Charged-lepton decays from soft flavour violation

We consider a two-Higgs-doublet extension of the Standard Model, with three right-handed neutrino singlets and the seesaw mechanism, wherein all the Yukawa-coupling matrices are lepton flavour-diagonal and lepton flavour violation is soft, originating solely in the non-flavour-diagonal Majorana mass matrix of the right-handed neutrinos. We consider the limit $m_R \to \infty$ of this model, where $m_R$ is the seesaw scale. We demonstrate that there is a region in parameter space where the branching ratios of all five charged-lepton decays $\ell_1^- \to \ell_2^- \ell_3^+ \ell_3^-$ are close to their experimental upper bounds, while the radiative decays $\ell_1^- \to \ell_2^- \gamma$ are invisible because their branching ratios are suppressed by $m_R^{-4}$. We also consider the anomalous magnetic moment of the muon and show that in our model the contributions from the extra scalars, both charged and neutral, can remove the discrepancy between its experimental and theoretical values.


Introduction
In this paper we resume an old idea of two of us [1]: in a multi-Higgs-doublet model furnished with three right-handed neutrino singlets and the seesaw mechanism [2], lepton flavour may be conserved in the Yukawa couplings of all the Higgs doublets and violated solely in the Majorana mass terms of the right-handed neutrinos ν R ( = e, μ, τ ), viz. in where C is the charge-conjugation matrix in Dirac space and M R is a non-singular symmetric matrix in flavour space. Since L ν R mass has dimension three, the violation of the individual lepton flavour numbers L and of the total lepton number L = L e + L μ + L τ is soft. Thus, in our framework L ν R mass is responsible for In this context, lepton flavour-violating processes were explicitly investigated at one-loop order in ref. [3] and the following property of our framework was discovered. Let m R denote the seesaw scale -the scale of the square roots of the eigenvalues of M R M * R -and n denote the number of Higgs doublets; it was found in ref. [3]  iii. however, if n ≥ 2, the amplitudes for lepton flavour-violating shell) neutral scalar, approach a nonzero limit when m R → ∞.
The non-decoupling of the seesaw scale in − an effect of the one-loop diagrams with neutrinos and charged scalars in the loop.
As a consequence, in our framework the amplitude of the process μ − → e − e + e − , which derives from μ − → e − S 0 b * followed by S 0 b * → e + e − , is unsuppressed in the limit m R → ∞. The same happens to the amplitudes of the four τ − decays of the same type.

Table 1
The experimental bounds on the branching ratios of some lepton flavour-changing decays. All the bounds are at the 90% CL. The first bound is from ref. [5], all the other bounds are from ref. [6].
It is important to stress that in our model the amplitude for μ − → e − e + e − is unsuppressed because of the penguin diagrams for neutral-scalar emission in the μ − → e − conversion; indeed, the penguin diagrams for either γ or Z 0 emission vanish in the limit m R → ∞. Thus, our model for lepton-flavour violation differs from, for instance, the scotogenic model discussed in ref. [4], wherein it is precisely the γ and Z 0 penguins that are instrumental in μ − → e − e + e − and in muon-electron conversion in nuclei. 1 Let us estimate a lower bound on m R by using the experimental bounds, given in Table 1, 2 on the radiative decays 1 → 2 γ . The amplitude for any such decay has the form where ε ρ is the polarisation vector of the photon, u 1 and u 2 are the spinors of ± 1 and ± 2 , respectively, and γ L and γ R are the projectors of chirality. The decay rate is given, in the limit Knowing that A L and A R are suppressed by m −2 R , one may estimate, just on dimensional grounds, that Using the first two bounds of Table 1 together with the experimental values for the masses and widths of the μ and τ , one may then derive the lower bounds m R 50 TeV from μ + → e + γ and m R 2 TeV from τ − → e − γ .
Thus, in the framework of ref. [3], if we take m R 500 TeV then the radiative decays 1 → 2 γ are invisible in the foreseeable future. On the other hand, because of the nonzero limit of the am- are unsuppressed when m R → ∞. It is the purpose of this paper to investigate those decays numerically in the framework of ref. [3], assuming m R to be so large that the radiative charged-lepton decays are invisible. Then, m R is also much larger than the masses of the scalars in the model, which we assume to be in between one and a few TeV. 1 In this paper we do not address muon-electron conversion in nuclei because in order to do it we would need to specify, through additional assumptions, the Yukawa couplings of the extra Higgs doublets to the quarks. This is so because in our model muon-electron conversion in nuclei occurs -in the limit m R → ∞through μ − → e − S 0 b * followed by the S 0 b * coupling to quarks. 2 Two new experiments are planned in search for lepton flavour-violation at the Paul Scherrer Institute. The MEG II experiment [7] plans a sensitivity improvement of one order of magnitude for μ + → e + γ . The Mu3e experiment [8], which is in the stage of construction, aims at a sensitivity for BR μ + → e + e − e + of order As a sideline, in this paper we also consider the contributions of both the neutral and charged scalars to the anomalous magnetic moment a of the charged lepton , with particular emphasis on a μ .
In order to keep the number of parameters of the model at a minimum, we restrict ourselves to just two Higgs doublets. Anticipating our results, we find that all five decays 1 → 2 + 3 − 3 may well be just around the corner, while at the same time the contributions of the non-Standard Model (SM) scalars of the model can make up for the discrepancy a exp μ − a SM μ of the anomalous magnetic moment of the muon. This paper is organised as follows. In section 2 we recall some results of ref. [3]. We then specialise to the case of just two Higgs doublets in section 3. We present the formulas for the contribution of the non-SM scalars to a in section 4. Section 5 is devoted to a numerical simulation. In section 6 we summarise and conclude.

The effective lepton flavour-violating interaction
The framework of ref. [3] assumes an n-Higgs-doublet setup wherein the violation of the family lepton numbers L is soft. The corresponding Yukawa Lagrangian has the form The basic assumption is the matrices k and k are diagonal, ∀ k = 1, . . . ,n, (6) as is already implicit in equation (5). In that equation, the Higgs doublets and the left-handed-lepton gauge doublets are given by respectively.
The scalar mass eigenfields S + a and S 0 b are related to the ϕ + k and ϕ 0 respectively [9]. The vacuum expectation values (VEVs) are v k √ 2.
The unitary n × n matrix U diagonalises the Hermitian mass matrix of the charged scalars. The 2n × 2n real orthogonal matrix Ṽ , which diagonalises the mass matrix of neutral scalar fields, is written as [9] The matrix V is n × 2n. We number the scalar mass eigenfields in such a way that S ± 1 = G ± and S 0 1 = G 0 are the Goldstone bosons. If there is only one Higgs doublet, i.e. when n = 1, the matrix V is simply V = (i, 1) in the phase convention where v 1 > 0, and S 0 2 is the Higgs field of the SM.
We define the diagonal matrices According to ref. [3], in the limit m R → ∞, where m R is the seesaw scale, the flavour-changing interactions of the physical neutral scalars S 0 b , induced by loops with charged scalars and neutrinos, are given by Note that the summation over b begins with b = 2, i.e. it excludes the Goldstone boson S 0 1 . The coefficients A b L,R 1 2 were computed in ref. [3]. Let us define the 3 where m 4,5,6 are, in the limit m R → ∞, the masses of the heavy neutrinos. We next define where μ is a mass scale which is arbitrary because of the unitarity of U R . Finally, we define the flavour space matrices A 1,2,3 as where m i is the mass of the charged lepton i and We note that, in every multi-Higgs-doublet model, it is possible to choose a basis for the scalar doublets such that only one of them, say φ 1 , has nonzero VEV: This basis is called the 'Higgs basis'. In it, from equation (10), * With equations (18) one finds that, in the sum over k in equation (16), the term with k = 1 gives a null contribution. Thus, in the Higgs basis, the contribution to A b 1 2 proportional to V * 1b is identically zero. In particular, if there is only one Higgs doublet, i.e. in the SM, A b 1 2 = 0, viz. when n = 1 there are no effective lepton flavourviolating interactions of the neutral scalar in the limit m R → ∞.

The decay rate
We write the decay amplitude for − where, from equations (11) and (19), In equations (22), M b is the mass of S 0 b . In the scalar propagators, we have neglected the four-momentum of the + 3 − 3 subsystem. With the amplitude in equation (21), the decay rate is given by We have neglected the masses of the final charged leptons in the kinematics.
In equation (21) one must antisymmetrise the amplitude with respect to − 2 and in the kinematics one must insert an extra factor 1/2. The final result is

Two Higgs doublets
From now on we assume n = 2, i.e. a two-Higgs-doublet model.
In the Higgs basis, the VEVs are given by where v ≈ 246 GeV is real and positive. Thus, according to equation (8), Moreover, the matrix U is the 2 × 2 unit matrix, i.e. ϕ + is the charged Goldstone boson and ϕ + 2 = S + 2 is the physical charged scalar. According to the notation of ref. [9], the 4 × 4 orthogonal matrix Ṽ of equation (9), which diagonalises the mass matrix of neutral scalar fields, is given bỹ for b = 2, 3, 4. We parameterise the flavour-diagonal Yukawa coupling matrices as Therefore, from equations (14) and (16), As demonstrated at the end of section 2.1, in A b 1 2 the term proportional to V * 1b vanishes.
We now make the further assumption that φ 1 is just identical with the Higgs doublet of the SM; this means that S 0 2 is exactly like the SM Higgs boson. This choice relieves us from having to take into account the experimental restrictions on the couplings of the SM Higgs boson, which become automatically fulfilled. We now have where S + 1 = G + and S 0 1 = G 0 are the Goldstone bosons. This means that we choose R 11 = 1, whence it follows that R can be written Thus, from equation (26), From equation (28), and, from equation (30a), The decay rates are then The decay rates depend on the masses M 3 and M 4 of the non-SM neutral scalar fields S 0 3 and S 0 4 , respectively. There is no dependence on the phase α. In equation (37a), 2 = 3 is understood. (38c) Lines (38a) and (38b) derive from a loop with and either S 0 3 or S 0 4 ; the photon line attaches to . Line (38c) comes from a loop with S ± 2 and light neutrinos, wherein the external photon attaches to S ± 2 ; in that line, μ 2 denotes the mass of S ± 2 . We have dropped all the terms proportional to m −2 R , including in particular the contributions from the loop with S ± 2 and heavy neutrinos. For the coupling of the charged scalars to the charged leptons we refer the reader to ref. [3].
The right-hand side of equation (40) is dominated by the two terms with logarithms. One readily sees that the terms with M 4 and μ 2 give negative contributions to a (S) μ (assuming γ 2 μ to be positive), while the term with M 3 gives a positive contribution; since a exp μ − a SM μ is positive, we would like the term with M 3 to dominate over the other two; this is achieved with M 3 < M 4 . Taking for instance M 3 = 1 TeV, M 4 = μ 2 = 2 TeV, 5 and γ μ = 1.7, we find a (S) μ = 258 × 10 −11 , which is of the right sign and absolute value to explain the discrepancy (39). We conclude that our model can, using reasonable parameters, fill the gap between a exp μ and a SM μ . The experimental AMM of the electron is in good agreement with the SM prediction for a e . We must therefore check that the non-SM scalars of our model give an a (S) e smaller than the experimental error 2.6 × 10 −13 [6] of a e . We might of course simply take γ e = 0, but this would eliminate e.g. the decay μ − → e − e + e − , which we would like to have close to its experimental upper limit. So we use instead the same scalar masses as before and choose 4 See also ref. [11] for a recent review. 5 Our choice M 4 = μ 2 has the advantage that it automatically leads to a zero oblique parameter T . Indeed, in our two-Higgs-doublet model with R 11 = 1, y) is a function [9,14] that is zero when x = y. Thus, T = 0 when γ e = 1.7, obtaining a (S) e = 1.0 × 10 −13 . Thus, even for a relatively large γ e , a (S) e can be below the experimental error. This is of course because of the tiny electron mass.

Numerics
In this section, we want to show that in the two-Higgs-doublet version of the framework of ref. [3], and assuming moreover  Table 1.
Notice that we only strive in this section to prove that something is possible; we do not attempt a full scan of the parameter space of our model, which is quite vast. On the contrary, we shall make many simplifying assumptions, for instance we assume that all the parameters of the model are real.
In the decay rates of equations (37) there are various unknowns: 1. the neutral-scalar masses M 3 and M 4 ; 2. the factors X 2 1 2 ; 3. the Yukawa couplings γ together with those in A .
In this section we also want to fit a In order to compute the factors X 2 1 2 we proceed in the following way. The mass matrix of the light neutrinos is obtained by the seesaw formula. In our notation, it reads Inverting equation (42), we obtain The matrix M ν is diagonalised as Using our simplifying assumption that all the parameters in the model are real, we set in equation (46) e iα = e iβ = 1 and we also We are now able to compute the matrix M R as a function of m 1 through equation (46); therefrom we compute the quantities X 2 1 2 by using equations (12) and (13). We obtain the result depicted in Fig. 1. Notice that X eμ has a zero for m 1 We then have X eμ In this way we have fixed the factors mentioned in point 2 above. Besides equations (49), we also obtain, from equation (48), heavyneutrino masses m 4 = 4.3 × 10 12 GeV, m 5 = 6.0 × 10 12 GeV, and m 6 = 2.2 × 10 14 GeV. These masses represent the seesaw scale, 7 which is so large that all the radiative charged-lepton decays are completely invisible. Actually, m R is this large partly because we chose the Yukawa couplings d close to one, cf. equation (43), in order to achieve large τ -lepton branching ratios. 8 Thus, the effect that we want to produce in our model can only occur for a large 6 We might alternatively have chosen e iδ = +1 we have checked that there is no qualitative difference between the two cases. 7 Actually, m 6 is two orders of magnitude larger than m 4 and m 5 and therefore there is no well-defined seesaw scale, but that is not relevant for our purposes. 8 (41) and (43). We now fix the remaining Yukawa couplings as γ e = γ τ = 1.7, δ e = 0, δ μ = 0.00007, δ τ = 0.2.

(50)
With all these input values, we obtain the branching ratios BR μ − → e − e + e − = 3.872 × 10 −13 , One sees that all these branching ratios are less than a factor of three away from the upper bounds of Table 1. We have thus demonstrated that in our model it is possible to suppress the radiative decays of the muon and tau lepton, while keeping the branching ratios of their decays into charged leptons very close to the experimental upper bounds.
Some remarks concerning the input values that we have utilised are in order: • All the experimental upper bounds on the branching ratios of the decays of the τ -lepton in Table 1 are quite similar. Therefore, if we want to have both τ − → − e + e − and τ − → − μ + μ − close to their experimental upper bounds, then γ e and γ μ will have to be similar -see the explicit factors γ 3 and γ 2 in the decay rates of equations (37a) and (37b), respectively. For definiteness we have chosen all three γ to be the same. In Fig. 2 we depict the way the five branching ratios vary as functions of some γ .
• In A 1 2 in equation (30b) the dominant terms have v 246 GeV in the numerator. For large γ e = γ μ = 1.7 and large d e = 0.6 and d μ = 0.1, these terms will give a much too large contribution to BR μ − → e − e + e − unless there is a delicate cancellation between the terms proportional to δ e and the terms proportional to δ μ . This cancellation is illustrated in Fig. 3 for δ μ of equation (50). For larger values of δ μ the curve is basically identical but shifted to the right.
• On the other hand, in the decays of the τ -lepton the terms with v in the numerator are just the relevant ones and we have needed, since we have chosen tiny δ e and δ μ , large parameters δ τ , d e , d μ , and γ ( = e, μ, τ ).
We may thus say that the branching ratios in equations (51) involve some finetuning.

Conclusions
It is now known, since the experimental observation of neutrino oscillations [16], 9 that there is lepton flavour-violation. However, that violation has not yet been observed in the charged-lepton sector and it is not quite certain where it is most likely to be observed first. In this context, the radiative decays ± 1 → ± 2 γ seem the best guess, and decays of the form ± 1 → ± 2 + 3 − 3 may be an option as well.
In this paper we have demonstrated, through an explicit numerical example, that there is a class of models where the radiative decays in the paragraph above may be so suppressed as to be utterly invisible, yet any of the five decays of the form ± 1 → ± 2 + 3 − 3 , or indeed -if one assumes some finetuning -all such five decays simultaneously, may be just around the corner.
Our class of models, first considered in ref. [1], has three righthanded neutrino singlets and has more than one Higgs doublet. The crucial assumption is that the lepton flavours are conserved in the Yukawa couplings and broken only in the Majorana mass terms of the right-handed neutrinos; this assumption is fieldtheoretically consistent because those mass terms have dimension three while the Yukawa couplings have dimension four. As demonstrated in ref. [3], the effect mentioned in the previous paragraph occurs if the seesaw scale is much larger than all other scales in this class of models. In the present paper we have shown that there is a relevant simplification of the effective flavour-violating couplings of the neutral scalars, emerging at the one-loop level, when one uses the Higgs basis, i.e. the basis for the Higgs doublets wherein only one of them has nonzero VEV.
We have explicitly computed the branching ratios of the five decays ± 1 → ± 2 + 3 − 3 in the case of a two-Higgs-doublet model assuming that the first doublet φ 1 coincides with the Higgs doublet of the SM, viz. it does not mix with the second doublet. Moreover, we have employed several simplifying assumptions in order to reduce the parameter space of the model. We have noted that some finetuning is needed in order that BR μ − → e − e + e − does not become too large when all other four branching ratios are simultaneously close to their experimental limits.
Flavour-diagonal Yukawa coupling matrices have no straightforward implementation in the quark sector, 10 so one has to admit non-diagonal Yukawa couplings there and avoid excessive flavourchanging neutral interactions by finetuning. Thus there is an asymmetry between the quark and the lepton sector. This may seem ugly, but, as pointed out in this paper, the intriguing consequences for charged-lepton decays make a consideration of such a framework worthwhile.