Lepton flavor violating Higgs boson decays in seesaw models: new discussions

The lepton flavor violating decay of the Standard Model-like Higgs boson (LFVHD), h->\mu\tau, is discussed in seesaw models at the one-loop level. Based on particular analytic expressions of Passarino-Veltman functions, the two unitary and 't Hooft Feynman gauges are used to compute the branching ratio of LFVHD and compare with results reported recently. In the minimal seesaw (MSS) model, the branching ratio was investigated in the whole valid range 10^{-9}-10^{15} GeV of new neutrino mass scale m_{n_6}. Using the Casas-Ibarra parameterization, this branching ratio enhances with large and increasing m_{n_6}. But the maximal value can reach only order of 10^{-11}. Interesting relations of LFVHD predicted by the MSS and inverse seesaw (ISS) model are discussed. The ratio between two LFVHD branching ratios predicted by the ISS and MSS is simply m^2_{n_6}\mu^{-2}_X, where \mu_X is the small neutrino mass scale in the ISS. The consistence between different calculations is shown precisely from analytical approach.

I.

INTRODUCTION
After the Higgs boson was observed by ATLAS and CMS [1], the LFVHD has been searched experimentally [2], where upper bounds for branching ratios (Brs) of the decays h → µτ, eτ are order of O(10 −2 ). Signals of LFVHD at future colliders have been discussed, where sensitivities for detecting these channel decays are shown to be 10 −5 in the near future [3]. Up to now, the lepton flavor violating (LFV) decays of the standard-model-like and new Higgs bosons have been investigated in many models beyond the standard model (SM) [4][5][6][7][8][9][10][11][12][13][14]. Among them, the MSS [15] is the simplest that can explain successfully the recent neutrino data. Naturally, the mixing between different flavor neutrinos leads to many LFV processes from loop corrections. But it predicts very suppressed branching ratios (Br) of LFV decays of charged leptons. Recent studies on the Br of LFVHD were also shown to be very small [6]. In contrast, the ISS [16], another simple extension of the SM, predicts much larger values of LFV branching ratios, including those of LFVHD [7,8]. In fact, the Br of LFVHD in the ISS were calculated in many different ways in order to guarantee the consistence of the LFVHD amplitudes.
We stress that understanding the mechanism for generating loop corrections to Brs of LFVHD in simple models like the MSS and ISS is very important for studying LFVHD processes in other complicated models. That is why LFVHD predicted by these two models were discussed in many works, for example [4][5][6][7][8][9]. In the ISS, recent results in [7] showed that branching ratios of LFVHD increase with increasing values of very heavy neutrino masses when the Casas-Ibarra method [17] was applied to formulating the Yukawa couplings of heavy neutrinos 1 . But the Brs are always constrained by upper bounds because of the perturbative limit of the Yukawa couplings. Using the mass insertion approximation, a recent study [8] also calculated the Br of LFVHD in the ISS model in both unitary and 't Hooft Feynman, where previous results in [7] were confirmed to be well consistent in the region of parameters containing large new neutrino mass scale m n 6 . The above discussions indicate that although one-loop contributions in both MSS and ISS arise from the same set of Feynman diagrams, the two models predict very different Br values. The reason is the appearance of a small mass scale µ X in the ISS, which gives tiny contributions to the heavy 1 We thank Dr. E. Arganda for this comment neutrino masses, but affects strongly on the neutrino mixing matrix. Hence there should exist simple relations between two expressions of Brs predicted by the two models. These interesting relations were not discussed previously, therefore will be focused in this work. We will show that if m n 6 is large enough, the ratio between Brs of LFVHD of the ISS and MSS is order of m 2 n 6 µ −2 X , enough to explain clearly the LFVHD difference between two models. Regarding the MSS, LFVHD was discussed mainly in ranges of 10 2 − 10 7 GeV [4,6], while the valid range of the new neutrino mass scale is from O(10 −9 ) GeV to O (10 15 ) GeV. In addition, a good estimation made in Ref. [4] suggested that the Br may enhance with increasing masses of heavy neutrinos, even when the Casas-Ibarra parameterization is used. We note that this parameterization are now still widely used to investigate the signal of seesaw models at recent colliders [18]. As a result, possibilities that large Brs of LFVHD may exist in ranges of new neutrino mass scales that were not mentioned previously.
Therefore, studies the LFVHD in the whole valid range as well as new approaches to compare well-known results and confirm consistent analytic formulas for calculating Br of LFVHD in seesaw models are still interesting and necessary. These are main scopes of this work.
In particular, in order to guarantee the stability of numerical results at very large values of m n 6 , LFVHD processes will be computed using analytic expressions of Passarino-Veltman functions (PV functions) given in ref. [13]. Using a mathematica code based on these functions, we found that it is much easier and more convenient to increase the precision than using available numerical packages such as Looptools [27]. This makes our calculation different from all previous works. In addition, the one-loop contributions to LFVHD in both unitary and 't Hooft Feynman gauges will be constructed using notations in [13]. Then we cross-check the consistence between total amplitudes calculated in two gauges, and the ones established in previous works [4,6,7]. A detailed checking divergence cancellation will be presented analytically. For the MSS, after showing that Br of LFVHD is suppressed with small m n 6 , we will pay attention mainly to the region with large m n 6 . To guarantee the consistence of our investigation on LFVHD in the MSS, the connection between analytic formulas of LFVHD amplitudes in the two models MSS and ISS will be discussed deeply.
In this work, Yukawa couplings of new neutrinos are only investigated following the Casas-Ibarra parameterization [17]. This parameterization was used to investigate independently LFVHD processes predicted by the MSS and ISS in Refs. [6,7], where other important properties of LFVHD were presented in details.
Our work is arranged as follows. Sec. II establishes notations and couplings of a general seesaw model needed for studying LFVHD. In Sec. III, we construct LFVHD amplitudes in two unitary and 't Hooft Feynman gauges using notations of PV functions given in [13].
Then we prove the divergent cancellation and the consistence between two expressions of the LFVHD amplitudes. In Sec. IV, we show the choice of parameterizing the neutrino mixing matrices. After that, the Brs of LFVHD are numerically investigated. We will focus on new results of LFVHD in the MSS, and interesting relations between the Brs predicted by two models MSS and ISS. Sec. V summarizes new results of this work.
In calculation, we will use a general notation of four-component (Dirac) spinor, n i (i = 1, 2, .., K + 3), for all active and exotic neutrinos. Specifically, a Majorana fermion n i is defined as n i ≡ (n L,i , (n L,i ) c ) T = n c i = (n i ) c . The chiral components are n L,i ≡ P L n i and n R,i ≡ P R n i = (n L,i ) c , where P L,R = 1±γ 5 2 are chiral operators. The similar definitions for the original neutrino states are ν a ≡ (ν L,a , (ν L,a ) c ) T , N I ≡ ((N R,I ) c , N R,I ) T , and ν = (ν, N ) T .
The relations in (4) are rewritten as follows, where more precise expressions are ν L,a = P L ν a = U ν * ai n L,i , (N R, We emphasize that the signs in D µ will result in signs of couplings hG ± W W ± and e a ν a W − . Correspondingly, the lepton flavor violating (LFV) couplings of W ± boson to leptons are, where a = 1, 2, 3; and j = 1, 2, ..., K + 3.
The Yukawa couplings that contribute to LFVHD are Using (M D ) aI = M ν a(I+3) , and N R,I = ν R,(I+3) , the last line in (7) changes in to the new form, 1 v hn i M ν a(I+3) U ν ai U ν (I+3)j P R + M ν * a(I+3) U ν * (I+3)i U ν * aj P L n j . It can be proved that which was given in [6,7]. A proof is as follows, based on the following properties of M ν and U ν defined in Eqs. (2) and (3), The first term in the left hand side of Eq. (8) will change exactly into the second term in the right hand side of Eq. (8), after mediate steps of transformation, namely From (10), the second term in the left hand side of (8) can be derived easily, Finally, the Feynman rule for the vertex (8) with two Majorana leptons hn i n j must be expressed in a symmet- [4,21] . The couplings relating with G ± W are proved the same way, namely The vertices relating to LFVHD are collected in Table I. We note that the coupling hG + W G − W in Table I is consistent with that given in [8,25].
The effective Lagrangian of the LFVHD is written as h.c., where ∆ L,R are scalar factors arising from loop contributions. The partial decay width where m h m 2 , m 3 and m 2 , m 3 being masses of muon and tau, respectively. The on-shell conditions for external momenta are p 2 a = m 2 a (a = 2, 3) and GeV. Next, ∆ L,R with be calculated at one-loop level, in two gauges of unitary and 't Hooft Feynman.

Vertex coupling
Vertex coupling The p 0 , p + and p − are incoming momenta of h, G + W and G − W , respectively.

A. Amplitude in the unitary gauge and divergence cancellation
In the unitary gauge, the Feynman diagrams for a decay h → e − a e + b (a < b) are presented in Fig. 1. The loop contributions are written as ∆ three terms come from private contributions of diagrams 1a), 1b), and sum of contributions from two diagrams c) and d), respectively. The analytic expressions of contributions from the three diagrams 1a), c), and d) can be derived directly from [13], except the diagram 1b) containing the coupling hn i n j . An analytic expression of ∆ L,R is derived in appendix C. We have used Form [23] to cross-check our results. In addition, the total ∆ L,R is consistent with the result calculated in the 't Hooft Feynman gauge, as we will show later. Expressions of LFVHD contributions in the unitary gauge are and Regarding ∆ L,R , the contributions from B (1) Divergence cancellation in the total amplitude is explained as follows. From divergent parts of the PV functions in Appendix A, the divergent parts of ∆ (a) where the unitary property of U ν is used to cancel the second term of Div[∆ In the 't Hooft Feynman gauge, there are ten form factors F (i) L,R , (i = 1, 2, .., 10) corresponding to ten diagrams shown in Fig. 1 of Refs. [6,7]. The total contribution is L,R in terms of PV functions defined in [13] are as follows, 1 , F L , 1 , F where B ai = U ν * ai , B * bj = U ν bj , C ij = 3 c=1 U ν ci U ν * cj , and D = 4−2 is the integral dimension defined in Appendix A. Although F L,R and F (9) L,R contain B-functions, they are finite because of the GIM mechanism. Hence it can be replaced with D = 4. Because B (12) L do not depend on m n i , therefore vanish because of the GIM mechanism. They will be ignored from now on.
Although our notations of PV functions are different from those in [6,7], transformations between two sets of notations are, (see a detailed proving in Appendix B) The PV functions used in our work were checked to be consistent with Looptools [27], see details in [14]. The differences between our results and those shown in [7] are minus signs in F L,R and F L,R . Our formulas are consistent with the results presented in Ref. [8] 3 , where the authors confirmed that these signs do not affect the results given in Ref. [7]. Now we will check the consistence between total amplitudes calculated in two gauges. Regarding to triangle diagrams with two internal neutrino lines, the deviation of contributions in two gauge are determined as follows, where useful equalities of B-functions are used [22]. In addition, C ij in the first line of (22) is simplified using the same trick given in (16). Similarly, other deviations are where B 0,1,2 ≡ B 0,1,2 (m 2 n i , m 2 W ). Then, it can be seen easily that δ 1 + δ 2 + δ 3 = 0. Hence, the total amplitudes calculated in two gauges are the same. IV.

A. Parameterization the neutrino mixing matrix
To start, we consider a general expression of the neutrino mixing matrix U ν [19], where O is a 3 × K null matrix, U and V are 3 × 3 and K × K unitary matrices, respectively.
The Ω is a (K + 3) × (K + 3) unitary matrix that can be formally written as where R is a 3 × K matrix where absolute values of al elements are smaller than unity. The unitary matrix U = U PMNS is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [30].
The mass matrices of neutrinos are written as follows, M N = diag(m n 4 , m n 5 , ..., m n K+3 ), where m n i is the physical masses of all neutrinos, where ∆m 2 a1 = m 2 na − m 2 n 1 (a = 2, 3). In this work, other parameters will be fixed as δ = α = β = 0.
The condition of seesaw mechanism for neutrino mass generation is |M D | |M N |, where |M D | and |M N | denote characteristic scales of M D and M N , resulting in useful relations 5 [19], Based on the second relation in (29), the matrix M D can be parameterized via a general K × 3 matrix ξ, which satisfies the only condition ξ T ξ = I 3 [6,17,19], namely The mixing matrix in the ISS model considered in ref. [7] can be found approximately using the above general discussion with K = 6. Relations of notations between two parameterizations in [7] and [19] are where O is the 3 × 3 matrix with all elements being zeros. From the definition of the inverse where M is defined as M = M R µ −1 X M T R [7]. From (29), we then find that [19] 5 We thank LE Duc Ninh for pointing out factors 1/2 in the last relation in (29).
These two expressions are consistent with those given in [7,19], giving a parameterization of m D as follows, where U M satisfies M = U * M diag(M 1 , M 2 , M 3 )U † M and ξ is a complex orthogonal matrix satisfying ξ ξ T = I 3 . The mixing matrix U ν now is a 9 × 9 matrix.
In order to compare and mark relations between LFVHD in two MSS and ISS models, we will pay attention to only simply cases of choosing parameters. In the MSS model, the choice is ξ = U N = I 3 , leading to following simple expressions of Eqs. in (29), namely In the ISS model, from (34) we see that m D is parameterized in terms of many free parameters, hence it is enough to choose that µ X = µ X I 3 . This parameter is a new scale making the most important difference between the neutrino mixing matrices in the ISS and MSS. We also assume that We can see that bothM R (ISS) and M N (MSS) play roles as exotic neutrino mass scales. Therefore, they are identified as neutrino masses in both models,M R = M N = diag(m n 4 , m n 5 , m n 6 ). The differences between two models now are two mixing matrix V in (36) and R, and the µ X scale, which does not appear in the MSS model. The µ X plays special roles in the ISS model via its appearance in the second sub-matrix of the mixing matrix R given in (33). A simple relation between largest elements of R matrices in two models is where m n 6 now is considered as exotic neutrino mass scale, m n 4 ≤ m n 5 ≤ m n 6 . The relation (37) is the main reason that explains why the Br of LFVHD predicted from the ISS is much larger than that from the MSS.
In the following, we will discuss on LFVHD in the MSS model. The results of LFVHD in the ISS model can be derived from discussion in the MSS model based on (37).

B. Discussion on LFVHD
In the MSS model, our investigation will use three physical masses of exotic neutrinos, m n 4,5,6 , as free parameters. The matrix M D can be derived from relations (30) (29), which suggest that m n 6 × m n 3 |M D | 2 < 6π × 174 2 , because of the perturbative limit of the Yukawa couplings Y ν,ij [7]. Combing with the active neutrino data given in (28), where at least one active neutrino mass is not smaller than ∆m 2 31 = 5×10 −11 GeV, we get an upper constrain, m n 6 < 8 × 10 15 GeV, when m n 1 ∆m 2 31 . The lower constrain is m n 6 > |M D | > m n 3 > 5 × 10 −11 GeV. Numerical illustrations are shown in The left panel of Fig. 2 presents Br(h → e a e b ) as functions of m n 6 . Unlike previous works such as [4,6], heavy neutrinos masses were not considered at the interesting scale above 10 10 GeV, where leptogenesis can be successful explained in the MSS frame work [29].  4 , when the matrix Ω is calculated up to O(R 2 ). This will lead to the maximal values of Br(h → µτ ) ≤ 10 −11 , the same order with large m n 6 ∼ O(10 15 ) GeV. If the matrix Ω is calculated more exactly, the Br(h → µτ ) will decrease significantly with small m n 6 , but will not change with large m n 6 . This can be explained from the conditions of the matrix Ω, which is written in terms of the power series in R. If m n 6 is small, R ∼ |m ν |/m n 6 will be large as m n 6 → |M D | → |m ν |. The calculation will be less accurate with smaller power k included in Ω. We consider more cases of U ν where the matrix Ω in (25) is considered up to order O(R 8 ). We conclude that the Br(h → µτ ) is very suppressed with small masses of exotic neutrinos. In contrast, large m n 6 results in |R| 1. Therefore, it is enough to consider the mixing matrix U ν with order of O(R 2 ) in the region where m n 6 ≥ 0.1 GeV. In conclusion, to find large Br(h → µτ ), we just consider the region with large m n 6 .
To explain why large Br(h → µτ ) corresponds to large m n 6 , we pay attention to the properties of the mixing matrix U ν , the PV-functions and factors relating with them in the expressions of ∆ will give dominant contributions. The PV functions containing m 2 n I will have the following properties: B 0,1,2 (m 2 n i ) = O(10), C 0,1,2 (m 2 n i ) ∼ ln(m 2 n 6 )/m 2 n 6 . Hence the largest contributions will come from m 2 n 6 B 0,1,2 ∼ m 2 n 6 in ∆ L,R will results in the following factors: . There are new factors in the ∆ Hence the largest contribution to the total gives ∆ L,R ∼ m n 6 with very large m n 6 , implying Br(h → µτ ) ∼ m 2 n 6 . The correlations between terms with and without factors m 2 n i are shown in the Fig. 3. Terms without factors m 2 n i are dominant with tiny m n 6 but they are very suppressed with large m n 6 .
The above discussions lead to new interesting results for LFVHD predicted by the MSS model, which were not concerned previously: i) the Br can reach values of order 10 −11 with large values of heavy neutrino masses satisfying the perurbative limit; ii) the Br enhances We realize that the property of Br(h → µτ ) ∼ m 2 n 6 agrees very well with the approximate expression shown in [4]. In particular, relating with active-heavy neutrino mixing elements in U ν . We believe that large values of the Br predicted in [4] arise from the reason that recent neutrino oscillation data could not be applied at that times. The numerical values of F N chosen in [4] may keep large contributions that should vanish because of the GIM mechanism.
Although the maximal Br of LFVHD predicted by the MSS is much smaller than the prediction from the ISS model given in [6,7], the behave of the curve presenting Br(h → µτ ) shown in Fig. 3  4. × 10 -9 6. × 10 -9 8. × 10 -9 Br(h→μτ) [10 -8 ] a+c+d mass scale m n 6 can be as large as O( 10 15 ) GeV, values of m n 6 in the ISS are constrained by relation (33), i.e. m 2 Hence, small µ X will give small upper bounds of m n 6 , and large Br(h → µτ ) will depend complicatedly on these two parameters. The left panel of Fig. 4 shows possible values of Br(h → µτ ) in the allowed regions of µ X and m n 6 . Our numerical results are well consistent with previous work [7]. In addition, by adding a factor There is an interesting relation between two LFVHD amplitudes calculated in the two models, as drawn in the right panel of Fig. 4. Here, |∆ ISS R |µ X m −2 n 6 and |∆ MSS R |m −1 n 6 are considered as functions of m n 6 . We have checked numerically that |∆ ISS R |µ X m −2 n 6 does not depend on µ X , and consistent with conclusion in [7]. It can be seen as follows. The dependence of m D and R ISS on M R and µ X can be separate into two parts. The first is the correlation between elements of these matrices in order to give correct experimental values of active neutrino data. And the second is the simple dependence on the scales of m n 6 and µ X . In the ISS, R ISS aI = U ν a(I+3) ∼ µ −1/2 X and do not depend on m n 6 . Now, if we pay attention to the region with large m n 6 , the terms like m 2 n i B 0,1,2 are dominant contributions to ∆ L,R because of the factors m 2 n i . As a result, ∆ (a+c+d) L,R containing a factor U ν * ai U ν bi ∼ µ −1 X will give an overall factor µ −1 X m 2 n 6 . Hence ∆ (a+c+d) L,R µ X m −2 n 6 may be constant, following the property of B-functions. On the other hand, ∆ L,R are still divergent, terms with µ −2 X must vanish in order to guarantee a finite ∆ L,R . This results in a common factor µ −1 X m 2 n 6 for ∆ L,R . In the right panel of Fig. 4, values of µ X m −2 L,R correspond to ∆ = 0. But we checked numerically that µ −1 X m 2 n 6 ∆ L,R is independent with ∆ . In addition, we can see that µ X m −2 L,R always have opposite signs, which is consistent with the fact that divergences contained in them are really canceled. Two absolute contributions from ∆ We can also estimate the maximal value of Br(h → µτ ) based on the numerical result shown in Fig. 4. If m n 6 ≥ 10 5 GeV, we have ∆ R 10 −24 µ −1 X m 2 n 6 , where small ∆ L is ignored. Equivalently, we have Br(h → µτ ) 10 −45 µ −2 X m 4 n 6 . The condition of perturbative limit gives m 2 n 6 × 5 × 10 −11 /µ X = |m D | 2 ≤ 174 2 × 6π, leading to µ −2 X m 4 n 6 ≤ O(10 36 ). Hence in the region of lagre m n 6 ≥ 10 5 GeV, Br(h → µτ ) can reach maximal value of O(10 −9 ). If m n 6 < 10 5 GeV, the allowed region in the left panel of Fig. 4 shows that Br(h → µτ ) can reach values of O(10 −7 ) only if m n 6 is few TeV, µ X is order of 10 −9 GeV, and m D gets values very close to the perturbative limit.

V. CONCLUSION
In this work, the LFVHD in the MSS and ISS models have been discussed where we have focused on new aspects that were not shown in previous works. We calculated the amplitude of the LFVHD using new analytical expressions of PV-functions discussed recently. From this we have checked the consistence of our results in many different ways: comparing them with results of previous works, calculating in two gauges of unitary and 't Hooft-Feynman, checking analytically the divergent cancellation of the total amplitude. In the MSS framework, we investigated numerically the Br(h → µτ ) in the valid and large range of exotic neutrino mass scale, from 10 −10 GeV to 10 16 GeV. When applying the Casas-Ibarra parameterization to Yukwa couplings of heavy neutrinos, we found a new result that Br(h → µτ ) ∼ m 2 n 6 with large m n 6 , because the mixing matrix elements affecting mostly the LFVHD amplitude by factors of m −1/2 n 6 . But in the valid region of perturbative requiring m n 6 < 10 16 GeV, the Br(h → µτ ) reaches maximal values of O(10 −11 ), still far from the recent experimental consideration. Anyway, this may be a hint to improve the MSS to more relevant models predicting higher values of Br(h → µτ ), for example the ISS. In this model, the largest mixing factors contributing to LFVHD amplitude do not depend on the exotic neutrino mass scale m n 6 but consist of a factor µ −1 X . Hence, if two models have the same neutrino mass scale, and the neutrino mixing matrices obey the Casas-Ibarra parameterization, there will be a very simple relation that BR ISS (h → µτ )/BR MSS (h → µτ ) m 2 n 6 µ −2 X . This explains why the signal of LFVHD in the ISS is extremely significant than that in MSS. But the perturbative condition does not allow both large m n 6 and small µ X , which can predict large Br(h → µτ ). Hence, maximal Br(h → µτ ) is still O(10 −7 ) with few TeV of heavy neutrino mass scale. Our discussion on LFVHD of the MSS suggests that Br(h → µτ ) may be large in the extended versions of the MSS which allow very large m n 6 .
Finally, although we presented here a different way to calculate the LFVHD, our numerical results for the ISS are well consistent with those noted in previous works [7,8].
Acknowledgments LTH thanks Professor Thomas Hahn for useful discussions on Looptools. We are especially thankful Dr. Ernesto Arganda and authors of Refs. [6][7][8] for important comments to correct our calculations and help us understand more deeply the LFVHD in the ISS. We if |y| ≥ 10.
The B-functions now can be expressed in terms of f n (y), namely f 0 (y ij ), Finally, the B In our work above use the following notations, m 1 ≡ m a , m 2 ≡ m b , p 1 ≡ p a and p 2 ≡ p b .

Appendix B: Matching with notations in previous works
This section will show the equivalence given in (21). We recall notations used in [6][7][8] as follows. The external momenta are p 1 ,(−p 2 ), and p 3 for ingoing Higgs boson, outgoing leptons e a and e b ,respectively. The prime is used to distinguish from the notions that were used in our work, especially those given in Sec. A. Three denominators of the propagators are D 0 = k 2 −m 2 1 , D 1 = (k+p 2 ) 2 −m 2 2 and D 2 = (k+p 1 +p 2 ) 2 −m 2 3 . The one-lopp-three-point functions are defined as, The equivalence between above notations with those given in Sec. A are p 1 = p 1 + p 2 , p 2 = −p 1 , m 1,2,3 = M 0,1,2 . As a result, we get D 0,1,2 = D 0,1,2 , leading to C 0 = C 0 and C µ = C µ . But the scalar factors C 11,12 and C 1,2 are different, namely C µ = C 11 (−p 1µ ) + C 12 (p 1µ + p 2µ ) = (C 12 − C 11 )p 1µ + C 12 p 2µ . Matching this with definition of C µ defined in Sec. A. We obtain the equivalence for C 1,2 in (21). Other B-functions is proved easily so we omit here.

Appendix C: Form factors in unitary gauge for LFVHD
The contribution from diagram in Fig. 1b) to the LFVHD amplitude is C ij m n i P L + m n j P R + C * ij m n j P L + m n i P R The final result is 1 + m 2 n j B 0 + 2 m 2 n i m 2 n j + m 2 W (m 2 n i + m 2 n j ) C 1 −(m 2 n i m 2 b + m 2 n j m 2 a )C 1 − m 2 n j m 2 W C 0 1 + m 2 n i B [1] The ATLAS Collaboration, Phys.Lett. B 716, 1 (2012); The CMS Collaboration, G. Aad et al, Phys. Lett. B 716, 30 (2012).