SM Higgs boson and $t\rightarrow cZ$ decays in the 2HDM type III with CP violation

We compute the contributions to rare top decays $t\rightarrow cZ$ and $t\rightarrow ch_1$ from the scalar sector in the 2HDM type III with CP violation, where $h_1$ is the Standard Model Higgs boson. The branching ratio for $BR(t\rightarrow c Z)$ and $BR(t\rightarrow c h_1)$ are obtained as a function of the model parameters. In particular, the $BR(t\rightarrow c Z)$ can increase its value up to $10^{-3}$ for $\tan\beta= 1$ and masses for the additional Higgs bosons of $m_{h_2,h_3,H^\pm}\sim 0.5$ TeV. Meanwhile $BR(t\rightarrow c h_1)$ can reach values of the order of $\sim10^{-2}$. We constrain the model parameters (mixing angles of the neutral scalar fields in the CP violation context and $\tan\beta$) using the reported values of the signal strengths $R_{XX}$ and $b \rightarrow s \gamma$ process.


I. INTRODUCTION
One of the goals of the Large Hadron Collider (LHC) was observing the Higgs Boson and to looking for physics beyond the Standard Model (SM). In 2012 the ATLAS and CMS collaborations took a big step with the observation of a SM-like Higgs boson with a mass of 125 GeV [1,2]; nevertheless it was the first step in the long search of the Higgs boson from its theoretical assumption by the Standard Model (SM). This theory originally incorporated only one Electroweak (EW) doublet scalar field where the Higgs boson particle arises when the symmetry SU (2) ⊗ U (1) is broken. Currently, there are no experimental and theoretical restrictions to suppose only one EW doublet scalar field, which suggests to consider models with more scalar fields in order to study physics Beyond Standard Model (BSM). One of the simplest models reported in the literature is the Two Higgs Doublet Model (2HDM) [3][4][5][6][7][8][9][10][11] which explains the hierarchy between the quark masses in the different families as a consequence of the hierarchy of Vacuum Expectation Values (VEVs). The classification of the 2HDM is reviewed in detail in the report [12]. In the literature, usually the discrete symmetry Z 2 is used to control the couplings and the models are classified according to their assignment of the values for the Z 2 charges in doublets and fermions. For instance, for the model named as 2HDM type I only one of the doublets give masses to the fermions [13], while in the 2HDM type II both doublets participate in the masses of the fermions where each doublet is assigned to give mass to each fermion sector, respectively. One for the up and the other for the down sector [14]. Without this Z 2 symmetry both doublet scalar fields give masses to the up and down sectors (Type-III).
There are many motivations to extend the scalar sector of the SM. To understand the relic density of Dark Matter (DM) of the Universe, one possibility, is the introduction of one scalar singlet which must have a vacuum expectation value (VEV) equal to zero, this in order to avoid faster decay in SM particles and have the abundance according to the stelar dynamic and lensing effects [15][16][17].
Another possibility for the introduction of DM candidates is to add an EW doublet scalar field with VEV equal to zero; this model is known as the Inert Doublet Model (IDM). Another interesting motivation for the extension of the scalar sector is the inclusion of CP-violation in order to incorporate leptogenesis and the matter antimatter excess in the Universe [18][19][20].
The observation of neutrino oscillation requires the neutrino masses to be incorporated in the SM [33]. In order to introduce masses to neutrinos an interesting proposal is to include right-handed or sterile neutrinos. New scalar fields are also necessary to generate masses through see-saw mechanism, which can be an EW doublet in order to generate Dirac masses or an EW singlet to give Majorana masses. As a consequence, we have to introduce a unitarity matrix which relates the mass and flavor eigenstates. The diagonalization of the mass matrices of charged leptons and neutrinos generates the so called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [21][22][23] . The PMNS matrix works similarly as the CKM matrix, plus two additional CP phases for Majorana neutrinos. CP-violation has been measured in the quark sector for the system K 0 −K 0 and B 0 s(d) −B 0 s(d) [34][35][36][37][38][39][40][41][42][43][44][45][46]. On the other hand, long baseline neutrino experiments, like NOvA, T2K and Minos have observed CP-violation in the neutrino sector [47], [48].
The study of models with new sources of CP-violation is very well motivated. In particular, the 2HDM type III, without including the Z 2 discrete symmetry, allows CP-violation simultaneously in the scalar sector [49] and in the Yukawa Lagrangian. Under this assumption, the neutral CP-even and CP-odd Higgs bosons are combined in a scalarpseudoscalar structure and their mass eigenstates do not have defined CP-parity. The pseudoscalar coupling depends of the CP-violation of the model and it must be strongly suppressed.
Indirect evidence of a new physics signal in rare processes mediated by Flavor-Changing Neutral Currents (FCNC) could give a crucial direction to BSM physics [50,51]. The main motivation for considering FCNC is that these processes are extremely suppressed in the SM while their extensions are improved by FCNC approaching the experimental limits. The rare decays with FCNC, which have the greatest increase, are associated with the top quark such as t → qV for q = u, c and V = γ, g, Z [53][54][55][56][57][58]. The current experimental limit is yet 10 orders of magnitude apart from SM; SM value is of the order of 10 −17 ∼ 10 −12 [53][54][55][56][57][58] meanwhile current limit is BR < 5.6 × 10 −3 [52]. In 2HDM with CP conserving the rare top decays present an increase in the branching ratio the order of 10 −7 − 10 −9 [5-7, 9, 59-62]. This means that a signal of rare top decays near the LHC experimental limits will be a clear evidence of new physics [63][64][65][66][67][68][69][70]. We analyze the FC in the context of 2HDM type III with CP violation which will introduce parameters, such as of α 1 , α 2 , α 3 , and they are absent in the usual models. In order to find allowed regions for {α 1 , α 2 , tan β} and Y tc , we consider the contributions of the pseudoscalar couplings between the fermions and h 1 to R XX by using the LHC measurements. Then we will find the values for the BR(t → cZ) and BR(t → ch 1 ) for this region of parameters.
In section II we present the model. In section III we find the allowed region for the parameter space based on experimental values and χ 2 analysis. The section IV is devoted to present our results for the rare top decay. Finally in section V we discuss the obtained result and the perspectives for the model and the conclusions in section VI.

II. MIXING AND FLAVOR-CHANGING NEUTRAL SCALARS IN 2HDM
Let us denote the two complex SU (2) L doublet scalar fields with hypercharge 1 as Φ 1 and Φ 2 . If the Φ 1,2 are included in the most general form in the scalar potential and Yukawa interactions, FC through neutral scalar fields can arise with the fermion interactions and a general mixing for the three physical states of the neutral scalar. Usually, the discrete symmetry Z 2 is introduced in order to suppress these features in the model. This suppression is motivated by the experimental limits for FC processes, however it could give signs of new physics and CP violation effects.
In the 2HDM, one linear combination of the Yukawa couplings is proportional to the mass fermion and this can be diagonalized by a bi-unitarity matrix. The other linear combination cannot be simultaneously diagonalized and this new coupling produces flavor change (FC). This kind of model is the so called 2HDM type III. In the scalar potential appear new bilinear and quartic interactions like Φ † 1 Φ 2 and Φ † 1 Φ 1 Φ † 1 Φ 2 which can induce CP violation explicitly. We study the 2HDM type III with explicit CP violation and FCNSI which is described below.
In the 2HDM type III the mixing of the neutral Higgs bosons, usually denoted as h 0 , H 0 and A 0 , can be parametrized by the three angles {α 1 , α 2 , α 3 } [71], but in all decay channels of the lightest neutral Higgs boson, h 0 1 , only α 1 and α 2 are required to describe the width decays. On the other hand, if α 2 = 0, the mixing of the neutral Higgs bosons recover the CP-parity and α 1 is the usual angle that mixes h 0 − H 0 in the CP-conserving 2HDM. In this scenario α 2 is an important parameter to analyze CP-violation because when it is zero all analytical expressions must be reduced to 2HDM type I or II. Using the signal strengths, R XX , reported by LHC [52] and the predicted by 2HDM-III, we will find the allowed region for α 1 and α 2 as a function of tan β, defined as a ratio of VEVs.

A. Yukawa interactions with FC
The most general structure for the Yukawa couplings among fermions and scalars is where Y u,d,l a are the 3 × 3 Yukawa matrices. q L and l L denote the left handed fermion doublets under SU (2) L , while u R , d R , l R correspond to the right handed singlets. The zero superscript in fermion fields and Yukawa matrices stands for the interaction basis and non diagonal matrices in the most general case, respectively. The doublets are written as The relation between the interaction and physical states is found through the Spontaneous Symmetry Breaking (SSB), where the most general U (1) EM -conserving VEVs can be taken as v 1 and v 2 are real and satisfy v 2 [72]. After getting a correct SSB, the Eq.(3) and Eq.(4) are used in Eq. (1) to obtain the mass matrices which are written as where M u,d,l = diagonal {m u,d,e , m c,s,µ , m t,b,τ } and The V f L,R matrices are used to diagonalize the fermion mass matrices and to relate the physical and interaction states for fermions. Note that in 2HDM-III the diagonalization of mass matrices does not imply the diagonalization of the Yukawa matrices, as it occurs in the 2HDM type I or II. An important consequence of non diagonal Yukawa matrices in physical states is the presence of FCNSI between neutral Higgs bosons and fermions.
We will only focus in the quarks, however the charged leptons can be included in an analogous form. The equations (5) not only establish the mass matrices but also provide relations to eliminate one of the Yukawa matrices in the physical states. In order to obtain the interactions in terms of only one Yukawa matrix, the equations (5) can be written in two possible forms where the quark sector label is q = u, d. The VEV's ratio defines the β parameter, tan β = v2 v1 , then v 1 = v sin β and v 2 = v cos β. By using Eq. (6) or Eq. (7) in the Yukawa Lagrangian L Y , Eq. (1), the 2HDM type III can be written in four different versions.
From Eq. (6) and Eq. (9) we can find Y U 2 and Y D 2 as a function of the other Yukawas and masses obtaining: Taking into account similar rotations for the lepton sector there are only two Feynman rules which correspond to 2HDM type I and II plus FC. The general structure for the interactions between the quarks and neutral scalars in any of them isq where P R,L = 1 2 (1 ± γ 5 ). The f (β) or g(β) can be written as sine, cosine, tangent or cotangent of β, which will depend on the model version. The A L,R contain all the information related with the mixing of the neutral scalars h k , which will be discussed below. The mass matrix must be diagonal, meanwhile the Yukawa matrix could be, in general, non diagonal. These elements of the Yukawa matrix are responsible for the FC mediated by neutral scalars.

B. Neutral scalar mixing from the scalar potential
Given Φ 1 and Φ 2 two complex SU (2) L doublet scalar fields, the most general gauge invariant and renormalizable Higgs scalar potential is [73,74] where m 2 11 , m 2 22 and λ 1 , λ 2 , λ 3 , λ 4 , λ 5 are real parameters and m 2 12 , λ 6 , λ 7 can be complex parameters. The neutral components of the doublets in the interaction basis are φ 0 a = 1 √ 2 (v a + η a + iχ a ), where a = 1, 2. As a result of the explicit CP symmetry breaking introduced, Eq. (13), a mixing matrix R relates the mass eigenstates h i with the η i as follows R matrix is parametrized in the usual form as [71]: Here, η 3 is the state orthogonal to the would-be Goldstone boson assigned to the Z gauge boson, explicitly it is written as η 3 = −χ 1 sin β + χ 2 cos β, where c i = cos α i , s i = sin α i for − π 2 ≤ α 1,2 ≤ π 2 and 0 ≤ α 3 ≤ π 2 . h i satisfy the mass relation m h1 ≤ m h2 ≤ m h3 [49,[75][76][77]. In the CP conserving case η 1 and η 2 are CP-even and mixed in a 2 × 2 matrix while η 3 is CP-odd decoupled from η 1 and η 2 . However, due to CP-symmetry breaking, in general, the neutral Higgs bosons h 1,2,3 do not have well defined CP eigenstates.
The focus is on the up-type quark Yukawa interactions that contain the Feynman rules for the rare top decay. Replacing Eq. (14) and Eq.(5) in the Yukawa Lagrangian of Eq.(1), the interactions between neutral Higgs bosons and fermions can be written as interactions of the CP conserving 2HDM (type I or II) plus additional contributions, which arise from any of the Y 1,2 Yukawa matrices. The relation among the mass matrix M f and the Yukawa matrices Y f 1,2 , for f = u, d, l, is used to write the Yukawa Lagrangian, Eq.(1), as a function of only one Yukawa matrix, Y f 1 or Y f 2 . We choose to write the interactions as a function of the Yukawa matrix Y 2 , as follows, We will replace Eq. (16) in Eq.
(1) for f = u, d. From now on, in order to simplify the notation, the subscript 2 in the Yukawa couplings will be omitted. Therefore, the interactions between quarks and neutral scalar bosons are explicitly written as where we define The fermion spinors are denoted as (u 1 , u 2 , u 3 ) = (u, c, t), where the indexes i, j = 1, 2, 3 denote the family generations in Eq. (17), while k = 1, 2, 3 is used for the neutral Higgs bosons. Note that a CP conserving case is obtained only if two neutral Higgs bosons are mixed with well-defined CP states, for instance α 2 = α 3 = 0 is the usual limit.

III. CONSTRAINTS FOR FC NEUTRAL SCALARS
Current observations in LHC impose restrictions on the neutral scalar h 1 , which we will chose to be the SM Higgs boson. The strongest constraints for the α 1,2 mixing angles come from its decay channels reported in the signal strength R XX for fermions and gauge bosons in the final state [52]. The first part of this section (A) is devoted to find bounds for tan β and charged Higgs mass using the measured b → sγ branching ratio. In the second part of this section, subsection B, we perform a χ 2 analysis on R XX with statistical errors only. We employ simultaneously R XX and b → sγ, and we to obtain an allowed region for the model parameters.

A. B physics constraints
The FC decay of the bottom quark b → sγ imposes the strongest constraint on tan β. In the 2HDM this decay has a one loop contribution from charged and neutral Higgs bosons. We will use the reported value of b → sγ to constrain tan β and Yukawa matrix element Y tc . In order to find allowed values for the parameters we first review the possible constraints that b → sγ decay can impose on the Y tc coupling, assuming the charged Higgs in the range 500 ∼ 900 GeV. New physics contributions can be parametrized in Wilson coefficients (WC). Following references [78][79][80][81][82], the branching ratio of the b → sγ decay is a function of the WC. The main contributions due to Wilson coefficients, beyond the charged current contribution, are given by the charged Higgs and FC Yukawa couplings, δC 7,

The charged-Higgs contribution is
while the FC contribution is (x) can be found in Ref. [78][79][80][81][82]. Using the hierarchy of the CKM matrix we have the following approximations ( In figure 1, the Cheng-Sher Ansatz for three different values of the Yukawa couplings Y bb is considered to explore the allowed region in the tan β-Y tc plane. For Y bb = 0 the allowed values for Y tc is around the zero value which reproduced the results for 2HDM with CP-conserving. The lightest neutral scalar h 1 , Eq. (14), is assumed to be the SM scalar H 0 observed by ATLAS and CMS collaborations [1,2]. The measured signal strength R X X for X = b, W, Z, τ, γ [52], see the table I, can be used to constrain the parameter space in 2HDM. The signal strength for a given final state XX is where H 0 is the observed Higgs in pp collisions at LHC, channel pp → gg → h → γγ. Similarly, we can define a signal strength for the new neutral boson h 1 in the 2HDM as The 2HDM is constructed in such a way that the quark-gluon interactions remain as in the SM. Therefore, the R 2HDM X X can be written only as the ratio of branching ratios of the 2HDM with CPV and SM multiplied by the ratio between the decay width h 1 → gg and H 0 → gg, We will use the limits on R XX reported by ATLAS and CMS to constrain the parameters in the model, by using Eq. (24).
The h 1 with a mass of the order of 125 GeV decays at tree level in the channel W + W − , ZZ, ff ; for f = e, µ, τ, b, c, s, d, meanwhile the channels at one loop are γγ, γZ, gg. A model with CPV as the 2HDM type III also introduces a pseudoscalar coupling between neutral Higgs and fermions which also appears inside the loop in h 1 → γγ, gg, γZ through the top quark vertex. In general, since all fermions can give loop contributions since these contributions are proportional to the fermion masses, the top quark gives the greatest contribution.
The Lagrangian used to calculate the Higgs decays at one loop level to γ γ, γ Z and gg is written as follows where a f,W,H andã f are the deviations from the SM which are given by a W = a Z = cos βR 11 + sin βR 12 , a H = 1.
It is important to note that if α 2 = 0 thenã f = 0. The three linear coupling in the charged and neutral Higgs bosons, H + H − h 1 , can be approximated to m 2 H ± in order to avoid the λ i parameters from the Higgs potential. In this case, we assume the scalar coefficient as a H = 1.
For the h 1 → W W * , ZZ * decays we will use the expressions reported in the literature [110], however, these expressions must be multiplied by additional factors denoted by a 2 W or a 2 Z and they are given in Eq. (28) respectively, which arise from the 2HDM-III. Note that if the matrix elements R 11 , R 12 and R 13 are independent of the mixing parameter α 3 , see Eq. (15), then all decays for h 1 do not depend on it. The CPV effects are only a function of α 1 and α 2 .
In order to obtain the branching ratios for h 1 , we calculate the partial widths for bb, W W * , ZZ * , cc, ττ , µμ at tree level, meanwhile gg, γγ, γZ at one loop in the 2HDM type III. We note that the h 1 does not have well-defined CP-parity and it couples to fermions with scalar and pseudoscalar interactions which contribute to partial width due to the Lagrangian Eq. (27). Similarly, the decays at one loop level with top quark as internal line in the loop will also have two contributions arising from scalar and pseudoscalar couplings. If the mixing angle α 2 = 0, the pseudoscalar couplings vanish and the partial decay widths are reduced to the 2HDM with CP conserving case. The radiative corrections due to QCD, QED and EW are considered in the numerical analysis from references [111,112]. The masses of the fermions are also running to the scale of the SM boson mass ∼ 125 GeV.
Following the measured at LHC of physical observables R bb , R W W , R ZZ , R τ τ and R γγ channels, we do an statistical analysis using a χ 2 function on the α 1 , α 2 and tan β parameters for m H ± ≈ 500 GeV. To implement the χ 2 analysis we take into account the reported observables by ATLAS and CMS for R XX shown in table I. Figures 2, 3 and 4 show the allowed values at 90% C. L. for the mixing parameters assuming fixed values of the Yukawa couplings.
The model with tan β = 0 corresponds to one VEV equal to zero and its doublet is inert. However in this case we have CPV couplings with fermion. For α 2 = 0 the allowed regions for α 1 is |α 1 | 1.2. We can see from Figure 2 that the allowed region is consistent with α 1 = α 2 = 0 which corresponds to SM.
There is another interesting scenario in this model when Y f f = 0. The Yukawa couplings Y f f are restricted to be smaller than one in order to have a perturbative theory. These Yukawa couplings can be parametrized by the assumption of certain structure or Ansatz which is motivated when some elements of the Yukawa matrix in the interaction basis are fixed to zero [113][114][115][116][117][118][119][120][121][122][123][124][125][126][127][128]. We consider for Yukawa couplings as limit values the Cheng-Sher Ansatz, 11]. Figure 2 shows the allowed regions for α 1 and α 2 with Yukawa couplings fixed in the extreme values, The last case, Y f f = 0, provides the greatest allowed region for mixing parameters.
The elements of the Yukawa matrices in Eq. (17) can be complex parameters in the most general case. However, the imaginary part of these couplings is strongly restricted by the Electric Dipole Moment (EDM) of the neutron [129]. In particular, restrictions over imaginary part of the Yukawa couplings are obtained in the 2HDM with CP violation [130]. Nevertheless, in this work the Yukawa couplings are assumed to be real parameters and Y ij = Y ji .
In Figure 3 and 4 we show regions in the plane α 1 -tan β and α 2 -tan β, respectively, for fixed values of Yukawa couplings, meanwhile Figure 4 is for α 2 and tan β. Note that α 2 = 0 is a particular case of 2HDM without CPV. In this case Figure 3 shows the values tan β > 0 and |α 1 | 1.5 are preferred. On the other side, when α 2 = 0 the allowed region is reduced, for instance, in the case of |α 2 | 0.2 the allowed region is bounded by tan β < 2 as shown in Figure 4.

IV. RARE TOP DECAYS
The observation of rare top decays with FCNC would be as a clear signal of physics beyond SM which can be understood in extended model. We will analyze the rare top decays t → cZ and t → cH 0 in this section, while the t → cγ and t → cg have been previously studied [63]. In the 2HDM type III, the neutral scalar field has a scalar and pseudoscalar coupling with the top quark which contributes inside the loop associated with the Higgs decay into two photons.
The contributions from neutral scalars with FC to the amplitude for t → c Z decay are shown in figure 5. In general, the amplitude associated to the Feynman diagrams in figure 5 is written as  Here the q µ contributions are not considered due to the gauge condition Z · q = 0. The form factors associated with γ µ and p µ can be related through the Gordon identity, In the amplitude M(t → c Z), the Gordon identity can be approximated as 2pµ mt P L,R ≈ γ µ P R,L and the amplitude of t → cZ at one loop can be written as: where D = a, b, c, d. The explicit values for all dimensionless form factorsṼ D L,R andF D L,R , after dimensional regularization, for Feynman diagrams are written in the appendix. All contributions are finite because there is not tcZ vertex at tree level and the Lagrangian can not be renormalized. The amplitude in Eq. (39) is used to obtain whereM Z = M Z mt . The dimensionless termsÃ andB arẽ In order to obtain the branching ratio BR(t → cZ) the SM width for the top quark can be approximated to tbW width as [52] Γ In the 2HDM-III with CPV, we include the FCNSI contributions, Eq. (40) in the total width for top quark, such that, Γ total = Γ top + Γ rare decays , where Γ rare decays = Γ(t → cZ) + Γ(t → ch 1 ). The dominant contribution is the Γ top ; however, the Γ rare decays contribution, which contains Γ(t → ch 1 ) at tree level, can reach up to ∼ 1% for specific values of the model parameters, as shows in the next subsection. When only the SM contribution is considered, the branching ratio for t → cZ can be approximated as The |Ã | 2 and |B | 2 are functions of the neutral scalar masses m h k , k = 1, 2, 3, and of the mixing parameters α 1,2,3 , β. h 1 is the SM Higgs mass, m h1 = 125 GeV [52], and we fix the masses of the neutral scalars h 2,3 of the order of 600 GeV .

B. t → cH
In this subsection, we will analyze the FC top decay t → ch 1 in this model which can occurs at tree level. The coupling for this decay is given in Eq. (17) and again the non vanishing Y tc is responsible for the flavor change through the neutral scalar mediation. The partial decay width is where is obtained from Eq. (13). Figures 9, 10 and 11 show the behavior of this BR(t → ch 1 ) as function of the mixing parameters.   Figure (a) for

V. RESULTS AND DISCUSSION
We consider a model with explicit CP violation in the scalar sector, known as 2HDM-III This model also contains neutral scalar fields that change flavor and have scalar-pseudoscalar interactions with the fermions, as we show in the equation (17). This type of interactions are confronted with the current experimental results, for Higgs decays, through a χ 2 analysis. In this statistical analysis was taken into account the following decay channels : bb, γγ, ZZ, W W and τ τ . R XX and the branching ratios were obtained in the 2HDM type III with CPV for h 1 .  Figure (a) for From the χ 2 analysis, allowed regions were found for mixing parameters with fixed values of Y ii . The results are shown in the Figures 2, 3 and 4. For large values of tan β, it is shown that the allowed region is significantly suppressed; while in the opposite case, for small or zero values, the regions have a significant increase. This means that large values for tan β, which can be consider from 10 approximately, are not viable to observe.
Two extreme cases were also considered for the Yukawa couplings, Y f f . One of the cases was to assume Y f f = 0, while the other case was to consider a maximum value determined by the Cheng-Sher parameterization. Under these assumptions a suppression was obtained when the Y f f couplings participate, as shown in the Figures 3 and 4. The region that shows the greatest suppression is for α 2 , which is bounded as |α 2 | 0.3.
To study the behavior of the FC parameters, Y ij , the reported value for the b → sγ was considered in an approximate scheme for its branching ratio expression in the 2HDM-III with CPV. Then, allowed values were obtained for Y tc with different values tan β and Y bb by assuming m H ± ≈ 500 GeV. In this scheme the charged Higgs contributes inversely proportional to tan β, as it shows the equation (19). The values obtained for Y tc are shown in Figure 1, which are used to analyze the t → cZ and t → ch 1 decays in the 2HDM-III with CPV. In the Γ(t → cZ), we obtain the analytical expression at the loop level, Eq. (40), while for Γ(t → ch 1 ) the expression was calculated at tree level, Eq (45), which are the highest contributions in the model.
In Figure 7, we show BR(t → cZ) as a function of α 1 for different values of tan β. The BR(t → cZ) for Y f f = 0 can reach values of the order of 10 −5 , while in the case Y f f = m f /m W the values increase to 8 × 10 −4 for tan β = 1.
The mixing angle α 2 was assigned with random values in the allowed region for SM Higgs and b → sγ decays. In figure 7, when we consider Y f f = 0 and the value of α 1 , in the allowed regions giving by the Higgs decays and bsγ, the values for BR(t → cZ) are between 10 −7 and 10 −5 ; but in the limit when α 1 goes to zero then BR(t → cZ) ∼ 10 −9 . On the other hand, when Y f f = m f /m W , the values for BR(t → cZ) are between 10 −5 and 10 −3 .
In figure 8, the behavior of BR(t → cZ) as a function of α 2 is analogous to figure 7, but in these cases for α 2 allowed regions of the BR(t → cZ) are not suppressed, in fact, in the case of α 3 allows the all range and the greatest region for α 2 . The α 3 is set as random values in the interval [0, π/2] and the neutral Higgs masses m h2,h3 ∼ 500 GeV. For the BR(t → ch 1 ), the results show a significant increase with respect to the results of BR(t → cZ). For specific regions, as for example tan β = 1 and Y f f = m f /m W , the BR(t → ch 1 ) reaches a maximum value of the order 10 −2 . Figure 12 shows a scatter plot considering these experimental limits and the allowed regions for the α 1,2 parameters that correlate the branching ratios for fixed values of tan β and Y f f . This correlation shows a behavior such that BR(t → cZ) is approximately 10 2 ∼ 10 3 times lower than BR(t → ch 1 ). This Figure also shows that for tan β < 1 the values of BR(t → ch 1 ) are close to the reported limit, while for Y f f = m f /m W the values of the BR(t → cZ) in the model are close to the experimental limit.
The experimental limits are BR(t → cZ) < 3.7% and BR(t → cH 0 ) < 0.46% [52]. We consider random values for the α 1,2 parameters in the allowed regions at 90 % C.L., obtained in the Section 3, to evaluate the BR(t → cZ) and BR(t → ch 1 ) in the 2HDM-III with CPV. The evaluation of these branching ratios has also been restricted by experimental limits. Figure 12 shows a scatter plot that correlates to BR(t → cZ) and BR(t → ch 1 ) for random values of α 1,2,3 and Y tc in the allowed regions when tan β is set.

VI. CONCLUSION
We obtain the analytical expressions for Γ(t → cZ) and Γ(t → ch 1 ) in the 2HDM type III with CPV at one loop and at tree level. Then, BR(t → cZ) and BR(t → ch 1 ) are analyzed in allowed regions for α 1 and α 2 parameters with different fixed values of tan β and Yukawa couplings. These allowed regions are obtained by applying an χ 2 analysis at 90 % C.L. into the Higgs decays. We also consider the contributions of 2HDM type III to BR(b → sγ) with its experimental value and we explore values of Y tc as a function of Y bb and tan β for m H ± ≈ 500 GeV. We find feasible scenarios for Γ(t → cZ) and Γ(t → ch 1 ) that can be comparable with the current experimental limits.
Appendix A: Form factors for t → cZ decay The useful notation for masses is introduced asx = x mt . Here, in order to simplify we introduce ij = 1 2 (sin βR i1 R k3 − cos βR i2 R k3 ). After the dimensional regularization of the integrals for the t → cZ amplitude, we obtain for the Feynman diagrams D = a, b, c, d, shown in figure 5, the following results: