MINISTRY OF
EDUCATION AND TRAINING
VIET NAM ACADEMY OF SCIENCE
AND TECHNOLOGY
INSTITUTE OF PHYSICS
LE THO HUE
LEPTON FLAVOR VIOLATING DECAYS IN SUPERSYMMETRIC
331 MODELS
Major subject: Theoretical and mathematical Physics
Code: 62 44 01 01
THE SUMMARY OF THE DOCTORAL DISSERTATION
HA NOI  2013
The research was finished at:
INSTITUTE OF PHYSICS,
VIET NAM ACADEMY OF SCIENCE AND TECHNOLOGY
Supervisor : Prof. Dr. Hoang Ngoc Long
Institute of Physics, Viet Nam Academy of Science and Technology
Referee 1: Prof. Dr Tran Huu Phat
Referee 2: Prof. Dr. Nguyen Xuan Han
Referee 3: Assoc. Prof. Dr. Phan Hong Lien
This dissertation will be defended in front of the evaluating assembly at academy
level
Place of defending: meeting room, floor , Institute of Physics, Viet Nam
Academy of Science and Technology.
No 10, Dao Tan, Ba Dinh, Ha Noi
Time: at…… … day… month……. year……
This thesis can be studied at the Vietnam National Library or
at Library of the National Academy of Public Administration
Introduction
Urgency of the topic
Particle physics are now strictly related with high energy col
liders. All models are waiting for experimental significance of
New Physics to compare with all predictions as well as limiting
parameter space of each model. Specially, in the last 2012 and
early 2013 the Large Hadron Collider (LHC), placed in CERN
Switzerland with two independent detectors CMS and ATLAS,
has discovered a new scalar particle inheriting many properties
of Higgs (Higgslike) predicted by the Standard Model (SM) with
mass around 125126 GeV. This is the last type of particle con
tained in SM which has already found by recent experiments. In
the coming time, when energy of LHC increases up to 14 TeV,
physicists hope the appearance of New Physics significance be
yond SM. In particular, there may be Lepton Flavor Violating
(LFV) decays of normal leptons. Until now, SM is still the most
successful model predicting all current experimental results ex
cept the experiments of neutrino oscillations. They prove that
neutrinos indeed have nonzero but very small masses and there
appear the conversions among them. This confirms that SM is
still an effective theory of some more general theory. The con
versions among neutrinos gives an clear evidence for the LFV
in the world of fundamental particles. Therefore, one hopes that
these effects may happen in the charged lepton sector. While the
Lepton Flavor (LF) numbers are absolutely conserved in the SM
1
framework, then cLFV decays, if happen, will be a clear evidence
of New Physics. It was proved that the simplest model beyond
SM, the Standard Model with righthanded neutrinos, has very
suppressed cLFV effects that cannot be detected by recent exper
iments. On the other hand, the minimal supersymmetric Stan
dard Model (MSSM) which is the most attractive model studied
by both theory and experiment, contains many interesting cLFV
effects. The model is also the simplest supersymmetric model
with the least new particles as well as parameters. Many publi
cations for MSSM confirm that cLFV significance can be happen
in colliders such as LHC. In addition, some cLFV decays, such
as cLFV decays of tauon , have limited the parameter space of
this model, excluded many regions containing light superpart
ners. This leads to the consequence that these superparticles are
out of the detection of recent experiments. For the supersym
meetric 331 (SUSY331) models, it is also necessary to predict
and investigate the parameter space in order to compare with
other models and experimental bounds. This is the main rea
son why we concentrate on the study of the cLFV decays in the
SUSY331 models and publish obtained results in this disserta
tion. We will consider two models supersymmetric economical
331 (SUSYE331) and supersymmetric reduced minimal 331
(SUSYRM331) models.
Research purpose
• We construct the SUSYRM331.
• We study the lepton flavor violation in the SUSYE331 model
through decays of Higgs, tau and Z boson.
Research object
• LFV verteices in the SUSYE331 and SUSYRM331.
2
• cLFV decays of H→ μτ , τ → μγ, τ → 3μ and Z → μτ in
the SUSYE331.
Research content
• The supersymmetric reduced minimal 331 model.
• Properties of LFV vertices in SUSY331 models
• The possibility of detection decay H→ μτ in recent colliders
• Discussing on regions of parameter space of SUSYE331 sat
isfying the experimental bounds.
Research method
• Quantum field theoretical method.
• Numerical investigation by Mathematica 7.0.
Dissertation Structure
The dissertation contains the introduction, four chapters present
ing main content and the conclusion listing all of new results
of our works. In addition, the dissertation contains three more
appendices which show in detail all necessary analytic formu
las and computations. The first chapter summaries properties of
the 331 models, discusses on the LFV property of these models
and the basis of the supersymmetric theory. The second chapter
concentrates on two models SYSYE331 and SUSYRM331. For
SUSYE331, we just discuss on the LFV vertices which are the
LFV sources creating all LFV decays considered in the next chap
ters. The remain part of this chapter involves with constructing in
detail the SUSYRM331 and the LFV vertices in the soft term of
the model. Two chapters three and four directly investigate some
particular LFV decays in the frame work of the SUSYE331.
3
Chương 1
Review of 331 models
The original 331 models are the minimal 331 (M331) and the
331 model with righthanded neutrinos. These models are con
structed by extending the SU(2)
L
gauge group in the SM to
SU(3)
L
group. A common property of these models are the ap
pearance of LFV effects in the original structures them selves.
1.1 The 331 model with righthanded neu
trinos
In this model, the SU(2)
L
group was extended SU(3)
L
by adding
a righthanded neutrino to the bottom component of each lepton
triplet. In the quark sector, new quarks appearing as third com
ponents of quark (anti) triplets are called exotic quarks, with the
lepton number L =2. One needs three Higgs (anti)triplets to gen
erate masses of particles. Because neutrinos and antineutrinos
are in the same triplets then the lepton numbers are not con
served. This is the common property of the 331 models. But
these models relate with a new conserved number called extended
lepton number, denoted L. It can be computed from the original
L number by a formula
L =
2
√
3
λ
8
+ LI. (1.1)
4
More detail, the table 1.1 lists particular values of L and baryon
number B = BI of multiplets contained in the considered model.
In addition, table 1.2 presents numerical values of components of
all multiplets.
Bảng 1.1: Values of B and L for 331 models with righthanded
neutrinos.
Multiplet χη ρQ
3L
Q
αL
u
aR
d
aR
T
R
D
αR
f
aL
l
aR
B 00 0
1
3
1
3
1
3
1
3
1
3
1
3
00
L
4
3
−
2
3
−
2
3
−
2
3
2
3
00−22
1
3
1
Bảng 1.2: Nonrezo values of L for fields in the 331 models with
righthanded neutrinos.
Field N
L
l
L
l
R
ρ
+
3
η
0
3
χ
0
1
χ
−
2
D
αL
D
βL
T
L
T
R
L −11 1−2 −22 2 22−2 −2
From table 1.2, we can see that neutral Higgses have zero val
ues of VEVs unless they have L =0. The number of neutral Hig
gses can be reduced if neutral Higgses with nonzero values of L
can get nonzero but enough small values of VEVs. Because LFV
is the natural property of the 331 models so this assumption is
allowed, provided VEVs are enough small to satisfy experimental
bounds. This is the idea of authors P.V Dong, H.N. Long, D.T.
Nhung and D.V. Soa to construct the economical 331 (E331)
model. This model contains many proper properties comparing
with other 331 models: The number of parameters are reduced,
VEV values are limited from experiment bounds, the LFV in the
neutral lepton sector can be explained, The simple property
of the Higgs sector in the E331 model is used to construct the
supersymmetric version which is widely investigated recently.
5
1.2 The minimal 331 model
The minimal 331 model (M331) model is constructed similar to
the cases of above 331 models. The difference is that all lepton
triplets are wellknown leptons in the SM and there need not
any new leptons. But the Higgs spectrum in this model is rather
complicated because of the appearance of a Higgs sextet. Table
1.3 lists values of B và L for multiplets in the model. Note that
Bảng 1.3: Values of B and L in the M331.
Đa tuyến χρ ηSQ
3L
Q
αL
u
aR
d
aR
T
R
D
αR
f
aL
Tích B 00 00
1
3
1
3
1
3
1
3
1
3
1
3
0
Tích L
4
3
−
2
3
−
2
3
2
3
−
2
3
2
3
00−22
1
3
formula (1.1) is still true for this model. Table 1.3 shows that all
Higgses with nonzero VEVs all have L =0.
Although the M331 does not need extra leptons such as right
handed neutrios, the Higgs spectrum is complicated and it is
hardly to find the exact mass eigenvalues of Higgses. Hence, su
persymmetric versions constructed from this model always has
problems of Higgs diagonalization. Recently, a new model with
two Higgs triplets, called the reduced minimal 331 (RM331)
model, was introduced. The Higgs sector in this model is as sim
ple as that of the E331, even the VEVs are fewer. The super
symmetric version, called the supersymmetric reduced minimal
(SUSYRM331) model, was also presented in 2013 by us and it is
reviewed in the chapter 2 of this dissertation.
6
Chương 2
Suppersymmetric 331
models
In this chapter, we concentrate on two supersymmetric models
constructed from two 331 models with the simplest Higgs spec
trums, namely the E331 and RM331 models. The general basis
of the supersymmetric theory is not summarized here.
2.1 Supersymmetric economical 331 model
The SUSYE331 was introduced in 2007 as the supersymmetric
version of the E331. Similar to the case of MSSM, this model con
tains the double number of Higgs multiplets as those in the non
supersymmetric model. In previous works studying the SUSYE331,
the LFV vertices in the soft terms are not considered. In this dis
sertation we assume that the LFV sources only appear in the soft
term of the lagrangian, namely
−L
˜μ˜τ
=(˜μ
∗
L
, ˜τ
∗
L
)
˜m
2
μ
L
˜m
2
L
μτ
˜m
∗2
L
μτ
˜m
2
τ
L
˜μ
L
˜τ
L
+(˜μ
c∗
L
, ˜τ
c∗
L
)
˜m
2
μ
R
˜m
2
R
μτ
˜m
∗2
R
μτ
˜m
2
τ
R
˜μ
c
L
˜τ
c
L
. (2.1)
7
Sleptons (˜μ
L
, ˜τ
L
) and (˜μ
c
L
, ˜τ
c
L
) are flavor eigenstates. We denote
mass eigenstates as
˜
l
L
2
,
˜
l
L
3
and
˜
l
R
2
,
˜
l
R
3
. The respective
masses are (˜m
2
L
2
, ˜m
2
L
3
) and (˜m
2
R
2
, ˜m
2
R
3
). The mass eigenstates
are mixing of flavor eigenstates ˜μ and ˜τ. The mixing is quan
titatively determined through new parameters s
L
and s
R
which
satisfy,
s
L
c
L
=
˜m
2
Lμτ
˜m
2
L
3
− ˜m
2
L
2
,s
R
c
R
=
˜m
2
Rμτ
˜m
2
R
3
− ˜m
2
R
2
. (2.2)
Eingenstates between two bases relate to each other by ˜μ
L
=
c
L
˜
l
L
2
−s
L
˜
l
L
3
, ˜τ
L
= s
L
˜
l
L
2
+ c
L
˜
l
L
3
,wherec
L
=cosθ
L
, s
L
=sinθ
L
;
μ
c
L
= c
R
l
R
2
−s
R
l
R
3
, τ
c
L
= s
R
l
R
2
+ c
R
l
R
3
. The lepton numbers are
conserved when s
L
= s
R
=0. Similar to the case of sneutrino
sector, the mixing between flavor eigenstates is parameterized by
s
ν
L
and s
ν
R
. Four parameters s
L
,s
R
,s
ν
L
and s
ν
R
are four inde
pendent LFV sources, while the MSSM has only two independent
LFV sources. These sources will generate oneloop Feynman di
agrams for cLFV decays considered in chapters 3 and 4.
2.2 Supersymmetric reduced minimal 33
1model
The SUSYRM331 was constructed in 2013 by D.T. huong, L.T.
Hue, M.C. Rodriguez and H.N. Long (Nuclear Physics B 870
(2013) 293). Comparing with previous SUSY331s, the lepton and
Higgs spectrums is simpler. The lepton and Higgs (anti) triplets
are as simple as those in the RM331. In addition, this supersym
metric version contains doubly charged bosons, instead of the
nonhermitian neutral bosons in the SUSYE331. Both SUSYE331
and SUSYRM331 have some massless leptons and quarks at tree
level. Fortunately, recent works have indicated that the lepton
and quark masses at oneloop are suitable with experiments.
Therefore, these models are realistic and they need to investi
gate more about phenomenology.
8
If we concentrate on the LFV sources as those in the SUSYE331,
we see that at tree level there is only one cLFV term m
2
Lab
˜
L
†
La
˜
L
Lb
.
Also, this term relates with the measures of the neutrino oscil
lation experiments so we can estimate some constrains to these
parameters. This problems are being studied and we will publish
in the near future.
In summary, in this chapter we concentrate on two results
1. We parameterize the LFV mixing in the soft term of the
SUSYE331 when assuming the existence of this mixing in
slepton sector.
2. We constructed the SUSYRM331 and discuss on the LFV
sources of the model.
9
Chương 3
H→ μτ decays in the
SUSYE331
3.1 Effective operators and branching ra
tios
The low energy effective operators in the general case are used
from previous works. The branching ratios of neutral Higgses are
determined as
BR(Φ
0
→ τ
+
μ
−
)=BR(Φ
0
→ τ
−
μ
+
)
=2(1+tan
2
γ)
 Δ
ρ
L

2
+  Δ
ρ
R

2
× BR(Φ
0
→ τ
+
τ
−
), (3.1)
where tan γ is the ratio of two Higgs VEVs t
γ
≡ tan γ = v/v
,
Δ
ρ
L
and Δ
ρ
R
are one loop effective LFV coefficients, Φ
0
denote
the mass eigenstates of Higgses in the SUSYE331, Φ
0
= ϕ
S
a36
or φ
S
a36
. Feynmann digrams presenting contributions to Δ
ρ
L
and
Δ
ρ
R
are shown in Fig. 3.1. The formulas are as follows,
Δ
ρ
L
=Δ
ρ
La
+Δ
ρ
Lb
+Δ
ρ
Lc
+Δ
ρ
Ld
+Δ
ρ
Le
+Δ
ρ
Lf
+Δ
ρ
Lk
, (3.2)
where Δ
ρ
La
, Δ
ρ
Lb
, Δ
ρ
Lc
, Δ
ρ
Ld
, Δ
ρ
Le
, Δ
ρ
Lf
và Δ
ρ
Lk
receive one loop
contributions from diagrams in Fig. 3.1. They are computed in
10
μ
τ
c
˜ρ
0
˜ρ
0
λ
B
ρ
0∗
˜
l
L
α
(a)
μ
τ
c
˜ρ
0
˜ρ
0
λ
3
A
λ
8
A
ρ
0∗
˜
l
L
α
(b)
μ
τ
c
˜ρ
−
1
˜ρ
+
1
˜
W
−
˜
W
+
ρ
0∗
˜ν
Lα
(c)
μ
τ
c
˜ρ
−
2
˜ρ
+
2
˜
Y
−
˜
Y
+
ρ
0∗
˜ν
Rα
(d)
μ
τ
c
˜ρ
−
2
˜ρ
+
2
ρ
0∗
˜ν
L
α
˜ν
R
β
(e)
μ
τ
c
˜ρ
−
1
˜ρ
+
1
ρ
0∗
˜ν
R
α
˜ν
L
β
(f)
τ
μ
c
˜ρ
0
˜ρ
0
λ
B
ρ
0∗
˜
l
R
α
(i)
μ
τ
c
λ
B
ρ
∗0
˜
l
L
α
˜
l
R
β
(k)
τ
μ
c
λ
B
ρ
∗0
˜
l
L
α
˜
l
R
β
(l)
Hình 3.1: Feynmann diagrams contribute to Δ
ρ
L
[(a), (b), (c), (d), (e), (f), (k)]and
Δ
ρ
R
[(i), (l)].
the dissertation. They depend on only the function I
3
(x, y, z)
where the precise formula is,
I
3
(x, y, z)=
xy ln(x/y)+yz ln(y/z)+zxln(z/x)
(x − y)(y − z)(z −x)
. (3.3)
3.2 Numerical investigation
We just investigate the case of maximal mixing. The LFV de
cays Br(H → μτ) are large when tan γ is large enough so we
choose tan γ =50. Other parameters are assumed correspond
ing to the values shown in Figs. refFDeltaRhoR1, 3.3, 3.4, 3.5
and 3.6. These Figs. show that Δ
ρ
R

2
obtains the maximal value
∼ 10
−3
when μ
ρ
/ ˜m
R
is very large, the maximal values of Δ
ρ
L

2
11
are ∼ 5.10
−3
when μ
ρ
/ ˜m
L
receive certain values. μ
ρ
/ ˜m
L
1
then Δ
ρ
L

2
slowly approaches to the value of 10
−3
. Fig. 3.4 draws
the correlative ratio of lefthanded and righthanded contribu
tions to the branching ratio (3.1). This plot shows that Δ
ρ
L
 is the
dominant contribution when μ
ρ
/ ˜m
L
is small. Δ
ρ
R
 gives large
contributions when μ
ρ
/ ˜m
L
are very large. To compare the cor
0 5 10 15 20 25 30
10
8
10
7
10
6
10
5
10
4
0.001
Μ
Ρ
m
R
50R
2
0 2 4 6 8 10
10
9
10
8
10
7
10
6
10
5
10
4
0.001
Μ
Ρ
m
R
50R
2
Hình 3.2: Δ
ρ
R

2
depends on μ
ρ
/ ˜m
R
. The respective parameters: 1) Blue–m
=
˜m
R
=˜m
L
; 2) green–3m
=˜m
R
=˜m
L
; 3) yellow m
=˜m
R
= 3 ˜m
L
; 4) red–m
=
˜m
R
=˜m
L
/3. Two black lines correspond to two values 10
−5
and 10
−3
of 50Δ
ρ
R

2
.
The right (left) panel draws for 0 ≤ μ
ρ
/m
SUSY
≤ 10 (0 ≤ μ
ρ
/m
SUSY
≤ 30).
0 5 10 15 20 25 30
10
8
10
7
10
6
10
5
10
4
0.001
0.01
Μ
Ρ
m
L
50L
2
0 2 4 6 8 10
10
7
10
6
10
5
10
4
0.001
0.01
Μ
Ρ
m
L
50L
2
Hình 3.3: Δ
ρ
L

2
depends on μ
ρ
/ ˜m
L
.The respective parameters: 1) blue–m
=
˜m
R
=˜m
L
; 2) green–3m
=˜m
R
=˜m
L
; 3) yellow– m
=˜m
L
= 3 ˜m
R
; 4) red–
m
=˜m
L
=˜m
R
/3. The black lines correspond to values 10
−3
of 50Δ
ρ
L

2
.
relation between Δ
L
and Δ
R
, we consider two Figs. 3.4 and 3.5.
In the left panel of the Fig. 3.4, it can be see that if μ
ρ
/ ˜m
L
≤ 8
then
Δ
ρ
R

2
Δ
ρ
L

2
is very small and the cLFV effects caused by only Δ
L
.
12
0 5 10 15 20 25 30
10
7
10
5
0.001
0.1
10
1000
Μ
Ρ
mSUSY
50R
2
50L
2
0 2 4 6 8
10
7
10
5
0.001
0.1
Μ
Ρ
mSUSY
50R
2
50L
2
Hình 3.4: Plots of Δ
ρ
R

2
/Δ
ρ
L

2
as function of μ
ρ
/ ˜m
L
. Numerical values of pa
rameters: 1) blue –m
=˜m
R
=˜m
L
; 2) green–3m
=˜m
R
=˜m
L
; 3)yellow–
m
=˜m
L
= 3 ˜m
R
; 4)red–m
=˜m
L
=˜m
R
/3. The black line in the left panel
presents
Δ
ρ
R

2
Δ
ρ
L

2
= 1. Two black lines in the right panel present values of 2 × 10
−3
and 0.1 for
Δ
ρ
R

2
Δ
ρ
L

2
.
In the right panel, there is a region satisfying 10 ≤μ
ρ
/ ˜m
L
≤ 30
which Δ
R
are much larger than Δ
R
. This is the region where dia
grams cancel to each other and there appears minima which each
minimum divides the plot into two regions. Therefore, both Δ
L
and Δ
R
are very suppressed. With μ
ρ
/ ˜m
L
≥ 30, the dominant
contributions to both Δ
L
and Δ
R
are pure gaugino diagrams
(diagrams (l) and (k) in the Fig. 3.1) and therefore the contri
butions of Δ
L
and Δ
R
to cLFV effects are the same. In general
the right contribution are large when μ
ρ
is very large. In the Fig.
3.5, the right panel shows that there do not exist any regions of
parameter space satisfying the condition
Δ
ρ
R

2
Δ
ρ
L

2
≥ 0.5.Theleft
panel shows that the ratio
Δ
ρ
R

2
Δ
ρ
L

2
is large when two following con
ditions are satisfied: 1) μ
ρ
is very large ;2) if ˜m
R
> ˜m
L
then the
ratio ˜m
R
/ ˜m
L
increases according to the increasing values of μ
ρ
,
if ˜m
R
< ˜m
L
then
Δ
ρ
R

2
Δ
ρ
L

2
increases when ˜m
R
→ ˜m
L
.
Table 3.1 presents interaction vertices of neutral Higgses in
the SUSYE331 comparing with those of the SM. If neutral Hig
gses are lighter than exotic quarks, they can not decay to the ex
otic quarks. Hence, light Higgses, which their masses depend on
13
0
.001
0.00
0.001
0.01
0.01
0.1
0.1
0.5
1
0 5 10 15 20 25 30
0
1
2
3
4
5
6
7
Μ
Ρ
mSUSY
m
Rm
L
0.0005
0.0005
0.0005
0.001
0.00
0
.001
0.01 0.1
0.1
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Μ
Ρ
mSUSY
m
Rm
L
Hình 3.5: Contour plots of
Δ
ρ
R

2
Δ
ρ
L

2
, ˜m
R
/ ˜m
L
vs μ
ρ
/m
SUSY
with ˜m
R
=˜m
ν
R
,
m
= m
λ
=˜m
L
=˜m
ν
L
= m
SUSY
. The red region correspond to values of
Δ
ρ
R

2
Δ
ρ
L

2
≥
0.5.
Bảng 3.1: Higgsfermionfermion vertices in the SUSYE331 comparing with those
in the SM .
Particle Upfermion Downfermion exotic upquark exotic downquark
SM Higgs 1 1 0 0
ϕ
Sa36
c
α
c
α
s
α
/s
γ
c
α
/s
γ
φ
Sa36
s
α
s
α
c
α
s
α
only VEVs v and v
, can satisfy this condition. Especially, the new
Higgs with mass 125 GeV discoverd recently by LHC can decay to
dominant decays of fermionantifermion b
¯
b and τ ¯τ. For example,
the branching ratio of light Higgs ϕ
Sa36
is Br(ϕ
Sa36
→ τ ¯τ ) 8%.
This leads to Br(ϕ
Sa36
→ μτ ) 8 ×10
−3
%. This is a significance
which can be detected in recent colliders. For heavy Higgses, the
main decays are decays to other heavy boson such as W
+
W
−
,
ZZ, so the LFV decays are very suppressed.
Numerical investigation in Fig. 3.6 shows the region where
the branching ratio BR(H → μτ)/BR(H → ττ) cực đại cỡ 10
−3
when 0.1 ≤μ
ρ
/M
SUSY
≤ 6 and 0.1 ≤˜m
g
/M
SUSY
≤ 7.This
branching ratio therefore can be detected by LHC in the coming
time.
14
0.001
0.003
0 2 4 6 8 10
0
1
2
3
4
5
6
7
Μ
Ρ
M
SUSY
m
g
M
SUSY
0.001
0.003
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Μ
Ρ
M
SUSY
m
g
M
SUSY
Hình 3.6: Contour plots of BR(H → μτ )/BR(H → ττ) as function of ˜m
g
and
μ
ρ
/m
SUSY
. Other parameters are fixed: m
= m
λ
=˜m
g
and ˜m
R
=˜m
ν
R
=˜m
L
=
˜m
ν
L
= m
SUSY
. In the right panel both green and yellow regions present the regions
satisfying BR(H → μτ)/BR(H → ττ) ≥O(10
−3
) .
15
Chương 4
cLFV decays of τ and Z
boson in the SUSYE331
model
Thank to the colliders such as LHC, BABAR, LEP, , many pro
cesses predicted by models beyond the SM are searching. One of
these is the class of cLFV decays which will be evidence of New
Physics if detected. We concentrate on only three cLFV decays
with upper bounds established by recent experiments
BR(τ
−
→ μ
−
γ) < 4.4 ×10
−8
, (4.1)
BR(τ
−
→ μ
−
μ
+
μ
−
) < 2.1 × 10
−8
, (4.2)
BR(Z → μ
+
τ
−
) < 1.2 × 10
−5
. (4.3)
These three processes are studied at the same time because the
experimental bounds are very clear and the analytic formulas
calculating them closely relate to one another. We will investigate
these three decays in the framework of the SUSYE331.
16
0.5
2.5
5
10
15
20
50
0.5
2.5
5
15
40
50
100 200 300 400 500
100
150
200
250
300
m
Λ
GeV
mL
3
GeV
D
L
Γ b
10
9
GeV
2
0.25
0.5
1
2.5
5
0.25
0.5
1
2.5
5
100 200 300 400 500
100
150
200
250
m
Λ
GeV
mL
3
GeV
D
L
Γ b
10
9
GeV
2
Hình 4.1: Contour plots of D
γ(b)
L
with tan γ = 3.0, m
˜
L
3
= m
˜ν
L
3
= m
˜ν
R
3
and
m
˜
L
2
= m
˜ν
L
2
= m
˜ν
R
2
= 300 GeV, θ
L
= θ
˜ν
L
= θ
˜ν
R
= π/4 and μ
ρ
= 140 GeV
(1TeV) for the left (right) panel. The black and dashed curves correspond to values
of m
B
= 300 GeV and m
B
= −300 GeV.
4.1 Effective operators and branching ra
tios
In the SUSYE331, the branching ratios of τ → μγ, Z → μτ,
Z
→ τμ and τ → 3μ in the limit of effective low energy theory
have the same formulas as those of MSSM.
Contributions from Hμτ effective vertices to the branching
ratio of τ → μμμ decays in our investigation is very suppressed.
Hence, they are ignored in this our calculation.
4.2 Numerical calculation and discussion
To guarantee the vacuum stability of the model constructed in
previous works, we add the B/μterm types in the soft term of
the original model. As the result, the charged Higgses naturelly
satisfy the lower bounds of experiments. Investigation of neutral
Higgs sector shows that t
γ
can get small values. These values
can allow the existence of parameter space with light sleptons
where limits of cLFV decays satisfy experimental bounds. In this
dissertation we only consider the small values of tan γ and light
sleptons which can be detected by LHC.
17
τ → μγ decay
The branching ratio τ → μγ trong (4.1) have rather small upper
bound, it corresponds to the condition D
γ
L,R
≤2.5×10
−9
[GeV
−2
].
Two Figs. 4.1 and 4.2 present numerical values of D
γ(b)
L
which is
the dominate contribution to D
γ
L,R
. Fig. 4.1 draws the case of
light sleptons m
˜
L
< 300 GeV. To satisfy the experimental bounds
of τ → μγ decay, masses of light sleptons approach to the degen
erate values.
Fig. 4.2 draws the regions of mass parameters containing
heavy sleptons, order of TeV. The results show that the satisfying
regions are expanded in the regions of heavier sleptons masses.
Fig. 4.3 presents leftright mixing A
τ
as function of μ
ρ
in the
20
30
40
50
80
100
20
30
40
50
80
100
100 200 300 400 500
100
150
200
250
300
m
Λ
GeV
mL
3
GeV
D
L
Γ b
10
9
GeV
2
3
5
10
15
2.5
3
5
10
15
100 200 300 400 500
100
150
200
250
300
m
Λ
GeV
mL
3
GeV
D
L
Γ b
10
9
GeV
2
Hình 4.2: Contour plots of D
γ(b)
L
with tan γ = 3.0, m
˜
L
2
= m
˜ν
L
2
= m
˜ν
R
2
and
m
˜
L
2
= m
˜ν
L
2
= m
˜ν
R
2
= 1 TeV, θ
L
= θ
˜ν
L
= θ
˜ν
R
= π/4 and μ
ρ
= 140 GeV (1TeV)
for left (right) panel. The black and dashed curves present for m
B
= 300 GeV and
m
B
= −300 GeV, respectively.
presence of three maximal mixing angles. With the very large
values of D
γ
L
, A
τ
must be large to satisfy experiments as well as
cancel tachyon sleptons. In general, the existence of three LFV
sources will exclude the mass parameter space of light sleptons.
If we consider the case of cLFV effects caused by only right
handed charged slepton sector, s
R
= c
R
=1/
√
2, then only D
γ
R
=
0. Numerical investigation in Fig. 4.4 indicates that the regions
of parameter space containing light sleptons allow D
γ
R
to easily
18
2.5
0
2.5
1000 500 0 500 1000
200
400
600
800
1000
A
Τ
GeV
Μ
Ρ
GeV
D
L
Γ
10
9
GeV
2
Hình 4.3: Contour plots of D
γ
L
with tan γ = 3., m
˜
L
2
= 1 TeV, θ
L
=
π/4,θ
R
= θ
˜ν
L
= θ
˜ν
R
=0, A
L
τμ
=0. Other parameters are chosen as
(m
B
,m
λ
,m
˜
L
3
,m
˜
R
)[GeV]: (200, 300, 300, 200)–black, (100, 400, 100, 200)–dashed,
(100, 500, 300, 100)–dotted. In particular, the center curves correspond to D
γ
L
=0
while two other curves limit D
γ
L
≤2.5 × 10
−9
[GeV
−2
].
satisfy the cLFV bounds of experiments. This case is similar to
that of MSSM, even the parameter space is wider. Hence, sleptons
still can be detected by LHC.
Correlation among effective operator coefficients
The τ → 3μ decay includes all contributions of Feynman di
agrams with photon, Higgs, Z, Z
exchanges and also box di
agrams. Contributions from Higgs exchanges are very suppress
because of small values of tan γ. Therefore, there will appear con
stant ratios of branching ratio of τ → 3μ to one of those of the
remain decays depending on which contributions are dominate.
Here we limit our work on two cases: i) LFV maximal mix
ing only in the lefthanded charged slepton sector, namely c
L
=
1/
√
2; ii) The same for the righthanded charged slepton sector.
To estimate relative contributions of effective vertices A
Z
L
and
D
γ
L
into the branching ratio (τ
−
→ μ
−
μ
+
μ
−
), we define two con
tribution coefficients f
A
Z
and f
D
γ
. The precise formulas of these
coefficients are shown in the dissertation. We denote A
Z
are dom
inant contribution if 1.05 ≥ f
A
Z
≥ 0.95. The regions of parameter
19
0.25
1
2
0
.25
1
300 200 100 0 100 200 300
100
150
200
250
m
B
GeV
m R
3
GeV
D
R
Γ a
10
9
GeV
2
5
2.
5
1
0.1
0
0.1
1
2.5
7.5
5
2.5
4.5
300 200 100 0 100 200 300
100
150
200
250
300
m
B
GeV
m R
3
GeV
D
R
Γ b
10
9
GeV
2
Hình 4.4: Contour plots of D
γ(a)
R
(left) and D
γ(b)
R
(right) as functions of two
parameters m
˜
R
3
and m
B
. Other parameters are fixed as follows: tan γ = 3., m
˜
R
2
=
1 TeV, θ
L
= θ
˜ν
L
= θ
˜ν
R
=0, θ
R
= π/4 and μ
ρ
= 150 GeV .
space satisfying this condditions are called the A
Z
domination
regions ( in contribution to F
μ
L(R)
L(R)
). If f
D
γ
≤ 0.05, the main con
tributions to the Br(τ → 3μ) is F
μ
L(R)
L(R)
and the respective regions
are called F −domination. In contrast, if 1.05 ≥ f
D
γ
≥ 0.95 then
the regions of parameter space are Ddomination.
Maximal mixing in lefthanded sector of (˜μ, ˜τ)
This case corresponds to s
L
= c
L
=
1
√
2
và s
R
= s
˜ν
L
= s
˜ν
R
=
0 and therefore leads to A
2Z
L
= A
2Z
R
=0. In the Fig. 4.5,
with small values of μ
ρ
the Ddomination regions in both two
panels are very narrow and they extend to the regions of large
values of parameters. The right panel shows that for large m
λ
the
Ddomination regions change rapidly according to the values of
m
˜
L
3
,andμ
ρ
> 900GeV. The experiment excludes these regions.
the left panel indicates that for m
λ
is small, μ
ρ
has both upper
and lower bounds because of the condition BR(τ → μγ) < 4.4 ×
10
−8
. There do not exist any A
Z
domination regions.
Now we come to two decays Z → μτ and τ → μμμ with
investigation shown in Fig. 4.6. The mass parameter of gaugino
m
λ
is chosen according to the limit bound of experiment. Other
parameters are chosen in order of O(100) GeV. From Fig. 4.6 the
20
0.05
0.05
0.3
0.3
0.9
5
1.05
1
1
4.4
0.9
0
.91
100 150 200 250 300 350 400
100
200
300
400
500
600
700
800
Μ
Ρ
GeV
mL
3
GeV
f
A
z
, f
D
Γ
,BRΤ ΜΓ 10
8
0.07
0
.07
0.2
0.2
0.85
1
1.05
1.05
1.2
1.2
1
2
4.4
0.0
5
0.1
0.7
0.75
200 400 600 800 1000
200
400
600
800
1000
1200
Μ
Ρ
GeV
mL
3
GeV
f
A
z
, f
D
Γ
,BRΤ ΜΓ 10
8
Hình 4.5: Correlations among A
Z
L
, F
μ
L(R)
L
and D
γ
L
with A
τ
=0. The contour
plots draw dashed, dotted and black corresponding to constant values of curves
f
A
Z
L
,f
D
γ
L
and BR(τ → μγ). Two numerical values chosen for (m
B
,m
λ
,m
˜
L
2
m
˜
L
R
)
are (100, 300, 1000, 100)[GeV] (left ) and (100, 500, 1000, 100) [GeV] (right).
maximal value of BR(Z → μτ ) is about 5.10
−10
and it is very
small comparing with the recent experimental bounds. While the
values of BR(τ → 3μ) can approach the limit detection of collid
ers.
Fig. 4.7 shows the numerical resulus of two branching ratios
as functions of m
B
−μ
ρ
with A
τ
=0. It is shown that the upper
experimental bound of BR(τ → μγ) directly affects to the values
of other cLFV decays. Specifically, BR(τ → 3μ) < 0.5×10
−9
and
BR(Z → μτ) < 10
−10
. On the other hand, if A
τ
=0and cancels
values of D
γ
,BRZ → μτ can be large but are still small than
the upper bound of experiment.
In the case the maximal mixing happens only in the right
handed charged sleptons, namely s
R
=1/
√
2,s
L
= s
˜ν
L
= s
˜ν
R
=
0 and the mass parameter space still is in the O(100) GeV scale.
The obtained results are the same in both model SUSYE331 and
MSSM. For SUSYE331,the BR(τ → μγ) can approach the exper
iment bounds and it excludes the large values of BR(τ → 3μ),
leading to BR(τ → 3μ) ≤O(10
−9
) with A
τ
=0. in addition,
numerical investigation shows that f
A
Z
as well as BR Z → μτ
are still very small.
21
100 200 300 400 500 600 700 800
1.0 10
10
5.0 10
10
2.0 10
10
3.0 10
10
1.5 10
10
m
B
BRZΜΤ
100 200 300 400 500 600 700 800
1 10
11
5 10
11
1 10
10
5 10
10
1 10
9
5 10
9
1 10
8
m
B
BRΤ
Μ
Μ
Μ
Hình 4.6: Branching ratios Z → μτ (left panel) and τ → 3μ (right panel) as
functions of m
B
. The parameter space including (m
λ
,μ
ρ
,m
˜
L
2
,m
˜
L
3
,m
˜
R
) [GeV] is
chosen for three cases : (300, 150, 1000, 100, 100)black,(400, 200, 1000, 100, 100)
green, (500, 150, 1000, 100, 100)blue.
Results and discussion
The main result of our work are summarized as follows
1. We constructed the 331 supersymmetric model with the
simplest particle content, called the SUSYRM331.
2. We established and parameterize analytic formulas present
ing cLFV vertices in the SUSYE331. This kind of vertices
in the SUSYRM331 are also discussed.
3. In the frame work of the SUSYE331, we constructed ana
lytic formulas of quantities needed for calculating branching
ratios of the cLFV decays at one loop level, namely effec
tive operators, Lagrangian and branching ratios of cLFV
decays.
4. We numerically investigated four particular LFV decays:
H → μτ, τ → μγ, Z → μτ and τ → 3μ. The results are
used to compare with bounds of experiments and find the
allowed regions of the parameter space in the SUSYE331.
We also predict that the branching ratios of H
0
→ μτ ,
τ → μγ and τ → 3μ are in the detectable limit of recent
colliders. In contrast the BRZ → μτ are very suppressed
22
0.5
0.5
2
2
1
1
4.4
4.4
0.1
0.5
1
200 400 600 800 1000
100
200
300
400
500
Μ
Ρ
GeV
m
B
GeV
ZΜΤ 10
10
, Τ
Μ
Μ
Μ
10
9
, ΤΜ Γ 10
8
Hình 4.7: Contour plots of branching ratios of τ
−
→ μ
−
μ
+
μ
−
(dotted), Z →
μτ ( dashed) with τ → μγ (black) with A
τ
=0and (m
λ
,m
˜
L
2
,m
˜
L
3
,m
˜
R
)=
(400, 150, 1000, 100, 200).
with maximal value of 10
−9
, out of the detection of exper
iments.
5. We discussed on the B/μ type terms which are not men
tioned in the previous works on SUSYE331 and SUSYRM331.
These terms are very necessary to cancel Higgs tachyons as
well as guarantee the vacuum stability of the SUSY mod
els. These terms also affect directly to the Higgs spectrums.
This is a suggestion for next studies on the SUSY331 mod
els. Many formulas established for cLFV branching ratios
in this dissertation can be used to calculate many other
cLFV decays which we are studying and will publish in the
near future.
23