#### The extended BLMSSM with a 125 GeV Higgs boson and dark matter

Eur. Phys. J. C
The extended BLMSSM with a 125 GeV Higgs boson and dark matter
Shu-Min Zhao 1
Tai-Fu Feng 1
Guo-Zhu Ning 1
Jian-Bin Chen 0
Hai-Bin Zhang 1
Xing Xing Dong 1
0 College of Physics and Optoelectronic Engineering, Taiyuan University of Technology , Taiyuan 030024 , China
1 Department of Physics, Hebei University , Baoding 071002 , China
To extend the BLMSSM, we not only add exotic Higgs superfields ( N L , ϕN L ) to make the exotic lepton heavy, but also introduce the superfields (Y ,Y ) having couplings with lepton and exotic lepton at tree level. The obtained model is called as EBLMSSM, which has difference from BLMSSM especially for the exotic slepton (lepton) and exotic sneutrino (neutrino). We deduce the mass matrices and the needed couplings in this model. To confine the parameter space, the Higgs boson mass mh0 and the processes h0 → γ γ , h0 → V V , V = (Z , W ) are studied in the EBLMSSM. With the assumed parameter space, we obtain reasonable numerical results according to data on Higgs from ATLAS and CMS. As a cold dark mater candidate, the relic density for the lightest mass eigenstate of Y and Y mixing is also studied.
1 Introduction
The total lepton number (L) and baryon number (B) are good
symmetries because neutrinoless double beta decay or proton
decay has not been observed. In the standard model (SM), L
and B are global symmetries [1,2]. However, the individual
lepton numbers Li = Le, Lμ, Lτ are not exact symmetries
at the electroweak scale because of the neutrino oscillation
and the neutrinos with tiny masses [3,4]. In the Universe,
there is matter-antimatter asymmetry, then the baryon
number must be broken.
With the detection of the light Higgs h0(m0h = 125.1 GeV)
[5,6], the SM succeeds greatly and the Higgs mechanism is
compellent. Beyond the SM, supersymmetry [7,8] provides a
possibility to understand the light Higgs. The minimal
supersymmetric extension of the SM (MSSM) [9] is one of the
favorite models, where the light Higgs mass at tree level is
mthree = m Z | cos 2β| [10–12]. The one loop corrections to
Higgs mass mainly come from fermions and sfermions, that
depend on the virtual particle masses and the couplings with
the Higgs.
There are many papers about the gauged B and L
models, although most of them are non-supersymmetric [13,14].
Extending MSSM with the local gauged B and L, one obtains
the so called BLMSSM, which was proposed by the authors
in Refs. [10–12]. The proton remains stable, as B and L are
broken at the TeV scale. Therefore, a large desert between the
electroweak scale and grand unified scale is not necessary. In
BLMSSM, the baryon number is changed by one unit, at the
same time the lepton number is broken in an even number.
R-parity in BLMSSM is not conserved, and it can explain the
matter-antimatter asymmetry in the Universe. There are some
works for Higgs and dark matters [15–17] in the BLMSSM
[18,19]. In the framework of BLMSSM, the light Higgs mass
and the decays h0 → γ γ and h0 → V V , V = (Z , W ) are
studied in our previous work [19]. Some lepton flavor
violating processes and CP-violating processes are researched
with the new parameters in BLMSSM [20,21].
In BLMSSM, the exotic leptons are not heavy, because
their masses just have relation with the parameters Ye4 υd ,
Ye5 υu . Here υu and υd are the vacuum expectation values
(VEVs) of two Higgs doublets Hu and Hd . In general, the
Yukawa couplings Ye4 and Ye5 are not large parameters, so
the exotic lepton masses are around 100 GeV. The light
exotic leptons may lead to that the BLMSSM is excluded
by high energy physics experiments in the future. To obtain
heavy exotic leptons, we add two exotic Higgs superfields to
the BLMSSM, and they are SU(
2
) singlets N L and ϕN L ,
whose VEVs are υN L and υ¯ N L [22]. The exotic leptons
and the superfields N L , ϕN L have Yukawa couplings, then
υN L and υ¯ N L give contributions to the diagonal elements
of the exotic lepton mass matrix. So the exotic leptons turn
heavy and should be unstable. In the end, the super fields
Y and Y are also introduced. At tree level, there are
couplings for lepton-exotic lepton-Y (Y ). It is appealing that
this extension of BLMSSM produces some new cold dark
matter candidates, such as the lightest mass eigenstate of Y
and Y mixing. The four-component spinor Y˜ is made up
of the superpartners of Y and Y . In this extended BLMSSM
(EBLMSSM), we study the lightest CP even Higgs mass with
the one loop corrections. The Higgs decays h0 → γ γ and
h0 → V V , V = (Z , W ) are also calculated here. Supposing
the lightest mass eigenstate of Y and Y mixing as a cold dark
matter candidate, we study the relic density.
After this introduction, in Sect. 2, we introduce the
EBLMSSM in detail, including the mass matrices and the
couplings different from those in the BLMSSM. The mass
of the lightest CP-even Higgs h0 is deduced in the Sect. 3. The
Sect. 4 is used to give the formulation of the Higgs decays
h0 → γ γ , h0 → V V , V = (Z , W ) and dark matter relic
density. The corresponding numerical results are computed
in Sect. 5. The last section is used for the discussion and
conclusion.
2 Extend the BLMSSM
The local gauge group of the BLMSSM [10–12] is SU (3)C ⊗
SU (
2
)L ⊗ U (
1
)Y ⊗ U (
1
)B ⊗ U (
1
)L . In the BLMSSM, the
exotic lepton masses are obtained from the Yukawa couplings
with the two Higgs doublets Hu and Hd . The VEVs of Hu
and Hd are υu and υd with the relation υu2 + υd2 = υ ∼ 250
GeV. Therefore, the exotic lepton masses are not very heavy,
though they can satisfy the experiment bounds at present.
In the future, with the development of high energy
experiments, the experiment bounds for the exotic lepton masses
can improve in a great possibility. Therefore, we introduce the
exotic Higgs superfields N L and ϕN L with nonzero VEVs
to make the exotic lepton heavy. The heavy exotic leptons
should be unstable, then the superfields Y, Y are introduced
accordingly. These introduced superfields lead to tree level
couplings for lepton-exotic lepton-Y (Y ).
In EBLMSSM, we show the superfields in the Table 1.
The superpotential of EBLMSSM is shown here
WE B L M SSM = WM SSM + WB + WL + WX + WY ,
WL = λL Lˆ 4 Lˆ 5ϕˆN L + λE Eˆ4c Eˆ5 ˆ N L
c
+ λN L Nˆ 4c Nˆ 5 ˆ N L + μN L ˆ N L ϕˆN L
+ Ye4 Lˆ 4 Hˆd Eˆ4 + Yν4 Lˆ 4 Hˆu Nˆ 4c
c
+ Ye5 Lˆ c5 Hˆu Eˆ5 + Yν5 Lˆ c5 Hˆd Nˆ 5
+ Yν Lˆ Hˆu Nˆ c + λN c Nˆ c Nˆ cϕˆL + μL ˆ L ϕˆL ,
WY = λ4 Lˆ Lˆ 5Yˆ + λ5 Nˆ c Nˆ 5Yˆ + λ6 Eˆ c Eˆ5Yˆ + μY Yˆ Yˆ . (
1
)
c
WM SSM is the superpotential of MSSM. WB and WX are
same as the terms in BLMSSM [19]. WY includes the terms
beyond BLMSSM, and they include the couplings of
leptonexotic lepton-Y (l I − L − Y ). Therefore, the heavy exotic
leptons can decay to leptons and mass eigenstates of Y and
Y mixing whose lighter one can be a dark matter candidate.
From WY , one can also obtain the coupling of lepton-exotic
slepton-Y˜ (l I − L˜ − Y˜ ), where Y˜ is the four component
spinor composed by the superpartners of Y and Y . The new
couplings of l I − L −Y and l I − L˜ −Y˜ can give one loop
corrections to lepton anormal magnetic dipole moment (MDM).
They may compensate the deviation between the experiment
value and SM prediction for muon MDM. The parameter
μY can be complex number with non-zero imaginary part,
which is a new source of CP-violating. Therefore, the both
new couplings produce one loop diagrams contributing to the
lepton electric dipole moment (EDM). Further more, if λ4 in
λ4 Lˆ Lˆ c5Yˆ is a matrix and has non-zero elements relating with
lepton flavor, this term can enhance the lepton flavor
violating effects. In the whole, WY enriches the lepton physics to
a certain degree, and these subjects will be researched in our
latter works.
Because of the introduction of the superfields N L , ϕN L ,
Y and Y , the soft breaking terms are written as
E B L M SSM
Lsof t
= LsBoLf Mt SSM − m2 N L ∗N L N L
−m2ϕN L ϕ∗N L ϕN L +
AL L λL L˜ 4 L˜ c5ϕN L
+ AL E λE e˜4e˜5 N L + AL N λN L ν˜4cν˜5 N L
c
+BN L μN L N L ϕN L + h.c.
+
A4λ4 L˜ L˜ c5Y + A5λ5 N˜ cν˜5Y
+ A6λ6e˜ce˜5Y + BY μY Y Y + h.c. .
(
2
)
B L M SSM is the soft breaking terms of BLMSSM,
Here Lsof t
whose concrete form is in our previous work [19]. The
SU (2)L doublets Hu , Hd acquire the nonzero VEVs υu , υd .
The SU (2)L singlets B , ϕB , L , ϕL , N L , ϕN L obtain the nonzero VEVs υB , υ B , υL , υ L , υN L , υ N L respectively.
Hu =
Hd =
1
B = √2
1
ϕB = √2
Hu+
√12 υu + Hu0 + i Pu0
√12 υd + Hd0 + i Pd0
Hd−
υB +
0
B + i PB0 ,
0 0
υ B + ϕB + i P B ,
U (
1
)B
1/3
−1/3
−1/3
0
0
0
B4
0
L + i PL0 ,
ϕN L = √12 υ N L + ϕ0N L + i P0N L .
0
N L + i PN0 L ,
Here, we define tan β = υu /υd , tan βB = υ¯ B /υB , tan βL =
υ¯ L /υL and tan βN L = υ¯ N L /υN L . The VEVs of the Higgs
satisfy the following equations
|μ|2 − g12 +8 g22 (υu2 − υd2) + m2Hd + Re[Bμ] tan β = 0,
(
3
)
(
4
)
|μ|2 + g12 +8 g22 (υu2 − υd2) + m2Hu + Re[Bμ] cot β = 0,
|μB |2 + g22B (υB2 − υ¯ B2 ) + m2 B − Re[BB μB ] tan βB = 0,
(
6
)
|μB |2 − g22B (υB2 − υ¯ B2 ) + m2ϕB − Re[BB μB ] cot βB = 0,
(
7
)
(
8
)
(
9
)
|μL |2 − 2g2L VL2 + m2 L − Re[BL μL ] tan βL = 0,
|μL |2 + 2g2L VL2 + m2ϕL − Re[BL μL ] cot βL = 0,
|μN L |2 − 3g2L VL2 + m2 N L − Re[BN L μN L ] tan βN L = 0,
(
10
)
(
11
)
|μN L |2 + 3g2L VL2 + m2ϕN L − Re[BN L μN L ] cot βN L = 0,
with VL2 = υ2L − υL2 + 23 (υ2N L − υN2 L ). Here, the Eqs. (
8
) and
(
9
) are similar as the corresponding equations in BLMSSM,
but Eqs. (
8
) and (
9
) have relation with the new parameters
υN L and υ¯ N L . We obtain the new Eqs. (
10
) and (
11
) through
∂ ∂VN L and ∂∂ϕVN L , with V denoting the Higgs scalar potential.
Here we deduce the mass matrices in the EBLMSSM.
Compared with BLMSSM, the superfields N L and ϕN L are
introduced and they give corrections to the mass matrices of
the slepton, sneutrino, exotic lepton, exotic neutrino, exotic
slepton and exotic sneutrino. That is to say, in EBLMSSM,
the mass matrices of squark, exotic quark, exotic squark,
baryon neutralino, MSSM neutralino, X and X˜ are same as
those in the BLMSSM, and their concrete forms can be found
in our previous works [23–25]. Though the mass squared
matrices of slepton and sneutrino in EBLMSSM are
different from those in BLMSSM, we can obtain the slepton and
sneutrino mass squared matrices in EBLMSSM easily just
2 2
using the replacement υ L − υL → VL2 for the BLMSSM
results.
In the BLMSSM, the issue of Landau pole has been
discussed in detail by the authors of Refs. [10–12]. Their
conclusion is that there are no Landau poles at the low scale
due to the new families. In EBLMSSM, the parts of quark
(squark), exotic quark (exotic squark) are same as those in
BLMSSM. Therefore, the Landau pole conditions for the
Yukawa couplings of quark (squark), exotic quark (exotic
squark) have same behaviors of BLMSSM. The added
superfields ( N L , ϕN L , Y, Y ) do not have couplings with the
gauge fields of SU (
3
)C , SU (
2
)L , U (
1
)Y and U (
1
)B . So the
characters of gauge couplings g1, g2, g3 and gB in BLMSSM
and EBLMSSM are same.
The different parts between BLMSSM and EBLMSSM
are the terms including N L , ϕN L , Y and Y . The new terms
in the superpotential WL are λL Lˆ 4 Lˆ 5ϕˆN L + λE Eˆ4c Eˆ5 ˆ N L +
c
λN L Nˆ 4c Nˆ 5 ˆ N L + μN L ˆ N L ϕˆN L and they have
corresponding relations with λQ Qˆ 4 Qˆ c5 ˆ B +λU Uˆ4cUˆ5ϕˆ B +λD Dˆ 4c Dˆ 5ϕˆ B +
μB ˆ B ϕˆ B in WB by the replacements Lˆ 4 ↔ Qˆ 4, Lˆ c5 ↔
Qˆ c5, Eˆ c 4 4 ↔ Dˆ 4c, Nˆ 5 ↔ Dˆ 5, ˆ N L ↔
4 ↔ Uˆ c, Eˆ5 ↔ Uˆ5, Nˆ c
ϕˆ B , ϕˆN L ↔ ˆ B . The corresponding relations for WY =
λ4 Lˆ Lˆ c5Yˆ + λ5 Nˆ c Nˆ 5Yˆ + λ6 Eˆ c Eˆ5Yˆ + μY Yˆ Yˆ and WX =
λ1 Qˆ Qˆ c5 Xˆ + λ2Uˆ cUˆ5 Xˆ + λ3 Dˆ c Dˆ 5 Xˆ + μX Xˆ Xˆ are
obvious with Lˆ ↔ Qˆ , Lˆ c5 ↔ Qˆ c5, Eˆ c ↔ Uˆ c, Eˆ5 ↔ Uˆ5, Nˆ c ↔
Dˆ c, Nˆ 5 ↔ Dˆ 5, Xˆ ↔ Yˆ , Xˆ ↔ Yˆ . From this analysis, the
Landau pole conditions of gauge coupling gL and Yukawa
couplings of exotic leptons should possess similar
peculiarities of gauge coupling gB and Yukawa couplings of exotic
quarks. In conclusion, similar as BLMSSM, there are no
Landau poles in EBLMSSM at the low scale because of the new
families. The concrete study of Landau poles for the
couplings should use renormalization group equation which is
tedious, and we shall research this issue in our future work.
2.1 The mass matrices of exotic lepton (slepton) and exotic
neutrino (sneutrino) in EBLMSSM
In BLMSSM, the exotic lepton masses are not heavy, because
they obtain masses only from Hu and Hd . The VEVs of N L
and ϕN L are υN L and υ¯ N L , that can be large parameters.
So, the EBLMSSM exotic leptons are heavier than those in
BLMSSM.
The mass matrix for the exotic leptons reads as
mass
− Le
=
e¯4R , e¯5R
− √12 λL υ N L , √12 Ye5 υu
− √12 Ye4 υd ,
√12 λE υN L
×
e4L
e5L
+ h.c.
×
ν4L
ν5L
+ h.c.
(
12
)
(
13
)
Obviously, υ N L and υN L are the diagonal elements of the
mass matrix in the Eq. (
12
). It is easy to obtain heavy exotic
lepton masses with large υ N L and υN L . If we take υ N L and
υN L as zero, the mass matrix is same as that in BLMSSM.
In fact, our used values of υ N L and υN L are at TeV order,
which produce TeV scale exotic leptons. Heavy exotic
leptons have strong adaptive capacity to the experiment bounds.
The exotic neutrinos are four-component spinors, whose
mass matrix is
mass
− Lν
=
ν¯4R , ν¯5R
√12 λL υ N L , − √12 Yν5 υd
√12 Yν4 υu ,
√12 λN L υN L
Similar as the exotic lepton condition, heavy exotic neutrinos
are also gotten.
In BLMSSM, the exotic sleptons of 4 generation and 5
generation do not mix, and their mass matrices are both 2×2.
In EBLMSSM, the exotic sleptons of 4 generation and 5
generation mix together, and their mass matrix is 4 × 4. With
the base (e˜4, e˜4c∗, e˜5, e˜5c∗), we show the elements of exotic
2 in the following form.
slepton mass matrix ME˜
M2E˜ (e˜5c∗e˜5c) = λ2L υ¯ N22 L + υ2u2 |Ye5 |2 + ML2˜ 5
2 2
− g1 −8 g2 (υd2 − υu ) − g2L (3 + L4)VL2,
2
M2E˜ (e˜5∗e˜5) = λ2E υN22 L + υ2u2 |Ye5 |2 + Me˜25
g2 2
+ 41 (υd2 − υu ) + g2L (3 + L4)VL2,
2
2 υ¯ N L
M2E˜ (e˜4∗e˜4) = λL 2
2 2
+ g1 −8 g2 (υd2 − υu )
2
υ2
+ 2d |Ye4 |2 + M 2
L˜ 4 + g2L L4VL2,
2
2 υN L
M2E˜ (e˜4c∗e˜4c) = λE 2
g2 2
− 41 (υd2 − υu )
υ2
+ 2d |Ye4 |2 + M 2
e˜4 − g2L L4VL2,
υN L
M2E˜ (e˜4∗e˜5) = υd Ye∗4 λE 2
M2E˜ (e˜5e˜5) = μ∗ √υd Ye5 + Ae5 Ye5 √ ,
c υu
2 2
+ λL Ye5
υ¯ N L vu ,
2
c υ¯ N L
M2E˜ (e˜4e˜5) = μ∗N L λE √2
c υN L
M2E˜ (e˜4e˜5) = −μ∗N L √2
− AL E λE υ√N2L ,
λL + AL L λL υ¯√N L ,
2
M2E˜ (e˜4e˜4) = μ∗ √υu Ye4 + Ae4 Ye4 √ ,
c υd
2 2
M2E˜ (e˜5ce˜4c∗) = −Ye5 λE
υu υN L − λL Ye∗4 υ¯ N L vd .
2 2
(
14
)
In Eq. (
14
), the non-zero terms M2E˜ (e˜4e˜5c), ME˜
2 (e˜4∗e˜5),
tMon2E˜m(e˜i5xcei˜n4c∗g)oafndgeMneEr˜ations 4 and 5. These mixing terms all
c
2 (e˜4e˜5) are the reason for the exotic
slepinclude the parameters υN L and υ¯ N L . It shows that this
mixing is caused basically by the added Higgs superfields N L
and ϕN L . Using the matrix Z E˜ , we obtain mass eigenstates
witIhntthheefsoarmmeulwa aZy†E,˜tMhe2Ee˜xZoE˜tic=sndeiuatgri(nmo2Em˜1 ,asms2Es˜ q2,umar2eE˜d3 ,mma2Et˜r4i)x.
is also obtained
2
2 υ¯ N L
M2N˜ (ν˜5c∗ν˜5c) = λL 2
2 2
− g1 +8 g2 (υd2 − υu )
2
υ2
+ 2d |Yν5 |2 + M 2
L˜ 5 − g2L (3 + L4)VL2,
2
2 υ¯ N L
M2N˜ (ν˜4∗ν˜4) = λL 2
2 2
+ g1 +8 g2 (υd2 − υu )
2
υ2
+ 2u |Yν4 |2 + M 2
L˜ 4 + g2L L4VL2,
2
M2N˜ (ν˜5∗ν˜5) = λ2N L υN2 L + g2L (3 + L4)VL2
2
M2N˜ (ν˜4c∗ν˜4c) = λ2N L υN2 L − g2L L4VL2
υ2
+ 2d |Yν5 |2 + Mν˜25 ,
υ2
+ 2u |Yν4 |2 + Mν˜24 ,
M2N˜ (ν˜5ν˜4c∗) = λN L Yν5
c
υN L υd − λL Yν∗4 υ¯ N L υu ,
2 2
M2N˜ (ν˜5ν˜5) = μ∗ √υu Yν5 + Aν5 Yν5 √ ,
c υd
2 2
υ¯ N L
M2N˜ (ν˜4cν˜5) = μ∗N L λN L √
2
υN L
− AL N λN √2
,
c υN L
M2N˜ (ν˜4ν˜5) = μ∗N L √2
υ¯ N L υd
M2N˜ (ν˜4∗ν˜5) = λL Yν5 2
λL − AL L λL υ¯√N L ,
2
−
υu υN L
2
λN L Yν∗4 ,
M2N˜ (ν˜4ν˜4c) = μ∗ √υd Yν4 + Aν4 Yν4 √υu . (
15
)
2 2
For the exotic sneutrino, the mixing of generations 4 and 5 is
similar as that of exotic slepton. In the base (ν˜4, ν˜4c∗, ν˜5, ν˜5c∗),
we get the mass squared matrix of the exotic sneutrino, and
foobrtmaiunlathZe†N˜mMas2sN˜ ZeiNg˜ e=nstdaiteasg(bmy2N˜th1,emm2N˜a2t,rimx 2N˜Z3N,˜ mth2N˜r4o)u.gh the
2.2 The lepton neutralino mass matrix in EBLMSSM
In EBLMSSM, the superfields ( L , ϕL , N L , ϕN L ) have
their SUSY superpartners (ψ L , ψϕL , ψ N L , ψϕN L ). They
mix with λL , which is the superpartner of the new lepton type
gauge boson Z μL. Therefore, we deduce their mass matrix in
the base (i λL , ψ L , ψϕL , ψ N L , ψϕN L )
ML = ⎜⎜⎜
⎜
⎝
⎛
2ML
2υL gL
−2υ¯ L gL
3υN L gL
−3υ¯ N L gL
2υL gL −2υ¯ L gL 3υN L gL −3υ¯ N L gL ⎞
0 −μL 0 0 ⎟
−μL 0 0 0 ⎟⎟ .
0 0 0 −μN L ⎠⎟
0 0 0
−μN L
The lepton neutralino mass eigenstates are four-component
spinors X 0Li = (K L0i , K¯ L0i )T , and their mass matrix is
diagonalized by the rotation matrix Z N L . The relations for the
components are
i λL = Z 1NiL K L0i , ψ L = Z 2NiL K L0i , ψϕL = Z 3NiL K L0i ,
ψ N L = Z 4NiL K L0i , ψϕN L = Z 5NiL K L0i .
In BLMSSM, there are no ψ N L , ψϕN L , and the base of lepton
neutralino is (i λL , ψ L , ψϕL ), whose mass matrix is 3 ×
3. EBLMSSM extends this matrix to 5 × 5 including the
BLMSSM results.
2.3 The Higgs superfields and Y in EBLMSSM
The superfields L , ϕL , N L , ϕN L mix together and form
4 × 4 mass squared matrix, which is larger than the corre
sponding 2 × 2 mass matrix in the BLMSSM.
Diagonalizing the mass squared matrix, four CP even exotic Higgs are
obtained.
M2φ ( 0L 0L ) = 21 g2L 6υL2 − 2υ¯ L2 + 3(υN2 L − υ¯ N2 L )
1 2 1 m2 ,
+ 2 μL + 2 L
M2φ (ϕL0 ϕL0 ) = 21 g2L 6υ¯ L2 − 2υL2 + 3(υ¯ N2 L − υN2 L )
(
16
)
(
17
)
+ 2 μL + 21 m2ϕL ,
1 2
,
We use Zφ˜L to diagonalize the mass squared matrix in
Eq. (
18
), and the relation between mass eigenstates and the
comments are
0L = Zφ1˜iL HL0i , ϕL0 = Zφ2˜iL HL0i ,
0 0
N L = Z 3i HL0i , ϕN L = Z 4i HL0i .
φ˜L φ˜L
In EBLMSSM, the conditions for the exotic CP odd Higgs
P0, P¯L0 are same as those in BLMSSM, and they do not mix
L
with the added exotic CP odd Higgs PN0 L , P¯N0 L . Here, we
show the mass squared matrix for the added exotic CP odd
Higgs PN0 L , P¯N0 L .
M2p( PN0 L PN0 L ) = 21 g2L 29 υN2 L − 29 υ¯ N2 L + 3(υL2 − υ¯ L2 )
1 2 1 m2
+ 2 μN L + 2 N L ,
M2p( P¯N0 L P¯N0 L ) = 21 g2L 29 υ¯ N2 L − 29 υN2 L + 3(υ¯ L2 − υL2 )
+ 2 μN L + 21 m2ϕN L ,
1 2
M2p( PN0 L P¯N0 L ) = BN L2μN L . (
20
)
The scalar superfields Y and Y mix, and their mass
squared matrix is deduced here. This condition is
similar as that of X and X , then the lightest mass eigenstate
of Y and Y can be a candidate of the dark matter. With
SY = g2L (2 + L4)VL2, the concrete form for the mass squared
matrix is shown here. To obtain mass eigenstates, the matrix
ZY is used through the following formula, with the supposi
tion m2Y1 < m2Y2 .
ZY†
|μY |2 + SY −μY BY
−μ∗Y BY∗ |μY |2 − SY
ZY =
m2Y1 0
0 m2Y2
Y1
Y2
= ZY†
Y
Y ∗
(
23
)
The superpartners of Y and Y form four-component Dirac
spinors, and the mass term for superfields Y˜ is shown as
− LmY˜ ass = μY Y˜¯ Y˜ , Y˜ =
ψY
ψ¯ Y
The spinor Y˜ and the mixing of superfields Y, Y are all new
terms beyond BLMSSM, that add abundant contents to lepton
physics and dark matter physics.
2.4 Some couplings with h0 in EBLMSSM
In EBLMSSM, the exotic slepton(sneutrino) of generations
4 and 5 mix. So the couplings with exotic slepton(sneutrino)
are different from the corresponding results in BLMSSM.
We deduce the couplings of h0 and exotic sleptons
4
− e2υ cos β 14−sW24csW2W2 (Z 1E˜i∗ Z 1E˜j − Z 4E˜i∗ Z 4E˜j )
AE4 Z 2i∗ Z 1 j μ∗
−υ cos β|Ye4 |2δi j − √2 E˜ E˜ − √2 Ye5 Z 4E˜i∗ Z 3E˜j
1
1 Z 1i∗Ye∗4 λE Z 3Ej υN L sin α .
− 2 Ye∗4 Z 2E˜j λL Z 4E˜i∗υ¯ N L + 2 E˜ ˜
In Eq. (
23
), different from BLMSSM, there are new terms
( 21 λL Ye5 Z 3E˜j Z 3E˜i∗υ¯ N L − 21 Ye∗5 Z 4E˜j λE Z 2E˜i∗υN L ) cos α−( 21 Z 1E˜i∗
iYne∗g4 λoEfZg3Ee˜jnυeNraLti−on21sYe4∗4 Zan2E˜djλ5L Zsl4E˜eip∗υt¯oNnL. )Osbinvαiobueslsyid,ethsethsee
mneixwterms include υN L and υ¯ N L , which are the VEVs of added
Higgs superfields N L and ϕN L . In the same way, the cou
plings of h0 and exotic sneutrinos are also calculated
4
(
29
)
3 The mass of h0
3
× vL Zφ1˜kL − v¯L Zφ2˜kL + 2 vN L Zφ3˜kN L
HL0k Yi∗Y j .
ing lepton Higgs HL0. Then the couplings of HL0Y Y ∗ and
χ¯ N0 χ N0 HL0 are needed
(
24
)
(
25
)
(
26
)
(
27
)
Similar as BLMSSM, in EBLMSSM the mass squared matrix
for the neutral CP even Higgs are studied, and in the basis
(Hd0, Hu0) it is written as
2
Meven =
M121 +
M122 +
11 M122 +
12 M222 +
,
where M121, M122, M222 are the tree level results, whose
concrete forms can be found in Ref. [19]
1L1,
B
11 +
B
12 +
B
22 +
M SSM
11
M SSM
12
M SSM
22
11 = +
12 = + 1L2,
22 = + 2L2. (
30
)
The MSSM contributions are represented by 1M1SSM ,
1M2SSM and 2M2SSM . The exotic quark (squark)
contributions denoted by 1B1, 1B2 and 2B2 are the same as those
in BLMSSM [19]. However, the corrections 1L1, 1L2 and
2L2 from exotic lepton (slepton) are different from those in
BLMSSM, because the mass squared matrices of exotic
slepton and exotic sneutrino are both 4 × 4 and they relate with
υN L and υ¯ N L . Furthermore, the exotic leptons and exotic
neutrinos are heavier than those in BLMSSM, due to the
introduction of N L and ϕN L .
L G F Yν44 υ4
11 = 4√2π 2 sin2 β ·
2.5 The couplings with Y
For the dark matter candidate Y1, the necessary tree level
couplings are deduced in EBLMSSM. We show the couplings
(lepton-exotic lepton-Y ) and (neutrino-exotic neutrino-Y )
L =
e¯I λ4WL1i ZY1 j∗ PR − λ6UL2i ZY2 j∗ PL Li+3Y j∗
2
i, j=1
6
−
α=1 i, j=1
3
I =1
3
2
6
X¯ Nα λ4 Z NIαν∗WN1i ZY1 j∗ PR
0
+ λ5 Z (NIν+3)αU N2i ZY2 j∗ PL Ni+3Y j∗ + h.c.
The new gauge boson Z L couples with leptons, neutrinos and
Y , whose concrete forms are
L = −
gL Z μLe¯I γμeI −
gL (2 + L4)Z μLYi∗i ∂μY j
2
ϕL gives masses to the light neutrinos trough the see-saw
mechanism and L , ϕL , N L , ϕN L mix together
produc
G F Ye45 υ4
+ 4√2π 2 sin2 β ·
At the LHC, h0 is produced chiefly from the gluon fusion
(gg → h0). The one loop diagrams are the leading order
(LO) contributions. The virtual t quark loop is the dominate
contribution because of the large Yukawa coupling.
Therefore, when the couplings of new particles and Higgs are large,
they can influence the results obviously. For h0 → gg, the
EBLMSSM results are same as those in BLMSSM, and are
shown as [26–28]
N P (h0 → gg) =
with xa = m2h0 /(4ma2). Here, q and q are quark and exotic
quark. While, q˜ and q˜ denote squark and exotic squark. The
concrete expressions for gh0qq , gh0q q , gh0q˜q˜ , gh0q˜ q˜ (i =
1, 2) are in literature [19]. The functions A1/2(x ) and A0(x )
are[28]
The decay h0 → γ γ obtains contributions from loop
diagrams, and the leading order contributions are from the one
loop diagrams. In the EBLMSSM, the exotic quark (squark)
and exotic lepton (slepton) give new corrections to the decay
width of h0 → γ γ . Different from BLMSSM, the exotic
leptons in EBLMSSM are more heavy and the exotic sleptons
of the 4 and 5 generations mix together. These parts should
influence the numerical results of the EBLMSSM theoretical
prediction to the process h0 → γ γ to some extent.
The decay width of h0 → γ γ can be expressed as [29]
(
33
)
(34)
where gh0W W = sin(β − α) and A1(x ) = − 2x 2 + 3x +
3(2x − 1)g(x ) /x 2.
The formulae for h0 → Z Z , W W are
,
,
(35)
with gh0 Z Z = gh0W W and F (x ) is given out in Refs. [30–
32]. The observed signals for the diphoton and Z Z , W W
channels are quantified by the ratios Rγ γ and RV V , V =
(Z , W ), whose current values are Rγ γ = 1.16 ± 0.18 and
RV V = 1.19+−00..2220 [33].
4.2 Dark matter Y
In BLMSSM, there are some dark matter candidates such
as: the lightest mass eigenstate of X X mixing, X˜ the
fourcomponent spinor composed by the super partners of X and
X . They are studied in Ref. [18]. In EBLMSSM, the dark
matter candidates are more than those in BLMSSM, because
the lightest mass eigenstate of Y Y mixing and Y˜ are dark
matter candidates. After U (
1
)L is broken by L and N L ,
Z2 symmetry is left, which guarantees the stability of the dark
matters. There are only two elements (
1, −1
) in Z2 group.
This symmetry eliminates the coupling for the mass
eigenstates of Y Y mixing with two SM particles. The condition
for X is similar as that of Y , and it is also guaranteed by the
Z2 symmetry.
In this subsection, we suppose the lightest mass
eigenstate of Y Y mixing in Eq. (
21
) as a dark matter candidate,
and calculate the relic density. So we summarize the relic
density constraints that any WIMP candidate has to satisfy.
The interactions of the WIMP with SM particles are deduced
from the EBLMSSM, then we study its annihilation rate and
its relic density D by the thermal dynamics of the Universe.
The annihilation cross section σ (Y1Y1∗ → anyt hi ng) should
be calculated and can be written as σ vrel = a + bvr2el in the
Y1Y1∗ center of mass frame. vrel is the twice velocity of Y1
in the Y1Y1∗ c.m. system frame. To a good approximation,
the freeze-out temperature (TF ) can be iteratively computed
from[15–17]
m D
xF = TF
ln
0.038MPl m D(a + 6b/xF )
√g∗xF
,
with xF ≡ m D/ TF and m D = mY1 representing the WIMP
mass. MPl = 1.22 × 1019 GeV is the Planck mass and g∗ is
the number of the relativistic degrees of freedom with mass
less than TF . The density of cold non-baryonic matter is
D h2 = 0.1186 ± 0.0020 [33], whose formula is
simplified as
(36)
D h2
1.07 × 109xF
√g∗ MP L (a + 3b/xF )GeV
.
To obtain a and b in the σ vrel , we study the Y1Y1∗ dominate
decay channels whose final states are leptons and light
neutrinos: (
1
) Y1Y1∗ → Z L → l¯I l I ; (
2
) Y1Y1∗ → Z L → ν¯ I ν I ;
(
3
) Y1Y1∗ → ϕL → ν¯ I ν I ; (
4
) Y1Y1∗ → L → l¯I l I ; (
5
)
Y1Y1∗ → N → ν¯ I ν I .
Using the couplings in Eqs. (
26
), (
27
), (
28
), we deduce
the results of a and b
(37)
a =
l=e,μ,τ
1
π |
2
i=1
0
χNα=νe,νμ,ντ
3 4
I =1 i=1
+
×
×
1
+ π
m Li
(m2D + m2 )
Li
g4L (2 + L4)2
8π
1
(4m2D − m2 i ) ×
λ4WL1i ZY11∗λ6UL2i Z 21∗ 2
Y |
(ZY11∗ ZY11 − Z 21∗ ZY21)
Y
λN c Z (NIν+3)α Z (NIν+3)α Zφ2iL
3 3
vL Zφ1˜iL − v¯L Zφ2˜iL + 2 vN L Zφ3˜iL − 2 v¯N L Zφ4˜iL
2
2 3 m Ni
i=1 I =1 (m2D + m2Ni )
×λ4 Z NIαν∗WN1i ZY11∗λ5 Z (NIν+3)αU N2i ZY21∗
2
5.1 h0 decays and m A0 , m H0
In this section, we research the numerical results. For the
parameter space, the most strict constraint is that the mass of
the lightest eigenvector for the mass squared matrix in Eq.
(
29
) is around 125.1 GeV. To satisfy this constraint, we use
mh0 = 125.1 GeV as an input parameter. Therefore, the CP
odd Higgs mass should meet the following relation.
m2A0 =
m2h0 (m2Z − m2h0 + 11 + 22) − m2Z A + 122 − 11 22
−m2h0 + m2Z cos2 2β + B
1000
Au5 GeV
1000
0
2000
3000
1000
0
2000
3000
where
A = sin2 β 11 + cos2 β 22 + sin 2β 12,
B = cos2 β 11 + sin2 β 22 + sin 2β 12 .
(40)
To obtain the numerical results, we adopt the following
parameters as
Yu4 = 1.2Yt , Yu5 = 0.6Yt , Yd4 = Yd5 = 2Yb,
gB = 1/3, λu = λd = 0.5,
Au4 = Ad4 = Ad5 = Ae4 = Ae5 = Aν4 = Aν5 = 1 TeV,
λQ = 0.4, gL = 1/6,
m Q˜ 4 = m Q˜ 5 = mU˜4 = mU˜5 = m D˜ 4
= m D˜ 5 = mν˜4 = mν˜5 = 1 TeV,
Ye5 = 0.6, υN L = υL = Ab = 3 TeV,
tan βN L = tan βL = 2,
λL = λN L = λE = 1, m L˜ = me˜ = 1.4δi j TeV,
AL˜ = AL˜ = 0.5δi j TeV (i, j = 1, 2, 3), μB = 0.5 TeV,
AB Q = ABU = AB D = μN L = AL L
= AL E = AL N = 1 TeV, Yν4 = Yν5 = 0.1,
m L˜ 4 = m L˜ 5 = m E˜4 = m E˜5 = m2 = 1.5 TeV,
m D˜ 3 = 1.2 TeV, B4 = L4 = 1.5.
(41)
Here Yt and Yb are the Yukawa coupling constants of top
quark and bottom quark, whose concrete forms are Yt =
√2mt /(υ sin β) and Yb = √2mb/(υ cos β) respectively.
To embody the exotic squark corrections, we calculate
the results versus Au5 which has relation with the mass
squared matrix of exotic squark. In the left diagram of
Fig. 1, Rγ γ and RV V versus Au5 are plotted by the solid
line and dashed line respectively with m Q˜ 3 = mU˜3 =
1.2 TeV, tan β = 1.4, At = 1.7 TeV, υB = 3.6 TeV, μ =
−2.4 TeV, tan βB = 1.5 and Ye4 = 0.5. In the left diagram
of Fig. 1, the solid line (Rγ γ ) and dashed line (RV V ) change
weakly with the Au5 . When Au5 enlarges, Rγ γ is the
increasing function and RV V is the decreasing function. During the
Au5 region (−1700 to 1000) GeV, both Rγ γ and RV V satisfy
the experiment limits. The dot-dashed line(dotted line) in the
right diagram denotes the Higgs mass m0A(m0H ) varying with
Au5 . The dot-dashed line and dotted line increase mildly with
Au5 . The value of m0A is a little bigger than 500 GeV, while
the value of m0H is very near 500 GeV.
For the squark, we assume the first and second generations
are heavy, so they are neglected. The scalar top quarks are
not heavy, and their contributions are considerable. At is in
the mass squared matrix of scalar top quark influencing the
mass and mixing. The effects from At to the ratios Rγ γ ,
RV V , Higgs masses m A0 and m H0 are of interest. As m Q˜ 3 =
2.4 TeV, mU˜3 = 1.2 TeV, tan β = tan βB = 2.15, υB =
4.1 TeV, μ = −2.05 TeV, Ye4 = 0.5 and Au5 = 1 TeV.
Rγ γ (solid line) and RV V (dashed line) versus At are shown
in the left diagram of Fig. 2. While the right diagram of Fig.
2 gives out the Higgs masses m A0 (dot-dashed line) and m H0
(dotted line). In the At region (2–4.8) TeV, the Rγ γ varies
from 1.25 to 1.34. At the same time, the RV V is in the range
(1.2–1.38). The dot-dashed line and dotted line are very near.
In the At region (3000–4000) GeV, the masses of Higgs A0
and H 0 are around 1000 GeV. In this parameter space, the
allowed biggest values of A0 and H 0 masses can almost reach
1350 GeV.
Ye4 is the Yukawa coupling constant that can influence
the mass matrix of exotic lepton and exotic slepton. We
use m Q˜ 3 = mU˜3 = 1.2 TeV, tan β = 2.3, tan βB =
1.77, At = 1.7 TeV, υB = 5.43 TeV, μ = −2.64 TeV,
Au5 = 1 TeV and obtain the results versus Ye4 in the Fig. 3.
In the left diagram, the Rγ γ (solid line) and RV V (dashed
line) are around 1.3 and their changes are small during the
Ye4 range (0.05–1). One can see that in the right diagram
m A0 (dot-dashed line) and m H0 (dotted line) possess same
behavior versus Ye4 . They are both decreasing functions of
Ye4 and vary from 1500 to 500 GeV. In general, Ye4 effect to
the Higgs masses m A0 and m H0 is obvious.
1.6
3000 3500
At GeV
01500
2000
2500
3000 3500
At GeV
4000
4500
5000
m Q˜ 3 and mU˜3 are the diagonal elements of the squark mass
squared matrix, and they should affect the results. Supposing
m Q˜ 3 = mU˜3 = MQ , tan β = 2.1, tan βB = 2.24, At =
1.7 TeV, υB = 3.95 TeV, μ = −1.9 TeV, Ye4 =
0.6, Au5 = 1 TeV, we calculate the results versus MQ and
plot the diagrams in the Fig. 4. It shows that in this figure the
solid line, dashed line, dotted line and dot-dashed line are all
stable. Rγ γ and RV V are around 1.2. At the same time m A0
and m H 0 are about 1 TeV.
5.2 Scalar dark matter Y1
Here, we suppose Y1 as a scalar dark matter candidate.
In Ref. [33] the density of cold non-baryonic matter is
D h2 = 0.1186 ± 0.0020. To obtain the numerical results
of dark matter relic density, for consistency the used
parameters in this subsection are of the same values as in Eq. (41)
if they are supposed. Therefore, we just show the values of
the parameters beyond Eq. (41). These parameters are taken
as
μY = 1500 GeV,
λ5 = 1,
μL = BL = BN L = 1 TeV, tan β = 1.4,
BY = 940 GeV,
m2 L = m2ϕL = m2 N L
= m2ϕN L = 3 TeV2,
Ye4 = 0.5.
With the relation λ4 = λ6 = L m, we study relic density D
and x F versus L m in the Fig. 5. In the right diagram of Fig. 5,
the grey area is the experimental results in 3 σ and the solid
line representing D h2 turns small with the increasing L m.
During the L m region (0.7–1.4), D h2 satisfies the experi
ment bounds of dark matter relic density. x F is stable and in
the region (23.5–24).
Taking Ye4 = 1.3, λ4 = λ6 = 1 and the other
parameters being same as Eq. (42) condition, we plot the relic
density(x F ) versus Ye5 in the left (right) diagram of the Fig.
6. In this parameter space, during Ye5 region (0.1–2.5), our
theoretical results satisfy the relic density bounds of dark
matter, and x F is very near 23.55. Generally speaking, both
the solid line and dashed line are very stable.
(42)
6 Discussion and conclusion
Considering the light exotic lepton in BLMSSM, we add
exotic Higgs superfields N L and ϕN L to BLMSSM in order
1.5
2hD0.10
F
x
F
x
to make the exotic leptons heavy. Light exotic leptons may be
excluded by the experiment in the future. On the other hand,
heavy exotic leptons should not be stable. So we also
introduce the superfields Y and Y to make exotic leptons decay
quickly. The lightest mass eigenstate of Y and Y mixing mass
matrix can be a dark matter candidate. Therefore, the exotic
leptons are heavy enough to decay to SM leptons and Y at tree
level. We call this extended BLMSSM as EBLMSSM, where
the mass matrices for the particles are deduced and compared
with those in BLMSSM. Different from BLMSSM, the exotic
sleptons of 4 and 5 generations mix together forming 4 × 4
mass squared matrix. EBLMSSM has more abundant content
than BLMSSM for the lepton physics.
To confine the parameter space of EBLMSSM, we study
the decays h0 → γ γ and h0 → V V , V = (Z , W ). The
CP even Higgs masses mh0 , m H0 and CP odd Higgs mass
m0A are researched. In the numerical calculation, to keep
mh0 = 125.1 GeV, we use it as an input parameter. In
our used parameter space, the values of Rγ γ and RV V both
meet the experiment limits. The CP odd Higgs mass m A0
is a little heavier than the CP even Higgs mass m H 0 .
Generally speaking, both m A0 and m H 0 are in the region (500–
1500) GeV. Based on the supposition that the lightest mass
eigenstate Y1 of Y and Y mixing possesses the character of
cold dark matter, we research the relic density of Y1. In our
used parameter space, D h2 of Y1 can match the experiment
bounds. EBLMSSM has a bit more particles and
parameters than those in BLMSSM. Therefore, EBLMSSM
possesses stronger adaptive capacity to explain the experiment
results and some problems in the theory. In our later work, we
shall study the EBLMSSM and confine its parameter space
to move forward a single step.
Acknowledgements Supported by the Major Project of NNSFC (no.
11535002, no. 11605037, no. 11705045), the Natural Science
Foundation of Hebei province with Grant no. A2016201010 and no.
A2016201069, and the Natural Science Fund of Hebei University with
Grants no. 2011JQ05 and no. 2012-242, Hebei Key Lab of
OpticElectronic Information and Materials, the midwest universities
comprehensive strength promotion project. At last, thanks Dr. Tong Li and
Dr. Wei Chao very much for useful discussions of dark matter.
Open Access This article is distributed under the terms of the Creative
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