Rare top decay and CP violation in THDM

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We discuss the formalism of the two Higgs doublet model of type III with CP violation from CP-even CP-odd mixing in the neutral Higgs bosons. The flavor-changing interactions among neutral Higgs bosons and fermions are presented at tree level in this type of model. These assumptions allow the study of rare top decays mediated by a neutral Higgs boson; particularly we are interested in ....... For this process we estimate the upper bounds of the branching ratios ...... of the order of ...... for a neutral Higgs boson mass equal to 125 GeV and ......, 1.5, 2, 2.5. For the case of ...... the number of possible events is estimated to range from 1 to 10 events, which could be observed in future experiments at LHC with a luminosity of 300 ...... and 14 TeV for the energy of the center of mass. Also we estimate that the number of events for the process ...... in different scenarios is of the order of 2,500.

Eur. Phys. J. C (2014) 74:2788DOI 10.1140/epjc/s10052-014-2788-4

Regular Article - Theoretical Physics

Rare top decay and CP violation in THDM

R. Gaitn1,a, R. Martinez2,b, J. H. Montes de Oca1,c, S. Rodriguez Romo3,d

1 Departamento de Fsica, FES-Cuautitln, Universidad Nacional Autonma de Mxico, C.P. 54770 Cuautitlan Izcalli, Estado de Mxico, Mexico

2 Departamento de Fsica, Universidad Nacional de Colombia, Bogot D.C., Colombia

3 Departamento de Qumica, FES-Cuautitln, Universidad Nacional Autonma de Mxico, C.P. 54770 Cuautitlan Izcalli, Estado de Mxico, Mexico Received: 6 December 2013 / Accepted: 18 February 2014 / Published online: 4 March 2014 The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract We discuss the formalism of the two Higgs doublet model of type III with CP violation from CP-even CP-odd mixing in the neutral Higgs bosons. The avor-changing interactions among neutral Higgs bosons and fermions are presented at tree level in this type of model. These assumptions allow the study of rare top decays mediated by a neutral Higgs boson; particularly we are interested in t cl+l.

For this process we estimate the upper bounds of the branching ratios Br(t c+) of the order of 109 107 for

a neutral Higgs boson mass equal to 125 GeV and tan = 1,

1.5, 2, 2.5. For the case of t c+ the number of possible

events is estimated to range from 1 to 10 events, which could be observed in future experiments at LHC with a luminosity of 300 fb1 and 14 TeV for the energy of the center of mass.

Also we estimate that the number of events for the process t cl+l in different scenarios is of the order of 2,500.

1 Introduction

The latest results from LHC have conrmed the observation of one scalar particle with a mass of the electroweak scale.The ATLAS [1] and CMS [2] collaborations have reported the observation of a new particle with mass of around 125 GeV.The observation has an important signicance of more than 5 standard deviations. Even with this research it is not yet possible for us to identify this particle as the Standard Model Higgs boson. However, if this result is conrmed by future analysis, it will be one of the greatest discoveries of mankind.On the other hand, the SM is often considered as an effective theory, valid up to an energy scale of O(GeV), which eventually will be replaced by a more fundamental theory,

a e-mail: [email protected]; [email protected]

b e-mail: [email protected]

c e-mail: [email protected]

d e-mail: [email protected]

which will explain, among other things, the physics behind electroweak symmetry breaking and perhaps even the origin of avor. Many examples of candidate theories, which range from supersymmetry [3,4] to strongly interacting models [5] as well as some extra dimensional scenarios [6], include a multi-scalar Higgs sector. In particular, models with two scalar doublets have been studied extensively [7], as they include a rich structure with interesting phenomenology.

The rst versions of the two Higgs doublet model (THDM) are known as THDM-I [8,9] and THDM-II [10]. These versions involve natural avor conservation and CP conservation in the potential through the introduction of a discrete symmetry. A general version which is named THDM-III allows the presence of avor-changing scalar interactions (FCNSI) at tree level [11]. There are also some variants (known as top, lepton, neutrino), where one Higgs doublet couples predominantly to one type of fermion [7], while in other models it is even possible to identify a candidate for dark matter [12,13]. The denition of all these models depends on the Yukawa structure and symmetries of the Higgs sector, whose origin is still not known. The possible appearance of new sources of CP violation is another characteristic of these models [14].

Within THDM-I only one Higgs doublet generates all gauge and fermion masses, while the second doublet only knows about this through mixing, and thus the Higgs phenomenology will share some similarities with the SM, although the SM Higgs couplings will now be shared among the neutral scalar spectrum. The presence of a charged Higgs boson is clearly a signal of physics beyond the SM. Within THDM-II one also has natural avor conservation [15], and its phenomenology will be similar to the THDM-I, although in this case the SM couplings are shared not only because of mixing, but also because of the Yukawa structure. The distinctive characteristic of THDM-III is the presence of FCNSI, which require a certain mechanism in order to suppress them, for instance one can impose a certain texture for the Yukawa

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couplings [16], which will then predict a pattern of FCNSI Higgs couplings [11]. Within all those models (THDM-I, -II, -III) [17], the Higgs doublets couple, in principle, with all fermion families, with a strength proportional to the fermion masses, modulo other parameters.

With higher energy, as planned, the LHC will also become an amazing top factory, allowing one to test the top properties, its couplings to SM channels, and rare decays [18]. One of the interesting rare decays for the top is t cl+l, which

is a clear signal of new physics. In the literature this type of top decay is often known as rare top decay and it could be mediated at tree level by neutral gauge bosons in the context of physics beyond SM. For instance, models with additional gauge symmetries introduce a neutral gauge boson Z , which allows the rare top decay [1921]. The results obtained for branching ratios with avor-changing neutral currents are extremely suppressed due to the mass of the additional gauge boson Z , which must be of the order of TeV. However, in the framework of the THDM-III these rare top decay are possible at tree level through neutral Higgs bosons in the framework of general THDM with upper bounds of branching ratio t

cl+l less suppressed.

In this work we discuss the avor-changing neutral Higgs interactions due to Yukawa couplings and a CP violation source from the Higgs sector in the framework of THDM-III.Our analysis is devoted to the study of the t cl+l decay

at tree level with the basic goal of identifying the effects of new physics. The organization of the paper goes as follows: Sect. 2 describes the CP violation source in the Higgs sector. The avor-changing interaction between neutral Higgs bosons and fermions are introduced in Sect. 3. Section 4 contains the analysis of the branching ratio for rare top decay.Finally, in Sect. 5 we present our conclusion and discussion.

2 Neutral Higgs bosons spectrum

Let 1 and 2 denote two complex SU(2)L doublet scalar elds with hypercharge 1. The most general gauge invariant and renormalizable Higgs scalar potential in a covariant form with respect to a global U(2) transformation is given by [22]

V = Yabab +

Yab and Zabcd. It is noted that Zabcd = Zcdab and hermiticity

of the potential implies Yab = (Yba) and Zabcd = (Zbadc).

The most general U(1)E M-conserving vacuum expectation values are

a =

1 2

0 va

, (2)

where (v1, v2) = (v cos , v sin ) and v = 246 GeV.

After spontaneous symmetry breaking, an orthogonal transformation R is used to diagonalize the squared mass matrix for the neutral Higgs elds. The mass eigenstates of the neutral Higgs bosons are

hi =

3

j=1

Ri jj , (3)

where i = 1, 2, 3 and R matrix can be written

R =

c1c2 s1c2 s2

(c1s2s3 + s1c3) c1c3 s1s2s3 c2s3

c1s2c3 + s1c3 (c1s1 + s1s2c3) c2c3

(4)

and ci = cos i, si = sin i for 2 1,2

2 and

0 3 2 . The 1,2 denote the real parts of the complex

scalar eld in a weak eigenstate, 0a =

12 (va + a + ia),

whereas 3 is written in terms of the imaginary parts and is orthogonal to the Goldstone boson, such as 3 = 1 sin +

2 cos . The neutral Higgs bosons hi are dened to satisfy the masses hierarchy given by the inequalities mh1 mh2

mh3 [23,24].

3 Yukawa interactions with neutral scalarpseudoscalar mixing

Now, we will describe the interactions between fermions and neutral Higgs bosons. The most general structure of the Yukawa interactions for fermions elds can be written as follows:

LYukawa =

3 2 q0LiY 0uaij

1

2 Zabcd ab cd , (1)

where a = +a, 0a T and a, b, c, d are labels with respect

to two dimensional Higgs avor space. The index convention means that replacing an unbarred index with a barred index is equivalent to complex conjugation and barredunbarred indices denote a sum. In the usual notation for the SM, the and parameters are associated with the terms () and ()2, respectively. In general, for the THDM there are six real parameters, 211, 222, 1,...,4, and four complex parameters, 212, 5,...,7 [7], which are rewritten as the parameters

au0Rj + q0LiY 0daijad0Rj

+l0LiY 0laijae0Rj + h.c. , (5)

where a = 1, 2 and i, j = 1, 2, 3 are summed over two

Higgs doublets and fermions families, respectively. The Y fa , with f = u, d, e denoting the different fermions families,

are the Yukawa matrices. The qL and lL are the left handed fermion doublets; meanwhile uR, dR, and eR correspond to the right handed singlets under SU(2)L. The 0 superscript in the fermion elds stands for weak eigenstates. After getting a correct spontaneous symmetry breaking by using (2), the

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lagrangian for the mass is obtained in the form

LYukawa =

3

i, j=1

2

a=1

meanwhile the interactions for charged leptons and neutral Higgs bosons are

L leptonshk =

1v sin

va2 u0LiY 0uaiju0Rj + d0LiY 0daijd0Rj

+e0LiY 0laije0Rj + h.c. . (6)

The mass matrices in weak eigenstates are dened as

M0 f =

2

a=1

i, j,k

(Rk2 + i5Rk3 cos ) ei Mli jhkej

1 2 sin

i, j,k

va2Y 0 fa . (7)

The mass eigenstates are related with weak eigenstates through the unitary matrices V fL(R),

fL(R) = V fL(R) f 0L(R). (8)

As a result, the Yukawa matrices cannot be diagonalized separately, giving rise to avor-changing neutral currents. However, the linear combination is diagonalizable and denes the fermion masses as

M f =

2

a=1

(Rk1 sin Rk2 cos

i5Rk3) eiYli jhkej . (12)

The fermion spinors are denoted (u1, u2, u3) = (u, c, t) and

(e1, e2, e3) = (e, , ). The down-type quarks are analo

gous to the charged lepton sector. We note that (11) and (12) generalize expressions obtained by [2326]. The CP conserving case is obtained if only two neutral Higgs bosons are mixed with well-dened CP states, for instance for 2 = 0

and 3 = /2 is the usual limit.

4 Rare top decay through neutral Higgs bosons

We assume that the avor-changing neutral Higgs interactions are responsible for rare top decay at tree level. The mass of the lightest physical Higgs boson h1 is identied with the particle observed by ATLAS and CMS with a mass value of the order of 125 GeV, meanwhile the masses of h2,3 are considered to be in the region of higher than 600 GeV. Then contributions of physical neutral Higgs bosons h2,3 are neglected in the amplitude for the width of rare top decay and only the contributions of the lightest neutral Higgs boson are taken into account. Therefore, the width for rare top decay at tree level is given by

dtcl+l

dxdy =

mt

va2Y fa , (9)

where Y fa = V fL Y 0 fa V f R for f = u, d, e. Therefore, THDM

type III contains the models type I and II plus the FCNSI terms. We solve for Y2 in order to obtain the THDM type II as follows [25]:

Y 0 f2 =

2 v2

Gu

23

2 Gl

2

1283

ii (1+cx) x +2c (1+c h x)2+2

,

(13)

V fL M f V fR v1v2 Y 0 f1 . (10)

In order to study the rare top decay we are interested in up-quarks and charged leptons elds. By using (3), the interactions between neutral Higgs bosons and fermions can be written in the form of the THDM type II with additional contributions which arise from Yukawa couplings Y1 and contain avor change. In order to simplify the notation we will omit the subscript 1 in the Yukawa couplings. Explicitly we write the interactions for up-type quarks and neutral Higgs bosons as

L up-quarkshk =

1v sin

where

u 23

Gu23 2 =

Y

2

2 sin2

(R11 sin R12 cos )2 + R213 (14)

and

Glii

2

=

1

2 sin2

Ylii (R11 sin R12 cos )+2miv R213 2

+

R213

2 sin2

i, j,k

(Rk2 i5Rk3 cos ) ui Mui jhku j

Ylii 2miv cos 2 . (15)

In the expression for the width of the decay (13) we have used the usual notation for dimensionless parameters, c =

m2c/m2t, h = m2h1/m2t, = Hmh1/m2t, x = 2Ec/mt and y = 2El/mt. We note that m2h1 can be of the same

order as the square of transferred momentum, then our result is computed without approximation in the propagator. By integrating the expression (13) we can estimate the branching

(Rk1 sin + Rk2 cos

i5Rk3) uiY ui jhku j ; (11)

1 2 sin

i, j,k

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2788 Page 4 of 7 Eur. Phys. J. C (2014) 74:2788

Fig. 1 Type III THDM branching ratio for t c+

as a function of 12 in regions R1 (left) and R2 (right) withtan = 1 and mH = 300 GeV

0.00

0.20

0.05

0.15

0.10

0.10

0.15

0.05

0.20

0.25

0.00 0.68 0.70 0.72 0.74 0.76 0.78 0.80

0.80 0.85 0.90 0.95 1.00 1.05 1.10

1.6 10 9 3.2 10 9 4.8 10 9 6.4 10 9 8.0 10 9

ratio for t cl+l. We use the experimental mean value for

the full width of the top quark given by t 1.6 GeV and

the width of the Higgs eld given by H 1.6 GeV [27].

Suppression for FCNC can be achieved when a certain form of the Yukawa matrices, reproducing the observed fermion masses and mixing angles, is implemented in the model. This could be done by studying a certain ansatz for the Yukawa matrices [16]. The rst proposal for the Higgs boson couplings, the so called ChengSher ansatz [11], was based on the Fritzsch six-texture form of the mass matrices, namely

M0 =

. (16)

Then, by assuming that each Yukawa matrix has the same hierarchy, one nds that A m3, B m2m3 and C

m1m2. Thus, if the structure is assumed to be based on the ChengSher ansatz, then the Yukawa couplings obey the following pattern: Y fi j mim j /v.

Therefore, the resulting branching ratio only has dependence on 1 and 2. The 3 mixing angle is absent in the physical state for h1. The allowed regions for the 12 parameter space are obtained through the bounds of R , dened by

R =

(gg h1)Br(h1 )

(gg hSM)Br(hSM )

0 C 0 C 0 B

0 B A

. (17)

For a charged Higgs boson with mass of the order of 100 300 GeV, Br(h1 ) contains an important contribution

from the charged Higgs boson at one loop level, which affects the allowed regions for 12. Thus, it is possible to nd

allowed values in the 12 parameter space if the parameters and mH are xed. A process used to set tan and charged

Higgs boson mass is, for instance, the avor-changing process B s [28], which receives a contribution from

THDM through the charged Higgs boson. This contribution

is comparable to the contribution of W from SM. For small values of tan this process gives a bound to the charged

Higgs boson mass of the order of 300 GeV [29,30]. Contributions from other processes such as B , B D ,

Z bb, Bd,s + and B0B0 set bounds for the

mass of H and tan as mH < 400 GeV and tan 10.

Therefore, the allowed regions for the 1,2 parameter space are obtained by experimental and theoretical constraints in the framework of the THDM type II with CP violation for xed tan and mH . For 0.5 R 2,

mH = 300 GeV and tan = 1, the 12 regions are [24] R1 = {0.67 1 0.8 and 0 2 0.23} (18)

and

R2 = {0.8 1 1.14 and 0.25 2 0}. (19) For the same settings but with mH = 500 GeV,

R3 = {1.18 1 1.55 and 0.51 2 0}. (20) In order to reduce the 12 parameter space we consider these regions as an approximation. In addition, we will assume that + and occur in the nal state. Figure 1 shows the branching ratio of rare top decay for regions R1 and R2; meanwhile Fig. 2 is obtained for R3. For 1 R 2,

mH = 350 GeV, and tan = 1.5 the allowed parameter

regions in the 12 plane in the framework of THDM with a potential but softly broken Z2 discrete symmetry are [31]

R4 = {1.57 1 1.3 and 0.46 2 0} (21) and

R5 = {0.93 1 1.57 and 0.61 2 0}. (22) For tan = 2 the regions are

R6 = {1.57 1 1.28 and 0.38 2 0}. (23)

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Eur. Phys. J. C (2014) 74:2788 Page 5 of 7 2788

0.0

0.1

1.05 10 7

0.2

8.40 10 8

0.3

6.30 10 8

4.20 10 8

0.4

2.10 10 8

0.5

1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55

Fig. 2 Type III THDM branching ratio for t c+ as a function

of 12 in region R3 with tan = 1 and mH = 500 GeV

and

R7 = {1.08 1 1.57 and 0.46 2 0}. (24)

Finally, for tan = 2.5 the region is

R8 = {1.39 1 1.3 and 0.13 2 0} (25)

and

R9 = {1.16 1 1.5 and 0.43 2 0.1}. (26)

Figures 3, 4, and 5 show the branching ratio for previous regions. We note that the branching ratio of rare top decay for tan = 1 and mH = 500 GeV is bounded as

Br(t c+) 5 107 for any 1,2. For a + and

pair in the nal state we nd that Br(t c+)

1.9 109 with the same tan = 1. If the mixing angle

is xed with values greater than tan = 1, the branch

ing ratio does not vary drastically over the whole 12 region; for instance if tan = 45, then Br(t c+)

2.8107. Table 1 contains the upper bounds for the regions

considered.

0.0

Fig. 3 Type III THDM branching ratio for t c+

as a function of 12 in

regions R4 (left) and R5 (right) with tan = 1.5 and

mH = 350 GeV

0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.6

1.55 1.50 1.45 1.40 1.35 1.30

1.0 1.1 1.2 1.3 1.4 1.5

1.6 10 9 3.2 10 9 4.8 10 9 6.4 10 9 8.0 10 9

Fig. 4 Type III THDM branching ratio for t c+

as a function of 12 in regions R6 (left) and R7 (right) withtan = 2 and mH = 350 GeV

0.00

0.0

0.05

0.1

0.10

0.15

0.2

0.20

0.25

0.3

0.30

0.35

0.4

1.55 1.50 1.45 1.40 1.35 1.30

1.1 1.2 1.3 1.4 1.5

1.6 10 9 3.2 10 9 4.8 10 9 6.4 10 9 8.0 10 9

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2788 Page 6 of 7 Eur. Phys. J. C (2014) 74:2788

Fig. 5 Type III THDM branching ratio for t c+

as a function of 12 in regions R8 (left) and R9 (right) with tan = 2.5 and

mH = 350 GeV

0.00

0.10

0.02

0.15

0.04

0.20

0.06

0.25

0.08

0.30

0.10

0.35

0.12

0.40

1.38 1.36 1.34 1.32 1.30

1.2 1.3 1.4 1.5

1.6 10 9 3.2 10 9 4.8 10 9 6.4 10 9 8.0 10 9

Table 1 Maximum numerical value of Br(t cl+l) for the consid

ered regions. The last column contains a naive estimation for the events that could be observed with a luminosity of the order of 300 fb1 and14 GeV for the center of mass energy

Regions Upper bound Estimated events

R1 2.52 109 0 R2 1.24 108 0

R3 1.93 107 10 R4 3.22 108 2

R5 8.46 108 4 R6 1.61 108 1

R7 2.84 108 1 R8 8.55 109 0

R9 1.66 108 1

5 Discussion and conclusion

From 2015 to 2017 the experiment is expected to reach 100 fb1 of data with an energy of the center of mass of14 TeV. In the year 2021 one expects to reach a luminosity of the order of 300 fb1 of data. Experiments with this luminosity could nd evidence of new physics beyond SM. Then

Run 3 in LHC could observe events for the neutral avor-changing process such that t ch cl+l, which can be

explained in a naive form as

Br(p p bWcl+l)

(p p t t)Br(t bW)Br(t cl+l). (27)

Then we estimate the number of events using the upper bound for a branching ratio with (p p t t) 176 pb

[32]. Table 1 contains this estimation for the considered regions.

Finally, we compare our result with reported results in other frameworks, such as effective theories and THDM type

I or II. Based on (11) we can write the branching ratio for t ch1 as

Br(t ch1)

=

mt

Gu

2

1 c h c

4t

23 (1, c, h)

(28)

where is the usual function. We nd that Br(t ch1)

5 103 with mh1 = 125 GeV and tan = 1. Despite the

absence of avor-changing neutral Higgs interactions in SM, t chSM decay can occur at one loop level. The reported

result for the branching ratio is of the order of 10141013

for mZ mSM 2mZ [33]. More recently, in the frame

work of the general THDM with CP-even (H0) and CP-odd (A0) neutral Higgs bosons the branching ratios are estimated as Br(t cH0) = 2.2 103 and Br(t cA0) =

1.2 104 for mH

0 = 150 GeV [34].

By using the effective operator formalism the avor changing neutral Higgs interactions are introduced. An upper bound is estimated as Br(t cH0) = 2.7 % for a neutral Higgs mass

of 125 GeV [35]. Top decay with effective theories is also studied, for the case of t ch Br(t cH0) = 5 103

for mh = 125 GeV is obtained [36]. In reference [37]

has been estimated an upper bound of Br(t cH) =

0.09 2.8 103 for 114 mH 170 GeV through

the one loop contributions of effective avor-changing neutral couplings tcH on the electroweak precision observables in SM. For the Yukawa complex couplings and CP effects in THDM type III Br(t cH0) 103 is predicted by [38].

From reference [31], Fig. 3, can be estimated the branching ratio of h1 into s, which is of the order of BR(h1

) 0.05 for any value of 1 and 2. Using this BR and

taking into account BR(t ch1) 103 for different sce

narios of the models, we obtain

BR(t ch1 c) 5 105, (29)

0 = 125 GeV and m A

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Eur. Phys. J. C (2014) 74:2788 Page 7 of 7 2788

which is two orders of magnitude larger than the value obtained by us for different regions of parameters; see Table 1. The number of events, in the best scenario, at LHC with 300 fb1 of luminosity and 14 TeV for the energy of the center of mass is of the order of 2,500.

Acknowledgments This work is supported in part by PAPIIT project IN117611-3, Sistema Nacional de Investigadores (SNI) in Mxico. J.H.Montes de Oca is thankful for support from the postdoctoral DGAPAUNAM grant. R. M. thanks Colciencias for nancial support.

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Funded by SCOAP3 / License Version CC BY 4.0.

References

1. ATLAS Collaboration, Phys. Lett. B 716, 1 (2012)2. CMS Collaboration, Phys. Lett. B 716, 30 (2012)3. M.S. Carena, H.E. Haber, Prog. Part. Nucl. Phys. 50, 63 (2003). http://arxiv.org/abs/arXiv:hep-ph/0208209

Web End =arXiv:hep-ph/0208209

4. See, for instance, recent reviews in Kane G L 1998 Perspectives on Supersymmetry (World Scientic Publishing Co.)

5. N. Arkani-Hamed, A. Cohen, H. Georgi, Phys. Lett. B 513, 232 (2001). http://arxiv.org/abs/arXiv:hep-ph/0105239

Web End =arXiv:hep-ph/0105239

6. W.F. Chang, J.N. Ng, A.P. Spray, Phys. Rev. D 82, 115022 (2010). http://arxiv.org/abs/arXiv:1004.2953

Web End =arXiv:1004.2953 [hep-ph]

7. G.C. Branco, P.M. Ferreira, L. Lavoura, M.N. Rebelo, M. Sher, J.P. Silva, Phys. Rep. 516, 1 (2012). http://arxiv.org/abs/arXiv:1106.0034

Web End =arXiv:1106.0034 [hep-ph]

8. H.E. Haber, G.L. Kane, T. Sterling, Nucl. Phys. B 161, 493 (1979)9. L.J. Hall, M.B. Wise, Nucl. Phys. B 187, 397 (1981)10. J.F. Donoghue, L.F. Li, Phys. Rev. D 19, 945 (1979)11. T.P. Cheng, M. Sher, Phys. Rev. D 35, 3484 (1987)12. E.M. Dolle, S. Su, Phys. Rev. D 80, 055012 (2009). http://arxiv.org/abs/arXiv:0906.1609

Web End =arXiv:0906. http://arxiv.org/abs/arXiv:0906.1609

Web End =1609 [hep-ph]

13. M. Cirelli, N. Fornengo, A. Strumia, Nucl. Phys. B 753, 178 (2006). http://arxiv.org/abs/arXiv:hep-ph/0512090

Web End =arXiv:hep-ph/0512090

14. I.F. Ginzburg, M. Krawczyk, Phys. Rev. D 72, 115013 (2005). http://arxiv.org/abs/arXiv:hep-ph/0408011

Web End =arXiv:hep-ph/0408011

15. S.L. Glashow, S. Weinberg, Phys. Rev. D 15, 1958 (1977)16. H. Fritzsch, Phys. Lett. B 70, 436 (1977)17. A.E. Carcamo Hernandez, R. Martinez, J.A. Rodriguez, Eur. Phys.J. C 50, 935 (2007). http://arxiv.org/abs/arXiv:hep-ph/0606190

Web End =arXiv:hep-ph/0606190 18. J.L. Diaz-Cruz, R. Martinez, M.A. Perez, A. Rosado, Phys. Rev. D 41, 891 (1990)

19. A. Arhrib, K. Cheung, C.W. Chiang, T.C. Yuan, Phys. Rev. D 73, 075015 (2006)

20. J.L. Diaz-Cruz, A. Diaz-Furlong, R. Gaitan-Lozano, J.H. Montes de Oca Y., Eur. Phys. J. C 72, 2119 (2012). http://arxiv.org/abs/arXiv:1203.6889

Web End =arXiv:1203.6889 [hepph]

21. R. Gaitan-Lozano, R. Martinez, J.H. Montes de Oca Y., Eur. Phys.J. Plus 127, 158 (2012)22. H.E. Haber, D. ONeil, Phys. Rev. D 74, 015018 (2006). http://arxiv.org/abs/arXiv:hep-ph/0602242

Web End =arXiv: http://arxiv.org/abs/arXiv:hep-ph/0602242

Web End =hep-ph/0602242

23. A. Arhrib, E. Christova, H. Eberl, E. Ginina, JHEP 1104, 089 (2011). http://arxiv.org/abs/arXiv:1011.6560

Web End =arXiv:1011.6560 [hep-ph]

24. L. Basso, A. Lipniacka, F. Mahmoudi, S. Moretti, P. Osland, G.M. Pruna, M. Purmohammadi, JHEP 1211, 011 (2012). http://arxiv.org/abs/arXiv:1205.6569

Web End =arXiv:1205. http://arxiv.org/abs/arXiv:1205.6569

Web End =6569 [hep-ph]

25. R.A. Diaz, R. Martinez, J.A. Rodriguez, Phys. Rev. D 63, 095007 (2001)

26. R. Martinez, J.A. Rodriguez, M. Rozo, Phys. Rev. D 68, 035001 (2003)

27. Aaltonen, Timo Antero and others, CDF Collaboration, FERMILAB-PUB-13-324-E (2013). http://arxiv.org/abs/arXiv:1308.4050

Web End =arXiv:1308.4050 [hep-ex]

28. T. Hermann, M. Misiak, M. Steinhauser, JHEP 1211, 36 (2012). http://arxiv.org/abs/arXiv:1208.2788

Web End =arXiv:1208.2788 [hep-ph]

29. M. Ciuchini, G. Degrassi, P. Gambino, G.F. Giudice, Nucl. Phys. B 527, 21 (1998). http://arxiv.org/abs/arXiv:hep-ph/9710335

Web End =arXiv:hep-ph/9710335

30. F. Mahmoudi, O. Stal, Phys. Rev. D 81, 035016 (2010)31. A. Arhrib, R. Benbrik, C.-H. Chen, http://arxiv.org/abs/arXiv:1205.5536

Web End =arXiv:1205.5536 [hep-ph]32. J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012)

33. B. Mele, S. Petrarca, A. Soddu, Phys. Lett. B 435, 401 (1998). http://arxiv.org/abs/arXiv:hep-ph/9805498

Web End =arXiv:hep-ph/9805498

34. C. Kao, H.-Y. Cheng, W.-S. Hou, J. Sayre, Phys. Lett. B 716, 2225 (2012)

35. Nathaniel Craig, Jared A. Evans, Richard Gray, Michael Park, Sunil Somalwar, Phys. Rev. D 86, 075002 (2012)

36. J.I. Aranda, A. Cordero-Cid, F. Ramirez-Zavaleta, J.J. Toscano, E.S. Tututi, Phys. Rev. D 81, 077701 (2010)

37. F. Larios, R. Martinez, M.A. Perez, Phys. Rev. D 72, 057504 (2005)38. E.O. Iltan, Phys. Rev. D 65, 075017 (2002)

123