Flavor violating Higgs signals in the Texturized Two-Higgs Doublet Model (THDM-Tx)

Flavor violating Higgs signals, such as the top FCNC decay t→ch0 and the LFV Higgs decay h0→τμ, have been studied at the LHC. These signals can arise within the general Two-Higgs Doublet Model (THDM), where each Higgs doublet couples to all fermion types through Yukawa matrices and . The Yukawa matrices can be assumed to have the same form or they could have different structures. In this paper we study the case when both and have completely different forms, but in such a way that they complement to produce a specific hermitian mass matrix. We find that for specific four-zero textures, the flavor violating Higgs couplings depend only on the free parameters tanβ, γf and the fermion masses. We use the current bounds on the low energy processes to derive constraints on the heavy Higgs boson mass, tanβ and γf. Then, we use these constraints to evaluate the LFV Higgs decays, which reach branching ratios that could be tested at the LHC.


I. INTRODUCTION
The recent discovery at the Large Hadron Collider (LHC) of a new particle with Standard Model (SM)-Higgs like properties, and mass m H ≈ 125 GeV [1,2], seems to confirm the linear realization of the mechanism of Electroweak symmetry breaking, needed to induce the masses of gauge bosons and fermions within the SM [3].Furthermore, the current experimental searches are starting to test the Higgs couplings, which can allow us to discriminate between the minimal SM Higgs doublet case and other extensions of the SM with more complicated Higgs sector [4].
For a while, finding a possible solution to the hierarchy problem suffered by the SM Higgs boson, has been the driving force behind the many proposals for extending the SM.It is expected that current and future data coming from LHC will give us the opportunity to test these scenarios.
One the simplest proposals for physics Beyond the Standard Model (BSM), is the so called Two-Higgs Doublet Model (2HDM), which was initially studied in connection with the search for the origin of CP violation [5], later on it was found that the model could be found in connection with the realization of new theoretical ideas, such as supersymmetry [6], extra dimensions [7] and strongly interacting models [8], [9].
Several possible realizations of the general 2HDM have been considered in the literature, which have been known as Type I, II and III, for a review see [10].There are also other models called X, Y, Z, but in some sense they can be considered variations of the above models.
Model I has a discrete symmetry Z 2 , which permits a possible dark matter candidate coming from the Z 2 −odd scalar doublet [11].Within Type-I models, a single Higgs doublet gives mass to the up, down quarks and leptons.The type II model [12] assigns one doublet to each fermion type, which suffices to avoid Flavor Changing Neutral Currents (FCNC) mediated by the Higgs bosons [13]; this type II model also arises in the minimal SUSY extension of the SM [14].
On the other hand, within the most general Two Higgs Doublet Model, both Higgs doublets could couple to all types of fermions.Therefore, the mass matrix for each fermion type f (= u, d, l) receives contributions from both Higgs doublets, which have vevs v 1 and v 2 after SSB, i.e.
where Y f 1,2 denote the Yukawa matrices associated with each Higgs doublet.Within this general model, the Yukawa matrices must have a structure that should reproduce the observed fermion masses and mixing angles, while at the same time the level of FCNC must satisfy current bounds [5,15,16].One possiblity to have FCNC at acceptable levels, is the assumption that the Yukawa matrices have a certain texture form, i.e. with zeroes in different elements.
In the past, this general model was called as 2HDM of type III.However, this naming scheme has become confusing, in part because it has been used to denote a different type of model [17], but also because some specific cases have acquired a relevance of their own.Among the relevant sub-cases of the general 2HDM, one can include the so-called Minimal Flavor violating THDM (MFV) [18], which is thought to provide precisely the minimal level of FCNC consistent with data; MFV could be studied from a pure phenomenological point of view [19] or as arising from flavor symmetries [20].Although the so-called 2HDM with Alignment does not contain flavor violation, it is another possibility one can use to obtain realistic models [21].Thus, in order to clarify the notation and to single out the use of textures within the THDM, from now on we shall call the two-higgs doublet model with textures as 2HDM-Tx.
The first study of the 2HDM with textures [22] considered the specific form with sixzeroes, but also the so-called cyclic model and other variations.In that work it was identified a specific pattern of FCNC Higgs-fermion couplings, known nowadays as the Cheng-Sher ansatz, which is of the form . It was found that such vertex could satisfy FCNC bounds with Higgs masses lighter than O(TeV).The extension of the THDM-Tx with fourzero texture was presented in [23,24].The phenomenological consequences of these matrix textures (Hermitian 4-textures or non-hermitian 6-textures) were considered in [25], while further phenomenological studies were presented in [26][27][28].However, the above studies usually assumed that both Y f 1 and Y f 2 have the same texture form, a pattern that can be called "Paralell textures" namely: However, it is also possible that Y f 1 and Y f 2 could have different forms, but in such a way that the resulting mass matrix has a specific type of texture [3].In particular, in this paper we shall discuss different combinations of Yukawa matrices that result in a mass matrix of the four-zero texture form.
The organization of our paper goes as follows.The different types of Yukawa matrices that result in a mass matrix with four-zero textures as well as the diagonalization of the mass matrix, are presented in section II.Section III discusses generalities of the 2HDM-III and includes the Yukawa interaction Lagrangian in terms of mass eigenstates.Low energy constraints are discussed in section IV, including K − K mixing, B s → µ + µ − and τ → 3µ.
Constraints from current Higgs searches at LHC are included in section V.The prediction of our model includes h → τ µ and t → ch, and are discussed in section VI. Conclusions of our work are shown in section VII, while specific expressions for FV couplings for all cases are included in appendix A.

II. CLASSIFICATION OF THE TEXTURES
As it was mentioned in the introduction, the THDM-Tx's that were studied previously usually make use of the so called "Parallel textures"".Here, we shall consider Yukawa matrices that have a different structure, but in such a way that they produce a hermitian mass matrix with four zero textures; we call this pattern as "Complementary textures".We shall assume that at most one element from each of the Yukawa matrices Y f 1,2 contributes to one entry of the full mass matrix.To discuss implications when this assumption is relaxed, we shall also considered one case where both Y f 1,2 contribute to the (3,3) element of the mass matrices (this case is called "Semi-parallel textures").Thus the cases that will be considered here are defined as follows: Case 1: In making these choices, we have included the 33 mass entry in Y 2 , except for case 7, where it appears in both Y 1 and Y 2 .
Then, the above combinations of Yukawa matrices produce a mass matrix with a 4-zero texture: which can be diagonalized by the matrix V given by, Furthermore, V can be simplified if we assume m 1 m 3 → 0, which seems to make sense, since the mass of the third generation is much larger than the first generation.The resulting expression fo V is given by, .
Although V diagonalises the matrix M , it does not necessarily diagonalize each of the Yukawa matrices that make up M , thus neutral flavor violating Higgs-fermion interactions will be induced.In finding the expression for mixing angles and mass eigenstates, it is useful to consider the following matrix invariants: Note that the use of the determinant forces us to take m 1 < 0. From these expressions we find a relation between the components of the 4-texture mass matrix and the physical fermion masses, which will be usefull to find the expressions for the Higgs-fermion interactions, namely: where r i = m i m 3 .In the above, we have expressed the relation between the top mass and the 33 entry of the mass matrix as: A = m 3 (1 − r 2 γ), which depends on a single parameter γ (0 < γ < 1) (for cases 1-6).On the other hand, the mass relation for case 7 includes an additional parameter, and thus find convenient to define the following scenarios: i) a = m 3 and a = −m 2 γ and ii) a = m 3 − m 2 γ, a = m 3 γ with 0 < γ, γ < 1.

III. THE TYPE-III 2HDM
The Yukawa Lagrangian in the THDM-III is given by [25,29] where writing all the terms explicitly we get As a matter of convenience we separate the Lagrangian into charged (L ch ) and neutral terms (L n ).We do this because in this work we restrict ourselves to the study of process that involve only neutral type Higgs bosons.Thus, the field φ 0 i is expanden as, with φ 0 i given in terms of mass eigenstates: The couplings of down type fermions to different neutral Higgs is written as follows In the above relations, we have implemented the condition that our Yukawa matrices are Hermitian . In order to obtain physical fermion masses we need to diagonalize the mass matrix; this is achieved through the introduction of a bi-unitary transformation; however, for the hermitian case one only needs a unitary matrix V. Namely, in order to transform to the quark mass eigenstate basis, the rotation , and we obtain: Finally, we can use a more compact notation to write the Higgs-fermion interactions, namely: where the η factors are defined as follows where The expressions for χ ij corresponding for each of the Yukawa cases are shown in the appendix A. For illustration we shall display here some of the elements of the χ factors for case 5.In general, each component has a complicated expression relating the masses of the particles and our parameter γ.For instance, the 12-Yukawa elements are given by, Thus, we find that the Cheng-Sher ansatz still holds, up to corrections, for case 5 of complementary Yukawa matrices; and this remains valid for all the cases we are considering here.

IV. CONSTRAINTS FROM LOW ENERGY
In order to find the allowed regions of parameter space we need to consider all relevant low energy and collider constraints.For the low energy constraints, we shall consider the contribution of neutral Higgses to the flavor violating processes K K-mixing, B s → μµ and τ → µµμ.

A. K K Mixing
The amplitude for K K Mixing receives contributions form the following Feynman diagram, The effective hamiltonian for the hadronic process is written as follows and the Wilson coefficients, Flavor change arises from the non-diagonal terms of the K (neutral kaon) mass matrix, in particular the component M K 12 , whose experimental value has been measured to be We will explore the following process, The Branching ratio for this decay is given by, where the form factors are, which depend on the Fermi constant G F , the lifetime of the B meson (τ B ) and the Wilson coefficients (C s,p , C 10 ) that appear in the effective Hamiltonian, where O s,p,10 are the effective operators, and the Wilson coefficients are, The (χ r ) ij − entries are associated with leptons (r = l) and quarks (r = q), and appear in appendix A. The index n labels the Yukawa matrix 1 or 2. G F , M b , V * tb(ts) , M H 0 (h 0 ) are the Fermi constant, bottom quark mass, CKM elements and heavy Higgs and SM-like Higgs masses, respectively.We do not include the contribution of the charged Higgs and so we do not need to consider the Wilson coefficient C 10 .

C. τ → µµμ
The Feynman diagram for this process is, and the differential decay width is given by [30], The integration gives, where α − β = n.Using the textures Y 1 and Y 2 , we obtain the element of the matrix

D. Allowed regions
We shall derive bounds on tan β and γ for the following choices (α − β) = 0, π 2 , π 3 , while for the masses of the heavy Higgs bosons we take: From the above low energy constraints we find that not all values of γ and tan(β) are allowed.In the following figures we show the regions which are allowed (blue) for each of our cases (thus the excluded regions are in white), after taking into account all the previous process (K K Mixing, B s → μµ and τ → µµμ).
In general, we find that values of tan β larger than about 50 are excluded.For case 1, we can identify that values of tan β below 20 are still allowed.This happens for all values of α − β considered here.For case 2, we notice that there is minimum value of γ for which one start to get constraints on the values the values of tan β and for γ → 1 we find that tan β must be below a number of order 10.Case 6 behaves similarle to case 2, while the other cases interpolate between cases 1 and 2. Finally, we find that case 7 is similar to case 2 although the constraints on γ are stronger; this hold for both cases 7a and 7b,where we need to include the γ parameter, as can be seen from 5

V. THE LHC SIGNALS
The next constraint that needs to be satisfied is the strength of the SM-like Higgs signal observed at the LHC.Currently one would need to include several production and decay Higgs channels, however for an estimate of the signal, we shall consider only the production of Higgs bosons by gluon fusion and the decays h → ZZ * , γγ.Then, in order to reproduce the signal rate for the SM-like Higgs signals with m h 125 GeV, one can consider the following ratios: for X = γ, Z.
Whithin the so-called Narrow-width approximation, we can write the expression for R XX as follows: We have evaluated the values of R γγ and R ZZ for a scanning of the parameters of the model, namely with a light Higgs mass m h = 125 GeV, 0 < tan β < 20 and 0 < γ < 1.
Then we looked at those values of parameters that are consistent with both LHC Higgs data R γγ = 1.56 ± 0.43 and R ZZ = 0.8 +0.35 −0.28 , as well as the low energy constraints derived in the previous section.Those set of parameters are then used for the predictions of the model, which are presented in the next section VI.

VI. PREDICTIONS FOR h → τ µ, t → ch
One interesting signal to probe FV Higgs couplings is provided by the decay h → τ µ, which was initially studied by [31,32].Subsequent studies on detectability of the signal appeared in [33], while improved calculations within SUSY and other models appeared in [34]; more recent discussions of LFV Higgs decays are presented in [35].
The other interesting signal to probe FV Higgs couplings is provided by the rare top decay t → ch, which has been studied within the 2HDM in [36][37][38][39], while the SUSY case was considered in [40].The search for this mode at LHC was considered in [41].
the Feynman diagrams for the 2-body process, h → τ µ and h → ch, are, where the width are given as follows, The decay width for the top FCNC process is given as, 2      Those entries in table V with dashes correspond to values of parameters that do not satisfy the low energy and/or collider constraints.

VII. CONCLUSIONS
We have studied the phenomena of flavor violation within the Two-Higgs doublet model with Textures (2HDM-Tx).We have defined the case of complementary textures, which consists of employing Yukawa matrices with different textures, but in such a way that their combination produce a mass matrix with a certain texture type; 4-zero texture in our case.
In this model, the flavor violating couplings are given by a set of factors χ ij , which depends on the parameter γ (0 < γ < 1).We have also explored another case (7), which is termed here as the semi-parallel case.
We find that in order to satisfy current experimental bounds from the low-energy proccesses: K − K mixing, B s → µ + µ − and τ → µµμ, only small values of γ and tan β(= v 2 v 1 ) are allowed.Furthermore we also study the constraints that current LHC results on the Higgs properties imposse on these parameters.The predictions of our study include the LFV Higgs decays (h → τ µ), as well as the rare top decay t → c + h, which could reach significant levels.For instance, in case 6 and for α − β = π 3 , tan β = 3 and γ = 0.1; we find Br(h → τ µ) = 6.9 × 10 −4 and Br(t → c h) = 2.8 × 10 −2 , which may be searched at the coming phases of LHC.

,
Since we are working in general THDM-Tx, the rotation matrix for neutral Higgs bosons is of the form

5 ×
5 where: α − β = π 3 and the first row corresponds to tan β = 10, the second row tan β = 15 and the third row tan β = 20 .The second part of the table has an analogous arrangement of tan β. 10 −4 2.6 × 10 −4 Some samples of numerical results for the Branching ratios B.R.(h → τ µ) and B.R.(t → ch) are presented in the tables I -VIII.The decay h → τ µ could have a B.R. as small as 10 −8 , and the largest value, within our model, is of order 10 −2 .On the other hand, for t → c h the largest B.R. is 10 −1 and the smallest B.R. is of order 10 −8 .
TABLE I. Branching ratios for Case 1 where: α

TABLE II .
Branching ratios for Case 2 where α−β = π 3 , the first two row corresponds to tan β = 1 and the last two rows to tan β = 2 .

TABLE V .
Branching ratios for Case