Lepton flavor violating decays of the SM-like Higgs boson $h\rightarrow e_ie_j$, and $e_i \rightarrow e_j\, \gamma $ in a flipped 3-3-1 model

In the framework of the flipped 3-3-1 model introduced recently [1], the lepton-flavor-violating (LFV) decay $\mu \rightarrow 3e$ was predicted to have a large branching ratio (Br) close to the recent experimental limit. We will show that the Br of LFV decays of the standard-model-like (SM-like) Higgs boson decays (LFVHD) Br$(h\rightarrow e_ae_b)$ may also be large. Namely, the Br$(h\rightarrow \mu\tau,e\tau)$ can reach values of $\mathcal{O}(10^{-4})-\mathcal{O}(10^{-5})$, which will reach the upcoming experimental sensitivities. On the other hand, for LFV decays of charged leptons (cLFV) $(e_b\rightarrow e_a\gamma)$, the branching ratios are well below experimental bounds.

Hence, there will appear couplings of new heavy leptons in the third components of these lepton representations with normal charged leptons and gauge or Higgs bosons. The mixing of these heavy leptons is an important source of LFV mediation at the one-loop level.
Therefore, LFV decays of charged leptons in the framework of 3-3-1 models have been widely investigated [73][74][75][76][77][78][79][80]. Many of the 3-3-1 models can explain the recent lower bounds on the decays Br(e b → e a γ) [81,82] Br(τ → µγ) < 4.4 × 10 −8 , Br(τ → eγ) < 3.3 × 10 −8 , In future projects, new sensitivities for these decay channels will be Br(µ → eγ) ∼ O(10 −14 ) [83] and Br(τ → µγ, eγ) ∼ O(10 −9 ) [84]. They will be used to determine allowed regions of the parameter spaces of the 3-3-1 models for further studying other LFV decays such as those of the SM-like Higgs boson h → e ± b e ∓ a . They just have been investigated in just a few specific 3-3-1 models [45,52], where the LFV sources come from the mixing of heavy neutrinos. In particular, the 3-3-1 model with inverse seesaw neutrinos [52] predicts very small regions of parameter space that give large Br(h → τ µ, τ e) O(10 −5 ) and also satisfy the current bounds of Br(µ → eγ). Recently, an interesting flipped 3-3-1model has been constructed [1], where the left-handed lepton is arranged in a lepton sextet, while the left-handed τ and µ are still the same as those known previously. In addition, all left-handed quarks are also arranged in the same SU (3) L triplets so that the model is anomaly free. The treel-level flavor neutral changing currents caused by the heavy neutral boson Z do not appear; hence m Z is not constrained by the corresponding experimental data. The active neutrino and electron masses can be produced consistent with experiments through loop corrections [85]. The effect of the Higgs sextet on fermion and Higgs boson couplings was discussed in ref. [86]. The Higgs potentials relating to the Higgs sextets were studied in refs. [87,88]. Based on these ingredients, our aim in this work is to investigate the LFV decays of charged leptons e b → e a γ and the SM-like Higgs boson h → e b e a in the framework of the flipped 3-3-1 model.
Our work is arranged as follows. In Sects II and III, we will collect the main content of the flipped 3-3-1 model, where masses, physical states, and needed couplings for calculating branching ratios of the LFV decays are presented. The analytic formulas of LFV branching ratios and the corresponding numerical investigations will be shown in section IV. We will summary main results in Sect V. Finally, there are two appendices showing the details of the one loop formulas contributing to the LFV decays of charged lepton (cLFV) amplitudes of the decays e b → e a γ and the equations for minimal conditions of the Higgs potential considered in this work.
These Higgs bosons develop vacuum expectation values (VEV) defined as where S k 1,2,3,S n 1,2,S in general [1]. In addition, it was shown that S and k S should be small to successfully generate neutrino mass consistent with experimental data. Hence, we can take k s = S 0 when solving the masses and physical states of Higgs and gauge bosons.
The Yukawa Lagrangian for the lepton sector is where the invariant term of the tensor product of the three sextets is expanded as (L e ) c L e S = abc ijk (L e ) c ai (L e ) bj S ck [73,89], (L e ) c ai ≡ C(L e ) ai T . Note that φ 3 only appears in the Yukawa part of the quark.
The fermions are presented as two-component spinors in the original version; see table I in Ref. [1]. In this work, we will use the Dirac (four-component) spinor notation, based on the equivalence given in detail in Ref. [90]. In particular, a Dirac spinor where f L,R is the respective left (right) component of a Dirac fermion, namely f L = P L f The mass term of all fermions at tree level is where we have used the identity ψ c a ψ c b = ψ b ψ a for leptons. According to the discussion on Ref. [1], in the basis Ψ L,R = (e α , E α , E e , e, Σ − ) T L,R the mass matrix of charged leptons always has one massless eigenstate at tree level, corresponding to the normal electron mass m e = 0. This is also the case for active neutrinos. However, when the loop corrections are included, the consistent masses of electrons and active neutrinos are obtained. The one-loop Feynman diagrams corresponding to these corrections are given in Fig. 1, and were pointed out in Ref. [1], along with a very detailed discussion active neutrinos (right panel) [1], where c means c α ≡ (e αR ) c on this property of the flipped 3-3-1 model. Accordingly, using the minimal Higgs sector given in Table I, the experimental data of an inverse hierarchy for active neutrinos can be fitted. Adding more scalar fields to the model will be another way to solve the problem of the neutrino oscillations that can be fitted with recent experimental data. As we will show, this problem does not affect significantly our discussion on LFV decays.
Because loop corrections are needed to generate masses of only very light leptons, namely electrons and active neutrinos, the other corrections to the lepton mass matrices are also smaller than other heavy masses appearing in the model. This is also because of another reason that one-loop corrections are suppressed by the two factors 1/(16π 2 ) and 1/M 2 relating respectively to the one-loop integral and new heavy masses M of a new particle running in the loop. In conclusion, loop corrections make tiny contributions to the lepton mass matrices. Hence, we will ignore loop corrections to the masses of heavy particles from now on.
For simplicity, in this work we will assume that only exotic charged leptons E e , E µ , E τ mix with each other to guarantee the existence of LFV couplings that contribute to oneloop amplitudes of the LFV decays. On the other hand, all of the original states of the SM charged leptons and Σ − are physical. This corresponds to the condition that , k S , n 1 , k 2 0.
The large Yukawa couplings of the physical states µ, τ and Σ − are Note that the masses of electrons and active neutrinos come from loop corrections.
The original basis (E µ , E τ , E e ) corresponds to the following mass term: where we have used the assumption that some of the Yukawa couplings in the Lagrangian (5) are zeros. The lepton mass matrix in Eq. (8) is arbitrary; hence it is diagonalized by the following transformation: where m E i are masses of the physical states E iL(R) , i = 1, 2, 3. For simplicity, in this work , and all Dirac and Majorana phases are set to be zeros. This matrix exactly satisfies the unitary property. We will use s E ij as free parameters. Other Yukawa couplings are non-zero for generating active neutrino masses and mixing consistent with experiments (see discussions in ref. [1]), but they are assumed to be suppressed in this work. We also note that the conditions in Eq. (7) still allow right SM quark masses and mixing consistent with experimental data. Similarly, there is one heavy Majorana neutrino Σ M = (Σ 0 , Σ 0 † ) T with the mass term −1/2(−2y n S )Σ 0 Σ 0 + h.c.. Three other active neutrinos get consistent masses and mixing from loop corrections, which prefer the inverted order of active neutrino data oscillation. Their physical states are denoted as n 1 , n 2 , n 3 [1]. The masses and mass eigenstates of heavy neutral leptons are Yukawa coupling terms in the Lagrangian (5) containing normal charged leptons are 13 + E τ R y 14 + E eR y 23 + E τ,R y 24 + E e,R y Corresponding to the above assumption that all charged leptons are diagonal, Yukawa couplings relating to one-loop corrections must guarantee that new Higgs bosons should couple to different SM charged leptons. As we will show later, the SM-like Higgs bosons will be h R 3 when we assume that k 1 k 3 . Combined with Lagrangian (12), we can see that tree-level couplings of the SM-like Higgs boson he i e j do not appear. The heavy neutral lepton n 4 does not couple with normal charged leptons. The couplings he i e i appear from the small mixing of R 3 and R 1 for e i = µ, τ and loop corrections for the electron. These couplings have small effects on the LFV decays so we omit them from now on.
After breaking, the masses and physical states of all gauge bosons are determined as follows.

A. Gauge boson
The covariant derivative of the SU (3) L × U (1) X is defined as where T a (a = 1, 2, .., 8) is the SU (3) generator with respective gauge boson W a µ , T 9 = I √ 6 is the U (1) X generator with the gauge boson X µ , and X is the U (1) X charge of the field acted by the covariant derivative. The particular forms of the generators are: • For an SU (3) L singlet: T a = 0 ∀a = 1, 2, .., 8, • For an SU (3) L triplet: T a = 1 2 λ a ∀a = 1, 2, .., 8, where λ a are Gell-Mann matrices. The covariant part can be written as: where we have defined the mass eigenstates of the charged gauge bosons as • For an SU (3) L sextet denoted as S ∼ (6, 2/3), given in table I, the action of an SU (3) L generator can be written in terms of the Gell-Mann matrix, T a S = Sλ a /2 + λ a /2S T [91]. Hence, the corresponding covariant derivative can be written in terms of the generators of the SU (3) triplet [91,92], namely The symmetry-breaking pattern is SU The covariant kinetic terms of the Higgs bosons are From this, the squared mass matrix of the charged gauge bosons in the basis (W ± µ , Y ± µ ) is given by It is enough to assume that k i /n i 1 for i = 2, S so that the non-diagonal term in the squared mass matrix (18) can be ignored. In this work we will accept that In particular, we will choose k 1,2,S ∼ O(10) GeV and n 2,s ∼ O(10 3 ) GeV, leading to the consequence that k i n i GeV 2 /(246GeV) 2 1. The non-zero values of k 1 still allow the reasonable Yukawa couplings of normal charged leptons given in Lagrangian (12). We note that this choice of VEV values are still allowed for generating consistent quark masses, as discussed previously [1]. The masses and physical states {W ± , Y ± } of charged gauge bosons are determined as Identifying the W ± with the SM one, we have v 174 GeV. If k 1,2,S = O(10) GeV, we have Using the assumption in Eq. (19) the neutral gauge boson mass can be determined as follows.
The non-Hermitian gauge bosons V 0 and V 0 * do not mix with the Hermitian ones. The masses and physical states are For simplicity in calculating the masses and mass eigenstates of the Hermitian neutral gauge bosons, we will safely use the limit that k 1 , k 2 , k S , S k 3 . Accordingly, these neutral gauge bosons will decouple with the ReV 0 . In the basis (X µ , W 3 µ , W 8 µ ), the squared mass matrix is where t = g X /g. This matrix will be diagonalized by a mixing matrix C defined by This mixing matrix C can be summarized in the three breaking steps as follows: corresponding to three physical gauge bosons. Two of them are identified with the massless photon A µ and the SM-like neutral gauge boson Z 1 found experimentally. After the first breaking step, the gauge couplings and U (1) Y charges are identified with the SM, leading to the following consequences: where g and s W are the well-known parameters defined in the SM, i.e., the SU (2) L gauge couplings and the sine of the Weinberg angle. In the first step, the two neutral gauge bosons W 8 µ and X µ mix, giving rise to the two bosons B µ and Z µ . The mixing angle is denoted by θ 331 and is given by [91] s 331 ≡ sin θ 331 = √ 6g The relation between the original and physical basis of the neutral gauge bosons are Using the limit 2 S k 2 α n 2 2,S , the mixing angle θ is determined as [93] The masses for the neutral gauge bosons in this limit are As usual for 3-3-1 models with non-zero Z − Z mixing, in the limit m 2 Z m 2 Z the tree-level contribution to the ρ parameter defined by is estimated approximately by the following formula [93] ∆ρ where s θ is given in Eq. (27). The recent experimental lower bound of m Z ≥ 4 TeV [96] results in that ∆ρ ≤ 7×10 −4 , which still satisfies 3σ allowed range of experimental data [94].
Previous studies of one-loop contributions from heavy gauge and Higgs bosons to the ρ parameter in some particular 3-3-1 models [79,93,99] suggest that these contributions from the heavy gauge bosons are very suppressed with m Z ≥ 4 TeV, while those from Higgs bosons can be negative and have the order of O(10 −4 ). Hence the total contributions to ∆ρ may satisfy the experimental constraint even with m Z smaller than 4 TeV, which was reported from the ATLAS experiment at LHC [96]. We will use this lower bound of m Z in the numerical investigation.
To determine the SM-like Higgs from its couplings to the gauge bosons W ± and Z, the relevant terms are In the limit k 1,2,S , S k 3 , we have k 3 v = √ 2m W /g. Then we can see that R 3 should be identified with the SM-like Higgs boson because it has the same couplings with the SM gauge bosons as those predicted by the SM.
As noted in previous works, m 2 Z 2 m 2 Z , so we get s θ 1 based on Eq. (27), hence the Z − Z mixing will be ignored in one-loop formulas involving with LFV decays. An interesting property of the heavy gauge bosons is that they get masses from two large vev n 2 and n S . Hence, in principle, n 2 can get low values of 1 TeV, even when m Z are constrained to be very heavy from recent experiments.

B. Higgs boson
The Higgs potential is 1 : where the invariant terms containing Higgs sextets were derived based on ref. [88], ijk is the total antisymmetric tensor. For simplicity in finding physical states and masses of neutral Higgs bosons, we use the following limit: We remind the reader of the other assumptions that we mentioned above that can be applied Inserting them into the Higgs potential (31), we will find the masses and mixing matrices of all physical Higgs bosons as follows.
There are six physical states of CP-even neutral Higgs bosons that are the original states themselves, namely with corresponding masses as follows: The squared matrix of the two states (R 2 , R S ) is which gives give two mass eigenstates corresponding to one Goldstone boson of V 0 and one physical state, which are denoted as G V and h 0 6 . Their masses and relations to the original states are We can see that the above assumptions of the VEV and Higgs self-couplings gives one Goldstone boson G V of the non-Hermitian gauge boson V and a light CP-even neutral Higgs boson h ≡ R 3 . It will be identified with the SM-like Higgs boson found by LHC through its couplings with fermions and gauge bosons, as we will show later.
The model contains only one pair of doubly charged Higgs bosons ∆ ±± with mass Regarding singly charged scalars, we have found two zero mass eigenvalues corresponding to two Goldstone bosons of W ± and Y ± . There are three original states that are also the mass eigeinstates, Corresponding to three other singly charged Higgs states (H ± 3 , σ ± , H ± S ), the squared mass matrix is It is easily seen that Det[M 2 3σS ] = 0, leading to a massless eigenstate that can be identified with the Goldstone boson of V ± .
In the CP-odd neutral Higgs spectrum, there are three massless eigenstates corresponding to three Goldstone bosons of gauge bosons Z, Z and V 0 . In particular, the three mass eigenstates and two Goldstone bosons are where G Z is the Goldstone boson absorbed by the gauge boson Z. Five remaining states divide into two sub-matrices of squared masses, corresponding to bases (I 2 , I S ) and (I σ 1 , I σ 2 , I σ S ), namely The first 2 × 2 matrix gives one Goldstone boson of V 0 denoted as G V , m G V = 0, and a physical CP-odd neutral Higgs a 6 . Their mass and mixing matrix is Regarding to the second matrix in Eq. (41), it is easy to check that Det[M 2 σ 1,2 ∆ ] = 0; equivalently, there exists one massless state that can be identified with the Golstone boson of Z . Because I σ 2 and I ∆ are irrelevant with the couplings in Eq. (12), which contribute to the one-loop amplitude of LFV decays, we choose a simple case that λ φS 12 = 0 so that I σ 1 is itself physical. The CP-odd neutral Higgs bosons relating to the one-loop contributions to LFV decays are I σ 1 and a 6 .
According to the above discussion on the Higgs sector, we can see that h 0 6 and a 6 can be considered as real and imaginary parts of a physical neutral complex Higgs boson denoted given in Eqs. (36) and (42).
Similarly, in the limit of the unknown parameterλ φ 12 = 0, R σ 1 and I σ 1 can be considered as the real and imaginary parts of a physical Higgs boson σ 1 with mass m 2 σ 1 =λ φ 12 n 2 2 +λ φS 1 n 2 S + µ 2 1 . More interesting, R σ 1 and I σ 1 give the same qualitative contributions to the amplitudes of the LFV decays. Therefore, we will use this limit for our numerical investigation to avoid unnecessary and lengthy private one-loop contributions of R σ 1 and I σ 1 to LFV decay amplitudes.
From the simple Higgs potential shown above, the Feynman rules for Higgs self-couplings of the SM-like Higgs boson that contribute to the LFVHD are shown in  gauge and Higgs bosons, the branching ratios of LFV decays h → e b e a and e b → e a γ can be computed in the next section.

IV.
LFV DECAYS e b → e a γ AND h → e a e b

A. Analytic formulas of branching ratios
In this section, we only pay attention to couplings that contribute to the LFV decay amplitudes h → e b e a and e b → e a γ at the one-loop level. We also apply the results introduced in Ref. [45] see the detailed explanation of the relations between these notations in Ref. [90]. The following terms are involved with LFV couplings: The f f s 0 couplings come from the Yukawa Lagrangian (12). In the physical basis, the Yukawa couplings involved to LFVHD are where the matrix Y is given in Eq. (8), which can be written in terms of heavy charged lepton masses and mixing parameters based on Eq. (9): For convenience in calculating the one-loop contributions of Higgs mediation to the LFV amplitudes, Lagrangian (45) is written in the following form: where the coupling Y s ji , i, j = 1, 2, 3, is defined as follows: where we have used s 2s = c 2s The corresponding one-loop Feynman diagrams that contribute to the LFVHD amplitude are shown in Fig. 2. Although the model under consideration contains charged Higgs ij ≡ m 2 i − m 2 j , with i = j and i, j = 1, 2, 3. This result can be derived using Taylor expansion in terms of the squared masses of the active neutrinos and applying the Glashow-Iliopoulos-Maiani (GIM) mechanism i V * ia V ib = 0 to cancel large contributions independent of m i , see previous discusion on LFV decays [46,100]. Hence these contributions from singly charged Higgs bosons are very suppressed so we then safely ignore them.
The partial decay width of the decays h → e a e b is defined as follows: with the condition m h m a,b and m a,b charged lepton, a, b = 1, 2, 3 corresponding to e, µ, τ .
The on-shell conditions for external particles are p 2 In the notations constructed in Ref. [45], the ∆ (ba)L,R can be written as where detailed calculations to derive analytic formulas of ∆ (i) (ba)L,R are given in Ref. [45]. In previous works [19,45], we can see that ∆ where b = 2, 3, and .
The ∆ σ 0 1 h 6 (32)L arises from the chirality flip in the Yukawa couplings of heavy fermions with σ 0 1 and h 0 6 given in Eq. (47), similar to the cases mentioned in Refs. [64,98], which relates to the Yukawa couplings with chirality flip. In our work, the ∆ In the unitary gauge, the one-loop three-point Feynman diagrams contributing to the decay amplitudes e b → e a γ (a < b) are shown in Fig. 3. For low energy, the branching ratios of the cLFV decays can be written in a more convinient form as follows: where Br(µ → eν e ν µ ) 100% [94]. The analytical forms of C (ba)L,R are derived based on previous results [77,95]. Accordingly, we can use the limit m 2 a , m 2 b 0, where the results are as follows, where and the functions g s (t s,i ), g v (t v,i ) are derived in Appendix A.
We note that σ 0 1 only contributes to LFV decays t → µγ and h → µτ . Because of the σ 0 1 couplings with only µ and τ . This is the proper property of the flipped 3-3-1 model, where left-handed electron is a component of a sextet, while the τ and µ are arranged in triplets as other usual 3-3-1 models. Consequently, the amplitudes of the two decays h → µτ and τ → µγ receive more one-loop contributions than the remaining decay amplitudes, hence we expect that the Br(h → τ µ) and Br(τ → µγ) will be large.

B. Numerical discussions
In this numerical discussion, the unknown input parameters are: the masses and mixing parameters of the heavy leptons s E ij and m E i ; heavy neutral Higgs masses and mixing m σ 0 1 , m h 6 and s 2s . In addition, the unknown VEVs in the model are k 1 and n 2 . From Eqs. (36) and (28), we have where t 2s ≡ s 2s /c 2s . This means that n 2 2 + 4n 2 S (2.15m Z ) 2 . For the latest lower bound of m 2 Z ≥ 4 TeV reported from experiment [96], we have n 2 2 + 4n 2 S ≥ 8.3 TeV. For our numerical investigation in this work, we will fix n 2 2 + 4n 2 S = 8.3 TeV, n 2 = 1 TeV, n S ≥ 4 TeV, leading to t 2s = √ 2n S /n 2 = 4 √ 2; equivalently s 2s 0.985. The large s 2s corresponds to the large Yukawa coupling Y h 6 given in Eq. (48). Because k 1 generates masses for the lepton τ at the tree level, it should not be too small. In addition, µ 2 12 given in Eq. (B1) is too large if k 1 is too small. Hence we will choose that 10 GeV ≤ k 1 ≤ 50 GeV. The In the first numerical investigation, the default values of the inputs are k 1 = 20 GeV, Br(h→τμ) Similarly, with s E 12 = s E 23 = 0 and s E 13 = 1 √ 2 , we have only two non-zero Br(h → µe) and Br(µ → eγ). Illustrations of these branching ratios as functions of m E 1 with different fixed k 1 are shown in Fig. 5. Accordingly, Br(µ → eγ) ≤ O(10 −15 ), which still satisfies the lower Br(h→μe) bound in Eq. (2). It is noted that although Br(h → µe) is sensitive to k 1 , the Br(µ → eγ) is not, because it does not receive contribution from Yukawa coupling of σ 0 1 . The case of s E 12 = s E 13 = 0 and s E 23 = 1 √ 2 correspond to the two non-zero Br(h → τ e) and Br(τ → eγ). Illustrations of these branching ratios as functions of m E 1 with different fixed k 1 are shown in Fig. 6. In this case, Br(h → τ e) has the same order as Br(h → τ µ) Br(h→τe) increasing m E 1 , which has an upper bound originating from the perturbative limit of the To estimate how large the LFV branching ratios can beome when m Z is large, we fix n 2 = m Z /4 ≥ 1 TeV, then t 2s and n S are determined from the relations given in Eq. (55).
The Br of LFV decays as functions of m Z are illustrated in Fig. 10. In this case we can see that all LFV branching ratios decrease with larger m Z , but Br(h → τ µ) and Br(h → τ e) are still close to the order of O(10 −5 ) or larger. Hence these decay channels are still interesting for experiments. On the other hand, all Br(e b → e a γ) decrease rapidly with increasing m Z .
They will not be detected by upcoming experiments.
These results are consistent with the formulas introduced in ref. [97], used to discuss on the muon anomalous magnetic moments.