#### Two-Higgs-doublet type-II and -III models and \(t\rightarrow c h\) at the LHC

Eur. Phys. J. C
Two-Higgs-doublet type-II and -III models and t → ch at the LHC
A. Arhrib 1 2
R. Benbrik 0 5
Chuan-Hung Chen 4
Melina Gomez-Bock 3
Souad Semlali 0
0 LPHEA, Semlalia, Cadi Ayyad University , Marrakech , Morocco
1 Physics Division, National Center for Theoretical Sciences , Hsinchu 300 , Taiwan
2 Département de Mathématiques, Faculté des Sciences et Techniques, Université Abdelmalek Essaadi , B. 416, Tangier , Morocco
3 DAFM, Universidad de las Américas Puebla, Ex. Hda. Sta. Catarina Mártir , 72810 Cholula, PUE , Mexico
4 Department of Physics, National Cheng-Kung University , Tainan 70101 , Taiwan
5 MSISM Team, Faculté Polydisciplinaire de Safi, Sidi Bouzid , B.P 4162, 46000 Safi , Morocco
We study the constraints of the generic twoHiggs-doublet model (2HDM) type-III and the impacts of the new Yukawa couplings. For comparisons, we revisit the analysis in the 2HDM type-II. To understand the influence of all involving free parameters and to realize their correlations, we employ a χ -square fitting approach by including theoretical and experimental constraints, such as the S, T, and U oblique parameters, the production of standard model Higgs and its decay to γ γ , W W ∗/Z Z ∗, τ +τ −, etc. The errors of the analysis are taken at 68, 95.5, and 99.7 % confidence levels. Due to the new Yukawa couplings being associated with cos(β − α) and sin(β − α), we find that the allowed regions for sin α and tan β in the type-III model can be broader when the dictated parameter χF is positive; however, for negative χF , the limits are stricter than those in the type-II model. By using the constrained parameters, we find that the deviation from the SM in h → Z γ can be of O(10 %). Additionally, we also study the top-quark flavor-changing processes induced at the tree level in the type-III model and find that when all current experimental data are considered, we get Br (t → c(h, H )) < 10−3 for mh = 125.36 and m H = 150 GeV, and Br (t → c A) slightly exceeds 10−3 for m A = 130 GeV.
1 Introduction
A scalar boson around 125 GeV was observed in 2012 by
ATLAS [1] and CMS [2] at CERN with more than 5σ
significance. The discovery of such particle was based on the
analyses of the following channels: γ γ , W W ∗, Z Z ∗ and τ +τ −
with errors of order of 20–30 % and bb¯ channel with an error
of order of 40–50 %. The recent updates from ATLAS and
CMS with 7 ⊕ 8 TeV data [3–7] indicate the possible
deviations from the standard model (SM) predictions. Although
the errors of the current data are still somewhat large, the
new physics signals may become clear in the second run of
the LHC at 13–14 TeV.
It is expected that the Higgs couplings to gauge bosons
(fermions) at the LHC indeed could reach 4–6 % (6–13 %)
accuracy when the collected data are up to the integrated
luminosity of 300 fb−1 [8–11]. Furthermore, the e+e− Linear
Collider (LC) would be able to measure the Higgs couplings
at the percent level [12]. Therefore, the goals of LHC at run
II are (a) to pin down the nature of the observed scalar and
see if it is the SM Higgs boson or a new scalar boson; (b)
to reveal the existence of new physics effects, such as the
measurement of flavor-changing neutral currents (FCNCs)
at the top-quark decays, i.e. t → qh.
Motivated by the observations of the diphoton, W W ∗,
Z Z ∗, and τ +τ − processes at the ATLAS and CMS, it is
interesting to investigate what sorts of models may naturally
be consistent with these measurements and what the
implications are for other channels, e.g. h → Z γ and t → ch.
Although many possible extensions of the SM have been
discussed [13–19], it is interesting to study the simplest
extension from a one-Higgs doublet to a two-Higgs-doublet model
(2HDM) [20–47]. According to the situation of Higgs fields
coupling to fermions, the 2HDMs are classified as type-I,
II, and -III models, lepton specific model, and flipped model.
The 2HDM type-III is the case where both Higgs doublets
couple to all fermions; as a result, FCNCs at the tree level
appear. The detailed discussions of the 2HDMs are shown
elsewhere [23].
After the scalar particle of 125 GeV was discovered, the
implications of the observed h → γ γ in the type-I and -II
models were studied [48–54] and the impacts on h → γ Z
are given [55–57]. As is well known, the tan β and angle
α are important free parameters in the 2HDMs, where the
former is the ratio of two vacuum expectation values (VEVs)
of Higgses and the latter is the mixing parameter between the
two CP-even scalars. It is found that the current LHC data
put rather severe constraints on the free parameters [24–29].
For instance, the large tan β ∼ mt /mb scenario in the type-I
and -II is excluded except if we tune the α parameter to be
rather small, α < 0.02. Nevertheless, both type-I and type-II
models can still fit the data in some small regions of tan β
and α.
In this paper, we will explore the influence of new Higgs
couplings on the h → τ +τ −, h → gg, γ γ , W W, Z Z , and
h → Z γ decays in the framework of the 2HDM type-III.
We will show what is the most favored regions of the
typeIII parameter space when theoretical and experimental
constraints are considered simultaneously. FCNCs of the heavy
quark such as t → qh have been intensively studied both
from the experimental and the theoretical points of view [58].
Such processes are well established in the SM and are
excellent probes for the existence of new physics. In the SM and
2HDM type-I and -II, the top-quark FCNCs are generated at
one-loop level by charged currents and are highly suppressed
due to the GIM mechanism. The branching ratio (BR) for
t → ch in the SM is estimated to be 3 × 10−14 [59]. If this
decay t → ch is observed, it would be an indisputable sign of
new physics. Since the tree-level FCNCs appear in the
typeIII model, we explore if the Br (t → ch) reaches the order
of 10−5–10−4 [60–62], the sensitivity which is expected by
the integrated luminosity of 3000 fb−1.
The paper is organized as follows. In Sect. 2, we
introduce the scalar potential and the Yukawa interactions in the
2HDM type-III. The theoretical and experimental constraints
are described in Sect. 3. We set up the free parameters and
establish the χ -square for the best-fit approach in Sect. 6.
In the same section, we discuss the numerical results when
all theoretical and experimental constraints are taken into
account. The conclusions are given in Sect. 7.
2 Model
In this section we define the scalar potential and the
Yukawa sector in the 2HDM type-III. The scalar potential
in SU (
2
)L ⊗ U (
1
)Y gauge symmetry and CP invariance is
given by [69]
V (
1, 2
) = m21 †1 1 + m22 †2 2 − (m122 1 2 + h.c.)
†
+ 21 λ1( 1† 1)2 + 21 λ2( 2† 2)2
+ λ3( 1† 1)( 2† 2) + λ4( 1† 2)( 1† 2)
+
λ25 ( 1† 2)2 + λ6 1† 1 + λ7 2† 2
†
1 2 + h.c. ,
(1)
where the doublets 1,2 have a weak hypercharge Y = 1, the
corresponding VEVs are v1 and v2, and λi and m212 are real
parameters. After electroweak symmetry breaking, three of
the eight degrees of freedom in the two Higgs doublets are the
Goldstone bosons (G±, G0) and the remaining five degrees
of freedom become the physical Higgs bosons: two CP-even
h, H , one CP-odd A, and a pair of charged Higgs H ±. After
using the minimized conditions and the W mass, the
potential in Eq. (
1
) has nine parameters, which will be taken as
(λi )i=1,...,7, m212, and tan β ≡ v2/v1. Equivalently, we can
use the masses as the independent parameters; therefore, the
set of free parameters can be chosen to be
(
2
)
{mh , m H , m A, m H± , tan β, α, m122},
where we only list seven of the nine parameters, the angle
β diagonalizes the squared-mass matrices of CP-odd and
charged scalars and the angle α diagonalizes the CP-even
squared-mass matrix. In order to avoid generating
spontaneous CP violation, we further require
λ6v12
λ7v22
2 − 2
with ζ = 1(0) for λ5 > (<)0 [69]. It has been well known
that by assuming neutral flavor conservation at the tree level
[63], we have four types of Higgs couplings to the fermions.
In the 2HDM type-I, the quarks and leptons couple only to
one of the two Higgs doublets and the case is the same as the
SM. In the 2HDM type-II, the charged leptons and down-type
quarks couple to one Higgs doublet and the up-type quarks
couple to the other. The lepton-specific model is similar to
type-I, but the leptons couple to the other Higgs doublet. In
the flipped model, which is similar to type-II, the leptons and
up-type quarks couple to the same double.
If the tree-level FCNCs are allowed, both doublets can
couple to leptons and quarks and the associated model is
called 2HDM type-III [23,64–66]. Thus, the Yukawa
interactions for the quarks are written as
(
3
)
LY = Q¯ L Y k dR φk + Q¯ L Y˜ k u R φ˜k + h.c.
where the flavor indices are suppressed, QTL = (u L , dL ) is
the left-handed quark doublet, Y k and Y˜ k denote the 3 × 3
Yukawa matrices, φ˜k = i σ2φk∗, and k is the doublet number.
Similar formulas could be applied to the lepton sector. Since
the mass matrices of the quarks are combined by Y 1(Y˜ 1) and
Y 2(Y˜ 2) for down- (up-) type quarks and Y 1,2(Y˜ 1,2)
generally cannot be diagonalized simultaneously, as a result, the
tree-level FCNCs appear and the effects lead to the
oscillations of K − K¯ , Bq − B¯q and D − D¯ at the tree level. To get
(
4
)
naturally small FCNCs, one can use the ansatz formulated
by Yikj , Y˜ikj ∝ √mi m j /v [64–66]. After spontaneous
symmetry breaking, the scalar couplings to the fermions can be
expressed as [67,68]
L2YHDM-III = u¯ Li
csoinsβα mvui δi j − co√s(2βsi−n βα) Xiuj u R j h
+ d¯Li − csoins αβ mvdi δi j + co√s2(βco−s βα) Xidj dR j h
+ u¯ Li
+ d¯Li
− i u¯ Li
+ h.c.,
+ i d¯Li − tan β mvdi δi j + √
ssiinn βα mvui δi j + si√n(2βsi−n βα) Xiuj u R j H
cos α mdi δi j − √
sin(β − α) Xidj dR j H
cos β v 2 cos β
1
mui δi j − √
tan β v
Xiuj
2 sin β
Xidj
2 cos β
u R j A
dR j A
(
5
)
(
6
)
where v =
χiqj are the free parameters. By the above formulation, if the
FCNC effects are ignored, the results are returned to the case
of the 2HDM type-II, given by
v1 + v22, Xi j = √mqi mq j /vχiqj (q = u, d) and
2 q
L2YHDM-II = u¯ Li
cos α mui
sin β v
δi j u R j h
sin α mdi
+ d¯Li − cos β v δi j dR j h + h.c.
The couplings of the other scalars to fermions can be found
elsewhere [67,68]. It can be seen clearly that if χiuj,d are of
O(10−1), the new effects are dominated by heavy fermions
and comparable with those in the type-II model. The
couplings of the h and H to the gauge bosons V = W, Z are
proportional to sin(β − α) and cos(β − α), respectively.
Therefore, the SM-like Higgs boson h is recovered when
cos(β − α) ≈ 0. The decoupling limit can be achieved if
cos(β − α) ≈ 0 and mh m H , m A, m H± are satisfied [69].
From Eqs. (
5
) and (
6
), one can also find that in the decoupling
limit, the h couplings to the quarks return to the SM case.
In this analysis, since we take α in the range −π/2 ≤ α ≤
π/2, sin α will have both a positive and a negative sign. In
the 2HDM type-II, if sin α < 0 then the Higgs couplings to
up- and down-type quarks will have the same sign as those
in the SM. It is worthwhile to mention that sin α in
minimal supersymmetric SM (MSSM) is negative, unless some
extremely large radiative corrections flip its sign [69]. If sin α
is positive, then the Higgs coupling to down quarks will have
a different sign with respect to the SM case. This is called the
wrong sign Yukawa coupling in the literature [69–71]. Later
we will explain whether the type-III model would favor such
a wrong sign scenario or not.
3 Theoretical and experimental constraints
The free parameters in the scalar potential defined in
Eq. (
1
) could be constrained by theoretical requirements and
the experimental measurements, where the former mainly
includes tree-level unitarity and vacuum stability when the
electroweak symmetry is broken spontaneously. Since the
unitarity constraint involves a variety of scattering processes,
we adopt the results [72–75]. We also force the potential to
be perturbative by requiring that all quartic couplings of the
scalar potential obey |λi | ≤ 8π for all i . For the vacuum
stability conditions which ensure that the potential is bounded
from below, we require that the parameters satisfy the
conditions [76–78]
(
7
)
(
8
)
λ1 > 0, λ2 > 0, λ3 +
λ1λ2 > 0,
λ1λ2 + λ3 + λ4 − |λ5| > 0,
1
2|λ6 + λ7| ≤ 2 (λ1 + λ2) + λ3 + λ4 + λ5.
In the following we state the constraints from the
experimental data. The new neutral and charged scalar bosons
in 2HDM will affect the self-energy of W and Z bosons
through the loop effects. Therefore, the involved
parameters could be constrained by the precision measurements of
the oblique parameters, denoted by S, T, and U [79]. Taking
mh = 125 GeV, mt = 173.3 GeV, and assuming that U = 0,
the tolerated ranges for S and T are found to be [80]
S = 0.06 ± 0.09,
T = 0.10 ± 0.07,
where the correlation factor is ρ = +0.91, S = S2HDM −
SSM, and T = T 2HDM − T SM, and their explicit
expressions can be found [69]. We note that in the limit m H± = m A0
or m H± = m H0 , T vanishes [81,82].
The second set of constraints comes from B physics
observables. It has been shown recently in Ref. [83] that
Br (B → Xs γ ) gives a lower limit on m H± ≥ 480 GeV in
the 2HDM type-II at 95 % CL. However, in 2HDM type-III
the situation is slightly different. In fact, the charged Higgs–
top-quark affects Br (B → Xs γ ) via the Wilson coefficients
C7,8 at leading-order (LO) as well as at the next-to-next LO
(NNLO). The Br (B → Xs γ ) constraint can get weakened
in the 2HDM type-III because of the off-diagonal element
that enters in the H +t b¯ coupling. This new term can lead to
a destructive interference with the SM and then reduce the
2HDM contribution. It was pointed out that in 2HDM
typeIII a light charged Higgs boson with a mass around 200 GeV
is still allowed at NLO level by the measured Br (B → Xs γ )
within 2σ [84–86]. While in type-II, such light charged Higgs
boson cannot be accommodated.
By precision measurements of Z → bb¯ and Bq –B¯q
mixing, the values of tan β < 0.5 have been excluded [87,88].
In this work we allow tan β ≥ 0.5. Except for some specific
scenarios, tan β cannot be too large due to the requirement
of perturbation theory.
By the observation of a scalar boson at mh ≈ 125 GeV,
the searches for Higgs boson at ATLAS and CMS can give
strong bounds on the free parameters. The signal events in the
Higgs measurements are represented by the signal strength,
which is defined by the ratio of the Higgs signal to the SM
prediction and is given by
μif = σiSσMi((hh)) ·· BBrrS(hM (→h →f ) f ) ≡ σ¯i · μ f ,
where σi (h) denotes the Higgs production cross section by
channel i and Br (h → f ) is the BR for the Higgs decay
h → f . Since several Higgs boson production channels
are available at the LHC, we are interested in the gluon
fusion production (gg F ), t t¯h, vector boson fusion (VBF) and
Higgs-strahlung V h with V = W /Z ; and they are grouped
into μgfgF+tt¯h and μVf B F+V h . In order to consider the
constraints from the current LHC data, the scaling factors which
show the Higgs coupling deviations from the SM are defined
as
κV = κW = κZ ≡
gh2HVDVM , κ
ghSVMV
f ≡
yh2HffDM
yhSMff ,
(
9
)
(
10
)
where ghV V and yh f f are the Higgs couplings to gauge
bosons and fermions, respectively, and f stands for top,
bottom quarks, and tau lepton. The scaling factors for the
loopinduced channels are defined by
κγ2 ≡
2
κZγ ≡
(h → γ γ )2HDM
(h → γ γ )SM
(h → Z γ )2HDM
(h → Z γ )SM
, κg2 ≡
, κh2 ≡
(h → g g)2HDM
(h → g g)SM
(h)2HDM ,
(h)SM
,
(
11
)
where (h → X Y ) is the partial decay rate for h → X Y . In
this study, the partial decay width of the Higgs is taken from
[89,90], where the QCD corrections have been taken into
account. In the decay modes h → γ γ and h → Z γ , we have
included the contributions of the charged Higgs and the new
Yukawa couplings. Accordingly, the ratio of the cross section
to the SM prediction for the production channels gg F + t t¯h
and VBF+V h can be expressed as
σ ggF+tt¯h =
κg2σSM (gg F) + κt2σSM (tth)
σSM (gg F) + σSM (tth)
,
σ V B F+V h
= κV2 σSMσ(SVM B(VFB) +F)κ+ZhσσSSMM((ZZhh))++σκSV2Mσ(SZMh()Z+h)σ+SMκ(V2WσShM) (W h) ,
(
12
)
(
13
)
where σSM (Z h) is from the coupling of Z Z h and occurs at
the tree level, and σSM (Z h) ≡ σSM (gg → Z h) represents
the effects of the top-quark loop. With mh = 125.36 GeV,
4 Parameter setting, global fitting, and numerical results
4.1 Parameters and global fitting
After introducing the scaling factors for displaying the new
physics in various channels, in the following we show the
explicit relations with the free parameters in the type-III
model. By the definitions in Eq. (
10
), the scaling factors for
κV and κ f in the type-III are given by
κV = sin(β − α),
cos α cos(β − α)
κU = κt = κc = sin β − χF √2 sin β ,
sin α cos(β − α)
κD = κb = κτ = − cos β + χF √2 cos β
.
Although the FCNC processes give strict constraints on the
flavor-changing couplings χifj with i = j , the constraints
are applied to the flavor-changing processes in the K , D,
and B meson systems. Since the couplings of the scalars to
the light quarks have been suppressed by mqi /v, the direct
limit on the flavor-conserved coupling χifi is mild.
Additionally, since the signals for top-quark flavor-changing
processes have not been observed yet, the direct constraint on
X3ui = √mt mqi /vχ3ui is from the experimental bound of
t → hqi . Hence, for simplifying the numerical analysis, in
u u d
Eq. (
15
) we have set χ22 = χ33 = χ33 = χ33 = χF . Since
X3u3 = mt /vχF , it is conservative to adopt the value of χF
(
15
)
to be O(
1
). In the 2HDM, the charged Higgs will also
contribute to h → γ γ decay and the associated scalar triplet
coupling h H + H − reads
λh H ± H ∓
1
= 2m2W
cos(α + β)
sin 2β
where we have used mh = 125.36 GeV and taken m H ± =
480 GeV. It is clear that the charged Higgs contribution to
h → γ γ and h → Z γ is small. In order to study the
influence of the new free parameters and to understand their
correlations, we perform the χ -square fitting by using the
LHC data for Higgs searches [1, 2, 4, 6]. For a given channel
f = γ γ , W W ∗, Z Z ∗, τ τ , we define the χ 2f as
χ 2f =
1 f f 2 1 f f 2
σˆ12(1 − ρ2) (μ1 − μˆ 1 ) + σˆ12(1 − ρ2) (μ2 − μˆ 2 )
2ρ f f
− σˆ1σˆ2(1 − ρ2) (μ1 − μˆ 1f )(μ2 − μˆ 2f ),
f , σˆ1,2 and ρ are the measured Higgs signal
where μˆ 1,2
strengths, their one-sigma errors, and their correlation,
respectively, and for their values one may refer to Table 1.
The indices 1 and 2 in turn stand for ggF + tth and VBF + Vh,
f
and μ1,2 are the results in the 2HDM. The global χ -square
is defined by
(
16
)
→
(
17
)
(
18
)
(
19
)
χ 2 =
χ 2f + χS2T ,
f
where the χS2T is related to the χ 2 for S and T parameters,
the definition can be obtained from Eq. (
18
) by replacing
μ1f → S2H D M and μ2f → T 2H D M , and the corresponding
values can be found from Eq. (
8
). We do not include the bb¯
channel in our analysis because the errors of the data are still
large.
In order to display the allowed regions for the parameters,
we show the best fit at 68, 95.5, and 99.7 % confidence levels
(CLs), that is, the corresponding errors of χ 2 are χ 2 ≤ 2.3,
5.99, and 11.8, respectively. For comparing with the LHC
data, we require the calculated results in agreement with those
shown by ATLAS (Fig. 3 of Ref. [3]) and by CMS (Fig. 5 of
Ref. [7]).
5 Numerical results
In the following we present the limits of the current LHC data
based on the three kinds of CL introduced in the last section.
In our numerical calculations, we set the mass of SM Higgs to
be mh = 125.36 GeV, and we scan the involved parameters
in the chosen regions as
480 GeV ≤ m H ± ≤ 1 TeV,
126 GeV ≤ m H ≤ 1 TeV,
(
20
)
100 GeV ≤ m A ≤ 1 TeV,
−1 ≤ sin α ≤ 1, 0.5 ≤ tan β ≤ 50,
− (1000 GeV)2 ≤ m212 ≤ (1000 GeV)2.
The main difference in the scalar potential between
typeII and type-III is that the λ6,7 terms appear in the type-III
model. With the introduction of the λ6,7 terms in the
potential, not only the mass relations of scalar bosons are modified
but also the scalar triple and quartic coupling receives
contributions from λ6 and λ7. Since the masses of the scalar bosons
are regarded as free parameters, the relevant λ6,7 effects in
this study enter the game through the triple coupling h– H +–
H −, which contributes to the h → γ γ decay, as shown in
Eq. (
16
) and the first line of Eq. (
17
). Since the contribution
of the charged Higgs loop to the h → γ γ decay is small,
expectedly the influence of λ6,7 on the parameter constraint
is not significant. To demonstrate that the contributions of
λ6,7 are not very important, we present the allowed ranges
for tan β and sin α by scanning λ6,7 in the region of [−1, 1]
in Fig. 1, where the theoretical and experimental constraints
mentioned earlier are included and the plots from left to right
in turn stand for χ 2 = 11.8, 5.99, and 2.3, respectively.
Additionally, to understand the influence of χF defined in
Eq. (
15
), we also scan χF = [−1, 1] in the plots. By
comparing the results with the case of λ6,7 1 and χF = 1,
which is displayed in the third plot of Fig. 2, it can be seen
that only a small region for positive sin α is modified and
the modifications happen only in the large errors of χ 2; the
plot with χ 2 = 2.3 shows almost no change. However, it
can be seen from Fig. 1 (right panel) that for small χ 2 the
values of positive sin α are excluded by LHC constraints on
h → τ +τ −, which exceed a 20 % deviation from SM values,
which is excluded by the LHC data. Therefore, to simplify
the numerical analysis and to reduce the scanned
parameters, it is reasonable in this study to assume λ6,7 1. Since
the influence of |χF | ≤ 1 should be smaller, to get the
typical contributions from the FCNC effects, we illustrate our
studies by setting χF = ±1 in the whole numerical analysis.
With λ6,7 1, we present the allowed regions for sin α
and tan β in Fig. 2, where the left, middle, and right panels
Fig. 2 The allowed regions in (sin α, tan β) constrained by theoretical
and current experimental inputs, where we have used mh = 125.36 GeV,
the left, middle, and right panels stand for the 2HDM type-II, type-III
with χF = −1 and type-III with χF = +1, respectively. The errors for
the χ-square fit are 99.7 % CL (black), 95.5 % CL (red), and 68 % CL
(green)
stand for the 2HDM type-II, type-III with χF = −1 and
type-III with χF = +1, respectively, and in each plot we
show the constraints at 68 % CL (green), 95.5 % CL (red),
and 99.7 % CL (black). Our results in type-II are consistent
with those obtained by the authors in Refs. [24–29,48–54]
when the same conditions are chosen. By the plots, we see
that in type-III with χF = −1, due to the sign of the coupling
being the same as type-II, the allowed values for sin α and
tan β are further shrunk; especially sin α is limited values
less than 0.1. On the contrary, for type-III with χF = +1,
the allowed values of sin α and tan β are broad.
As discussed before, the decoupling limit occurs at α →
β − π/2, i.e. sin α = − cos β < 0. Since we regard the
masses of new scalars as free parameters and scan them in
the regions shown in Eq. (
20
), the three plots in Fig. 2 cover
lower and heavier mass of charged Higgs. We further check
that sin α > 0 could be excluded at 95.5(99.7) % CL when
m H± ≥ 585(690) GeV. The main differences between
typeII and type-III are the Yukawa couplings as shown in Eq. (
5
).
In order to see the influence of the new effects of type-III,
we plot the allowed κg as a function of sin α and tan β in
Fig. 3, where the three plots from left to right correspond to
type-II, type-III with χF = −1, and type-III with χF = +1,
the solid, dashed, and dotted lines in each plot stand for the
decoupling limit (DL) of SM, 15 % deviation from DL and
20 % deviation from DL, respectively. For comparison, we
also put the results of 99.7 % in Fig. 2 in each plot. By
the analysis, we see that the deviations of κg from DL in
χF = +1 are clear and significant, while the influence of
χF = −1 is small. It is pointed out that a wrong sign Yukawa
coupling to down-type quarks could happen in type-II 2HDM
[69–71]. For understanding the sign flip, we rewrite the κD
defined in Eq. (
15
),
sin α
κD = − cos β
1 −
χF sin β
√2
+
χF cos α
√2
In the type-II case, we know that in the decoupling limit
κD = 1, but κD < 0 if sin α > 0. According to the results in
the left panel of Fig. 2, sin α > 0 is allowed when the errors
of best fit are taken by 2σ or 3σ . The situation in type-III is
more complicated. From Eq. (
21
), we see that the factor in
the brackets is always positive, therefore, the sign of the first
term should be the same as that in type-II case. However, due
to α ∈ [−π/2, π/2], the sign of the second term in Eq. (
21
)
depends on the sign of χF . For χF = −1, even sin α < 0,
κD could be negative when the first term is smaller than the
second term. For χF = +1, if sin α > 0 and the first term
is over the second term, κD < 0 is still allowed. In order to
deviation from DL and 20 % deviation from DL, respectively. The
dotted points are the allowed values of parameters resulting from Fig. 2
understand the available values of κD when the constraints
are considered, we present the correlation of κU and κD in
Fig. 4, where the panels from left to right stand for type-II,
type-III with χF = −1 and type-III with χF = +1. In each
plot, the results obtained by χ -square fitting are applied. The
similar correlation of κV and κD is presented in Fig. 5. By
these results, we find that comparing with type-II model, the
negative κD gets a stricter limit in type-III, although a wider
parameter space for sin α > 0 is allowed in type-III with
χF = +1.
Besides the scaling factors of tree-level Higgs decays,
κD,U and κV , it is also interesting to understand the allowed
values for loop-induced processes in 2HDM, e.g. h → γ γ ,
gg, and Z γ , etc. It is well known that the differences in the
associated couplings between h → γ γ and gg are the
colorless W -, τ -, and H ±-loop. By Eq. (
17
), we see that the
contributions of τ and H ± are small, therefore, the main
difference is from the W -loop in which the κV is involved.
By using the χ -square fitting approach and with the inputs of
the experimental data and theoretical constraints, the allowed
regions of κγ and κg in type-II and type-III are displayed in
Fig. 6, where the panels from left to right are type-II, type-III
with χF = −1 and +1; the green, red, and black colors in
each plot stand for the 68, 95.5, and 99.7 % CL, respectively.
We find that except a slightly lower κγ is allowed in type-II,
the first two plots have similar results. The situation can be
Fig. 8 Correlation between μγggγF+tth and μgZgγF+tth at √s = 13 TeV
after imposing theoretical and experimental constraints. Left, middle,
and right panels represent the allowed values in type-II, type-III with
χF = −1 and type-III with χF = +1, respectively, and the results in
Fig. 2 obtained by χ -square fitting are applied
understood from Figs. 4 and 5, where the κU in both models
is similar, while κV in type-II could be smaller in the region
of negative κD ; that is, a smaller κV will lead to a smaller κγ .
In the χF = +1 case, the allowed values of κγ and κg are
localized in a wider region.
It is well known that except for the different gauge
couplings, the loop diagrams for h → Z γ and h → γ γ are
exactly the same. One can understand the loop effects by
the numerical form of Eq. (
17
). Therefore, we expect the
correlation between κZ γ and κγ should behave like a linear
relation. We present the correlation between κγ and κZ γ in
Fig. 7, where the legend is the same as that for Fig. 6. From
the plots, we see that in most of the region κZ γ is less than
the SM prediction. The type-III with χF = −1 gets a stricter
constraint and the change is within 10 %. For χF = +1, the
deviation of κZ γ from unity could be over 10 %. From run
I data, the LHC has an upper bound on h → Z γ , at run II
this decay mode will be probed. We give the predictions at
13 TeV LHC for the signal strength μγggγF+tth and μgZgγF+tth
in Fig. 8. Hence, with the theoretical and experimental
con
Z γ
straints, μgg F+tth is bounded and could be O(10 %) away
from SM at 68 % CL.
In this section, we study the flavor-changing t → ch
process in the type-III model. Experimentally, there have been
intensive activities to explore the top FCNCs. The CDF, D0,
and LEPII collaborations have reported some bounds on top
FCNCs. At the LHC with rather large top cross section,
ATLAS and CMS search for top FCNCs and put a limit on
the branching fraction, which is Br (t → ch) < 0.82 %
for ATLAS [60,61] and Br (t → ch) < 0.56 % for CMS
[62]. Note that the CMS limit is slightly better than the
ATLAS limit. CMS search for t → ch in different
channels: h → γ γ , W W ∗, Z Z ∗, and τ +τ −, while ATLAS used
only a diphoton channel. With the high luminosity option of
the LHC, the above limit will be improved to reach about
Br (t → ch) < 1.5 × 10−4 [60,61] for the ATLAS detector.
From the Yukawa couplings in Eq. (
5
), the partial width
for the t → ch decay is given by
(t → ch) =
cos(β − α)X2u3
sin β
2 mt
32π
(xc + 1)2 − xh
2
× 1 − (xh − xc)2 1 − (xh + xc)2
(
22
)
where xc = mc/mt , xh = mh /mt , and X2u3 is a free
parameter and dictates the FCNC effect. It is clear from the above
expression that the partial width of t → ch is proportional to
cos(β −α). As seen in the previous section, in the case where
h is SM-like, cos(β − α) is constrained by LHC data to be
rather small and the t → ch branching fraction is limited. As
we will see later in 2HDM type-II with flavor conservation the
rate for t → ch is much smaller than type-III [92–95]. Since
we assume that the charged Higgs is heavier than 400 GeV,
the total decay width of the top contains only t → ch and
t → bW . With mh = 125.36 GeV, mt = 173.3 GeV, and
mc = 1.42 GeV, the total width can be written
t = tSM + 0.0017
cos(β − α)X2u3 2
sin β
GeV
(
23
)
where tSM is the partial decay rate for t → W b, given by
SM
t
G F mt3
= 8π √2
1 −
m2W
mt2
2
1 + 2 mt2
m2W
×
1 − 2 αs3(πmt ) 2 π32 − 25
= 1.43 GeV,
in which the QCD corrections have been included. By the
above numerical expressions together with the current limit
from ATLAS and CMS, the limit on the t ch FCNC coupling
is foun:
(
24
)
in agreement with [96].
We perform a systematic scan over the 2HDM
parameters, as depicted in Eq. (
20
), taking into account LHC and
theoretical constraints. Although X2u3 is a free parameter,
in order to suppress the FCNC effects naturally, as stated
earlier we adopt X2u3 = √mt mc/vχ2u3. Since the current
experimental measurements only give an upper limit on
t → hc, basically χ2u3 is limited by Eq. (
24
) and could be
as large as O(
1 − 102
), depending on the allowed value of
cos(β − α). In order to use the constrained results which
are obtained from the Higgs measurements and the
selfconsistent parametrization X3u3 = mt /vχF , which was used
u
before, we assume χ23 = χF = ±1. In our numerical
analysis, the results under the assumption should be
conservative. In Fig. 9(left) we illustrate the branching fraction of
t → ch in 2HDM-III as a function of cos(β − α). The
LHC constraints within 1σ restrict cos(β − α) to be in the
range [−0.27, 0.27]. The branching fraction for t → ch is
slightly above 10−4. The actual CMS and ATLAS constraint
on Br (t → ch) < 5.6 × 10−3 does not restrict cos(β − α).
The expected limit from the ATLAS detector with high
luminosity 3000 fb−1 is depicted as the dashed horizontal line. As
one can see, the expected ATLAS limit is somehow similar
to LHC constraints within 1σ . In the right panel, we show the
allowed parameters space in the (sin α, tan β) plane where we
apply the ATLAS expected limit Br (t → ch) < 1.5 × 10−4.
This plot should be compared to the right panel of Fig. 2. It
is then clear that this additional constraint only acts on the
3σ allowed region from the LHC data.
In Figs. 10 and 11, we show the fitted branching fractions
for t → ch (left), t → c H at m H = 150 GeV (middle),
and t → c A at m A = 130 GeV (right) as a function of κU ,
where Fig. 10 is for χF = +1, while Fig. 11 is χF = −1.
In the case of χF = +1 the fitted value for κU at the 3σ
level is in the range [0.6, 1.18] and the branching fraction
for t → ch, c H are less than 10−3, while Br (t → c A)
slightly exceeds the 10−3 level. Similarly, for χF = −1 the
fitted value for κU at the 3σ level is in the range [0.85, 1.25]
and the branching fraction for t → ch, c H, c A are the same
size as in the previous case.
7 Conclusions
For studying the constraints of the 8 TeV LHC
experimental data, we perform a χ -square analysis to find the most
favorable regions for the free parameters in the
two-Higgsdoublet models. For comparison, we focus on the type-II and
type-III models, in which the latter model not only affects the
flavor conserving Yukawa couplings but also generates the
scalar-mediated flavor-changing neutral currents at the tree
level.
Although the difference between type-II and type-III is the
Yukawa sector, however, since the new Yukawa couplings in
type-III are associated with cos(β − α) and sin(β − α), the
modified couplings t t h and bt H ± will change the constraint
of the free parameters.
In order to represent the influence of modified Yukawa
couplings, we show the allowed values of sin α and tan β
in Fig. 2, where the LHC updated data for pp → h → f
with f = γ γ , W W ∗/Z Z ∗ and τ +τ − are applied and other
bounds are also included. By the results, we see that sin α and
tan β in type-III get an even stronger constraint if the dictated
parameter χF = −1 is adopted; on the contrary, if we take
χF = +1, the allowed values for sin α and tan β are wider.
It has been pointed out that there exist Yukawa couplings to
down-type quarks of the wrong sign in the type-II model,
i.e. sin α > 0 or κD < 0. By the study, we find that except
that the allowed regions of parameters are shrunk slightly,
the situation in χF = −1 is similar to the type-II case. In
χF = +1, although the κD < 0 is not excluded completely
yet, the case has a strict limit by current data. We show the
analyses in Figs. 4 and 5. In these figures, one can also see
the correlations with a modified Higgs coupling to the top
quark, κU , and to the gauge boson, κV .
When the parameters are bounded by the observed
channels, we show the influence on the unobserved channel
h → Z γ by using the scaling factor κZ γ , which is defined
by the ratio of the decay rate to the SM prediction. We find
that the change of κZ γ in type-III with χF = −1 is less than
10 %; however, with χF = +1, the value of κZ γ could be
lower from the SM prediction by over 10 %. We also show
our predictions for the signal strengths μγ γ and μγ Z and
their correlation at 13 TeV.
The main difference between type-II and -III model
is that the flavor-changing neutral currents in the former
are only induced by loops, while in the latter they could
occur at the tree level. We study the scalar-mediated t →
c(h, H, A) decays in the type-III model and find that when
all current experimental constraints are considered, Br (t →
c(h, H )) < 10−3 for mh = 125.36 and m H = 150 GeV, and
Br (t → c A) slightly exceeds 10−3 for m A = 130 GeV. The
detailed numerical analyses are shown in Figs. 9, 10, and 11.
Acknowledgments The authors thank Rui Santos for useful
discussions. A.A would like to thank NCTS for warm hospitality where part
of this work has been done. The work of CHC is supported by the
Ministry of Science and Technology of R.O.C under Grant #:
MOST103-2112-006-004-MY3. The work of M. Gomez-Bock was partially
supported by UNAM under PAPIIT IN111115. This work was also
supported by the Moroccan Ministry of Higher Education and Scientific
Research MESRSFC and CNRST: “Projet dans les domaines
prioritaires de la recherche scientifique et du développement technologique”:
PPR/2015/6.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. G. Aad et al. (ATLAS Collaboration) , Phys. Lett. B 716 , 1 ( 2012 ). arXiv: 1207 .7214 [hep-ex]
2. S. Chatrchyan et al. (CMS Collaboration) , Phys. Lett. B 716 , 30 ( 2012 ). arXiv: 1207 .7235 [hep-ex]
3. G. Aad et al. (ATLAS Collaboration) , ATLAS note: ATLASCONF-2015-007
4. G. Aad et al. (ATLAS Collaboration) , ATLAS note: ATLASCONF-2013-012
5. ATALS Collaboration , ATLAS-CONF- 2013-034
6. S. Chatrchyan et al. (CMS Collaboration) , CMS-PAS-HIG-13-001
7. S. Chatrchyan et al. (CMS Collaboration) , CMS-PAS-HIG-14-009
8. S. Dawson et al., arXiv:1310 .8361 [hep-ex]
9. D. Zeppenfeld , R. Kinnunen , A. Nikitenko , E. Richter-Was, 013009 ( 2000 ). arXiv:hep-ph/0002036
10. F. Gianotti , M. Pepe-Altarelli , Nucl. Phys. Proc. Suppl . 89 , 177 ( 2000 ). arXiv:hep-ex/0006016
11. C. Englert et al., J. Phys. G 41 , 113001 ( 2014 ). arXiv: 1403 .7191 [hep-ph]
12. G. Moortgat-Pick et al., arXiv:1504 .01726 [hep-ph]
13. A. Arhrib , R. Benbrik , C.-H. Chen , arXiv: 1205 .5536 [hep-ph]
14. A. Arhrib , R. Benbrik , M. Chabab , G. Moultaka , L. Rahili, arXiv: 1202 .6621 [hep-ph]
15. A. Arhrib , R. Benbrik , N. Gaur , Phys. Rev. D 85 , 095021 ( 2012 ). arXiv: 1201 .2644 [hep-ph]
16. A. Arhrib , R. Benbrik , M. Chabab , G. Moultaka , L. Rahili, JHEP 1204 , 136 ( 2012 ). arXiv: 1112 .5453 [hep-ph]
17. C.W. Chiang , K. Yagyu , Phys. Rev. D 87 ( 3 ), 033003 ( 2013 ). arXiv: 1207 .1065 [hep-ph]
18. C.S. Chen , C.Q. Geng , D. Huang , L.H. Tsai , Phys. Lett. B 723 , 156 ( 2013 ). arXiv: 1302 .0502 [hep-ph]
19. C.S. Chen , C.Q. Geng , D. Huang , L.H. Tsai , New scalar contributions to h → Z γ . Phys. Rev. D 87 , 075019 ( 2013 ). arXiv: 1301 .4694 [hep-ph]
20. T.D. Lee , Phys. Rev. D 8 , 1226 ( 1973 )
21. T.D. Lee , Phys. Rep . 9 , 143 ( 1974 )
22. J.F. Gunion , H.E. Haber , G. Kane, S. Dawson , The Higgs Hunters Guide. Frontiers in Physics Series (Addison-Wesley , Reading, 1990 )
23. G.C. Branco , P.M. Ferreira , L. Lavoura , M.N. Rebelo , M. Sher , J.P. Silva , Phys. Rep . 516 , 1 ( 2012 ). arXiv: 1106 .0034 [hep-ph]
24. P.M. Ferreira , R. Santos , M. Sher , J.P. Silva , Phys. Rev. D 85 , 077703 ( 2012 ). arXiv: 1112 .3277 [hep-ph]
25. D. Carmi , A. Falkowski , E. Kuflik, T. Volansky, JHEP 1207 , 136 ( 2012 ). arXiv: 1202 .3144 [hep-ph]
26. H.S. Cheon , S.K. Kang , JHEP 1309 , 085 ( 2013 ). arXiv: 1207 .1083 [hep-ph]
27. W. Altmannshofer, S. Gori , G.D. Kribs , Phys. Rev. D 86 , 115009 ( 2012 ). arXiv: 1210 .2465 [hep-ph]
28. Y. Bai , V. Barger , L.L. Everett , G. Shaughnessy, Phys. Rev. D 87 , 115013 ( 2013 ). arXiv: 1210 .4922 [hep-ph]
29. C.-Y. Chen , S. Dawson, Phys. Rev. D 87 , 055016 ( 2013 ). arXiv: 1301 .0309 [hep-ph]
30. C.-W. Chiang , K. Yagyu , JHEP 1307 , 160 ( 2013 ). arXiv: 1303 .0168 [hep-ph]
31. M. Krawczyk , D. Sokolowska , B. Swiezewska , J. Phys. Conf. Ser . 447 , 012050 ( 2013 ). arXiv: 1303 .7102 [hep-ph]
32. B. Grinstein , P. Uttayarat, JHEP 1306 , 094 ( 2013 ) [Erratum-ibid . 1309 , 110 ( 2013 ) ] . arXiv: 1304 .0028 [hep-ph]
33. A. Barroso , P.M. Ferreira , R. Santos , M. Sher , J.P. Silva , arXiv: 1304 .5225 [hep-ph]
34. B. Coleppa , F. Kling , S. Su, JHEP 1401 , 161 ( 2014 ). arXiv: 1305 .0002 [hep-ph]
35. P.M. Ferreira , R. Santos , M. Sher , J.P. Silva , arXiv: 1305 .4587 [hepph]
36. O. Eberhardt , U. Nierste , M. Wiebusch , JHEP 1307 , 118 ( 2013 ). arXiv: 1305 .1649 [hep-ph]
37. S. Choi , S. Jung , P. Ko, JHEP 1310 , 225 ( 2013 ). arXiv: 1307 .3948 [hep-ph]
38. V. Barger , L.L. Everett , H.E. Logan , G. Shaughnessy, Phys. Rev. D 88 , 115003 ( 2013 ). arXiv: 1308 .0052 [hep-ph]
39. D. López-Val, T. Plehn , M. Rauch , JHEP 1310 , 134 ( 2013 ). arXiv:1308 . 1979 [hep-ph]
40. S. Chang , S.K. Kang , J.-P. Lee , K.Y. Lee , S.C. Park , J. Song, arXiv: 1310 .3374 [hep-ph]
41. K. Cheung , J.S. Lee , P.-Y. Tseng, JHEP 1401 , 085 ( 2014 ). arXiv: 1310 .3937 [hep-ph]
42. A. Celis , V. Ilisie , A. Pich , JHEP 1312 , 095 ( 2013 ). arXiv: 1310 .7941 [hep-ph]
43. L. Wang , X.F. Han , JHEP 1404 , 128 ( 2014 ). arXiv: 1312 .4759 [hepph]
44. K. Cranmer , S. Kreiss , D. López-Val , T. Plehn, arXiv: 1401 .0080 [hep-ph]
45. F.J. Botella , G.C. Branco , A. Carmona , M. Nebot , L. Pedro , M.N. Rebelo , JHEP 1407 , 078 ( 2014 ). arXiv: 1401 .6147 [hep-ph]
46. F.J. Botella , G.C. Branco , M. Nebot , M.N. Rebelo , arXiv: 1508 .05101 [hep-ph]
47. S. Kanemura , K. Tsumura , K. Yagyu , H. Yokoya, arXiv: 1406 .3294 [hep-ph]
48. A. Celis , V. Ilisie , A. Pich , arXiv: 1302 .4022 [hep-ph]
49. P.M. Ferreira , H.E. Haber , R. Santos , J.P. Silva , arXiv: 1211 .3131 [hep-ph]
50. A. Barroso , P.M. Ferreira , R. Santos , J.P. Silva , Phys. Rev. D 86 , 015022 ( 2012 ). arXiv: 1205 .4247 [hep-ph]
51. P.M. Ferreira , R. Santos , M. Sher , J.P. Silva , Phys. Rev. D 85 , 035020 ( 2012 ). arXiv: 1201 .0019 [hep-ph]
52. P.M. Ferreira , R. Santos , M. Sher , J.P. Silva , Phys. Rev. D 85 , 077703 ( 2012 ). arXiv: 1112 .3277 [hep-ph]
53. A.E. Carcamo Hernandez , R. Martinez , J.A. Rodriguez , Eur. Phys. J. C 50 , 935 ( 2007 ). arXiv:hep-ph/0606190
54. A. Crivellin , J. Heeck , P. Stoffer, arXiv: 1507 .07567 [hep-ph]
55. A. Arhrib , M. Capdequi Peyranere , W. Hollik , S. Penaranda, Phys. Lett. B 579 , 361 ( 2004 ). arXiv:hep-ph/0307391
56. D. Fontes , J.C. Romo , J.P. Silva , Phys. Rev. D 90 , 015021 ( 2014 ). arXiv: 1406 .6080 [hep-ph]
57. G. Bhattacharyya, D. Das , P.B. Pal , M.N. Rebelo , JHEP 1310 , 081 ( 2013 ). arXiv: 1308 .4297 [hep-ph]
58. J.A. Aguilar-Saavedra , Acta Phys. Polon. B 35 , 2695 ( 2004 ). arXiv:hep-ph/0409342
59. G. Eilam, J.L. Hewett , A. Soni , Phys. Rev. D 44 , 1473 ( 1991 ) [Erratum-ibid . D 59 , 039901 ( 1999 )]
60. ATLAS Collaboration, ATL- PHYS-PUB- 2013- 012 ( 2013 )
61. ATLAS Collaboration, ATLAS-CONF- 2013-081
62. CMS Collaboration, CMS-PAS-HIG- 13- 034 ( 2014 )
63. S.L. Glashow , S. Weinberg, Phys. Rev. D 15 , 1958 ( 1977 )
64. T.P. Cheng, M. Sher, Phys. Rev. D 35 , 3484 ( 1987 )
65. D. Atwood , L. Reina , A. Soni , Phys. Rev. D 55 , 3156 ( 1997 ). arXiv:hep-ph/9609279
66. C.-H. Chen , C.-Q. Geng , Phys. Rev. D 71 , 115004 ( 2005 ). arXiv:hep-ph/0504145
67. M. Gomez-Bock , R. Noriega-Papaqui , J. Phys . G 32 , 761 ( 2006 ). arXiv:hep-ph/0509353
68. M. Gomez-Bock , G. Lopez Castro , L. Lopez-Lozano , A. Rosado , Phys. Rev. D 80 , 055017 ( 2009 ). arXiv: 0905 .3351 [hep-ph]
69. J.F. Gunion , H.E. Haber , Phys. Rev. D 67 , 075019 ( 2003 ). arXiv:hep-ph/0207010
70. I.F. Ginzburg , M. Krawczyk , P. Osland, In 2nd ECFA/DESY Study 1998-2001 , pp. 1705 - 1733 . arXiv: hep-ph/0101208
71. I.F. Ginzburg , M. Krawczyk , P. Osland , Nucl. Instrum. Methods A 472 , 149 ( 2001 ). arXiv:hep-ph/0101229
72. A.G. Akeroyd , A. Arhrib , E.M. Naimi , Phys. Lett. B 490 , 119 ( 2000 ). arXiv:hep-ph/0006035
73. A. Arhrib, arXiv:hep-ph/0012353
74. S. Kanemura, T. Kubota , E. Takasugi, Phys. Lett. B 313 , 155 ( 1993 ). arXiv:hep-ph/9303263
75. S. Kanemura, T. Kubota , E. Takasugi, Eur. Phys. J. C 46 , 81 ( 2006 ). arXiv:hep-ph/0510154
76. M. Sher , Phys. Rep . 179 , 273 ( 1989 )
77. S. Kanemura, T. Kasai , Y. Okada , Phys. Lett. B 471 , 182 ( 1999 ). arXiv:hep-ph/9903289
78. P.M. Ferreira , R. Santos , A. Barroso , Phys. Lett. B 603 , 219 ( 2004 ). arXiv:hep-ph/0406231
79. M.E. Peskin , T. Takeuchi, Phys. Rev. D 46 , 381 ( 1992 )
80. M. Baak et al. (Gfitter Group Collaboration), Eur. Phys. J. C 74 , 3046 ( 2014 ). arXiv: 1407 .3792 [hep-ph]
81. J.-M. Gerard , M. Herquet , Phys. Rev. Lett . 98 , 251802 ( 2007 ). arXiv:hep-ph/0703051 [HEP-PH]
82. E. Cerver, J.M. Gerard , Phys. Lett. B 712 , 255 ( 2012 ). arXiv:1202 . 1973 [hep-ph]
83. M. Misiak et al., Phys. Rev. Lett . 114 ( 22 ), 221801 ( 2015 ). arXiv: 1503 .01789 [hep-ph]
84. A. Crivellin , A. Kokulu , C. Greub , Phys. Rev. D 87 ( 9 ), 094031 ( 2013 ). doi: 10 .1103/PhysRevD.87.094031. arXiv: 1303 .5877 [hep-ph]
85. F. Mahmoudi , O. Stal , Phys. Rev. D 81 , 035016 ( 2010 ). doi: 10 . 1103/PhysRevD.81.035016. arXiv: 0907 .1791 [hep-ph]
86. Z.J. Xiao , L. Guo, Phys. Rev. D 69 , 014002 ( 2004 ). doi:10.1103/ PhysRevD.69.014002. arXiv:hep-ph/0309103
87. M. Baak , M. Goebel , J. Haller , A. Hoecker , D. Ludwig , K. Moenig , M. Schott , J. Stelzer , Eur. Phys. J. C 72 , 2003 ( 2012 ). arXiv: 1107 .0975 [hep-ph]
88. H.E. Haber , H.E. Logan , Phys. Rev. D 62 , 015011 ( 2000 ). arXiv:hep-ph/9909335
89. A. Djouadi, Phys. Rep . 457 , 1 ( 2008 ). arXiv:hep-ph/0503172
90. A. Djouadi, Phys. Rep . 459 , 1 ( 2008 ). arXiv:hep-ph/0503173
91. CMS Collaboration, CMS-PAS-HIG- 13-005
92. D.M. Asner , T. Barklow , C. Calancha , K. Fujii , N. Graf , H.E. Haber , A. Ishikawa , S. Kanemura et al., arXiv:1310 .0763 [hep-ph]
93. A. Arhrib, Phys. Rev. D 72 , 075016 ( 2005 ). arXiv:hep-ph/0510107
94. I. Baum, G. Eilam, S. Bar-Shalom , Phys. Rev. D 77 , 113008 ( 2008 ). arXiv: 0802 .2622 [hep-ph]
95. S. Bejar , J. Guasch , J. Sola , Nucl. Phys. B 600 , 21 ( 2001 ). arXiv:hep-ph/0011091
96. B. Altunkaynak , W.S. Hou , C. Kao , M. Kohda , B. McCoy , arXiv: 1506 .00651 [hep-ph]