Lepton flavor violating decays of Standard-Model-like Higgs in 3-3-1 model with neutral lepton

The one loop contribution to the lepton flavor violating decay $h^0\rightarrow \mu\tau$ of the SM-like neutral Higgs (LFVHD) in the 3-3-1 model with neutral lepton is calculated using the unitary gauge. We have checked in detail that the total contribution is exactly finite, and the divergent cancellations happen separately in two parts of active neutrinos and exotic heavy leptons. By numerical investigation, we have indicated that the one-loop contribution of the active neutrinos is very suppressed while that of exotic leptons is rather large. The branching ratio of the LFVHD strongly depends on the Yukawa couplings between exotic leptons and $SU(3)_L$ Higgs triplets. This ratio can reach $10^{-5}$ providing large Yukawa couplings and constructive correlations of the $SU(3)_L$ scale ($v_3$) and the charged Higgs masses. The branching ratio decreases rapidly with the small Yukawa couplings and large $v_3$.


I. INTRODUCTION
The observation the Higgs boson with mass around 125.09 GeV by experiments at the Large Hadron Collider (LHC) [1][2][3][4][5] again confirms the very success of the Standard Model (SM) at low energies of below few hundred GeV. But the SM must be extended to solve many well-known problems, at least the question of neutrino masses and neutrino oscillations which have been experimentally confirmed [6]. Neutrino oscillation is a clear evidence of lepton flavor violation in the neutral lepton sector which may give loop contributions to the rare lepton flavor violating (LFV) decays of charged leptons, Z and SM-like Higgs bosons.
The LFVHD of the neutral Higgses have been investigated widely in the well-known models beyond the SM [10][11][12], including the supersymmetric (SUSY) models [13][14][15]. The SUSY versions usually predict large branching ratio of LFVHD which can reach 10 −4 or higher, even up to 10 −2 in recent investigation [13], provided the two following requirements: new LFV sources from sleptons and the large tan β-ratio of two vacuum expectation values (vev) of two neutral Higgses. At least it is true for the LFVHD h 0 → µτ under the restrict of the recent upper bound of Br(τ → µγ) < 10 −8 [16]. In the non-SUSY SU(2) L × U(1) Y models beyond the SM such as the seesaw or general two Higgs doublet (THDM), the LFVHD still depends on the LFV decay of τ lepton. The reason is that the LFVHD is strongly affected by Yukawa couplings of leptons while the SU(2) L × U(1) Y contains only small Yukawa couplings of normal charged leptons and active neutrinos. Therefore, many of non-SUSY versions predict the suppressed signal of LFVHD.
Based on the extension of the SU(2) L × U(1) Y gauge symmetry of the SM to the SU(3) L × U(1) X , there is a class of models called 3-3-1 models which inherit new LFV sources. Firstly, the particle spectra include new charged gauge bosons and charged Higgses, normally carrying two units of lepton number. Secondly, the third components of the lepton (anti-) triplets may be normal charged leptons [17,18] or new leptons [19][20][21][22][23] with non-zero lepton numbers. These new leptons can mix among one to another to create new LFV changing currents, except the case of normal charged leptons. The most interesting models for LFVHD are the ones with new heavy leptons corresponding to new Yukawa couplings that affect strongly to the LFVHD through the loop contributions. This property is different from the models based on the gauge symmetry of the SM including the SUSY versions. In the 3-3-1 models, if the new particles and the SU(3) L scale are larger than few hundred GeVs, the one-loop contributions to the LFV decays of τ always satisfy the recent experimental bound [24]. While this region of parameter space, even at the TeV values of the SU(3) L scale, favors the large branching ratios of LFVHD. The one-loop contributions on LFV processes in SUSY versions of 3-3-1 models were given in [14,25], but the non-SUSY contributions were not mentioned. The 3-3-1 models were first investigated from interest of the simplest expansion of the SU(2) L gauge symmetry and the simplest lepton sector [17]. They then became more attractive by a clue of answering the flavor question coming from the requirement of anomaly cancellation for SU(3) L ×U(1) X gauge symmetry [18]. The violation of the lepton number is a natural property of these models, leading to the natural presence of the LFV processes and neutrino oscillations. Many versions of 3-3-1 models have been constructed for explaining other unsolved questions in the SM limit: solving the strong CP problem [26] with Peccei-Quinn symmetry [27]; allowing the electric charge quantization [28],... More interesting, the neutral heavy leptons or neutral Higgses can play roles of candidates of dark matter (DM) [23]. Besides, the models with neutral leptons are still interesting for investigation of precision tests [19].
From the above reasons, this work will pay attention to the LFVHD of the 3-3-1 with left-handed heavy neutral leptons or neutrinos (3-3-1LHN) [23]. It is then easy to predict which specific 3-3-1 models can give large signals of LFVHD. As we will see, the 3-3-1 models usually contain new heavy neutral Higgses, including both CP-even and odd ones. But the recent lower bound of the SU(3) L scale is few TeV, resulting the same order of these Higgs masses. At recent collision energies of experiments, the opportunity to observe these heavy neutral Higgses seems rare. We therefore concentrate only on the SM-like Higgs.
Our work is arranged as follows. The section II will pay attention on the formula of branching ratio of LFVHD which can be also applied for new neutral CP-even Higgses, listing the Feynman rules and the needed form factors to calculate the amplitudes for general 3-3-1 models. In the section III, the model constructed in [23] will be improved including adding new LFV couplings; imposing a custodial symmetry on the Higgs potential to cancel large flavor neutral changing currents in the Higgs sector and simplify the Higgs self-interactions.

II. FORMULAS FOR DECAY RATES OF NEUTRAL HIGGSES
For studying the LFVHD, namely h 0 → τ ± µ ∓ , we consider the general form of the corresponding LFV effective Lagrangian as follows − L LF V = h 0 (∆ L µP L τ + ∆ R µP R τ ) + h.c., (1) where ∆ L,R are scalar factors arisen from the loop contributions. In the unitary gauge, the one-loop diagrams contributing to ∆ L,R are listed in the figure 1. They can be applied for the models beyond the SM where the particle contents include only Higgses, fermions and gauge bosons. The amplitude decay is [10]: where u 1 ≡ u 1 (p 1 , s 1 ) and v 2 ≡ v 2 (p 2 , s 2 ) are respective Dirac spinors of the µ and τ . The partial width of the decays is where m h 0 , m 1 and m 2 are the masses of the neutral Higgs h 0 , muon and tauon, respectively.
They satisfy the on-shell conditions for external particles, namely p 2 i = m 2 i (i=1,2) and is an arbitrary even-CP neutral Higgs in the 3-3-1 models, including the SM-like one.
In the unitary gauge, the relevant Feynman rules for the LFV decay of h 0 → l ± 1 l ∓ 2 are represented in the figure 2. For each diagram, there is a corresponding generic function expressing its contribution to the LFVHD. These functions are defined as The notations are introduced as follows. The set of the form factors (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) was calculated in details in the appendix B which we find them consistent with calculations using Form [29]. These form factors are simpler than those calculated in the appendix because they contain only terms contributing to the final amplitude of the LFVHD. The excluded terms are come from the two reasons: i) those do not contain the neutral leptons in the loop so they vanish after summing all virtual leptons, reflecting the GIM mechanism; ii) the divergent terms defined by (A3). The second is true only when the final contribution is assumed to be finite. This is right for the models having no tree level LFV couplings of µ − τ . The 3-3-1 LHN model we will consider in this work satisfies this condition and the divergent cancellation is checked precisely in the appendix B.
Another remark is that the divergent term (A3) contains a conventional choice of ln µ 2 /m 2 h in which m h can be replaced by an arbitrary fixed scale. We find that only the contributions of the diagram 1d) and sum of two diagrams 1g) and 1h) are finite. Now the form factors ∆ L,R can be written as the sum of all E L,R functions. The one loop contributions to the LFV decays such as ∆ L,R are finite without using any renormalization procedure to cancel divergences. In addition, ∆ L,R do not depend on the µ parameter arising from the dimensional regularization method used to derive all above scalar E L,R functions in this work. But in general contributions from the separate diagrams in the figure 1 do contain the divergences and therefore the particular finite parts E L,R do depend on µ, so it will be nonsense for computing separate contributions. There is another simple analytic expressions given details in [15], updated from previous works [30]. It can be applied for not only SUSY models but also the models predicting new heavy scales including 3-3-1 models. The point is that this treatment uses the C-functions with approximation of zero-external momentums of the two charged leptons, i.e. p 2 1 = p 2 2 = 0. Unlike the case of LFV decays of τ → µγ, the LFVHD contains a large external momentum of neutral Higgs: 2p 1 , which should be included in the C-functions, as discussed in the appendix A. This is consistent with discussion on C-functions given in [31].

III. 3-3-1 MODEL WITH NEW NEUTRAL LEPTON
In this section we will review a particular 3-3-1 model used to investigate the LFVHD, namely the 3-3-1LHN [23]. We will keep most of all ingredients shown in ref. [23], while add two new assumptions: i) in order to appear the LFV effects, we assume that apart from the oscillation of the active neutrinos, there also exists the maximal mixing in the new lepton sector; ii) The Higgs potential satisfies a custodial symmetry shown in [22] to avoid large loop contributions of the Higgses to precision tests such as ρ-parameter and flavor neutral changing currents. More interesting, the latter results a very simple Higgs potential in the sense that many independent Higgs self-couplings are reduced and the squared mass matrix of the neutral Higgses can be solved exactly at the tree level. The following will review the needed ingredients for calculating the LFV decay of h 0 → l + i l − j .
A. Particle content • Fermion. In each family, all left-handed leptons are included in the SU(3) L triplets while right-handed ones are always singlets, where the numbers in the parentheses are the respective representations of the SU(3) C , SU(2) L and U(1) X gauge groups. The prime denotes the lepton in the flavor basis.
Recall that as one of the assumption in [23], the active neutrinos have no right-handed components and their Majorana masses are generated from the effective dimension-five operators. There is no mixing among active neutrinos and exotic neutral leptons.
• Gauge boson. The SU(3) L ×U(1) X includes 8 gauge bosons W a µ (a=1,8) of the SU(3) L and the X µ of the U(1) X , corresponding to eight SU(3) L generators T a and a U(1) X generator T 9 . The respective covariant derivative is Denote the Gell-Mann matrices as λ a , we have T a = 1 2 λ a , − 1 2 λ T a or 0 depending on the triplet, antitriplet or singlet representation of the SU(3) L that T a acts on. The T 9 is defined as T 9 = 1 √ 6 and X is the U(1) X charge of the field it acts on.
• Higgs. The model includes three Higgs triplets, As normal, the 3-3-1 model has two breaking steps: The non-zero U(1) G charged field η 0 2 and χ 0 1 have zero vacuum expectation (vev) values: η 0 2 = χ 0 Others neutral Higgs components can be written as As shown in ref. [22], after the first breaking step, the corresponding Higgs potential of the 3-3-1 model should keep a custodial symmetry to avoid large FCNCs as well as the large deviation of ρ-parameter value obtained from experiment. This only involves to the ρ and η Higgs scalars which generate non-zero vevs in the second breaking step. Applying the Higgs potential satisfying the custodial symmetry given in [32], we obtain a Higgs potential of the form, where f is assumed to be real. Minimizing this potential leads to v 1 = v 2 and two additional conditions, We stress that if the custodial symmetry is kept in this 3-3-1 model, the model automatically satisfies most of the conditions assumed in ref. [23] for purpose of simplifying or reducing independent parameters in the Higgs potential. For this work, which especially concentrates on the neutral Higgses, the most important consequence is that all of the mass basis of Higgses, including the neutral, can be found exactly without reduction of the number of Higgs multiplets.
In the following, we just pay attention to those used directly in this work, i.e. the mass spectra of leptons, gauge bosons and Higgses. Other parts have been mentioned in [23].
B. Mass spectra

Leptons
We use the Yukawa terms shown in [23] for generating masses of charged leptons, active neutrinos and heavy neutral leptons, namely where the notation ( T being the Dirac spinor and its charge conjugation, respectively. The Λ is some high energy scale. Remind that ψ L = P L ψ, ψ R = P R ψ where are the right-and left-chiral operators. The corresponding mass terms are This means that the active neutrinos are pure Majorana spinors corresponding to the mass Λ . This matrix can be proved to be symmetric [33] (chapter 4), therefore the mass eigenstates can be found by a single rotation expressed by a mixing matrix U that where V L ab , U L ab and V R ab are transformations between flavor and mass bases of leptons. Here unprimed fields denote the mass eigenstates. Remind that ν ′c aR = (ν ′ aL ) c = U ab ν c aR . The fourspinors representing the active neutrinos are ν c a = ν a ≡ (ν aL , ν c aR ) T , resulting the following equalities: ν aL = P L ν c a = P L ν a and ν c aR = P R ν c a = P R ν a . The upper bounds of recent experiments for the LFV processes in the normal charged leptons are very suppressed [7], therefore suggest that the two flavor and mass bases of charged leptons should be the same.
The relations between the mass matrices of leptons in two flavor and mass bases are where Y ν and Y N are Yukawa matrices defined as (Y ν ) ab = y ν ab and (Y N ) ab = y N ab . The Yukawa interactions between leptons and Higgses can be written according to the lepton mass eigenstates, where we have used the Marojana property of the active neutrinos: ν c a = ν a with a = 1, 2, 3. In addition, using the equality e c b P L ν a = ν a P L e b for this case the term relating with η ± in the last line of (31) is reduced to √ 2η + ν a P L e b .

Gauge bosons
It is simpler to write the charged gauge bosons in the form of W a T a with T a being the gamma matrices, namely The masses of these gauge bosons are: where we have used the relation v 1 = v 2 = v √ 2 and the matching condition of the W boson mass in 3-3-1 model with that of the SM.
The covariant derivatives of the leptons contain the lepton-lepton-gauge boson couplings,

Higgs bosons
• Singly charged Higgses. There are two Goldstone bosons G ± W and G ± V of the respective singly charged gauge bosons W ± and V ± . Two other massive singly charged Higgses have masses Denoting s θ ≡ sin θ, c θ ≡ cos θ, we get some useful relations The relation between two flavor and mass bases of the singly Higgses are  • CP-odd neutral Higgses. There are three Goldstone bosons G Z , G Z ′ and G ′ U 0 , and two massive CP-odd neutral Higgses H A 1 and H A 2 with the values of squared masses are The relations between the two bases are: • CP-even neutral Higgses. Apart from the three exactly massive Higgses shown in the ref. [22], the model predicts one more Goldstone boson G U and another massive Higgs.
The masses and egeinstates of these Higgses are The transformations among the flavor and the mass bases are where s α = sin α, c α = cos α defining by In the limit t ≪ 1 the expression of the lightest neutral even-CP Higgs is where both λ 1 and λ 2 must be positive to guarantee the vacuum stability of the potential (25). This Higgs is easily identified with the SM-like Higgs observed by LHC.

C. Couplings for LFV decay of the SM-like Higgs and the amplitude
From the detailed discussions on the particle content of the 3-3-1LHN, the couplings of SM-like Higgs needed for calculating LFVHD are collected in the table I.

Vertex
Coupling Vertex Couplinḡ Here we only consider the couplings the unitary gauge.
Matching the Feynman rules in the figure 2, we have the specific relations among the vertex parameters and the couplings in the 3-3-1LHN, namely for the exotic leptons and the active neutrinos, The expression of ∆ L is separated into two parts, namely from neutral exotic leptons and Similarly for the ∆ R we have Before going to the numerical calculation we remind that the divergent cancellations in two separate sectors of neutrinos and exotic leptons are presented precisely in the second subsection of the appendix B. Higgs self-couplings in the scalar potential: λ 1 , λ 2 , λ 12 and f . The first two free parameters we choose are the v 3 and mass of the H 2 given in (35). Then the f parameter can be determined by Another parameter that can be fixed is the mass of the neutral SM-like Higgs [5]  shown in (40) are roots of the equation giving a relation among λ 2 , λ 1 and λ 12 : Because the λ 1 , λ 2 and λ 12 are factors of quartic terms in the Higgs potential (25), they must satisfy the unbounded from below (UFB) conditions that guarantee the stability of the vacuums of the considering model. According to the ref. [42], these conditions are easily found as follows. Defining ρ † ρ + η † η = h 2 1 and χ † χ = h 2 2 , the quartic part of the Higgs potential (25) In the basis (h 2 1 , h 2 2 ) the V 4 corresponds to the 2 × 2 matrix that must satisfy the conditionally positive conditions as follows: In our calculation, apart from positive λ 1 and λ 2 we will choose λ 12 > 0 so that all conditions given in (49) are always satisfied.
To identify h 0 1 with the SM Higgs, the h 0 1 must satisfy new constrains from LHC, as discussed in [43]. Namely, the mixing angle α of neutral Higgses, defined in (42), should be constrained from the h 0 1 W + W − coupling. Following [43] the we can identify that −c α ≡ 1 + ǫ W where ǫ W = −0.15 ± 0.14 is the universal fit for the SM Higgs. This results the constraint of c α as By canceling a factor of t in (42), we have a simpler expression which shows that c α < 0 when m H 2 > can be written as If the lower constraint in (50) is not considered, m 2 H 2 can be arbitrary large when |c α | → 1. In contrast, the constraint (50) gives a consequence GeV and λ 1 is large enough. On the other hand, this relation will not hold if the custodial symmetry assumed in the Higgs potential (25) is only an approximation. Hence in the numerical calculation, for the general case we will first investigate the LFVHD without the constraint (50). This constraint will be discussed in the final.
Regarding to the parameters of active neutrinos we use the recent results of experiment.
In particularly, if the mixing parameters in the active neutrino sector are parameterized by Because U L has a small deviation from the well-known neutrino mixing matrix U M N P S so we ignore this deviation [34]. We will use the best-fit values of neutrinos oscillation parameters given in [35], ∆m 2 21 = 7.60 × 10 −5 eV 2 , ∆m 2 31 = 2.48 × 10 −3 eV 2 , sin 2 θ 12 = 0.323, sin 2 θ 23 = 0.467, sin 2 θ 13 = 0.0234, and mass of the lightest neutrino will be chosen in range 10 −6 ≤ m ν 1 ≤ 10 −1 eV, or 10 −15 ≤ m ν 1 ≤ 10 −10 GeV. This range satisfies the condition b m ν b ≤ 0.5 eV obtained from the cosmological observable. The remain two neutrino masses are m 2 ν b = m 2 ν 1 + ∆m 2 ν b1 . We note that the above case corresponds to the normal hierarchy of active neutrino masses. In the 3-3-1LHN, the inverted case gives the same result so we do not consider here.
The mixing matrix of the exotic leptons is also parameterized according to (52). In particularly it is unknown and defined as V L ≡ U L (θ N 12 , θ N 13 , θ N 23 ). If all θ N ij = 0, all contributions from exotic leptons to ∆ L,R will be exactly zero. In the numerical computation, we consider only the cases of maximal mixing in the exotic lepton sector, i.e. each θ N ij gets only the value of π/4 or zero. There are three interesting cases: i) θ N 12 = π/4 and θ N 13 = θ N 23 = 0; ii) θ N 12 = θ N 13 = θ N 23 = π/4; and iii) θ N 12 = θ N 13 = π/4 and θ N 23 = −π/4. The other cases just change minus signs in the total amplitudes, and do not change the final results of LFVHD branching ratios. For example the mixing matrix of first case is Our numerical investigation will pay attention to the first case, where the third exotic lepton does not contribute to the LFVHD decays. The two other cases are easily deduced from this investigation.
From the above discussion, we chose the following unknown parameters as free parameters: v 3 , m H 2 , λ 1 , λ 12 , m ν 1 and m Na (a = 1, 2, 3). The vacuum stability of the potential (25) results the consequence λ 1,2 > 0. In order to be consistent with the perturbativity property of the theory, we will choose λ 1 , |λ 12 | < O(1). The numerical check shows that the LFVHD branching ratio depends weakly on the changes of these Higgs self-couplings in this range. Therefore we will fix λ 1 = λ 12 = 1 without loss of generality. These values of λ 1 and λ 12 also satisfy all UFB conditions (49). In addition, the Yukawa couplings in the Yukawa term (27) should have a certain upper bound, for example in order to be consistent with the perturbative unitarity limit [36]. Because the vev v 3 generates masses for exotic leptons from the Yukawa interactions (28), following [10] we assume the upper bound of the lepton masses as follows After investigating the dependence of the LFVHD on the Yukawa couplings through the ratio m Na v 3 we will fixed m N 2 /v 3 = 0.7 and 2 corresponding to the two cases of lower and larger than 1 of the Yukawa couplings.
Unlike the assumption in [23] where f = v 3 /2, we treat f as a free parameter relating  [39], addressing directly for 3-3-1 models [19,40], where m Z ′ must be above 2.5 TeV. It is enough using an approximate relation of m Z ′ and v 3 :  The figure 5 shows the dependence of LFVHD on the mass of m H 2 . The first property we can see is that the LFVHD branching ratio always has an upper bound that decreases with increasing v 3 . In other word, it has an maximal value depending strictly on the constructive correlation of v 3 and m H 2 . But if the Yukawa couplings are small, this maximum seems never reach the value of 10 −6 . The case of the large Yukawa couplings is more interesting because maximal LFVHD can be asymptotic 10 −5 , provided that v 3 is small enough, see the right panel. general for many other models beyond the SM with the same class of particles. In numerical investigation the LFVHD in the case of maximal mixing between the first two exotic neutral leptons, we find that the branching ratio Br(h 0 1 → µτ ) depends the mostly on Yukawa couplings of neutral exotic leptons and the SU(3) L scale v 3 . For small y N ij ≃ 1, equivalently m N 2 /v 3 ≃ 0.7, this branching ratio is always lower than 10 −6 , and even that of about 10 −7 , the parameter space is very narrow. In contrast, with large Yukwa couplings, for example leptons, such as [20], can predict large LFVHD. So when calculating the LFVHD in SUSY versions, the non-SUSY contributions must be included. In contrast, the 3-3-1 models with light leptons [21] give suppressed signals of LFVHD, and the SUSY-contributions in [44] are dominant.
where i = 1, 2. In addition, D = 4 − 2ǫ ≤ 4 is the dimension of the integral. The notations M 0 , M 1 , M 2 are masses of virtual particles in the loops. The momenta satisfy conditions: i and C 0,1,2 are PV-functions. It is well-known that C i is finite while the remains are divergent. We define where γ E is the Euler constant and m h is the mass of the neutral Higgs. The divergent parts of the above scalar factors can be determined as 1 ] = Div[B 1 ] = Div[B We remind that the finite parts of the PV-functions such as B-functions depend on the scale of µ parameter with the same coefficient of the divergent parts.
The analytic formulas of the above PV-functions are: 0,1,2 = Div[B The b (1) 0 can be found in a very simple form in the limit p 2 where x k , (k = 1, 2) are solutions of the equation The final expression of b The B i 1 , B (12) i are calculated through the B 0 and A 0 functions, namely The C i functions can be found through the equation The C 0 function was generally calculated in [45], a more explicit explanation was given in [46]. In the limit p 2 1 , p 2 2 → 0, we get the following expression where both δ and δ ′ are positive and extremely small, x 0 and x 3 are defined as and x 1 , x 2 are solutions of the equation (A9). The limit of p 2 1 , p 2 2 = 0 will be used in our work, even when the loops contain active neutrinos with masses extremely smaller than these quantities, because of the appearance of heavy virtual particles. The explanation is as follows. The denominator in the first line of (A13) has the general form of D = Our calculation relates to the two following cases: • Only M 0 is the mass of the active neutrino, • M 1 = M 2 is the mass of the neutrino: We use the following result given in [45] R where i = 1, 2, 3 and Li 2 (z) is the di-logarithm defined by We also use the real values of x 0 to give the result η(−x i , 1 leading to Using the following equalities with any real A, B, δ, δ ′ positive real and extremely small; and This results the very simple expression of C 0 function where x 1,2 are solutions of the equation (A9), and x 0,3 are given in (A14). This result is consistent with that discussed on [31].
For simplicity in calculation we will also use other approximations of PV-functions where where x k is the two solutions of the equation (A9),

Appendix B: Calculations the one loop contributions
In the first part of this section we will calculate in details the contributions of particular contributions of diagrams shown in the figure 1 which involve with exotic neutral lepton N a , a = 1, 2, 3. From this we can derive the general functions expressing the contributions of particular diagrams.

Amplitudes
It is needed to remind that the amplitude will be expressed in terms of the PV-functions, so the integral will be written as where µ is a parameter with dimension of mass. This step will be omitted in the below calculation, the final results are simply corrected by adding the factor i/16π 2 . As an example in the calculation of contribution from the first diagram, we will point out a class of divergences that automatically vanish by the GIM mechanism. More explicitly for any terms which do not depend on the masses of virtual leptons, they will vanish because of the appearance of the factor a V L 1a V L * 2a = 0. The contribution from diagram 1a) is: where We can see that P 1 does not contain any divergent terms. The formula of P 2 is We can see that the terms like B The contribution from P 3 is Again all terms in the first and third lines do not contribute to the amplitude. But the four terms m 2 2 B (2) 1 , do contain divergences. The first two terms have divergent parts having the corresponding forms of (−m 2 2 ∆ ǫ ) and m 2 1 ∆ ǫ , which do not depend on the masses m a of the virtual leptons. Hence they also vanish by the GIM mechanism. The finite parts of these terms still contribute to the amplitude. The remain two terms include the most dangerous divergent parts. They have factors m 2 a which can not cancel by the GIM mechanism. We remark them by the bold and will prove later that they finally vanish after summing all diagrams. From now on we can exclude all terms that do not depend on the masses of virtual leptons.
the expression of the total contribution from the diagram 1a) is simply where E F V V L,R is defined in (4)  The contribution from diagram 1b) is: The contribution to the total amplitude is The contribution from diagram 1c) is: .
The contribution to the total amplitude is The contribution from diagram 1d) is:  (10) and (11), the contribution to the amplitude is The contribution from diagram 1e) is: The final result is written as where E V F F L,R are defined in (12) and (13).
The contribution from diagram 1f) is The final result is written as where E HF F L,R are defined in (14) and (15).
The contribution from diagram 1g) is: The contribution from diagram 1h) is: 1 − (2 − d)B The total amplitude from the two diagrams 1g) and 1h) is: We note that the divergence part in the above expression is zero. The final result is where E F V L,R are defined in (16) and (17).
The contribution from the diagram 1i) is: The final result is written as where E F H L,R are defined in (18) and (19). After calculating contributions from all diagrams with virtual neutral leptons N a we can prove that all divergent parts containing the factor m 2 a will be canceled in the total contribution. The details are shown below. For active neutrinos the calculation is the same.

Particular calculation for canceling divergence
In this section, for contribution of exotic neutral leptons N a we use the following relations And we concentrate on the divergent parts which are bolded in the expressions of the amplitudes calculated above. With the notations of the divergences shown in the appendix A, all of divergent parts are collected as follows, where It is easy to see that the sum over all factors is zero. Furthermore, it is interesting to see that the sums of the two parts having factor c α and √ 2s α independently result the zero values. From (41), the factor c α arises from the contributions of neutral components of η and ρ, while the s α factor arises from the contribution of χ.
For contribution of the active neutrinos, the two diagrams (b) and (c) of the fig.1 do not give contributions due to absence of the H − 2 H + 2 W couplings. Using the following properties We see again that sum of all divergent terms is zero.