.4
0.6
0.8
1.0
Λnp (GeV)
C1,C
2,C
3,C
4
C1
C3
Fig. 2.5.— The running of the Wilson coefficients. Assuming that at the TeV scale,
C1 = 1, C2 = C3 = C4 = 0.
n
n
p
p
e−
e−
p1
p2
Fig. 2.6.— The process of the two pion-two electron operator in Eq. 2.49.
36
Where M0 is the NME and is given in Ref. [63] as
M0 = 〈Ψf |∑
i,j
R
ρij[F1~σi · ~σj + F2Tij ] τ
+i τ
+j |Ψi〉 (2.52)
where Tij = 3~σi · ρij~σj · ρij − ~σi · ~σj, R = r0A1/3, ~ρij is the distance between nucleons i
and j, and the functions F1,2(|~ρij|) are all given in Ref. [63].
After computing the contractions and traces in the squared amplitude, we have
the following expression for the width of the neutrinoless double beta decay:
Γ = − 1
2Mi
∫
d3p1(2π)3
1
2E1
∫
d3p2(2π)3
1
2E2
∫
d3pf(2π)3
1
2Ef
1
144π2
g4AG4FΛ
4HM
2
R2Λ2ββ
·|M0|2 · 4(
m2e + p1 · p2 − 2E1E2
)
(2π)4δ(4)(pi − p1 − p2 − pf ). (2.53)
We can then compute the integral in the width, and use the relation 1/T1/2 =
Γ/ ln 2, we can then obtain the half-life formula for the neutrinoless double beta decay:
1
T1/2=
~c2
288π5 ln 2· g
4AG
4FΛ
4H
R2Λ2ββ
∫ Eββ−me
me
dE1 · F (Z + 2, E1)F (Z + 2, E2)
·(
p1E1p2E2 − p1p2m2e
)
|M0|2. (2.54)
To find the numerical value of the integral in the above equation, we can use the
value of G(A,Z)0ν , which is defined as below and its value is tabulated in Ref. [93]:
G(A,Z)0ν ≡ (GF cos θcgA)
4
(
~c
R
)21
32π5~ ln 2
·∫ Eββ−me
me
dE1F (Z + 2, E1)F (Z + 2, E2)p1E1p2E2, (2.55)
where for 76Ge, it is computed as(
GGe
0ν
)−1
= 4.09× 1025 eV2 yrs.
The value for M0 was calculated by Ref. [90], using the quasiparticle random
phase approximation (QRPA). In QRPA, the particle numbers, isospin, and angular
momentum are not good quantum numbers in the basis states but they are conserved
37
on average. Some of these symmetries are partially restored after the equations of
motion are solved [4]. For 76Ge, it is MGe0 = −1.99. However, one thing we need to
note is that both ΛH and theM0 are subject to theoretical uncertainties. For the 0νββ-
decay mediated by light Majorana neutrinos, as an example, the NME computations
using the nuclear shell model are often a factor of two smaller than the value from
QRPA. To include the impact of both sources of uncertainty, we will later show the
results for the two different values of the product M0Λ2H which differ by a factor of
two.
Combining Eqs. 2.54 and 2.55, and considering the effect of the running of the
coefficient, we have the simplified expression for the half-life:
1
T1/2= G(A,Z)
(
ΛH
TeV
)4(1
18
)
( v
TeV
)8
×(
1
gA cos θC
)4
|M0|2[
C2eff
(Λ/TeV)10
]
, (2.56)
We can then use the above equation and the experimental values of the half-life
to constrain the effective LNV coupling constant Ceff . Then we can use the constrained
Ceff in collider studies.
2.4 Collider studies
We implemented the lepton number violating model in Eq. 2.3 using FeynRules,
and generated events using MadGraph and MadEvent for pp collisions at 14 TeV, car-
rying out showering, jet matching, and hadronization with Pythia and detector simu-
lation with PGS. For every process in the signal and backgrounds, one million events
were generated using the Titan cluster of the University of Massachusetts Amherst.
38
Because we are looking for the dd → uuee process, the signal we look for is the
same sign dilepton plus jets process:
pp→ e−e− + jets (2.57)
There are three categories of backgrounds [94, 26]. The first category of back-
grounds is the diboson, in which the events contains two gauge bosons. It contains
the following three backgrounds:
• WW + jets
• WZ + jets
• ZZ + jets
The diboson backgrounds have very small cross sections at 14 TeV, which is 10−6 ∼
10−5 fb. So they are subdominant backgrounds.
The second category of backgrounds is the charge-flip. Their cross sections can
be hundreds of fb, so they are one of the dominant backgrounds. In these events,
the opposite charged leptons are misreconstructed as same-sign lepton events by the
detector, which looks like the same-sign lepton signals. This often occurs when a
lepton undergoes bremsstrahlung, emitting a photon and converts to lepton and anti-
lepton pairs. The photon passes the majority of its momentum to the lepton with the
opposite charge. The overall effect is that the actual original lepton was identified as
having the opposite charge. There are two such processes:
• Z/γ∗ → e+e−,
• tt semileptonic decays.
39
The graph illustration of the charge-flip process Z/γ∗ → e+e− is illustrated in
Fig. 2.7. In this process, the e+ transfers most of its transverse momentum via the
photon to the e−, making the e+ looks like a e−. The graph illustration of the charge-
flip process tt is illustrated in Fig. 2.8. In this process, the e+ transfers most of its
pT to the e− via the photon, looking like a e−. The b’s are not tagged, making this
process look like the signal pp→ e−e− + jets.
The charge-flip background is the largest in the ee channel, and is also present
in the eµ channel. It does not appear in the µµ channel because of the near absence of
photon decaying to two muons. It is also possible to have a charge-flip background even
when the bremsstrahlung photon does not carry the larger share of the momentum; in
this case, the misidentification happens when the lepton ID track is misreconstructed.
This is also considered to be charge-flip.
In experiment, the lepton charge misidentification probability can be measured
and is dependent on the pseudorapidity of the lepton. Here we use the ATLAS lepton
charge misidentification probability which is shown in the left panel of Fig. 23 in
Ref. [96]. The zeros in the regions 1.37 < |η| < 1.52 are indicative of a physical hole
in the sensitivity of the ATLAS detector in the transition region between barrel and
endcap EM calorimeters.
As a cross check, we calculated the global misidentification probability on the
charge-flip events we generated and obtained the following value, which is in agreement
with the value in the left panel of Fig. 23 in Ref. [96]:
Global misidentification probability =∑
bins
NiPi
N≈ 2.41%, (2.58)
where Pi is the misidentification probability in the i-th η bin, Ni is the number of
40
g
g
Z
e+
e−
e−
e+
γ
Fig. 2.7.— The charge-flip Z/γ∗ → e+e− process.
gt
t
W
W
b
e+
ν
ν
e−
e−
e+
γb
Fig. 2.8.— The charge-flip tt process. The b’s are not tagged.
41
events with the lepton having η within the i-th η bin, and N is the total number of
events.
In our collider simulation analysis, we implemented the charge-flip probabilities
in this way. For every charge-flip event, we bin it according to its lepton pseudorapidity
and the η bins in the left panel of Fig. 23 in Ref. [96]. When making histograms of the
physics variables, if an event has η falling into any of the bins in the left panel of Fig. 23
in Ref. [96], the event weight is then multiplied by the corresponding misidentification
probability in that η bin. When making the cut-flow, we also multiply the cross
sections and the event weight by the misidentification probability in the corresponding
η bin.
The third category of backgrounds is the jet-fake. They are also dominant back-
grounds because the cross sections can be from several fb to several hundred fb. Jet-
fake is when a jet is identified as a lepton. This occurs when a lepton originates from
a jet, and the lepton is observed instead of the jet. There are four jet-fake processes:
• tt semileptonic decays,
• Single t decay,
• W + jets,
• QCD multijet.
Similarly to charge misidentification probability, a jet also has some probability
to be misidentified as a lepton. For electrons, medium criteria are used, instead of
tight criteria [96]. The medium cuts can be seen as a conservative choice. There is
no strong kinematic dependence of the jet-fake probabilities over the η region. In an
42
event with multiple jets, it is impossible to determine which jet will fake a lepton. To
account for all possibilities, for each physical variable we perform an event-by-event
average over all jets that could have faked the electron in a given sample, which will
be explained later.
In our analysis, we implemented jet-faking by recalculating the cross sections
and averaging the physics variables when making the distributions. The jet-fake cross
sections were calculated using the following formula:
σJF, before cuts = σJF, MG+Pythia+PGS ×(
1
5000× 1
2
)# of jet fakes
×(
# of jets
# of jet fakes
)
,
(2.59)
where 1/5000 is the medium jet-fake probability [96] and the factor of 1/2 is because
a jet fakes an electron and a positron with equal probability. The combinatorial factor
means the number of ways to choose the jets which will fake leptons among all the
jets.
To average the physical variables, we need to distinguish the ordinary jet-fake
backgrounds (tt, single t, W + jets) and the QCD multijet backgrounds. For the
ordinary jet-fake backgrounds, we do the following because, in these backgrounds, we
only need one jet to fake lepton and we required at least three jets in the signal selection
cuts, so we select the leading three jets as the potential jets which may fake leptons,
and we call their transverse momentum p(1)T , p
(2)T , p
(3)T . For HT , we first calculate it
by summing all the jet pT , and we call it H(raw)T , and calculate the average HT as in
the following way:
H(average)T =
1
3
((
H(rawT − p
(1)T
)
+(
H(raw)T − p
(2)T
)
+(
H(raw)T − p
(3)T
))
, (2.60)
where in the above equation, we subtract the pT of the leading jets from H(rawT because
43
these jets are faking leptons, so they should not be included in the HT value. The
factor of 1/3 is because each of the three leading jets have equal probability of faking
leptons. We can do similar things for other variables, like the lepton invariant mass
mll, leading lepton pT , etc. The MET is not affected by jet-faking, so we can calculate
it in the ordinary way.
For the QCD multijet background, the difference is that we need two jets to fake
leptons and we required four jets in the signal selection cuts. So we select the leading
four jets, from these four jets we select pairs of two jets and subtract their pT from
H(raw)T in a similar way as Eq. 2.60, and divide the result by 6, because there are six
different ways to choose two jets from four jets. We can do similar things for other
variables, like mll, and leading lepton pT , etc. Again, the MET is not affected by
jet-faking so we can calculate it in the ordinary way.
In order to make sure every background behaves like the signal, we imposed the
signal selection for the various types of backgrounds and the signal, as shown below:
• For signal: Njet ≥ 2, Ne− ≥ 2, Nb = 0,
• For diboson: Njet ≥ 2, Ne− ≥ 2, Nb = 0,
• For charge-flip: Njet ≥ 2, Ne− ≥ 1, Ne+ ≥ 1, Nb = 0,
• For tt, t+ jets, W + jets jet-fake: Njet ≥ 3, Ne− ≥ 1, Nb = 0,
• For QCD jet-fake: Njet ≥ 4, Nb = 0,
When generating events at MadGraph level, we also required that pTj,b,ℓ±
>
20 GeV, |ηj| < 2.8, |ηℓ± | < 2.5.
44
For the events which pass the signal selections, we find that the cuts on HT ,
ml1l2 , and MET are very effective in reducing the background while still maintaining
the signal. These variables are defined and calculated as below:
HT ≡∑
jets
pT (2.61)
ml1l2 : invariant mass of the two leading leptons (2.62)
MET : the missing transverse energy. (2.63)
We then made the distributions of these variables for events which passed the
signal selections at 14 TeV, which are shown in Figs. 2.9, 2.10, and 2.11.
From the HT distributions in Fig. 2.9, we see that there is a very good separation
between the signal and the backgrounds, and a numerical analysis on the relation
between the S/√S + B value and the location of cut shows that a cut at HT > 650
GeV is the optimal cut to suppress the backgrounds while maintaining the signal.
From the ml1l2 distribution in Fig. 2.10, we see that the distributions of processes
which have a Z as the intermediate particle have sharp peaks around the mZ ∼ 91
GeV. We can do a Z veto, which is to select the events which have ml1l2 falling outside
a region near mZ : [mZ −∆mZ , mZ +∆mZ ], but after studying the relation between
the S/√S + B value and the cuts, we find that a cut like ml1l2 > 130 GeV works
better than a Z veto cut.
From the MET distribution in Fig. 2.11, we see that there is not as clear a
separation between the signal and the backgrounds as with the HT distribution, but
the MET is still useful, because the shapes of the MET distribution are divided into
two groups. The signal and a few processes without neutrinos in the final states (like
jjzz, and QCD) have small MET, so their MET distributions are peaked at very small
45
σ(fb) Signal Backgrounds
Diboson Charge Flip
W−W−+2j W−Z+2j ZZ+2j Z/γ∗+2j tt
Before Cuts 0.142 0.541 6.682 0.628 903.16 68.2
Signal Selection 0.091 0.358 4.66 0.435 721.7 28.9
HT (jets) > 650 GeV 0.054 0.04 0.187 0.015 5.6 0.266
mℓ1ℓ2 > 130 GeV 0.039 0.029 0.105 0.008 0.163 0.127
MET < 40 GeV 0.036 0.005 0.036 0.007 0.126 0.014
Table 2.1: Cut-flow part 1/2. Designed for optimizing signal relative to backgrounds.
The backgrounds include diboson and charge-flip. For the cut flow of jet-fake back-
grounds, see Table 2.2.
value. In contrast, the other processes, which have neutrinos in the final states, have
more flat MET distributions. So MET can still be very effective in suppressing the
latter group of processes. A similar study on the relation between the S/√S + B value
and the cuts shows that a cut on MET < 40 GeV is an optimal cut.
After determining the cuts from the distributions of the variables, we can make
a cut flow of all the processes. The cut flow is shown in Table 2.1 and Table 2.2. In
this cut flow, the signal is generated for MS = MF = 1 TeV and the coupling in the
model in Eq. 2.2 is C1 = C2 = 0.176, which corresponds to a neutrinoless double beta
decay rate consistent with the present GERDA upper bound.
From the cut flow we see that before the cuts, the dominant backgrounds, which
are the charge-flip Z/γ∗+2j and tt, and the jet-fake W−+3j and QCD 4j, have cross
sections several orders larger than the signal. However, after the cuts, they are com-
parable or smaller than the signal. This means that the cuts based analysis is very
effective in suppressing the backgrounds. After the cuts, the charge-flip Z/γ∗+2j back-
ground still dominates, but is much smaller and acceptable for the current luminosity.
46
0 500 1000 1500 20000.00
0.05
0.10
0.15
0.20
HT [GeV]
Nu
mb
er
of
ev
en
ts(n
orm
ali
zed
) Signal
jjww
jjwz
jjzz
CF Zjets
CF tt
JF W+jets
JF t+jets
JF tt
QCD
Fig. 2.9.— The HT distribution for signal and backgrounds at 14 TeV.
0 200 400 600 800 10000.0
0.1
0.2
0.3
0.4
ml1, l2[GeV]
Nu
mb
er
of
ev
en
ts(n
orm
ali
zed
) Signal
jjww
jjwz
jjzz
CF Zjets
CF tt
JF W+jets
JF t+jets
JF tt
QCD
Fig. 2.10.— The ml1l2 distribution for signal and backgrounds at 14 TeV.
47
σ(fb) Signal Backgrounds S√S+B
(√
fb)
Jet Fake
tt t+3j W−+3j 4j
Before Cuts 0.142 6.7 0.45 15.09 362.352 0.0038
Signal Selection 0.091 2.37 0.22 11.73 72.03 0.0031
HT (jets) > 650 GeV 0.054 0.025 0.0003 0.102 0.027 0.0213
mℓ1ℓ2 > 130 GeV 0.039 0.024 3× 10−4 0.101 0.027 0.0493
MET < 40 GeV 0.036 0.005 3× 10−5 0.03 0.017 0.0684
Table 2.2: Cut-flow part 2/2. Designed for optimizing signal relative to backgrounds.
The backgrounds include jet-fake. For the cut flow of diboson and charge-flip back-
grounds, see Table 2.1.
Using the cross sections of the signal and backgrounds after cuts, we can make
the curves of the significance of the e−e− + dijet signal as a function of integrated
luminosity, as shown in Fig. 2.12. Here, for the C1/Λ5 constraint, we used the GERDA
0νββ half-life limit, which is T1/2 (76Ge) < 3 × 1025 years. The two dashed curves
correspond to values of the NME of M0 = −1.0 and M0 = −1.99, respectively. From
Fig. 2.12, we see that a nonobservation with luminosity at ∼ 735 fb−1 for M0 = −1.99
and ∼ 70 fb−1 for M0 = −1.0 would imply exclusion at a level consistent with the
present GERDA limit. The requirement for discovery is S/√S + B ≥ 5, which means a
requirement of luminosity of >∼ 435 fb−1 for M0 = −1.99 and >∼ 4.6 ab−1 for M0 = −1.0.
It is striking to see that a difference of a factor of 2 in M0, after transferal to the limit
on C1/Λ5, implies an order of magnitude of difference in the required LHC luminosity
for exclusion and discovery.
We then did a parameter scan for the signal model, for 0 TeV ≤ Λ ≤ 5 TeV and
0 ≤ geff ≤ 1.5, where geff ≡ C1 = C2 and C1, C2 are from Eq. 2.2. In Fig. 2.13 and
Fig. 2.14, we show in the scanned region, the exclusion and discovery reache curves
48
0 50 100 150 2000.00
0.05
0.10
0.15
0.20
0.25
0.30
MET [GeV]
Nu
mb
er
of
ev
en
ts(n
orm
ali
zed) Signal
jjww
jjwz
jjzz
CF Zjets
CF tt
JF W+jets
JF t+jets
JF tt
QCD
Fig. 2.11.— The MET distribution for signal and backgrounds at 14 TeV.
0 1 2 3 40.5
1
5
10
ℒ [ab-1]
S/
S+
B
M0=-1
M0=-1.99
Discovery
Exclusion
Fig. 2.12.— Significance of the e−e− + dijet signal as a function of integrated lumi-
nosity, assuming that the C1/Λ5 is consistent with the GERDA 0νββ half-life limit.
49
for both the LHC at different luminosities (100 fb−1, 300 fb−1, 1000 fb−1). We also
show the exclusion and discovery reach curves for the low energy 0νββ decay GERDA
experiment and the future 1 Tonne experiment. For the 1 Tonne experiment, we used
a prospective sensitivity of T1/2 (76Ge) = 6×1027 years. For the GERDA and 1 Tonne
experiments, the solid and dotted curves indicate the impact of varying M0 by a factor
of 2.
From Fig. 2.13, we observe that with luminosity ≥ 100 fb−1, the LHC would
begin to extend the present GERDA exclusion limit for Λ between 1 TeV and 3 TeV.
And from Fig. 2.14, we see that the opportunities for discovery with a luminosity of
300 fb−1 appear more limited, for both the smaller and bigger nuclear and hadronic
matrix elements. However, at a higher luminosity of 3 ab−1, it could open the chance
for discovery in a range of Λ which depends on the value of M0.
We see that the reach of the 1 Tonne scale 0νββ decay experiments tend to ex-
ceed the reach of the high-luminosity LHC reaches over almost the entire range of the
parameter space. Therefore, in terms of the reach of the LHC, the above conclusion is
not as optimistic as obtained in Ref. [97] and Ref. [98]. These papers delineated the
simplified models and did a first round of collider analysis. They found that for the
simplified model we use, the 14 TeV LHC with an integrated luminosity of 300 fb−1 is
much more sensitive than the future low energy 0νββ decay experiments. We expect
that our findings regarding the three perspectives, which are the full background stud-
ies, the QCD running, and the long-range NME contributions, to generalize to other
simplified LNV models. However, it is always interesting to compare the prospects
for the high energy LHC and the low energy 0νββ decay experiments, because the
observation of an LNV signal in both these experiments is possible, and may point to
50
1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Λ [TeV]
ge
ff
GERDA(M0
=1) GERDA(M0
=2)
1 Tonne (M0=1)
1 Tonne (M0=2)
100fb
-1
300fb
-1
3000fb
-1
Fig. 2.13.— Current and future exclusion reach of 0νββ decay and LHC searches for
the TeV LNV interaction as a function of the coupling geff and mass scale Λ. The
coupling is defined as geff = C1/41 , where the C1 is the coupling in Eq. 2.45.
1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Λ [TeV]
ge
ff
GERDA(M0
=1) GERDA(M0
=2)
1 Tonne (M0=1)
1 Tonne (M0=2)
100fb
-1
300fb
-1
3000fb
-1
Fig. 2.14.— Current and future discovery reach of 0νββ decay and LHC searches for
the TeV LNV interaction as a function of the coupling geff and mass scale Λ. The
coupling is defined as geff = C1/41 , where the C1 is the coupling in Eq. 2.45.
51
the existence of LNV interactions.
52
Chapter 3
Lepton number violation collider study at
100 TeV
53
3.1 LHC signal and backgrounds
As shown in the Introduction, the future FCC-pp is designed for a center-of-mass
energy of 100 TeV and an integrated luminosity of 10 − 20 ab−1. Therefore, here we
extend our previous study of 14 TeV lepton number violation to a 100 TeV collider wth
integrated luminosity of 30 ab−1, which is within the design of the FCC-pp collider.
The model will be the same as our previous 14 TeV study, which is shown in
Eq. 2.45. We used this model to generate one million events for the signal and every
process in backgrounds, using the UMass Titan Cluster [95].
For the signal, we found that it is useful to include the e+e+ in the final state
leptons as well, because the protons contain more positively charged u-quarks than d-
quarks, making the e+e+ state in the proton-proton collisions enhanced relative to the
e−e− state. Therefore, it is better to search for the pp→ e±e± + 2j signal, rather than
the pp → e−e− + 2j signal. We ran MadGraph at 100 TeV and found the following
ratio in the cross sections:
σ(pp→ e±e± + 2j)
σ(pp→ e−e− + 2j)≃ 3.5, (3.1)
which is larger than a simple factor of 2. We will also apply the e±e± signal to the 14
TeV LHC study.
The backgrounds are similar to the 14 TeV case. The only difference is that we
now should generate e±e± instead of e−e−. So we have the following backgrounds:
The diboson backgrounds are shown below:
• WW + jets
54
• WZ + jets
• ZZ + jets
The charge-flip backgrounds are shown below. We use same charge misidenti-
fication probability as shown in the left panel of Fig. 23 in Ref. [96]. We calculated
the global misidentification probability in the same way as Eq. 2.58, and our results
were consistent with Ref. [96]. So it is reasonbale to continue using the misidentifica-
tion probability there. We also do the same as in the 14 TeV case to implement the
charge-flip misidentification probability in our collider analysis and calculate the cross
sections in the cut-flow.
• Z/γ∗ → e+e−,
• tt semileptonic decays.
The jet-fake backgrounds are shown below. We use the same jet-fake probabil-
ity as in the 14 TeV study, and the same formula to calculate the cross section as
in Eq. 2.59. We also use the same averaging scheme when calculating the physical
variables like HT , ml1l2 , and MET, which are defined in Eq. 2.63.
• tt semileptonic decays,
• Single t decay,
• W + jets,
• QCD multijet.
55
3.2 Cut analysis
Similar to the 14 TeV studies, we impose the signal selections for all the signal and
background processes. However, since we are now searching for the e±e± + 2j signal,
the signal selections will be different from the 14 TeV studies, where we searched for
the e−e− + 2j signal. The signal selections are shown below:
• For signal: Njet ≥ 2, Ne− ≥ 2, Ne+ ≥ 2, Nb = 0,
• For diboson: Njet ≥ 2, Ne− ≥ 2, Ne+ ≥ 2, Nb = 0,
• For charge-flip: Njet ≥ 2, Ne− ≥ 1, Ne+ ≥ 1, Nb = 0,
• For tt, t+ jets, W + jets jet-fake: Njet ≥ 3, Ne− ≥ 1, Ne+ ≥ 1, Nb = 0,
• For QCD jet-fake: Njet ≥ 4, Nb = 0,
When generating events at MadGraph level, we also required that pTj,b,ℓ±
>
20 GeV, |ηj| < 2.8, |ηℓ± | < 2.5.
Using the events which pass the signal selections, we can make the distributions
of variables like HT , ml1l2 , and MET, which are defined in Eq. 2.63, and determine
the optimal cuts on them to suppress the backgrounds. We also find that the leading
lepton pT is a good variable in separating the signal from the backgrounds. The
distributions of the variables are shown in Fig. 3.1, Fig. 3.2, Fig. 3.3, and Fig. 3.4.
From these distributions, we again see that the signal jets are more energetic
than the backgrounds, making HT a very good variable to select the signal from the
backgrounds. The ml1,l2 are good at suppressing the processes which have Z in the
backgrounds. The MET can suppress the backgrounds which have no neutrinos in the
56
σ(fb) Signal Backgrounds
Diboson Charge Flip
W−W−+2j W−Z+2j ZZ+2j Z/γ∗+2j tt
Before cuts 28.3 49.2 591 26.5 8× 106 3× 106
Signal selections 4.5 6.3 68 5.8 1.3× 103 137
HT > 500GeV 3.1 2.2 13 0.97 72 8.6
pT (lead e) > 150 GeV 2.1 0.8 7.1 0.4 22 3.0
MET < 40GeV 2.0 0.8 5.7 0.37 1.3 2.8
Z veto 1.2 0.09 1.1 0.16 0.7 0.8
Table 3.1: Cut-flow part 1/2 for 100 TeV. Designed for optimizing signal relative to
backgrounds. The backgrounds include diboson and charge-flip. For the cut flow of
jet-fake backgrounds, see Table 3.2.
final state, and the signal has a very different distribution of leading lepton pT than
the backgrounds, making this another useful variable in selecting the signal.
3.3 LHC results and comparison to 0νββ decay experiments
After determining the best cuts from the distributions, we can do the cut flow;
and the result is shown in Table 3.1 and Table 3.2. From the cut flow, we see that
similarly to the 14 TeV case, before the cuts, the dominant backgrounds are the charge-
flip and jet-fake backgrounds, especially the charge-flip Z/γ∗+2j and tt backgrounds,
which are several orders larger than the signal. After the cuts, their cross sections
are smaller than the signal. After the cuts, the cross sections of W−Z+2j, W−+3j,
and the QCD 4j backgrounds have comparable or even larger cross sections than the
signal, but not too much larger than the signal. So the cuts are still very effective in
suppressing the backgrounds and maintaining the signal.
We can then scan the parameters of the signal model and do the cuts on the
models of all the scanned parameter values, and use the cross sections of all the
57
0 500 1000 1500 2000 2500 30000.00
0.05
0.10
0.15
0.20
HT [GeV]
Nu
mb
er
of
ev
en
ts(n
orm
ali
zed
) Signal
jjww
jjwz
jjzz
CF Zjets
CF tt
JF W+jets
JF t+jets
JF tt
QCD
Fig. 3.1.— The HT distribution for signal and backgrounds at 100 TeV.
0 500 1000 1500 20000.00
0.05
0.10
0.15
0.20
0.25
0.30
ml1, l2[GeV]
Nu
mb
er
of
ev
en
ts(n
orm
ali
zed
) Signal
jjww
jjwz
jjzz
CF Zjets
CF tt
JF W+jets
JF t+jets
JF tt
QCD
Fig. 3.2.— The ml1l2 distribution for signal and backgrounds at 100 TeV.
58
0 50 100 150 200 250 3000.00
0.02
0.04
0.06
0.08
MET [GeV]
Nu
mb
er
of
ev
en
ts(n
orm
ali
zed) Signal
jjww
jjwz
jjzz
CF Zjets
CF tt
JF W+jets
JF t+jets
JF tt
QCD
Fig. 3.3.— The MET distribution for signal and backgrounds at 100 TeV.
0 200 400 600 800 10000.00
0.05
0.10
0.15
Leading lepton PT [GeV]
Nu
mb
er
of
ev
en
ts(n
orm
ali
zed
) Signal
jjww
jjwz
jjzz
CF Zjets
CF tt
JF W+jets
JF t+jets
JF tt
QCD
Fig. 3.4.— The leading lepton pT distribution for signal and backgrounds at 100 TeV.
59
σ(fb) Signal Backgrounds
Jet Fake
tt t+3j W−+3j 4j
Before cuts 28.3 573 25.5 585 3× 103
Signal selections 4.5 38.9 1.4 120 264
HT > 500GeV GeV 3.1 2.4 0.03 9.9 4.3
pT (lead e) > 150 GeV GeV 2.1 2.2 0.03 9.7 4.3
MET < 40GeVGeV 2.0 2.2 0.03 9.7 4.3
Z veto 1.2 0.3 3× 10−3 1.8 1.6
Table 3.2: Cut-flow part 2/2 for 100 TeV. Designed for optimizing signal relative
to backgrounds. The backgrounds include jet-fake. For the cut-flow of diboson and
charge-flip backgrounds, see Table 3.1.
backgrounds after the cuts in the cut flow tables, to make the curves of exclusion and
discovery reaches of future 100 TeV colliders. The curves are shown in Fig. 3.5 and
Fig. 3.6.
From Fig. 3.5, we can see that a luminosity of 1fb−1 of the collider would begin
to exceed the present GERDA exclusion reach for the smaller value of M0. For a
luminosity of 100fb−1, it is comparable for the future 1 Tonne experiment for smaller
M0 values in most Λ regions, and is even comparable to the 1 Tonne experiment for
larger M0 values near Λ ∼ 2.7 TeV. For the future designed larger luminosity, which
is 3ab−1 and 30ab−1, they are all beyond the exclusion reach of the future 1 Tonne
experiment for Λ > 3 TeV, making the future high energy collider more effective than
the future low energy 0νββ decay experiment in excluding lepton number violation if
no signals are observed.
From Fig. 3.6, we see that a 10fb−1 of luminosity of the collider is beyond the
discovery reach of the present GERDA experiment for the smaller M0, and is com-
60
parable to the GERDA experiment for the larger M0 for Λ in the range 3.5− 4 TeV.
A 100fb−1 of luminosity of the collider is beyond the discovery reach of the GERDA
experiment, but is still less effective than the future 1 Tonne experiment. For the
designed luminosity of 3ab−1 for the future designed collider, the discovery reach is
comparable to the future 1 Tonne experiment for the larger value of M0 at Λ ∼ 3
TeV, but is slightly less sensitive in other Λ regions. For the designed luminosity
of 30ab−1 for the future designed collider, the discovery reach exceeds that of the 1
Tonne experiment for the larger value of M0 for Λ > 3 TeV. Therefore, a future 100
TeV collider (like FCC-pp) with the designed luminosity of 30ab−1 is more effective
for both excluding and discovering the lepton number violation than the future low
energy 0νββ experiments.
3.4 Comparison to machine learning results
We can also compare the discovery reaches from the cut based analysis with that
from the Machine Learning analysis, which is shown in Figure 3.7.
The machine learning analysis was done by Peter Winslow. In this machine
learning analysis, a random forest classifier is trained to classify the events. For the
random forest, we first build a collection of decision trees. A decision tree is very
similar to a cut based analysis. Each node of a decision tree is a feature used to
separate the events. A feature can be a single or a combination of several physical
variables, like Ht, MET, etc. On a node, the events are divided into different groups
according to the value of the physical variable. Each group is a child of the original
node. The leaves of the tree are the result of decisions, i.e., whether an event is a
signal or background.
61
0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
Λ [TeV]
ge
ff
GERDA
1 Tonne
1 fb-1
10 fb-1
100 fb-1
3 ab-1
30 ab-1
Fig. 3.5.— Current and future exclusion reach of 0νββ decay and 100 TeV LHC
searches for LNV interaction as function of the coupling geff and mass scale Λ. The
requirement for exclusion is S/√S + B ≥ 2. The coupling is defined as geff = C
1/41 ,
where the C1 is the coupling in Eq. 2.45. The blue shaded areas are for the uncertainty
of M0, whose borders correspond to M0 = −1.0 and M0 = −1.99.
62
0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
Λ [TeV]
ge
ff
GERDA
1 Tonne
1 fb-1
10 fb-1
100 fb-1
3 ab-1
30 ab-1
Fig. 3.6.— Current and future discovery reach of 0νββ decay and 100 TeV LHC
searches for LNV interaction as function of the coupling geff and mass scale Λ. The
requirement for discovery is S/√S + B ≥ 5. The coupling is defined as geff = C
1/41 ,
where the C1 is the coupling in Eq. 2.45. The blue shaded areas are for the uncertainty
of M0, whose borders correspond to M0 = −1.0 and M0 = −1.99.
63
In this way, we can build many decision trees, each differing in the structure
or how we separate events in the nodes. The collection of all decision trees forms an
ensemble, which is often called a forest. There are several ways to utilize the forest as
a classifier. One way is to randomly pick up several trees from the forest each time,
and either let the trees vote on the decision of a particular event or take the average
of the decision of the picked trees. This is called the random forest method. Another
ensemble method is boosting, in which we make a distribution of the events in each
iteration, and the distribution gives higher weight (or probability) to the events which
we classified wrong, so that in the next iteration we can place more emphasis on the
“harder" problems. The right figure of Figure 3.7 is the result using a random forest
as the classifier.
From the right figure of Figure 3.7, we observe that a luminosity of 1fb−1 of a
100 TeV collider can exceed the discovery reach of the present GERDA experiment
for Λ greater than 2-3 TeV. A luminosity of 10fb−1 of a 100 TeV collider can be close
to the discovery reach of the future 1 Tonne experiment, and a luminosity of 100fb−1
can be comparable to the future 1 Tonne experiment.
By comparing the two figures in Figure 3.7, we see that using the machine
learning methods makes the collider experiments much more effective. This is because
of two reasons. First, the machine learning method uses many more physics variables
to construct the features, so it uses much more information. Many of these variables
are not very useful in the cut based analysis, but by combining them together using the
ensemble algorithms, they can be much more effective. Second, the cut based analysis
is equivalent to using only one decision tree, while the random forest method used in
the machine learning analysis uses a collection of many trees. Each tree can make a
64
decision, and the final decision is the average or vote of all the trees. This makes the
final result more reliable and less likely to suffer from over-fitting. However, as it uses
many features and constructs and trains many trees, the machine learning analysis is
much more computationally expensive, especially when we have large volumes of data.
As a summary, both cut based and machine learning analysis show that a future
100 TeV collider with high designed luminosity is more effective in searching for lepton
number violation than the future low energy 0νββ experiments.
65
0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
Λ [TeV]
ge
ff
GERDA
1 Tonne
1 fb-1
10 fb-1
100 fb-1
3 ab-1
30 ab-1
Fig. 3.7.— Comparison of the discovery reaches using cut-based analysis (left) and
machine learning analysis (right). The random forest method is used in the machine
learning analysis. The machine learning analysis was done by Peter Winslow. In the
right figure, the notation M means Λ and yeff means geff .
66
Acknowledgement
I need to thank Professor Michael Ramsey-Musolf and Michael Graesser for the
helpful discussion on the LNV signal at 100 TeV, especially the discussion on searching
for the e±e± signal.
I also thank Dr. Peter Winslow for his machine learning analysis, for sharing his
result figures, and for generating the events at the UMass Titan Cluster.
67
Chapter 4
LHC Signatures of Non-Abelian Kinetic
Mixing
68
4.1 The model
In this chapter, we study the non-Abelian kinetic mixing between the Standard
Model SU(2)L and a dark sector U(1)′ gauge group, with the presentence of a scalar
triplet, and the corresponding phenomenology. This discussion is based on work pub-
lished in Ref [30]. We consider the Standard Model fields with the scalar triplet Σ,
and the scalar triplet-doublet potential, which were described in Section 1.8:
Σ =1
2
(
Σ0√2Σ+
√2Σ− −Σ0
)
, DµΣ = ∂µΣ + ig
[
3∑
a=1
W aµT
a,Σ
]
, (4.1)
V (H,Σ) = −µ2H†H + λ0(
H†H)2 − µ2
ΣG+ b4G2 + a1H
†ΣH + a2H†HG, (4.2)
where G ≡ TrΣ†Σ =(Σ0)
2
2+ Σ+Σ−, and T a is the SU(2)L generator.
We focus on the dimension-five operator
O(5)WX = −β
ΛTr (WµνΣ)X
µν , (4.3)
where Xµν is the U(1)′ vector boson and Wµν the SU(2)L vector boson fields. Λ
is the mass scale of the intermediate fields which are integrated out. Σ is defined as
Σ = ΣaT a. After Σ acquires its vacuum expectation value 〈Σ0〉 ≡ vΣ, the U(1)′ boson
Xµ and the neutral SU(2)L gauge boson W 3µ will mix, with the mixing parameter
ǫ = β(vΣΛ
)
sin θW , (4.4)
69
with θW being the usual weak mixing angle. The coupling between Xµ and other
fields are the same as the photon and other SM fields, except for a universal rescaling
factor ǫ. As can be seen from Eq. 4.4, the magnitude of ǫ is controlled by the ratio
vΣ/Λ. As we saw in Section 1.8, ǫ is constrained by experiments to be smaller than
O(10−3), which was hard to explain in other models. Here we see that ǫ will satisfy
the experimental bounds for β ∼ O(1) and Λ larger than 1 TeV.
In the model, we add to the Standard Model Lagrangian the dimension four and
five operators, which involve the dark photon and the real triplet fields:
L = LSM +∆L(d=4) +∆L(d=5) + . . . . (4.5)
In the above Lagrangian, the dimension four and five operators are in the form:
∆L(d=4) = −1
4XµνX
µν +ǫ0
2 cWBµνX
µν + Tr[
(DµΣ)†DµΣ
]
− V (Σ, H) + ∆L(d=4),
∆L(d=5) = − 1
ΛTr (WµνΣ) (αB
µν + βXµν) ≡ O(5)WB +O(5)
WX . (4.6)
The dimension four operator ∆L(d=4) contains the abelian kinetic mixing term
(the XB term), and cW is the cosine of the usual weak mixing angle. The terms in
∆L(d=4) which breaks the dark U(1)′ gauge group are not explicitly shown here.
Here we wrote the O(5)WX as an effective theory, leaving the model-dependent
details unspecified and focusing on the corresponding collider phenomenology. This
effective operator O(5)WX can be generated in several ways, as shown in Fig. 4.1. It
can be generated via loops, in which the mediators in the loop can be either fermions
or scalars. Or it can be generated from other degrees of freedom in non-perturbative
theories. These can be considered as possible UV complete theory. After the heavy
intermediate states are integrated out, we have the effective operator O(5)WX . Similar
graphs as shown in Fig. 4.1 may also generate the same effective operator.
70
The effective operator O(5)WX can contribute to the S parameter after electroweak
symmetry breaking (EWSB):
αemS = 4cW sWαvΣΛ. (4.7)
This sets a confidence level bound of 90% for αvΣ/Λ ∼< 0.0008. So we set α = 0 and
focus on the phenomenology of the operator O(5)WX .
In Eq. 4.6, after the hypercharge zero field Σ acquires a vev, we have the mixing
term in the Lagrangian:
∆Lmixing = −1
2W 0,3
µν
(
ǫWBB0,µν + ǫWXX
0,µν)
− 1
2ǫBXB
0µνX
0,µν , (4.8)
where ǫWB = αvΣ/Λ, ǫWX = βvΣ/Λ. We can then rewrite the B0, W 0,3 fields in
terms of A0, Z0, and we have
LSM +∆Lmixing ⊃ −1
4(1 + αA)A
0µνA
0,µν − 1
4(1 + αZ)Z
0µνZ
0,µν − 1
4X0
µνX0,µν
−1
2αAZA
0µνZ
0,µν − 1
2αAXA
0µνX
0,µν − 1
2αZXZ
0µνX
0,µν , (4.9)
where the α parameters are defined as
αA = 2cW sW ǫWB, αZ = −2cW sW ǫWB, αAZ = ǫWB
(
c2W − s2W)
,
αAX = ǫBXcW + ǫWXsW , αZX = −ǫBXsW + ǫWXcW . (4.10)
We can then diagonalize the kinetic terms in the Lagrangian into the following
form:
−1
4AµνA
µν − 1
4ZµνZ
µν − 1
4XµνX
µν , (4.11)
71
using the following transformation matrix:
A
Z
X
=
√1 + αA αAZ
√1 + αZ αAX
0√
1− α2AZ
−αAX αAZ+αZX√1−α2
AZ
0 0
√(1−α2
AX)(1−α2
AZ)−(αZX−αAX αAZ)2√
1−α2AZ
A0
Z0
X0
.
(4.12)
The above transformation has the inverse form:
A0
Z0
X0
=
1√1+αA
− αAZ√1+αA
√1−α2
AZ
−αAX+αAZ αZX√1+αA
√1−α2
AZ
√(1−α2
AX)(1−α2
AZ)−(αZX−αAX αAZ)2
0 1√1−αAZ
√1+αZ
αAX αAZ−αZX√1+αZ
√1−α2
AZ
√(1−α2
AX)(1−α2
AZ)−(αZX−αAX αAZ)2
0 0
√1−α2
AZ√(1−α2
AX)(1−α2
AZ)−(αZX−αAX αAZ)2
A
Z
X
,
(4.13)
where the α parameters are defined as
αAZ ≡ αAZ√1 + αA
√1 + αZ
, αAX =αAX√1 + αA
, αZX =αZX√1 + αZ
. (4.14)
After spontaneous symmetry breaking, the mass terms for the gauge boson are
∆Lmass =1
2m2Z0
µZ0,µ +
1
2m2
XX0µX
0,µ, (4.15)
and after being transformed to the fields {A, Z, X}, the mass matrix becomes off-
diagonal:
∆Lmass =1
2(M)2ij V
iµV
jµ, (4.16)
where V 1,2 = Z, X and
M2 =
m2Z
(1−α2AZ
)(1+αZ)
−m2Z(αZX−αAX αAZ)
(1−α2AZ
)(1+αZ)√
(1−α2AX
)(1−α2AZ
)−(αZX−αAX αAZ)2
−m2Z(αZX−αAX αAZ)
(1−α2AZ
)(1+αZ)√
(1−α2AX
)(1−α2AZ
)−(αZX−αAX αAZ)2
m2X(1−α2
AZ)+m2Z
(−αAXαAZ+αZX )2
(1−α2AZ
)(1+αZ )
(1−α2AX
)(1−α2AZ
)−(αZX−αAX αAZ)2
.
(4.17)
Now we can concentrate on the non-Abelian mixing part, and we can set the
Abelian mixing parameters α = ǫBX = 0. Then the following two terms in the
72
Standard Model Lagrangian becomes
eQfγµA0,µf → eQfγµ
(
Aµ − ǫWX sW√
1− ǫ2WX
Xµ
)
f, (4.18)
gZ fγµ(I3L−Qs2W )Z0,µf → gZ fγµ(I3L−Qs2W )
(
Zµ − ǫWX sW√
1− ǫ2WX
Xµ
)
f.
Here we see that with the new fields, the actual value of the electron charge and the
Weinberg angle in the Standard Model are shifted from experimentally known values
of e, sW , cW .
We can then derive the following Feynman rules for the interactions between
the dark bosons fields, SM leptons, gauge bosons, the neutral and charged Higgs
bosons. The new Feynman rules of the non-Abelian kinetic mixing scenario of the form
W±H∓X, ZH1X, ZH2X, AH1X, AH2X dictate novel collider signatures, which are
listed below.
Interaction Feynman rule
Xl+l− ie(
ǫ0 − βv∆sWΛ
)
W±H∓X iβΛ(gµνpp′ − pνp′µ) c∓
ZH1XiβΛ(gµνpp′ − pνp′µ) cW s0
ZH2XiβΛ(gµνpp′ − pνp′µ) cW c0
AH1XiβΛ(gµνpp′ − pνp′µ) sW s0
AH2XiβΛ(gµνpp′ − pνp′µ) sW c0
W+µ (p1)W
−ν (p2)H1Xα(p3)
iβ gΛ
(pµ3gνα − pν3g
µα) s0W+
µ (p1)W−ν (p2)H2Xα(p3)
iβ gΛ
(pµ3gνα − pν3g
µα) c0W±
µ (p1)Zν(p2)H∓Xα(p3) ∓ iβ g
Λ(pµ3g
να − pν3gµα) cW c∓
W±µ (p1)Aν(p2)H
∓Xα(p3) ∓ iβ gΛ
(pµ3gνα − pν3g
µα) sW c∓
In the above Feynman rules, c∓ ≡ cos θ∓ and c0 ≡ cos θ0 are as defined in Ref. [31]. In
the table above, all the momentum of the particles flow into the vertices. The LHC sig-
nature of the SM with an additional scalar field have been studied in Ref. [31], in which
all the Feynman rules needed for the production of scalar particles like H1, H2, H±
at colliders are listed.
73
We also comment here that the kinetic mixing of gauge bosons can arise from
non-abelian gauge groups. For instance, in an SU(N)×SU(M) theory in which the
gauge fields are W and Y , we can introduce a scalar field ∆ab, which transfroms as
the adjoint representation under both the SU(N)×SU(M) groups. Here the indices
“a” and “b” correspond to the indices of the SU(N) and SU(M) groups, respectively.
We can then build the d = 5 operator W aµνY bµν∆
ab similar to the operator O(5)WX . The
kinetic mixing between W and Y occurs after the ∆ab acquires a non-zero vacuum
expectation value. There can also be some renormalizable models that generate this
effective operator at the one-loop level.
4.2 Collider phenomenology
Ref. [31] studied the predictions of a simple extension to SM, where the Higgs
sector inlcudes the usual SU(2)L and the scalar triplet mentioned in Section 4.1. This
model predicts a pair of charged scalars and a dark matter candidate for vanishing
triplet vev. This model predicts a significant excess of the two-photon events compared
to that in SM. With the existence of the non-Abelian kinetic mixing operator O(5)WX ,
the collider phenomenology related to the real triplet can be very different from those
studied in Ref. [31]. Here we make the assumption that the doublet-triplet mixing
angle, which is proportional to vΣ, to be some small but non-zero value. From Eq. 1.16,
we see that in this case, the neutral scalar sector has two mass eigenstates H1,2, where
H1 is mostly the Standard Model Higgs boson andH2 is mostly Σ0. And from Eq. 1.17,
we see that the charged scalars H± are not pure triplet states, instead they are the
mixtures of Σ± and the charged component of the doublet scalar, with the other
mixtures being the longitudinal components of weak gauge bosons. We can also see
74
that if the doublet-triplet mixing angle is zero, then the neutral component Σ0 does
not couple to the Standard Model fermions and can be a dark matter candidate.
When the triplet gets a vev, as needed for the non-Ablelian mixing mechanism, Σ0
can no longer be a DM candidate. The coupling of the mass eigenstates H± and H2
to the SM fermions through the Yukawa interactions is enabled by the presence of a
non-vanishing doublet-triplet mixing angle.
If we have a zero vΣ value instead, then the triplet states have a common mass
m2Σ = −µ2
Σ + a2v2/2. In this case, the loop effect increases the mass of the charged
components and makes the mass splitting between it and the neutral component to
be ∼ 166 MeV. This makes the decay H+ → H2π+ possible. In our studies, which are
under the assumption that vΣ 6= 0, the choice of parameter values will not alter this
mass splitting substantially.
To do the collider studies, We implemented in FeynRules the triplet given in
Eq. 4.1 and the model given in Eq. 4.5. In the model file, we choose the parame-
ters {vΣ, λ0, b4, a1, a2, v0} in Eq. 4.2 as the fundamental parameters, where v0 is
the Higgs vev. However, in the collider studies, our input is the scalar masses like
{
M2H1, M2
H2, M2
H±
}
. So we solved the fundamental parameters in terms of the input
parameters as following, using the relations in Ref. [31]:
a1 =M2
H+
vΣ (1 + v20/(4x20)),
λ0 =M2
H1+M2
H2±√
p
4v20,
b4 =M2
H1+M2
H2∓√
p− a1v20/(2vΣ)
4x20, (4.19)
75
where
p =1
2
(
r ±√
r2 − 4qr)
,
r =(
M2H1
−M2H2
)2,
q = (a1v0 − 2vΣa2v0)2 (4.20)
We can then consider the production and decays of the triplet-like scalars. The
LHC production and decay mechanisms of interest are shown in Fig. 4.2. Diagrams
(a) and (b) are the Drell-Yan pair production process: pp → V ∗ → φφ, where the
symbol φ denotes any of the physical scalars (H1,2, H±), with the subsequent scalar
decays φ→ XV . These graphs have the topology XXV V . As shown in the discussion
above, when the mixing angle is small, the φ states are mostly triplet-like. Diagrams
(c) and (d) are the production pp→ V ∗ → φX which is mediated by the non-Abelian
mixing operator O(5)WX , with a scalar decay φ→ XV . These graphs have the topology
XXV .
We applied the model files in MadGraph to obtain the LHC cross sections for
the production of the neutral and charged scalars. Fig. 4.3 and Fig. 4.4 shows the
LHC production cross sections at√s = 8 for different channels, for mφ = 130 GeV
and mφ = 300 GeV, respectively. For both the mφ values we see that the Drell-Yan
pair production dominates for β/Λ ∼< 1 /TeV, and the O(5)WX-mediated production
dominates for β/Λ ∼> 1 /TeV. For an LHC energy of√s = 14 TeV, we did the similar
study and it shows that the corresponding transition between the Drell-Yan process
and the O(5)WX-mediated production process dominates for almost the same β/Λ values.
76
W µaXν
Σa
F
W µaXν
Σa
S
W νa
Σa
Xν
(a) (b) (c)
Fig. 4.1.— Feynman diagrams that may generate the non-abelian mixing effective
operator O(5)WX . The intermediate particles in the loops are (a) fermions, (b) scalars,
or (c) other sources from non-perturbative dynamics.
(a) (b) (c) (d)
Fig. 4.2.— The Feynman diagrams for LHC production and the subsequent decay of
the particles in the non-Abelian mixing model with the triplet scalars. Diagrams (a)
and (b) are the scalar pair productions, followed by the scalar decays mediated by the
non-Abelian mixing operator O(5)WX . Diagrams (c) and (d) are production and decays
of H and X. In all four diagrams, the incoming vector bosons are all virtual.
77
10- 2 0.1 1 10 100
10- 4
10- 2
1
100
� / � [TeV-1
]
�[pb]
pp � H ± X
pp � H 2 X
pp � H + H -
pp � H ± H 2
Fig. 4.3.— LHC production cross sections for pp → V → φφ and pp → V → Xφ
at√s = 8 TeV, where φ = H+, H2. The mφ = 130 GeV, and mX = 0.4 GeV. For
the processes with final states of a single charged scalar and one neutral boson, we
summed the cross sections for both charges, for example: σ(H+H2) + σ(H−H2).
78
10- 2 0.1 1 10 100
10- 4
10- 2
1
100
� / � [TeV-1
]
�[pb]
pp � H ± X
pp � H 2 X
pp � H + H -
pp � H ± H 2
Fig. 4.4.— LHC production cross sections for pp → V → φφ and pp → V → Xφ
at√s = 8 TeV, where φ = H+, H2. The mφ = 300 GeV, and mX = 0.4 GeV. For
the processes with final states of a single charged scalar and one neutral boson, we
summed the cross sections for both charges, for example: σ(H+H2) + σ(H−H2).
79
4.3 Triplet-like scalar decay branching ratios
Besides the final states considered in Ref. [31], the triplet-like scalar H± can also
decay to W±X and H2 can decay to Z/γ X. For illustrative purposes, we show the
decay width of the tree level H± → W±X below, which is sufficient for the analysis
we consider below.
Γ(H± → W±X) (4.21)
=
√
1− 2(m2X+M2
W±)
M2H±
+(m2
X−M2
W±)2
M4H±
16πMH+
[
1
2
(
M2H± −m2
X −M2W±
)2+M2
XM2W±
]
β2
Λ2c2∓ ,
where c∓ is the mixing angle between the charged scalar fields. Combining other H+
decay channels given in Ref. [31], we calculated the branching ratios of all the channels,
which is shown in Fig. 4.5, Fig. 4.6, which are for vΣ = 1 GeV and vΣ = 10−3 GeV,
respectively. Both figures have mX = 0.4 GeV. In Fig. 4.5, the top plot corresponds
to mH+ = 130 GeV, and the bottom plot corresponds to mH+ = 300 GeV, and the
same applies to Fig. 4.6.
From Fig. 4.5, we see that for vΣ = 1GeV, which is near the maximum allowed
by electroweak precision tests, the H+ → W+X channel has almost 100% branching
ratio for ǫ ∼> 10−4. This means a range of β/Λ ∼> 0.1/TeV. For smaller values of
β/Λ, H+ → W+X can have any branching ratio from zero to one. From Fig. 4.6,
we see that for smaller value of vΣ, the branching ratio of H+ → W+X is essentially
100% for all values of ǫ. From these two figures, we have the observation that when
β/Λ ∼> 0.1/TeV, the branching ratio of H+ → W+X is near 100% and is independent
on vΣ, while for lower β/Λ values, the branching ratio can be any value and depends
strongly on vΣ.
80
10- 5
10- 4
10- 3
10- 2
10- 1
10- 5
10- 4
10- 3
10- 2
0.1
110
- 810
- 710
- 610
- 5
� / � [TeV-1
]
Br(H
+)
�
W+X
W+Z
W+H 1
c s
t b
� + �
10- 5
10- 4
10- 3
10- 2
10- 1
10- 6
10- 4
10- 2
110
- 810
- 710
- 610
- 5
� / � [TeV-1
]
Br(H
+)
�
W+X
W+Z
W+H 1
c s
t b
� + �
Fig. 4.5.— Branching ratios for H+ decays as a function of ǫ (upper horizontal axis)
and β/Λ (bottom horizontal axis), for the triplet vev vΣ = 1 GeV. The dark photon
mass is chosen as mX = 0.4 GeV. The top plot corresponds to mH+ = 130 GeV, and
the bottom plot corresponds to mH+ = 300 GeV. The solid black line is the branching
ratio for H+ → W+X. Branching ratios for other final states are as indicated by other
colors.
81
10- 2 0.1 1 10 100
10- 8
10- 5
10- 2
1010
- 810
- 710
- 610
- 5
� / � [TeV-1
]
Br(H
+)
�
W+X
W+Z
W+H 1
c s
t b
� + �
H 2 � +
10- 2 0.1 1 10 100
10- 7
10- 5
10- 3
0.1
1010
- 810
- 710
- 610
- 5
� / � [TeV-1
]
Br(H
+)
�
W+X
W+Z
W+H 1
c s
t b
� + �
Fig. 4.6.— Branching ratios for H+ decays as a function of ǫ (upper horizontal axis)
and β/Λ (bottom horizontal axis), for the triplet vev vΣ = 10−3 GeV. The dark photon
mass is chosen as mX = 0.4 GeV. The top plot corresponds to mH+ = 130 GeV, and
the bottom plot corresponds to mH+ = 300 GeV. The solid black line is the branching
ratio for H+ → W+X. Branching ratios for other final states are as indicated by other
colors.
82
From the production cross sections in Fig. 4.3, Fig. 4.4 and the decay branching
ratios in Fig. 4.5, and Fig. 4.6, we see that the LHC signatures and thus the detection
strategies vary depending on the value of β/Λ. Therefore, we divide the search regions
into the following three parts, which lead to different phenomenology for the 8 TeV
LHC search.
For β/Λ ∼ 1/TeV. In this region, the Drell-Yan pair production pp → φφ
dominates in the LHC production rates. The branching ratio of φ→ XV can be any
value from zero to one, depending on the value of vΣ.
For β/Λ <∼ 0.1/TeV. In this region, the Drell-Yan pair production pp → φφ
continues to dominate. The decay φ → XV also dominates in all the channels and
has almost 100% branching ratio.
For β/Λ ∼> 1/TeV. In this region, the pp→ Xφ process dominates. This process
is mediated by the non-Abelian kinetic mixing operator O(5)WX , which is our interest.
In addition, the branching ratio of φ → XV is close to one. In this case, the LHC
production final states are those in Fig. 4.2 (c) and (d).
We did the similar study for the production at 14 TeV, which shows that the
transition between the non-Abelian mixing O(5)WX operator mediated production and
the Drell-Yan pair production pp → φφ also happens at β/Λ ∼ 1/TeV. While all the
above three regions are interesting to explore in the future, for illustrative purposes
and interest in the O(5)WX operator mediated production, we focus below on the third
region, which is β/Λ ∼> 1/TeV.
83
4.4 ATLAS recast
Now we focus on the third region, which is β/Λ ∼> 1/TeV, and pp → Xφ domi-
nates and Br(φ→ XV ) is close to one. ATLAS has done the dark photon search and
has their constraints [99], which is shown in the left panel of Figure 16 in Ref. [99]. We
can recast the ATLAS result into constraints in our scenario. The ATLAS analysis in
Ref. [99] assumes the SM Higgs boson decays to 2 γd and 4 γd, leading to displaced
vertices and lepton jets in the final states. This is similar to our process, which is
shown in Fig. 4.2 (c), where an intermediate off-shell vector boson becomes two X
bosons and an on-shell vector boson. We need to note that the ATLAS analysis only
used cuts to isolate events with lepton jets and displaced vertices. They did not recon-
struct the Higgs boson invariant mass, nor did they apply cuts on the missing energy.
They did not require the presence of a final state vector boson either. Therefore, their
analysis is inclusive enough to accommodate the scenario in our case, although it as-
sumed different underlying X-boson productions. In future studies, we can improve
the LHC sensitivity to the non-Abelian kinetic mixing operator O(5)WX by including
more criteria and cuts to identify the final state vector boson.
ATLAS used the limits in the left panel of Figure 16 in Ref. [99] to obtain their
constraints in the parameter space {ǫ,mX}. However, there are several distinctions
between their analysis and our scenario. For examples, the left panel of Figure 16 in
Ref. [99] is the 95% exclusion limits for σ(H)×Br(H → 2X+· · · ) and the dark photon
lifetime cτ . In our case, however, the 95% limits is for σ(φX)×Br(φ→ V X) instead.
Moreover, in the ATLAS analysis, the σ(H) and Br(H → 2X + · · · ) are independent
form the ǫ value and their dependence on mX is negligible when mX is small. In our
84
scenario, however, the corresponding production cross section and branching ratios do
depend on the parameters that govern ǫ, σ(HX) ∼ (β/Λ)2 ∼ 1/(τv2Σ), where τ is the
lifetime of the dark photon X. Therefore, we need to recast the ATLAS limits in the
left panel of Figure 16 in Ref. [99] to the constraints to the parameters in our case.
To make the constraints of our parameters, we first used the ATLAS 95% CL
limit on the process σ(φX)× Br(φ→ V X) in the left panel of Figure 16 in Ref. [99],
where mX = 0.4GeV, we then plotted in the same figure the lines of several constant
cross sections of our process σ(pp→ φX) summed over all φ for three different values
of the triplet vev: vΣ = 1MeV, vΣ = 1.5MeV and vΣ = 2.5MeV. The plots are shown
in Fig. 4.7. In each of our lines, Br (φ→ V X) ≈ 100%. The region above the ATLAS
curve is excluded. So for each of our constant vΣ curves, the points of intersection with
the ATLAS curve determine the boundaries of the excluded region of cτ , which is the
decay length of X. We see that the ATLAS exclusion then applies to the exclusion of
vΣ in the MeV range, which is well below the ρ-parameter limit. The cτ can be related
to Λ/β and vΣ in the following way. We first calculated the decay width (life-time) of
X:
Γ(X → ff) ≡ 1
τ=g2Xff
Nc
12πmX(1 + 2r2)
√1− 4r2, (4.22)
where r ≡ mf/mX , and the coupling is
gXff = ie
(
ǫ0 −βvΣsW
Λ
)
. (4.23)
And then from the constraint of cτ , we can make the constraints on the {Λ/β, vΣ}
plane, which is shown in Fig. 4.8. We made this exclusion for mX = 0.4 GeV and
mX = 1.5 GeV. From the figure, we see that Λ/β can be excluded up to several hundred
GeV, for proper values of mX and vΣ. In Fig. 4.8, we observe that the exclusion limits
85
mX=0.4 GeV
mH+=mH2
=130 GeV
5 10 50 100 500 1000
0.1
1
10
100
cΤ HmmL
Σ*BrHpbL
ATLAS 95% CL limit on Σ*BrΣHvS=1.0 MeV,cΤLΣHvS=1.5 MeV,cΤLΣHvS=2.5 MeV,cΤL
Fig. 4.7.— Constraints on the cτ of X from the ATLAS exclusion. The ATLAS
exclusion in the (cτ , σ × BR) plane [99], where the region above the parabola is
excluded. The diagonal curves are the dependence of σ×BR on cτ for different values
of vΣ. This figure was made by our collaborator G. Ovanesyan.
on Λ/β in the right panel is weaker with smaller values of vΣ. This is because from
Eq. 4.22 we see that for a given value of Λ/β, a smaller value of vΣ means larger value
of cτ , which may fall below the ATLAS exclusion curve in Fig. 4.7.
4.5 More on the UV completion
Our analysis has been as model-independent as possible. But it is also inter-
esting to consider the possible dynamics which can generate the non-Abelian kinetic
mixing operator O(5)WX that we are interested in, and analyze the implications for the
86
ATLAS
mH+=mH2
=130 GeV
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
vS HMeVL
L�ÈΒÈHTeVL
mX=0.4 GeVmX=1.5 GeV
Fig. 4.8.— Constraints on the non-Abelian kinetic mixing model parameters, recasted
from the ATLAS results in Ref. [99]. The curves give the exclusion regions in the(vΣ,
Λ/β) parameter plane for mX = 0.4 GeV (the red region) and mX = 1.5 GeV (the
yellow region). This figure was made by our collaborator G. Ovanesyan.
current and future LHC searches. We have shown in Fig. 4.1 some possible diagrams
that generate this mixing operator, which involves fermion loops, scalar loops, and
non-perturbative dynamics. For a new vector-like fermion F with mass MF , the di-
mensional analysis estimates that Λ/β ∼ 16π2MF/y, where y is the FFΣ coupling,
and we took the gauge couplings to be O(1). The fermion needs to be vector-like
because it can have gauge invariant mass term without the Higgs which means it
can be arbitrarily heavy. If the fermion F is sufficiently light, it would likely have
been observed because it carries the SU(2) charge. In Ref. [100], the non-observation
87
of the pairs of new charged particles like the vector-like leptons at LHC indicates a
lower bound of MF>∼ 200 GeV, which translates to the bound of Λ/β >∼ 3.2 TeV for
y ∼ O(1). To reach this level of sensitivity, we need much larger integrated luminosity
or a search which analyzes the final state gauge boson reconstruction. In the scalar
case, for a new electroweak scalar S with mass MS, the similar dimensional analysis
gives Λ/β ∼ 64π2M2S/aS, where aS is the SSΣ coupling which has the dimension of
mass. To evade the LEP II limits, we assume MS>∼ 100 GeV and take aS ∼MS. This
also gives Λ/β ∼ 6.4 TeV. But aS can be a few times larger than MS. Therefore, we
can anticipate the Λ/β to be the upper end of the exclusion region in Fig. 4.8.
The dynamics in Fig. 4.1 are all loop or non-perturbative processes. The reason
why they are more important than the tree-level processes is the following. It is
possible that the charged triplet state H+ decay to a pair of scalar mediators S at
tree-level. So in order to have the H+ → SS tree level decay not allowed and the
H+ → W+X decay dominate, if there is a mediator S in the loop in Fig. 4.1, the mass
of S needs to be greater than half of the triplet mass.
In our foregoing analysis, we discussed two cases. The first case is mH+ =
130 GeV, which satisfies this constraint of mass and it is also what we used in Fig. 4.1
when recasting the ATLAS results. So in this case, we do not need to worry about
the on shell decays to the S mediators. The second case is mH+ = 300 GeV. It was
used in some of our branching ratio plots. But we did not use it to do the recast in
Fig. 4.1 in deriving our bounds on Λ/β. Without the tree-level process, we can also
see that our assumption of a ∼ 100% branching of H+ → W+X is consistent.
For an internal particle mass near 100 GeV, we may be near the border of the
region of a valid pure effective theory to treat the collider phenomenology. To do
88
a more quantitatively realistic study, we need to either study an explicit model that
generates the mixing operator coefficient or include a form factor. This is similar to the
application of the Higgs effective theory when studying the Higgs boson observables.
For example, Ref. [101] discussed in the case of Standard Model di-Higgs production
together with an additional high pT jet. Currently, the lepton jet reconstruction
efficiency of ATLAS peaks at the region close to pXT ∼ 40 GeV, which is larger than
the masses of the intermediate H±/H2 scalars and the W -bosons in the final state.
So in a more realistic model-dependent analysis, we would expect a little degradation
of the signal strength in the collider. However, we still consider our results about
the current 8 TeV LHC reach as more an indication of the sensitivity than a definite
quantitative result.
As a summary, we see that the mixing between the SU(2)L and the dark U(1)′
gauge groups, which is mediated by the non-Abelian kinetic mixing operator O(5)WX ,
can provide a natural small value of the mixing parameter ǫ, whose value is at the
scale of vΣ/Λ if we have the Wilson coefficient β at the order of O(1). This model
has very distinctive collider phenomenology, because the O(5)WX mediated production
of X dominates the production of all final states when Λ/β >∼ 1 TeV at both the
LHC energy of√s = 8 TeV and
√s = 14 TeV. The current ATLAS bounds, which is
based on the inclusive search for the displaced vertices and lepton jets, can be recast
to our case and indicates an exclusion of Λ/β up to about several hundreds of GeV,
depending on the value of the triplet vev vΣ and the dark photon mass mX . The
future collection of additional data of the ATLAS run will extend the reach of the
search. The additional search criteria associated with the vector bosons in the final
state can help to distinguish this non-Abelian mixing from other scenarios.
89
Chapter 5
QCD corrections for dark matter effective
interactions
90
5.1 Purpose of this study
As shown in the Section. 1.10, it is useful to study the dark matter operators
in a model independent way, using the effective theory, and it is also important to
study the effect of loop corrections and the mixing of dark matter operators. In the
following, we study the short-distance effective operators and the beyond leading order
QCD effects from renormalization group running and mixing of these operators. These
short-distance QCD effects may be useful in extracting the dark matter couplings, to
consider ahead the era of precision dark matter phenomenology after discovery and
identification of the dark matter’s particle nature and interaction features. While new
physics may be generated at a high scale and probed at the collider scale, the direct
detection of dark matter may probe the Wilson coefficients of the effective operators
at a different low scale. Since we have such different scales and we may like to compare
experiments results in these scales, we should consider the effect of the variations of
corresponding interactions due to QCD evolution. The one-loop anomalous dimension
matrix due to QCD corrections of the dark matter effective operators have been studied
in Refs. [56, 57, 72, 102].
In this study, besides doing the full one-loop QCD corrections for all the possible
effective operators, we consider the following two additional issues with short-distance
QCD corrections. First, the QCD running of one set of effective operators, which
was not considered in previous studies. This issue involves dark matter interactions
with quark scalar and pseudoscalar densities. In many of these interactions in the
dark matter models, there are explicit factors of quark Yukawa couplings, which leads
91
to quark mass factor in the operators, as shown in Refs. [52, 53, 54, 55, 58]. In the
following, we discuss a scenario in which such factor suppression in the Yukawa does
not appear. We will also show that the differences in the QCD evolution of the two
scenarios can be noticeable.
The second additional issue is more subtle and it is about the fact that one can
factorize the Standard Model and dark matter components of the effective operators
when computing the αs corrections. This issue involves dark matter fields which are
charged in the Standard Model electroweak gauge group [103, 104]. In this scenario,
the Wilson coefficients of the dark matter operators can have substantial contribu-
tions from box graphs with the exchange of two standard model gauge bosons. From
Ref. [105], we see that the QCD corrections need not factorize for low scale semi-
leptonic weak processes. In this case, we have to calculate the αs corrections by
computing the entire box graphs. In this work, we compute these entire box graphs in
scenarios where the dark matter fields are Dirac fermions, Majorana fermions, inelastic
fermions, and scalars. Our results show that there are non-factorizable contributions,
which corresponds to a matching correction when integrating out the standard model
gauge bosons. Although the matching correction is not in general universal, it still
involve a universal factor of the form (1−αs/π). This correction does not apply when
the Wilson coefficient is generated by the tree-level interactions.
5.2 The effective operators
Here we study the set of operators as our basis shown in Table 5.1. They are
from Ref. [53], with the addition of operators involving scalar and pseudoscalar quark
operators having the factor of quark mass mq in the coefficient.
92
i OΓi ni
1 mqχΓχ qq 7
2 mqχΓχ qγ5q 7
3 χΓχ qq 6
4 χΓχ qγ5q 6
5 χΓαχ qγαq 6
6 χΓαχ qγαγ5q 6
7 χΓαβχ qσαβq 6
8 αsχΓχ (Gaαβ)
2 7
9 iαs χΓχGaαβG
aαβ 7
Table 5.1: Operator basis and their corresponding dimensions.
The effective Lagrangian, written in terms of the operator basis, is
Leff =∑
i
CΓi (µ)
Λni−4OΓ
i (µ) + h.c., (5.1)
where Λ is the scale at which the heavy intermediate particles are integrated out, which
is taken to be of the order of electroweak scale or higher. µ is the renormalization
scale and CΓi (µ) is the Wilson coefficient at scale µ. Γ denotes the scalar, pseudoscalar,
vector, axial vector, and tensor bilinears. For fermion dark matters, they are the Dirac
matrix form χΓχW for Γ = 1, γ5, σµν , γµ, γµγ5, respectively. For scalar dark matters,
these bilinears become χ†χ form, and only appear for i = 1− 4, 8, 9 in Table. 5.1.
In Table. 5.1, for the scalar and pseudo-scalar operators (i = 1 − 4), we have
included both the operators with and without the quark mass operator. To our knowl-
edge, the literature does not typically consider the scalar DM-quark effective operators
without quark mass factor [52, 53, 54, 55, 58]. To motivate the operators without quark
mass factor (O3 and O4 in Table. 5.1), we consider an illustrative scenario, which in-
volves the SU(2)L fermion doublets Ψ(1),Ψ(2). To make the Lagrangian U(1) invariant,
they should carry hypercharge ∓1/2. We also consider a gauge singlet fermion field
93
χR. We can then write down the Lagrangian, which has the following dimension four
operators that can generate the S and P operators without the quark mass factor:
LDM = ǫij
(
c1 Ψ(1)i dR Qj χR + c2 Qi dR Ψ
(1)j χR
)
+ǫij
(
c3 Ψ(2)i uR Qj χR + c4 Qi uR Ψ
(2)j χR
)
⊃(c14− c2
2
)
Ψ(1)1 χR
(
dd+ dγ5d)
+(c34− c4
2
)
Ψ(2)2 χR (uu+ uγ5u) , (5.2)
where Q is the Standard Model quark SU(2)L doublet. Ψ(1)i and Ψ
(2)i (i = 1, 2) are the
components of the SU(2)L doublets Ψ(1) and Ψ(2). Ψ(1)1 is neutral and Ψ
(1)2 is charged,
while Ψ(2)1 is charged and Ψ
(2)2 is neutral. The neutral components of Ψ(1), Ψ(2), and
χR, can mix into the neutral mass eigenstates:
χ0j = Nj1Ψ
(1)1 +Nj2Ψ
(2)2 +Nj3χR (5.3)
where j = 1, 2, 3. The parameters leading to the Nj1,2,3 must be chosen to make
the DM-Z boson couplings small enough to evade the direct detection limits on the
Z-exchange cross section. The lowest mass eigenstate χ01 can be the dark matter
candidate, since its stability is guaranteed by using a Z2 symmetry. We can then invert
Eq. 5.3 and substitute it into Eq. 5.2 to obtain the interaction operator between DM
and quarks. Note that in the operators obtained, there is no quark mass factor. These
are exactly the form of operators O3 and O4 in Table. 5.1. Therefore, we include O3
and O4 in our set of operators.
We next consider the effective Lagrangian in Eq. 5.2 at low energies, which ap-
pears due to the renormalization group running associated with next-to-leading order
QCD corrections, and the possible phenomenological consequences of the corrections.
94
5.3 Loop corrections and the anomalous dimension matrix
To calculate the anomalous dimension matrix, we first compute the one loop
QCD corrections to the operators in Table. 5.1. There are seven possible diagrams
for the one loop correction, as shown in Fig. 5.1. The diagrams with two gluons in
the final state also include the crossed graphs and the symmetry factors. We then
extracted the ultraviolet poles of these diagrams using dimensional regularization.
From the operators in Table. 5.1, we have the following Feynman Rule for the
χχgg vertex is shown in Fig. 5.2.
Extracting the divergent term from the diagrams, we have
iMdivergent =g2
16π2ǫ
1
(µ2)ǫ· 13(χΓ1χ) (qγ
µγνΓ2γνγµq) , (5.4)
where ǫ ≡ 2− d/2, and µ is the energy scale.
By inserting Γi = 1, γ5, σµν , γµ, γµγ5 in the above result, we have
O1,3 → g2
16π2ǫ
1
(µ2)ǫ
(
16
3O1,3
)
,
O2,4 → g2
16π2ǫ
1
(µ2)ǫ
(
16
3O2,4
)
,
O5 → g2
16π2ǫ
1
(µ2)ǫ
(
4
3O5
)
,
O6 → g2
16π2ǫ
1
(µ2)ǫ
(
4
3O6
)
,
O7 → 0. (5.5)
We did similar calculation to other diagrams and other operators.
The way to calculate the anomalous dimension matrix is similar to Section. 2.2,
so we do not repeat the procedure here. The result of the anomalous dimension matrix
95
χ
χ
q
q
g
χ
χ
g
g
χ
χ
q
q
χ
χ
g
g
χ
χ
g
g
χ
χ
g
g
χ
χ g
g
Fig. 5.1.— The diagrams for the one loop QCD corrections to the operators in Ta-
ble. 5.1. The grey dot represents insertion of the operators in Table. 5.1.
96
is
γO =αs
4πγ(0), where
γ(0)33 = γ
(0)44 = −6CF , γ
(0)77 = 2CF ,
γ(0)81 = γ
(0)92 = 24αsCF , (5.6)
where for QCD CF = 4/3. The other elements of the anomalous dimension matrix
are zero. All of the anomalous dimension entries except for entries for the O3 and O4
have been listed in Ref. [72], and for those listed, we have exact agreement except for
the tensor operator anomalous dimension. This is because the tensor operator in [72]
is defined differently from us: imq qσµνγ5q.
Once we have the anomalous dimension matrix, we can know the running of the
Wilson coefficients by solving the equation
dC
d lnµ= γTO · C (5.7)
where C is the vector of the Wilson coefficients and µ is the renormalization scale.
From Eq. 5.6, we see that most of the entries of the anomalous dimension matrix are
zero, so it is relatively straightforward to solve the renormalization group equation
analytically. When solving the equations, we recall that there are additional quark
mass factors in operators O1 and O2, and the entire mq qq and mq qγ5q do not run.
We also need to consider the scale dependence of the αs coupling in operators O8 and
O9. The effect of this scale dependence is that there is no diagonal running of αsGG
97
and αsGG. The full solution to the RG equations is
c1,2(µ) = c1,2(Λ) + c8,9(Λ)Y (r),
c3,4(µ) = c3,4(Λ)X(r),
c5,6,8,9(µ) = c5,6,8,9(Λ),
c7(µ) = c7(Λ)S(r) . (5.8)
where r is the ratio as r = αs(µ)/αs(Λ), and the functions X, Y, S are defined as
X(r) = r3CFβ0 ,
Y (r) = −12CF
β0(αs(µ)− αs(Λ)) ,
S(r) = r−CF
β0 . (5.9)
From the above solution, we see that the operators with quark vector current O5 and
axial vector current O6 do not run, and the operators O8 and O9 with the square
of gluon fields αsGG do not run either. We also see that the operators with quark
scalar and pseudoscalar part and explicit of quark mass factor O1 and O2 mix with
the gluonic operators O8 and O9 when running, while the scalar and pseudoscalar
operators without the quark mass factor O3 and O4 and the tensor operators O7 run
diagonally and do not mix with other operators.
To get more ideas about the effect of the running of the Wilson coefficients, we
show in Fig. 5.3 the dependence of the functions X(r), Y (r), S(r) on the scale Λ from
the new physics scale Λ ∼ 100GeV to the lowest perturbative scale ∼ 1GeV, where
we fix the low energy scale µ = 1 GeV. From the S(Λ) curve, we see that the running
of the tensor operator O7 is mild. And from the X(Λ) curve, we see that the diagonal
running of the scalar and pseudoscalar operators with quark mass factor O3 and O4
98
can be significant when Λ runs from low to above 100 GeV. A similar significant effect
also happens to the mixing of the scalar and pseudoscalar operators without quark
mass factor O1 and O2 into the gluonic operators.
5.4 Phenomenological effects of QCD corrections
We can then use the result of the evolution to do the phenomenological analyses.
Here we focus on the direct detection experiment results. The running and mixing
can also affect the collider phenomenology, but a robust collider study needs to con-
sider more of other higher-order corrections, like the corrections in soft and collinear
resummations. So we will not do the collider study here.
For the direct detection experiments to search for dark matter, the relevant
scale is µ ∼ q < mπ, which is the scale of the momentum transfer in the elastic
scattering between a dark matter particle and a nucleus. We do not evolve the Wilson
coefficients at this scale for two reasons. First, in this scale, the running of the strong
couplings is no longer perturbative. Second, to evaluate the cross sections, we match
our operators onto nucleon level operators at the scale 1 GeV, but at lower scales, the
nucleons, instead of the quark operators, become the effective degrees of freedom. To
have an idea of the theoretical uncertainty of the result, we vary the low scale in a
range µ = 1− 2 GeV. As we see in Fig. 5.3, the Wilson coefficients can change by an
order of 50% depending on the heavy scale Λ. To illustrate the running and mixing
effects, we consider the effect of running on the direct detection cross section σ from
the operators O8,9. At LO, there is no direct effect from O8,9 since they do not run
diagonally. But at NLO, they contribute to the cross section directly as well as by
mixing into O1,2.
99
χ
χ
p
q
a, µ
b, ν
= 4iδab(
gaνgβµ − gµνgαβ)
pαpβχχǫaµǫ
bν
Fig. 5.2.— The Feynman rule for the χχgg vertex.
20 40 60 80 100-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Λ [GeV]
X,Y
,S
X
Y
S
Fig. 5.3.— The dependence of the functions X(r), Y (r), S(r) on the new physics scale
Λ, with r = αs(µ)/αs(Λ), in the RG equation solutions in Eq. 5.8 and Eq. 5.9. Here
we used the low energy scale µ = 1 GeV.
100
To be concrete, we take the fermionic dark matter and choose Γ = 1 as an
example. To evaluate the cross sections, we first consider the operators O1,2,8,9 and
match them into the χ-nucleon interactions:
LNeff = χΓχ
(
CN1 NN + CN
2 Niγ5N)
, (5.10)
where the Wilson coefficients are given in Ref. [106]:
CN1 =
∑
q=u,d,s
cq1mN f(N)Tq
+2
27fTG
·(
∑
q=c,b,t
cq1mN − 12πcg8mN
)
,
CN2 =
∑
q=u,d,s
mN [(−i)(cq2 − Ctot) + 4πicg9m] ∆(N)q ,
(5.11)
where m = (1/mu+1/md+1/ms)−1, and Ctot =
∑
q cq2 m/mq. The numerical values of
the constants f(N)Tq
, f(N)TG
,∆(N)q are also given in Ref. [106]. There are some differences
between Eq. 5.11 and the corresponding equations Eqs. (45a) and (45b) in Ref. [106].
This is because we included the quark mass factor in our definition of O1,2, and our
overall normalization factor in operators O8,9 are also different from the factors in
Ref. [106]. Moreover, Ref. [106] does not have the prefactor of 1/2 in the definition of
the dual tensor Gµνa = (1/2)ǫµνρσGa
ρσ.
From the above χ-nucleon Lagrangian and the solution to the RG equation, we
can calculate the ratio of the direct detection cross sections σNLO/σLO. Here σLO is
the leading order DM-nucleus cross section generated by O8,9, and σNLO is the next-
to-leading order cross section, which includes the mixing of O8,9 into O1,2. We assume
that at the new physics scale Λ, there are only O8,9. At low scale µ, the O8,9 mix into
O1,2.
The results for OΓ=18 and OΓ=1
9 are that for these operators, the overall effect is
101
modest: roughly a 10-15% increase (decrease) in the OΓ=18 (OΓ=1
9 ) with mild depen-
dence on the DM mass at it varies in the range 20 GeV − 1TeV . We have used for
Λ the lower bound on the new physics scale derived from LHC monojet data Λ(mDM)
from Ref. [], hence the mild dependence of σNLO/σLO on mDM, with the effect becom-
ing smaller as mDM approaches 1TeV. The range of the σNLO/σLO corresponds to
varying the low scale µ in 1− 2GeV. So we have c1(Λ) = 0 and
c1,2(µ) = c1,2(Λ) + c8,9(Λ)Y (r) = c8,9(Λ)Y (r)
c8,9(µ) = c8,9(Λ). (5.12)
And the Wilson coefficients in the χ-neucleon Lagrangians at different scales are
CN1 (Λ) = − 2
27fTG
· 12πc8(Λ)mN
CN1 (µ) =
∑
q=u,d,s
c1(µ)mN f(N)Tq
+2
27fTG
(
∑
q=c,b,t
c1(µ)mN − 12πc8(µ)mN
)
(5.13)
And then we have the ratio
σNLO
σLO=
∣
∣
∣
∣
CN1 (µ)
CN1 (Λ)
∣
∣
∣
∣
2
. (5.14)
By plugging in CN1 (µ) and CN
1 (Λ) from Eq. 5.13, we can calculate the ratio σNLO/σLO
numerically as a function of the high scale Λ for operators O8,9. We then use the
lower bound on the new physics scale from the LHC monojet result from Ref. [58] as
a function of Λ(mDM), to derive the numerical relation between the ratio σNLO/σLO
and the dark matter mass mDM. Our result shows a modest dependence of the cross
section ratio on the dark matter mass: a variation of mDM in the range 20 GeV−1TeV
causes approximately a 10-15% increase (decrease) in operator O8 (O9), with the effect
becoming even milder as mDM approaches 1TeV. In this numerical study, we varied
the low scale µ in 1− 2GeV.
102
We then consider the operators O3,4 as another illustration. From Eq. 5.8 and
Fig. 5.3, we see that there is a significant diagonal running effect on the cross section,
which is described by the factor X(r)2. To our knowledge, currently there is no LHC
bound for these operators. However, one would have a stronger bound on O3,4 than
O1,2, for which the scale Λ is very loosely bounded from LHC for Λ >∼ 20−30GeV [58].
Here we take Λ in 10 − 100 GeV as an example. We make a numerical plot of the
ratio σNLO/σLO as a function of the new physics scale Λ, which is shown in Fig. 5.4.
For O3,4, the NLO cross section can be nearly three times larger than the LO cross
section. The nucleon matrix elements cancel in the ratio of the cross sections. So the
entire ratio in Fig. 5.4 is given by only X(µ,Λ)2, as can be seen from the RG running
Eq. 5.8.
We can also study the effect of the operator running on the dark matter relic
density constraints. Here we again consider the operators O3,4 because their diagonal
running effects are the most significant. We take the low energy scale µ to be the order
of the dark matter mass, since the energy released to the dark matter annihilation
products is governed by this energy scale. More specifically, we take µ = mDM and
µ = mDM/2 as two examples. We can write the LO and the NLO annihilation cross
section in the following form
〈σv〉LO = f(mDM)/Λ4,
〈σv〉NLO = f(mDM)/Λ4 ×X[r(mDM,Λ)]2, (5.15)
where the v is the relative velocity of the dark matter particles, and the function
f(mDM) depends on the type of the operator (OS,P3 , OS,P
4 ). It is given in Appendix A
of Ref. [59]. Because we did not include the quark mass factor in operators O3,4, we
103
need to remove a factor of m2q/Λ
2 from the equations of f(mDM) there.
From Eq. 5.15, we have
ΛLO =
(
f(mDM)
〈σv〉LO
) 14
, (5.16)
and we can solve ΛNLO numerically in the same way.
We then solve the scale of new physics ΛLO and ΛNLO in terms of µ and mDM
by setting the LO or NLO order annihilation cross sections 〈σv〉LO and 〈σv〉NLO to be
equal to the cross section corresponding to the relic abundance of the dark matter,
which is 3× 10−26cm/s. The ratio of ΛLO and ΛNLO for the operators OS3 and OP
3 is
shown in Fig. 5.5. The curves for operators type OP3 and OP
4 , which are also identical to
each other, are approximately larger by a factor of 1.025 from the curves in Figure 5.5.
5.5 Box graph corrections and factorizability
Next we study the QCD corrections in the case when dark matter is charged
under SU(2)L ⊗U(1)Y . The effective operators in Section 5.3 may be generated from
the box graphs with electroweak gauge bosons as the intermediate fields. These graphs
shown in Fig. 5.6. This makes it necessary to consider the QCD corrections at the
scale of mW . These corrections will affect the evolution of the operators from the
electroweak scale to the hadronic scale, so they will be different from the corrections
in Section 5.3.
To be concrete, we consider the Standard Model with an additional electroweak
n-tuplet field. The neutral component of the n-tuplet is the lowest mass state, and can
thus be a candidate of the dark matter particle [103, 104]. Here we consider both the
scalar and fermion dark matter fields. For the fermion dark matter, we also consider
104
20 40 60 80 1000.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Λ [GeV]
σN
LO/σ
LO
μ = 1 GeV
μ = 2 GeV
Fig. 5.4.— Ratio of the NLO order to LO DM-nucleon cross sections from the opera-
tors O3 and O4. There is no quark mass factor in operators O3 and O4.
10 15 20 25 301.00
1.05
1.10
1.15
1.20
1.25
mDM [GeV]
ΛN
LO
/ΛL
O
μ = mDM/2
μ = mDM
Fig. 5.5.— The QCD running effect on the relic abundance curve for the operators
O3 and O4. There is no quark mass factor in operators O3 and O4.
105
the Dirac, Majorana, and the inelastic dark matter. We do the box graph calculation
for each of these cases, respectively.
For the QCD corrections for the loop mediated χ-quark interactions, we focus
on the box graphs with gauge bosons as intermediate fields. Ref. [105] studied the
graphs of the semileptonic weak interactions. These graphs have a non-factorizable
pattern. This suggests that one should explicitly check whether the graphs in our case
have a non-factorizable pattern or not. The non-factorizability is what we are going to
study. The loop interactions mediated by other fields such as Higgs bosons have been
studied and the corresponding QCD corrections for the spin-independent interactions
are given in Ref. [75]. Next we consider the box graphs for each of the dark matter
types.
5.5.1 Fermion dark matter
The loop level interactions of dark matter with quarks are more phenomeno-
logically favorable than the tree level interactions because of the strong constraints
from the direct detection experiments of dark matter. In this part we consider the
case when the dark matter is a fermion field which is charged under the SM gauge
group. It can be achieved by taking the hypercharge Y = 0 or allowing Y 6= 0 for
both Majorana fermion or inelastic fermion dark matter. Some examples are given in
Ref. [65]. Fig. 5.6 shows the W and Z box graphs which generate the leading effective
χ-quark Lagrangian. The leading effective Lagrangian, separating the spin-dependent
106
χ χχ
q
W,Z W,Z
χ χχ
q
W,Z W,Z
Fig. 5.6.— Box graphs which describe the χ-quark interactions at the one loop level.
107
and independent terms, is in the following form:
LW/Zeff =
∑
q=u,d,s
dq χγλγ5χ qγλγ5q + fqmq χχ qq + hq χγλχ qγ
λq
+g(1)q
mχ
χi∂µγνχOqµν +
g(2)q
m2χ
χ(i∂µ)(i∂ν)χOqµν , (5.17)
where the effective coefficients dq, fq, hq, g(1)q , g
(2)q are from the loop integration, and
the DM field χ can be either Dirac or Majorana fermions. The Oqµν is the twist-2
operator defined in Ref. 65:
Oqµν ≡ 1
2qi
(
Dµγν +Dνγµ −1
2gµν /D
)
q. (5.18)
5.5.1.1 Dirac dark matter
In the Dirac case, we consider an n-tuplet field and the dark matter is the neutral
component. So it has T 3 = −Y . The corresponding Lagrangian is
LDM = χ(
i /D +M)
. (5.19)
Then we can derive the Feynman rules shown in Fig. 5.7. Using those Feynman
rules, we can calculate the box graphs in Fig. 5.6. The left graph in Fig. 5.6 with W
as the intermediate particle as an example, can be written down as
iM =
∫
d4k
(2π)4ig√2χγµT− i(/p2 + /k +M)
(p2 + k)2 −M2
ig√2γνT+χ
· ig√2uLγ
µ′ i(/p4 − /k)
(p4 − k)2ig√2γν
′
uL · −igµµ′
k2 −m2W
−igνν′(p1 − p2 − k)2 −m2
W
. (5.20)
And then do the loop integrals and the matrix contractions. We can do the same
thing for other graphs and the Z bosons. The result is the effective Lagrangian shown
in Eq. 5.17, and the coefficients are
108
χ χ
W+µ
=ig√2γµT−
χ χ
W−µ
=ig√2γµT+
χ χ
Aµ
= ieQγµ
χ χ
Zµ
=ig
cos θW
(
T 3 − sin2 θWQ
)
γµ
Fig. 5.7.— Feynman rules for the Dirac DM and gauge boson interaction.
109
dLOq =
n2 − (4Y 2 + 1)
4
α22
m2W
gAV(w)
+8[
(aVq )2 + (aAq )
2]
Y 2
cos4 θW
α22
m2Z
gAV(z), (5.21)
fLOq =
4[
(aVq )2 − (aAq )
2]
Y 2
cos4 θW
α22
m3Z
gS(z), (5.22)
hLOq = ± Y α2
2
4m2W
gV(w), (5.23)
g(1),LOq =
n2 − (4Y 2 − 1)
4
α22
m3W
gT1(w)
+8(
(aVq )2 + (aAq )
2)
Y 2
cos4 θW
α22
m3Z
gT1(z),
g(2),LOq =
n2 − (4Y 2 − 1)
4
α22
m3W
gT2(w)
+8(
(aVq )2 + (aAq )
2)
Y 2
cos4 θW
α22
m3Z
gT2(z), (5.24)
where the ratios are defined as w = m2W/m
2χ, z = m2
Z/m2χ; and the couplings aVq =
12T3q − Qq sin2 θW , a
Aq = −1
2T3q are the vector and axial-vector couplings of quarks
with Z boson, respectively; the + and − signs correspond to the up and down type
quarks, respectively.
Our result of the functions gAV(x) and gS(x) agree exactly with Ref. [65]. To our
knowledge, the function gV(x) is new, which is
gV(x) =
√x[2 + (2− x)x]
2 bxarctan
(
2bx√x
)
+2− 2x+ x2 ln x
2, (5.25)
where bx =√
1− x/4. For completeness, we also list the results for the following
110
functions
gAV(x) =
√x(8− x− x2)
24bxarctan
(
2bx√x
)
− x [2− (3 + x) ln x]
24,
gS(x) =(4− 2x+ x2)
4bxarctan
(
2bx√x
)
+
√x (2− x ln x)
4.
gT1(x) =1
3bx(2 + x2) arctan
(
2bx√x
)
(5.26)
+1
12
√x (1− 2x− x(2− x) ln(x)) ,
gT2(x) =1
4bxx(2− 4x+ x2) arctan
(
2bx√x
)
,
−1
4
√x (1− 2x− x(2− x) ln(x)) .
We then proceed to find the next-to-leading QCD corrections, which is the order
O(αs), to the leading results of the coefficients. We use the method explained in
Ref. [105], which is to express the amplitudes of the box graphs in terms of the time
ordered products of two currents. The terms, which can be written in the form of
the Ward identity receive the O(αs) correction. For illustration purpose, we show
the procedure of computing the O(αs) correction for the W exchange box graph. For
the other graphs and other DM scenarios, we give our final answer for the correction
directly.
Written in terms of the currents, the W exchange box graph in Fig. 5.6 can be
written as
M(±)box
W =g4
4N∓
∫
d4l
(2π)41
(l2 −m2W + iε)2
· uχγµ1
/l + /k −mχ + iεγν uχ T
±µνW (l),
(5.27)
where k is the momentum of the incoming dark matter particle. The superscripts ±
means the charges of the dark matter component in the dark matter multiplet. If the
quark is up-type, then M(+)box
W is the amplitude of the upper graph in Fig. 5.6, and
111
M(−)box
W is the lower graph. If the quark is down-type, then M(+)box
W is the amplitude
of the lower graph in Fig. 5.6, and M(−)box
W is the upper graph. The factor N± is
defined as
N± ≡(
T± · T∓)
χ0χ0=n2 − (1± 2Y )2
4, (5.28)
where T± is the raising and lowering generator that acts on the dark matter multiplet.
The TW (l) is defined as
T±λρW (l) =
∫
d4x eilx〈p′T(
J±λW (x)J∓ρ
W (0))
〉p, (5.29)
and the current J±µW (x) is
J±µW (x) = QT±γµPLQ, (5.30)
where Q is the quark SU(2)L doublet. We then use the identity
γµγνγρ = γµgνρ + γρgµν − γνgµρ − iǫσµνργσγ5 ,
to rewrite the γµΓγν terms in terms of single γ matrices, and use the equal time
commutator identity
[
J±0 (x), J
∓ν (0)
]
x0=0= ±2J3
ν (x) δ(3)(~x) , (5.31)
and the result of rewriting Eq. 5.27 becomes
M(±)box
W =g4
4N∓
∫
d4l
(2π)41
[l2 −m2W + iε]2
1
(l + k)2 −m2χ + iε
·{
uχγλuχ
[
± 4i〈p′J3λ(0)〉p+
(
kνgµλ + kµgνλ − (l + k)λgµν
)
T±µνW (l)
]
+[
mχuχγµγνuχ + iǫσµνα(l + k)αuχγσγ5uχ
]
T µνW (l)
}
.
(5.32)
112
By summing up the two graphs in Fig. 5.6, we have
M(+)box
W +M(−)box
W =4Y 2 + 1− n2
2
α22
m2W
gAV(w) uχγλγ5uχ uγλPLu
± α22 Y
2m2W
uχγλuχ uγλPLu
[
g1V (w) + g2V (w)]
+3α2
2
2m3W
n2 − 1− 4Y 2
4uχuχ u /p PLu [g1T(w) + g2T(w)] ,
(5.33)
where p is the momentum of the external quarks, and in the ± α22 Y
2m2W
term, the + sign
means the quark is up-type and the − means the quark is down-type. The functions
g1V,T, g2V,T are calculated as
g2V(x) =(2− x)
√x arctan 2bx√
x
bx+ x ln x ,
g2T(x) =
√x
6(2− x ln x) +
x(x− 2) arctan 2bx√x
6bx,
g1i(x) = gi(x)− g2i(x). (5.34)
Now we can figure out the O(αs) corrections. According to Ref. [105], the terms
in Eq. 5.32 which are proportional to the structure kνTµν(k), kµT
µν(k) do not receive
the O(αs) correction. These terms are identified as the g2V and g1T terms. The other
terms receive a factor of 1 − αs/π correction at next-to-leading order. They are the
gAV, g1V and g2T terms.
We then do the same calculation to the Z exchange box graphs in Fig. 5.6, and
113
the coefficients with the next-to-leading order correction are
dNLOq = dLO
q
(
1− αs
π
)
,
fNLOq =
4α22
[
(aVq )2 − (aAq )
2]
Y 2
m3Z cos4 θW
·[(
1− αs
π
)
g1S(z) + g2S(z)]
,
hNLOq = ± Y α2
2
4m2W
[(
1− αs
π
)
g1V(w) + g2V(w)]
,
g(1),NLOq = g(1),LO
q , (5.35)
g(2),NLOq = g(2),LO
q
(
1− αs
π
)
.
where the extra loop functions other than shown in Eq. 5.34 are defined as
g2S(x) = −g2T(x) ,
g1S(x) = gS(x)− g2S(x) . (5.36)
We can see the non-factorizable terms in Eq. 5.36, which are independent from the
(1− αs/π) factor.
5.5.1.2 Majorana dark matter
We then consider the case when we have a Majorana multiplet λ of the SM
gauge group. The dark matter particle χ is one of the Majorana components of this
multiplet. The corresponding Lagrangian is
LMajoranaDM = λiγµD
µλ/2. (5.37)
There is no tree-level coupling between the Z boson and the dark matter field, so
all the Z exchange box graphs in Fig. 5.6 vanish. Also, note a factor of 1/2 difference
between the Majorana Lagrangian Eq. 5.37 and the Dirac Lagrangian Eq. 5.19. This
is a difference in the standard normalization factors between Dirac and Majorana
114
effective Lagrangians. This factor brings an overall factor of 2 difference in the effective
coefficients compared to the Dirac dark matter. Considering these two effects, the
effective coefficients in this case are
dLOq =
n2 − (4Y 2 + 1)
4
α22
2m2W
gAV(w),
fLOq = hLO
q = 0,
g(1),LOq =
n2 − (4Y 2 + 1)
4
α22
m3W
gT1(w),
g(2),LOq =
n2 − (4Y 2 + 1)
4
α22
m3W
gT2(w). (5.38)
Following the same procedure for the O(αs) correction in Section 5.5.1.1, we
can compute the O(αs) correction for the Majorana case. What is different in the
Majorana case is that the Z box graphs do not contribute, and for the Majorana fields
the term χγµχ in the effective Lagrangian vanishes. So the results are
dNLOq = dLO
q
(
1− αs
π
)
,
fNLOq = hNLO
q = 0, (5.39)
g(1),NLOq = g(1),LO
q ,
g(2),NLOq = g(2),LO
q
(
1− αs
π
)
.
5.5.1.3 Inelastic dark matter
We also consider the case when there is a Dirac multiplet and the neutral com-
ponent of the multiplet is split into two Majorana states:
ψ0 =1√2(χ0 + iη0), (5.40)
where χ0 is the lightest mass state and can be the dark matter candidate. Because
the Majorana feature χγµχ = 0, the χχZ vertex does not exist. But the vertex χηZ
115
is allowed at tree level. Here we assume that the splitting among χ, η is large enough,
so that the Z exchange tree level inelastic scattering vanishes. The Z exchange box
graph with the χηZ vertex, however, is non-zero. In this case, the effective coefficients
are
dLOq =
n2 − (4Y 2 + 1)
4
α22
2m2W
gAV(w)
+8[
(aVq )2 + (aAq )
2]
Y 2
cos4 θW
α22
2m2Z
gAV(z), (5.41)
fLOq =
4[
(aVq )2 − (aAq )
2]
Y 2
cos4 θW
α22
2m3Z
gS(z), (5.42)
hLOq = 0,
g(1),LOq =
n2 − (4Y 2 − 1)
8
α22
m3W
gT1(w)
+4(
(aVq )2 + (aAq )
2)
Y 2
cos4 θW
α22
m3Z
gT1(z),
g(2),LOq =
n2 − (4Y 2 − 1)
8
α22
m3W
gT2(w)
+4(
(aVq )2 + (aAq )
2)
Y 2
cos4 θW
α22
m3Z
gT2(z), (5.43)
The above results agree exactly with Ref. [75]. We then use the same method
described in Section 5.5.1.1 to compute the O(αs) correction. The results are
dNLOq = dLO
q
(
1− αs
π
)
,
fNLOq =
4[
(aVq )2 − (aAq )
2]
Y 2
cos4 θW
α22
2m3Z
[(
1− αs
π
)
g1S(z) + g2S(z)]
,
hNLOq = 0 , (5.44)
g(1),NLOq = g(1),LO
q ,
g(2),NLOq = g(2),LO
q
(
1− αs
π
)
.
116
5.5.2 Scalar dark matter
We now consider a complex scalar dark matter. The corresponding Lagrangian
is
LDM = |Dµχ|2 −m2χ|χ|2. (5.45)
The W and Z exchange box graphs generate the effective Lagrangian in the
following form
LW/Zeff =
∑
q=u,d,s
hq χ†i∂µχ qγ
µq + fqχ†χmq qq ++
g(2)q
m2χ
χ(i∂µ)(i∂ν)χOqµν ,
(5.46)
To our knowledge, our following results of the LO and NLO effective coefficients
are not computed by other people and are new:
hLOq = ∓ α2
2 Y
4m2W
gscalarV (w),
fLOq =
α22 Y
2[
(aVq )2 − (aAq )
2]
m2Z cos4 θW
gscalarS (z),
g(2),LOq =
α22
8m2W
n2 − (4Y 2 + 1)
4gscalarT (w)
+α22 Y
2[
(aVq )2 + (aAq )
2]
m2Z cos4 θW
gscalarT (z)] , (5.47)
where in hLOq , the − sign is for the up type quark and the + is for the down type
quark. The functions gscalarV , gscalar
S , gscalarT are
gscalarV (x) = −
(4− x)(1− x)√x arctan 2bx√
x
2bx− (3− x)x ln x+ 2x+ 4
2,
gscalarS (x) = x ln x+ 4 +
(4− x)(2 + x) arctan 2bx√x
bx√x
,
gscalarT (x) = x ln x− 2 +
(4− x)(2 + x) arctan 2bx√x
bx√x
. (5.48)
117
Similarly to the Dirac case, we compute the NLO QCD corrections to the coef-
ficients. The results are
hNLOq = ∓ α2
2 Y
4m2W
[(
1− αs
π
)
gscalar1V (w) + gscalar
2V (w)]
,
fNLOq =
α22Y
2[
(aVq )2 − (aAq )
2]
cos4 θW m2Z
·[(
1− αs
π
)
gscalar1S (z) + gscalar
2S (z)]
,
g(2),NLOq = g(2),LO
q
(
1− αs
π
)
,
(5.49)
where the loop functions are
gscalar2V = −
√x
2bx[8 + (x− 7)x] arctan
2bx√x+x[−2 + (x− 5) ln x]
2,
gscalar2S = x ln x− 4 +
√x(2− x) arctan 2bx√
x
bx, (5.50)
g1i ≡ gi − g2i, where i = V, S.
As a summary for this section, we considered fermion and scalar DM. For
fermion, we considered Dirac, Majorana, and inelastic cases. For each case, we calcu-
lated the LO effective Lagrangian from the W and Z box graphs and the NLO O(αs)
corrections using the method in Ref. [105]. In several cases we found a non-factorizable
O(αs) correction, which is well known in electron-quark interactions. This correction
occurs in the effective coefficients fq for Dirac, inelastic and scalar DM cases; in hq for
Dirac and scalar DM; and in g(2)q for all fermion DM and scalar DM. For fermion DM
and scalar DM, the coefficient dq has the correction in the form of an overall factor of
(1− αs/π).
118
Chapter 6
SUSY radiative corrections
119
6.1 Parameter scans in MSSM
From Section 1.12, we see that ∆CKM, ∆e/µ and the weak charge are very useful
in searching for the MSSM. So below we extend the numerical studies in Ref [78]. We
make broader constraints in the MSSM parameter space from values of ∆CKM and
∆e/µ, with additional constraints from the weak charge values and LHC constraints
on the sparticle masses. When doing the numerical calculations, we used the codes
by the authors of Ref. [78] and Ref. [79]. From Eq. 1.22 and Eq. 1.24, we see that
the experimental precision on ∆CKM and ∆e/µ can be O(10−4) - O(10−3). So if we
can find some region in the parameter space in MSSM which can make the correction
to ∆CKM and ∆e/µ to be of order O(10−4) - O(10−3), then it will be convenient for
experiments to exclude these parameter regions.
To evaluate the magnitude of the MSSM corrections to ∆CKM and ∆e/µ, we
scanned several variables in the MSSM parameter space, while we took random values
for other parameters. We scanned or randomized the following parameters: the super-
symmetric Higgs-Higgsino mass parameter µ, as defined in the following superpotential
for the MSSM [107]:
W = uyuQHu − dydQHd − eyeLHd + µHuHd, (6.1)
where the µ term is the supersymmetric version of the mass of the Standard Model
Higgs boson. The the gaugino mass parameters M1, M2, M3, M1 and M2 are the bino
and wino mass parameters, which are defined in the following soft supersymmetry
120
breaking interaction for gauginos in MSSM [107]:
Lsoft = −1
2
(
M3gg +M2WW +M1BB + c.c.)
+ ..., (6.2)
and if mZ ≪ |µ±M1| , |µ±M2|, then the neutralinos are nearly a bino-like, a wino-
like, and two higgsino-like mass eigenstates. We also scanned or randomized the ratio
between the up and down type Higgs v.e.v. tan β, the left and right handed slepton
mass m2L and m2
R, and the squark masses m2Q, m
2U , m
2D.
6.2 Correction results
We studied the dependence of ∆CKM on MSSM parameters M1, M2 and µ,
which are defined in the previous section. The dependence of ∆CKM and ∆e/µ on
other parameters, like the sfermion masses, have been studied by Ref. [78].
We varied M1, M2, and µ values, and studied how ∆CKM changes with them.
From Ref. [78], we see that to have large corrections, many parameters, like the second
generation slepton mass, should be small. So we took ml2= 120 GeV and M1 =M2 =
80 GeV when making the plots. The plots of ∆CKM versus M2 and µ are in Fig. 6.1.
From these figures we can see that ∆CKM are mostly O(10−4) and can reach the order
O(10−3) when M2 and µ are smaller than roughly 100 GeV. In Fig. 6.2, we plotted the
dependence of ∆CKM on M1, and we see that ∆CKM almost has a very mild dependence
on M1.
Fig. 6.3 shows the dependence of ∆CKM and ∆e/µ vs µ and M2 when µ = M2.
We see that both ∆CKM and ∆e/µ have similar behavior in terms of their magnitudes.
And again, for most regions, ∆CKM and ∆e/µ are mostly O(10−4) and can reach the
order O(10−3) when M2 and µ are smaller than roughly 100 GeV. We also studied
121
0 100 200 300 400 500
- 0.0014
- 0.0012
- 0.0010
- 0.0008
- 0.0006
- 0.0004
- 0.0002
0.0000
M 2 [GeV ]
�CKM
0 100 200 300 400 500
- 0.0014
- 0.0012
- 0.0010
- 0.0008
- 0.0006
- 0.0004
- 0.0002
0.0000
� [GeV ]
�CKM
Fig. 6.1.— Upper: ∆CKM vs M2, the parameter values are ml2= 120 GeV and
µ = M1 = 80 GeV. Lower: ∆CKM, the parameter values are ml2= 120 GeV and
M1 = M2 = 80 GeV. The resulting charginos are sufficiently heavy as to obey the
LEP limits.
122
the case when µ 6= M2, and the behaviors of ∆CKM and ∆e/µ are very similar to this
figure.
We then studied the effect of adding the constraints on the weak charge and the
collider constraints on the slepton masses. For the weak charge, we selected the data
sets where δ(QeW )SUSY/(Q
eW )S > 2.4% and the S, T , U parameters ranging in 2σ.
For the squark masses, we use the constraints from [109], which searched for
the squark and gluino final states characterized by high-pT jets, missing energy and
absence of electrons and muons at the energy of√s = 8 TeV and a luminosity of
20.3 fb−1 in LHC. For the chargino and slepton masses, we use the constraints from
Ref. [108], which searched for the neutralinos, charginos, and sleptons final states
characterized by the presence of two leptons and missing energy at the same energy
and luminosity in LHC.
After imposing the constraints from the weak charge and the sparticle masses,
the ∆CKM and ∆e/µ values are shown in the scatter plots in Fig. 6.4. For comparison,
we also show the points without constraints from the weak charge and the sparticle
masses. For those points, the sparticle masses are chosen to be random. From the
figure, we see that without the weak charge and sparticle mass constraints, ∆CKM and
∆e/µ can reach near O(10−3), while with these constraints, ∆CKM and ∆e/µ can only
reach O(10−4).
From the figures in this section, we can make the conclusion that the SUSY
correction to ∆CKM and ∆e/µ are or order O(10−4) for most parameter regions. Future
experimental measurements of them also need to reach the precision of O(10−4) in
order to have a reliable exclusion of the MSSM parameter space.
123
0 100 200 300 400 500
- 0.0014
- 0.0012
- 0.0010
- 0.0008
- 0.0006
- 0.0004
- 0.0002
0.0000
M 1 [GeV ]
�CKM
Fig. 6.2.— ∆CKM vs M1, the parameter values are ml2= 120 GeV and µ = M2 =
80 GeV.
124
0 100 200 300 400 500- 0.002
- 0.001
0.000
0.001
0.002
� =M 2 [GeV ]
� CKM
� e / �
M 1 = 500 GeV
ml
�
2
= 120 GeV
0 100 200 300 400 500- 0.002
- 0.001
0.000
0.001
0.002
� =M 2 [GeV ]
� CKM
� e / �
M 1 = 1 TeV
ml
�
2
= 120 GeV
Fig. 6.3.— ∆CKM and ∆e/µ vs µ and M2, where µ = M2. The difference between the
two figures is that the upper figure has M1 = 500 GeV, while the lower figure has
M1 = 1 TeV.
125
-0.0002 -0.0001 0.0000 0.0001 0.0002-0.0002
-0.0001
0.0000
0.0001
0.0002
Δe/μ
ΔCKM
-0.0010 -0.0005 0.0000 0.0005 0.0010-0.0010
-0.0005
0.0000
0.0005
0.0010
Δe/μ
ΔCKM
With constraints
No constraint
Fig. 6.4.— The ∆CKM and ∆e/µ scatter plots for parameters constrained by weak
charge and LHC results. The upper one shows only the constrained plot. The lower
one shows the comparison between the constrained plot and the plot with completely
random parameters.
126
Acknowledgement
We thank S. Bauman for his codes in calculating the ∆CKM and ∆e/µ values. We
also thank W. Chao in imposing the weak charge and LHC constraints, and thank S.
Su for her codes in imposing these constraints.
127
Chapter 7
Summary
As a summary, we considered several symmetries and interactions beyond the Standard
Model, and studied their phenomenology in both high energy colliders and low energy
experiments.
In particular, we studied the lepton number violation. We did this study because
the lepton number conservation is not a fundamental symmetry in Standard Model
(SM). The nature of the neutrino depends on whether or not lepton number is violated.
Leptogenesis also requires lepton number violation (LNV). To test LNV, we compared
the sensitivity of high energy collider and low energy neutrinoless double-β decay
(0νββ) experiments. We included the QCD running effects to obtain the constraints of
the effective coefficient in the operator at the LHC scale. We included the long-distance
contributions to nuclear matrix elements and matched the quark level operators to the
hadronic operators using the SU(2)L × SU(2))R chiral transformation properties of
the operators and obtained the 0νββ decay half life time. We also did the background
analysis for the collider search at both 14 TeV and 100 TeV. We used simplified models,
which allows for the case when one or more heavy particle goes on shell. Our result
128
shows that the reach of future tonne-scale 0νββ decay experiments generally exceeds
the reach of the 14 TeV LHC for a class of simplified models. For a range of heavy
particle masses at the TeV scale, the high luminosity 14 TeV LHC and tonne-scale
0νββ decay experiments may provide complementary probles. The 100 TeV collider
with a luminosity of 30 ab−1 exceeds the reach of the tonne-scale 0νββ experiments
for most of the range of the heavy particle masses at the TeV scale. These results can
show how the future colliders and low energy experiments can improve sensitivity and
do the complementary tests for LNV.
We then studied the non-Abelian kinetic mixing and its LHC signatures. We
introduced a U(1)′ vector boson to the Standard Model with an extra scalar triplet.
The U(1)′ vector boson can mix with the SU(2)L gauge boson via non-Abelian kinetic
mixing. The benefit of this scenario is that it can explain the smallness value of the
coupling ǫ naturally. We then studied the collider phenomenology of the non-Abelian
kinetic mixing operator. In particular, we studied the production rates of the charged
and neutral Higgs scalars and the dark photon. We see that in some regions, the Drell-
Yan pair production pp→ φφ dominates in the LHC production rates, while in other
regions the production of pp→ Xφ dominates. We also studied the decay width of the
charged scalar, and we see for some regions the branching ratio of H+ → W+X can be
nearly 100%. We then outlined the possible LHC signatures and recasted the current
ATLAS dark photon experimental results into our non-Abelian mixing scenario. Our
result shows that the exclusion can reach Λ/β to several hundreds of GeV depending
on the values of the dark photon mass and the triplet vev. The future experiments
may improve sensitivity and may distinguish this scenario from the Abelian mixing by
searching for the lepton jets and displaced vertices and the vector boson signatures.
129
We then studied the QCD corrections for dark matter effective interactions. we
studied the QCD running for a list of dark matter effective operators. This will be
useful to connect the new physics at high scale and the effect of the direct detection
of DM at low scale. We then studied the phenomenological effects of QCD corrections
by analyzing the effects on direct detection experiments and the dark matter relic
density constraints. These results are important in precision DM physics. Currently
little is known about the short-distance physics of DM. We calculated the box graphs
that can generate the dark matter effective operators for several different types of
dark matter fields. We then calculated the next to leading order QCD corrections,
and studied the non-factorizability. The results shows that the short-distance QCD
corrections generate a finite matching correction when integrating out the electroweak
gauge bosons.
Finally, we studied the supersymmetry. The high precision measurements of
electroweak precision observables can provide crucial input in the search for super-
symmetry (SUSY) and play an important role in testing the universality of the SM
charged current interaction. We calculated the impact of the radiative corrections to
the charged current universality in MSSM, which involves the observables ∆CKM and
∆e/µ, with the experimental constraints from the weak charge and the LHC constraints
on the sparticle masses. The order of magnitude of the corrections implies that future
experiments on charged current universality need to improve the precision to an order
of 10−4 in order to make reliable exclusions in the MSSM parameter space.
130
References
[1] G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976).
[2] J. Schechter and L. Valle, Phys. Rev. D 25, 2951 (1982).
[3] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986).
[4] A. de Gouvea and P. Vogel, arXiv: 1303.4097 (2013).
[5] G. Prezeau, M. Ramsey-Musolf, P. Vogel, Phys. Rev. D 68, 034016 (2003).
[6] S. Pascoli and S.T. Petcov, Phys. Lett. B 544, 239 (2002).
[7] J.D. Vergados, Phys. Rep. 361, 1 (2002).
[8] R.N. Mohapatra and J.C. Pati, Phys. Rev. D 11, 2558 (1975).
[9] G. Senjanovic and R.N. Mohapatra, Phys. Rev. D 12, 150 (1975).
[10] R.N. Mohapatra, Phys. Rev. D 34, 3457 (1986).
[11] J.D. Vergados, Phys. Lett. B 184, 55 (1987).
[12] M. Hirsch, H.V. Klapdor-Kleingrothaus, and S.G. Kovalenko, Phys. Rev. D 53,
1329 (1996).
[13] H. Klapdor-Kleingrothaus, A. Dietz, L. Baudis, et al. Eur Phys J A 12, 147
(2001). doi:10.1007/s100500170022.
[14] H. Klapdor-Kleingrothaus, and I. Krivosheina, Mod. Phys. Lett. A 21 1547-1566
(2006).
131
[15] J. B. Albert et al. (EXO-200 Collaboration), Nature (London) 510, 229 (2014).
[16] M. Agostini et al. (GERDA Collaboration), Phys. Rev. Lett. 111, 122503 (2013).
[17] U. Bellgardt et al. (SINDRUM Collaboration), Nucl. Phys. B 299, 1 (1988).
[18] E. Goudzovski, Moriond EW 2011 conference, arXiv:1105.5957.
[19] MAJORANA Collaboration, arXiv:1109.4154 [nucl-ex].
[20] A. Gando et al. (KamLAND-Zen Collaboration), Phys. Rev. Lett. 117, 082503
(2016).
[21] ATLAS Collaboration, JHEP 03, 041 (2015).
[22] CMS Collaboration, JHEP 01, 163 (2014).
[23] CMS Collaboration, Eur. Phys. J. C 76, 439 (2016).
[24] R.N. Mohapatra, Nucl. Phys. Proc. Suppl. 77, 376 (1999).
[25] ATLAS Collaboration, Eur. Phys. J. C 72, 2244 (2012).
[26] ATLAS Collaboration, Phys. Rev. Lett. 108 24, 241802 (2012).
[27] V. Shiltsev, arXiv:1509.08369 [physics.acc-ph].
[28] M. Benedikt, B. Goddard, D. Schulte, F. Zimmermann, M.J. Syphers, FCC-hh
Hadron Collider - Parameter Scenarios and Staging Options, Proceedings of
IPAC2015.
[29] CEPC-SPPC Preliminary Conceptual Design Report, The CEPC-SPPC Study
Group, IHEPCEPC-DR-2015-01 (2015).
132
[30] C.A. Argüelles, X.-G. He, G. Ovanesyan, T. Peng, M.J. Ramsey-Musolf, Phys.
Lett. B770, 101.
[31] P. F. Perez, H. H. Patel, M. J. Ramsey-Musolf, and K. Wang, Phys. Rev. D 79,
055024 (2009).
[32] F. Chen, J. M. Cline, and A. R. Frey, Phys.Rev. D79, 063530 (2009), 0901.4327.
[33] F. Chen, J. M. Cline, and A. R. Frey, Phys.Rev. D80, 083516 (2009), 0907.4746.
[34] J. M. Cline and A. R. Frey, (2014), 1408.0233.
[35] G. Barello, S. Chang, and C. A. Newby, (2015), 1511.02865.
[36] R. Bernabei et al., DAMA Collaboration, Eur.Phys.J., C56:333-355 (2008),
arXiv:0804.2741.
[37] G.W. Bennett et al., Phys. Rev. D73, 072003 (2006).
[38] R. Essig et al., Working Group Report: New Light Weakly Coupled Particles,
in Community Summer Study 2013: Snowmass on the Mississippi (CSS2013)
Minneapolis, MN, USA, July 29-August 6, 1311.0029 (2013).
[39] D. Curtin, R. Essig, S. Gori, and J. Shelton, JHEP 02, 157 (2015), 1412.0018.
[40] B. Holdom, Phys.Lett. B166, 196 (1986).
[41] R. Foot and X.-G. He, Phys.Lett. B267, 509 (1991).
[42] P. Fayet, Phys.Rev. D70, 023514 (2004), hep-ph/0403226.
133
[43] H. Davoudiasl, H.-S. Lee, and W. J. Marciano, Phys.Rev. D85, 115019 (2012),
1203.2947.
[44] H. Davoudiasl, H.-S. Lee, and W. J. Marciano, Phys.Rev. D89, 095006 (2014),
1402.3620.
[45] R.D. Peccei, CP Violation, C. Jarlskog, ed. (World Scientific, 1989).
[46] S. Baek, P. Ko, and W.-I. Park, JCAP 1410, 067 (2014), 1311.1035.
[47] R. Agnese et al., CDMS Collaboration, Phys.Rev.Lett. (2013), arXiv:1304.4279.
[48] D.S. Akerib et al., CDMS Collaboration. Phys.Rev., D82:122004 (2010),
arXiv:1010.4290.
[49] J. Angle et al., XENON10 Collaboration. Phys.Rev.Lett., 107:051301, (2011),
arXiv:1104.3088.
[50] E. Aprile et al., XENON100 Collaboration. Phys.Rev.Lett., 109:181301, (2012),
arXiv:1207.5988.
[51] D.S. Akerib et al., LUX Collaboration, Phys. Rev. Lett. (2013), arXiv:1310.8214.
[52] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T.M.P. Tait, et al, Phys.Lett.,
B695:185-188, (2011), arXiv:1005.1286.
[53] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T.M.P. Tait, et al, Phys.Rev.,
D82:116010, (2010), arXiv:1008.1783.
[54] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T.M.P. Tait, et al, Nucl.Phys.,
B844:55-68, (2011), arXiv:1009.0008.
134
[55] A. Rajaraman, W. Shepherd, T.M.P. Tait, A.M. Wijangco, Phys.Rev., D84:095013
(2011), arXiv:1108.1196.
[56] R.J. Hill and M.P. Solon, Phys.Lett., B707:539-545 (2012), arXiv:1111.0016.
[57] L. Vecchi, arXiv:1312.5695.
[58] N. Zhou, D. Berge, and D. Whiteson, Phys.Rev.D87(9):095013 (2013),
arXiv:1302.3619.
[59] M.R. Buckley, Phys.Rev.D88:055028 (2013), arXiv:1308.4146.
[60] M. Papucci, A. Vichi, and K.M. Zurek, arXiv:1402.2285.
[61] J. Abdallah et al, arXiv:1409.2893.
[62] S. Baek, P. Ko, M. Park, W.-I. Park, and C. Yu, arXiv:1506.06556.
[63] G. Prezeau, A. Kurylov, M. Kamionkowski, and P. Vogel, Phys.Rev.Lett.,
91:231301 (2003), arXiv:astro-ph/0309115.
[64] M. Freytsis and Z. Ligeti, Phys.Rev., D83:115009 (2011), arXiv:1012.5317.
[65] J. Hisano, K. Ishiwata, N. Nagata, and T. Takesako, JHEP, 1107:005 (2011),
arXiv:1104.0228.
[66] V. Cirigliano, M.L. Graesser, and G. Ovanesyan, JHEP, 1210:025 (2012),
arXiv:1205.2695.
[67] U. Haisch, F. Kahlhoefer, and J. Unwin, JHEP, 1307:125 (2013), arXiv:1208.4605.
[68] R.J. Hill and M.P. Solon, arXiv:1309.4092.
135
[69] V. Cirigliano, M.L. Graesser, G. Ovanesyan, and I.M. Shoemaker,
arXiv:1311.5886.
[70] R.J. Hill and M.P. Solon, arXiv:1401.3339.
[71] A. Crivellin, F. D’Eramo, and M. Procura, arXiv:1402.1173.
[72] R. J. Hill and M.P. Solon, Phys. Rev. D91:043505 (2015), arXiv:1409.8290.
[73] G. Ovanesyan and L. Vecchi, JHEP 07:128 (2015), arXiv:1410.0601.
[74] F. D’Eramo and M. Procura, JHEP 04:054 (2015), arXiv:1411.3342.
[75] J. Hisano, K. Ishiwata, and N. Nagata, JHEP, 06:097 (2015), arXiv:1504.00915.
[76] J.L. Feng, J. Kumar, D. Marfatia, and D. Sanford, Phys.Lett. B703:124-127
(2011), arXiv:1102.4331.
[77] A. Kurylov and M. J. Ramsey-Musolf, Phys. Rev. Lett. 88, 071804 (2002).
[78] S. Bauman, J. Erler, and M. J. Ramsey-Musolf, Phys. Rev. D 87, 035012 (2013).
[79] M. Ramsey-Musolf, S. Su, and S. Tulin, Phys. Rev. D 76, 095017 (2007).
[80] I. S. Towner and J. C. Hardy, Rept. Prog. Phys. 73, 046301 (2010).
[81] V. Cirigliano and I. Rosell, JHEP 0710, 005 (2007) [arXiv: 0707.4464 [hep-ph]].
[82] J. Erler, A. Kurylov, and M.J. Ramsey-Musolf, Phys. Rev. D68, 016006 (2003).
[83] A. Kurylov, M. J. Ramsey-Musolf and S. Su, Phys. Rev. D 68, 035008 (2003)
[hep-ph/0303026].
136
[84] A. Czarnecki and W. Marciano, Phys. Rev. D53, 1066 (1996).
[85] G. Degrassi, B. A. Kniehl, and A. Sirlin, Phys. Rev. D48, 3963 (1993).
[86] T. Peng, M.J. Ramsey-Musolf, and P. Winslow, Phys. Rev. D93, 093002 (2016).
[87] M. Hirsch, talk BLV-2013, Heidelberg, 2013.
[88] N. Bell, V. Cirigliano, M.J. Ramsey-Musolf, P. Vogel, and M. Wise, PRL 95
151802 (2005).
[89] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory,
Addison-Wesley, Reading, 1995.
[90] A. Faessler, S. Kovalenko, and F. Simkovic, Phys. Rev. D 58, 115004 (1998),
hep-ph/9803253.
[91] J. Donoghue, E. Golowich, and B. Holstein, Dynamics of the Standard Model
(Second Edition) Cambridge Univ. Press (2014).
[92] Particle Data Group, K. A. Olive et al., Chin. Phys. C 38, 090001 (2014).
[93] M. Doi, T. Kotani, and E. Takasugi, Prog. Theor. Phys. Suppl. 83, 1 (1985).
[94] ATLAS Collaboration, JHEP 1110, 107 (2011) [arXiv:1108.0366].
[95] UMass Cluster: https://twiki.cern.ch/twiki/bin/view/Main/UMassCluster
[96] ATLAS Collaboration, Eur. Phys. J. C72, 1909 (2012) [arXiv:1110.3174].
[97] J. Helo, M. Hirsch, S. Kovalenko, and H. Pas, Phys. Rev. D 88, 011901 (2013),
1303.0899.
137
[98] J. Helo, M. Hirsch, H. Pas, and S. Kovalenko, Phys. Rev. D 88, 073011 (2013),
1307.4849.
[99] ATLAS, G. Aad et al., JHEP 11, 088 (2014), 1409.0746.
[100] N. Kumar and S. P. Martin, Phys. Rev. D92, 115018 (2015), 1510.03456.
[101] M. J. Dolan, C. Englert, and M. Spannowsky, JHEP 10, 112 (2012), 1206.5001.
[102] B. Grinstein and L. Randall, Phys.Lett. B217:335 (1989).
[103] M. Cirelli, N. Fornengo, and A. Strumia, Nucl.Phys. B753:178-194 (2006),
arXiv:hep-ph/0512090.
[104] M. Cirelli, A. Strumia, and M. Tamburini, Nucl.Phys. B787:152-175 (2007),
arXiv:0706.4071.
[105] A. Sirlin, Rev.Mod.Phys. 50:573 (1978).
[106] M. Cirelli, E.D. Nobile, and P. Panci, JCAP 1310:019 (2013), arXiv:1307.5955.
[107] S.P. Martin, A Supersymmetry Primer, arXiv:hep-ph/9709356.
[108] G. Aad et al. [ATLAS Collaboration], JHEP 1405, 071 (2014) [arXiv:1403.5294
[hep-ex]].
[109] G. Aad et al. [ATLAS Collaboration], arXiv:1405.7875 [hep-ex].