Charged Lepton Flavor Violating processes in Neutrinophilic Higgs+Seesaw model

We investigate charged lepton flavor violating (CFLV) processes in the `neutrinophilic Higgs+seesaw model', in which right-handed neutrinos couple only with an extra Higgs field which develops a tiny VEV and the right-handed neutrinos also have Majorana mass. The model realizes a seesaw mechanism around TeV scale without extremely small Dirac Yukawa couplings. A phenomenological feature of the model is CFLV processes induced by loop diagrams of the charged scalar particles and heavy neutrinos. Therefore, first we constrain the model's parameter space from the search for $\mu\to e\gamma$. Next, we predict the branching ratios of other CFLV processes including the $\mu\to3e$, $\mu+{\rm Al}\to e+{\rm Al}$, $\mu+{\rm Ti}\to e+{\rm Ti}$, $Z\to e\mu$, $Z\to e\tau$, $Z\to \mu\tau$, $h\to e\tau$, $h\to\mu\tau$ processes, and discuss their detectability in future experiments.


Introduction
The origin of the smallness of the neutrino mass is one of the prime open questions in particle physics. One candidate solution to the above mystery is the neutrinophilic Two Higgs Doublet Model [1], where there is an extra Higgs doublet called 'neutrinophilic Higgs') that couples to the lepton doublets and right-handed neutrinos while the coupling of the Standard Model (SM) Higgs doublet to right-handed neutrinos is forbidden by a Z 2 symmetry, and the smallness of the vacuum expectation value (VEV) of the neutrinophilic Higgs explains the smallness of the neutrino mass. In the original proposal [1], Majorana mass for right-handed neutrinos is absent and the neutrinos are purely Dirac particles. However, the Z 2 symmetry that forbids the coupling of the SM Higgs and right-handed neutrinos, does not exclude the possibility that right-handed neutrinos have Majorana mass term. If Majorana mass for right-handed neutrinos is introduced, the model becomes a low-scale realization of the seesaw mechanism [2]- [5], where the smallness of the neutrino mass is accounted for by the seesaw mechanism in addition to the tininess of the neutrinophilic Higgs VEV. We call the new model 'neutrinophilic higgs+seesaw model' for the obvious reason.
Important experimental signatures of the neutrinophilic Higgs+seesaw model are (i) the presence of new scalar particles H ± , H, A originating dominantly from the neutrinophilic Higgs field, and (ii) charged lepton flavor violating (CLFV) processes, such as µ → eγ, mediated by a loop of the charged scalar H ± and a heavy neutrino. In this paper, we investigate CFLV processes in the neutrinophilic Higgs+seesaw model in detail. First, we constrain the parameter space of the neutrinophilic Higgs+seesaw model from current experimental bounds on CFLV processes, the most stringent bound coming from the µ → eγ decay. Next, we predict branching ratios (or conversion rates) of various CFLV processes and discuss whether it is possible to detect these processes in the future.

Neutrinophilic Higgs+Seesaw Model
The model contains two Higgs doublet fields, H 1 , H 2 , left-handed leptons, ℓ α L , right-handed charged leptons, e α R , and right-handed neutrinos, ν i R , where α = e, µ, τ is the flavor index for charged leptons and i = 1, 2, 3 is another flavor index. It also contains quarks, q k L , u k R , d k R , but they play no role in this study. The fields are charged under the SM SU(3) C × SU(2) L × U(1) Y gauge group and a Z 2 symmetry as Table 1.
1 −1/3 + Note that the above Z 2 charge assignment allows Majorana mass for right-handed neutrinos, while it forbids the Yukawa couplings of SM fermions with H 2 and the Yukawa coupling of righthanded neutrinos with H 1 .
We assume that the Z 2 symmetry is softly broken in the scalar potential. The most general scalar potential and Yukawa couplings are then where ǫ g denotes the antisymmetric tensor acting on SU(2) L indices and ǫ s denotes that acting on spinor indices. Here, we have taken the flavor basis in which the Majorana mass for righthanded neutrinos is diagonal, and we have made m 2 We turn our attention to the lepton mass. The Dirac and Majorana mass terms are given by Then, the mass matrix for neutrinos is obtained as The above mass matrix is diagonalized by a unitary matrix, U, as where m ν 1 , m ν 2 , m ν 3 correspond to the tiny active neutrino masses, and m ν 4 , m ν 5 , m ν 6 to the masses of heavy neutrinos. We assume v 2 ≪ M N j . The unitary matrix U is then approximated by where U P M N S denotes the PMNS mixing matrix [8,9] and I 3 denotes the 3-dimensional identity matrix, and we obtain the following seesaw formula: Inverting the relation Eq. (14), one can express the neutrino Dirac Yukawa coupling Y D as where R 3×3 is an arbitrary complex-valued 3 × 3 rotation matrix [10]. The masses of heavy neutrinos are approximated as and the mass eigenstates belonging to m ν 4 , m ν 5 , m ν 6 are mostly given by the right-handed neutrinos, namely, we find 5

Branching Ratios of Charged Lepton Flavor Violating Processes
The limits with m β /m α → 0 and m α /M Z → 0 are taken throughout this section. We only consider the dominant contribution coming from one-loop diagrams of the charged scalar H ± and heavy neutrinos ν 4 , ν 5 , ν 6 .
3.1 e α → e β γ CLFV decays of a charged lepton into a charged lepton and a photon, e α → e β γ, arise from the following dipole term, induced by loop diagrams of the charged scalar H ± and heavy neutrinos ν 4 , ν 5 , ν 6 : The branching ratio is given by 3.2 e α → e βēβ e β CLFV decays of a charged lepton into three charged leptons, e α → e βēβ e β , arise from the following dipole, non-dipole and box-induced terms, induced by loop diagrams of the charged 6 scalar and heavy neutrinos: .
The branching ratio is given by Br(e α → e β ν ανβ ).
Here, the contribution from the Z-penguin diagram is neglected because it is suppressed by m α m β /M 2 Z compared to the contribution from the photon-penguin diagram.

µN → eN
µ → e conversion processes in a muonic atom arise from the dipole term A D and the non-dipole term A N D . The conversion rate divided by the muon capture rate, CR(µ → e), reads where p e and E e are the momentum and energy of the final state electron, and Z and N are the number of protons and neutrons, respectively. Z ef f is the effective atomic charge, F p is the nuclear matrix element, and g (0) LV are effective charges. Γ capture denotes the muon capture rate. Here, the contribution from the Z-penguin diagram is again neglected, and that from the Higgs-penguin diagram is neglected because the up and down quark Yukawa couplings are tiny.
Also, since cos β ≃ 0, box diagrams involving two quarks and two leptons do not contribute.
3.4 Z →ē α e β CLFV decays of a Z boson arise from the non-dipole term A N D . In the leading order of M 2 Z /M 2 H ± , the effective Lagrangian contributing to Z →ē α e β decay is given by The branching ratio for Z →ē α e β is

h →ē α e β
In the leading order of m 2 h /M 2 H ± , the effective Lagrangian contributing to h →ē α e β decay is given by where λ 3 is the scalar quartic coupling that appears in Eq. (1). G H is a novel function different The branching ratio for h →ē α e β is Br(h →ē α e β ) = Br(h →ē α e α ) 1 2 (36) 8

Numerical Study
We investigate CLFV processes in the neutrinophilic Higgs+seesaw model, based on the branching ratio formulas in Section 3. First, we use current experimental upper limits on CLFV branching ratios to constrain the parameters of the model. Next, under the above constraint, we predict the branching ratios of various CFLV processes including µ → 3e, µ + Al → e + Al,

Assumptions on the Model Parameters
The branching ratio formulas of CLFV processes depend on the neutrino Dirac Yukawa matrix Eq. (15), the charged scalar mass m H ± and the right-handed neutrino Majorana masses There are too many parameters and it is not easy to gain physical insight on phenomenology of the model. So, we reduce the number of parameters by considering the following situation: For the charged scalar mass m H ± , the most phenomenologically interesting situation is when the charged scalar particle is detectable at the LHC. Hence, we assume For the tiny active neutrino masses m ν 1 , m ν 2 and m ν 3 , we consider both Normal Hierarchy (NH) and Inverse Hierarchy (IH) cases, while focusing on the case where the lightest neutrino mass is 0, namely, we assume The values of m ν 2 and m ν 3 (m ν 1 and m ν 2 ) in NH (IH) case are obtained from the mass differences measured in neutrino oscillation experiments. In this paper, we employ the central values of the mass differences in NuFIT 4.1 [11,12].
For the parameters of U P M N S , we employ the central values of the three mixing angles in NuFIT 4.1 [11,12]. As benchmark values of the Dirac phase δ, we take the 3σ bounds and central value in the NuFIT 4.1 result [11,12] as We set the Majorana phase to be 0.
For the Majorana masses of right-handed neutrinos, we assume them to be degenerate as where M N is taken real positive by a phase redefinition. We have found numerically that the branching ratios of CFLV processes do not change significantly even when the Majorana masses For the neutrinophilic Higgs VEV v 2 , we take it to be proportional to These values of v 2 ensure |Y D | ∼ 0.05 in each hierarchy, where we have defined |Y D | as the minimum absolute value of the Yukawa matrix components when Imθ 1 =Imθ 2 =Imθ 3 = 0.
Note that the motivation for the neutrinophilic Higgs+seesaw model is to realize low-scale seesaw without taking very small values for the neutrino Dirac Yukawa coupling, and hence it is essential to have |Y D | not much smaller than 1.
For R 3×3 , we parametrize it in terms of three complex rotation angles θ j = Reθ j + iImθ j (j = 1, 2, 3) as For the sake of simplifying the analysis, we vary each θ j separately while fixing the other complex angles at zero. When we vary each θ j , its real part Reθ j does not affect the branching ratios of CFLV processes. Therefore, we only regard the imaginary parts Imθ 1 , Imθ 2 and Imθ 3 as the parameters of R 3×3 . The larger the absolute value of Imθ j is, the larger Y D becomes.

Constraints on the Neutrinophilic Higgs+Seesaw Model from Charged Lepton Flavor Violating Processes
The CLFV processes experimentally searched for are e α → e β γ, e α → 3e β , µN → eN, Z → e α e β , h →ē α e β . For each process, the upper limit on the branching ratio (or conversion rate) is obtained by experiments and it constrains the model parameter space. At present, the strongest constraint comes from the upper limit on the µ → eγ branching ratio, Br(µ → eγ) < 4.2 × 10 −13 [13], in the entire parameter space. Therefore, in the study of the current experimental constraints, we can concentrate on the µ → eγ process while neglecting bounds from other CFLV processes [14]- [17].
The constraint on the (M N , Imθ j )-parameter space from the bound Br(µ → eγ) < 4.2 × 10 −13 is displayed by the blue solid line in every figure, for both NH and IH, for m H ± = 0.3 TeV in NH (IH), and for the benchmark values of the Dirac phase δ. Additionally, we show the constraint when v 2 is multiplied by 1/3 and thus Y D is uniformly multiplied by 3, by the dashed blue line.
We observe that the constraint tends to be weaker for smaller M N and |Imθ j |. This is because Y D is proportional to √ M N and R 3×3 (see Eq. (15)), and so Br(µ → eγ) is suppressed for small M N and |Imθ j |.

Prediction on Charged Lepton Flavor Violating Processes
Among the e α → e βēβ e β processes, the future sensitivity for the µ → 3e decay reaches Br(µ → 3e) = 10 −16 [18] and so there is a large chance that this mode is detected even when the model satisfies the current experimental bound on Br(µ → eγ). Therefore, we show in figure 1 (Normal Hierarchy) and figure 2 (Inverse Hierarchy) the prediction on Br(µ → 3e), along with the value of Br(µ → eγ).
In figure 1, the blue solid line agrees with Br(µ → eγ) = 4.2 × 10 −13 for NH and v 2 in Eq. (42), and the region on the left of the blue solid line is excluded by the search for µ → eγ.
The green solid line agrees with Br(µ → 3e) = 10 −16 , the future sensitivity. Therefore, in the region between the blue solid line and the green solid line, the µ → 3e process can be detected in the future. Figure 2 is the corresponding figure for IH and v 2 in Eq. (43).
In the same figures, the blue and green dashed lines are contours of Br(µ → eγ) = 4.2×10 −13 and Br(µ → 3e) = 10 −16 in the case when v 2 is multiplied by 1/3 and thus Y D is uniformly multiplied by 3 according to Eq. (15). Since the dipole and non-dipole terms A D , A N D are proportional to Y 2 D whereas the box-induced term B is proportional to Y 4 D , reducing v 2 affects Br(µ → 3e) and Br(µ → eγ) differently. However, such an effect is not clearly seen in the figures, as the region between the blue and green dashed lines has a similar size to that between the blue and green solid lines.

µ-e Conversions
The processes whose sensitivity will be improved in the future are the µ + Al → e + Al and µ + Ti → e + Ti processes. The future sensitivity for CR(µ + Al → e + Al) is 2 × 10 −17 [19], and that for CR(µ + Ti → e + Ti) is 10 −18 [6]. Therefore, we study whether the µ + Al → e + Al and µ + Ti → e + Ti processes can be detected in the future. In the numerical calculation of the conversion rates, we employ the values of Z ef f , F p , Γ capture in Ref. [20].
We comment that a peculiar property of the conversion rates CR (µN → eN) is that they are H ± = 1. Therefore, the plots of CR(µ+Al → e+Al) and CR(µ + Ti → e + Ti) show a different behavior from other processes around the region M N ≃ m H ± = 0.3 TeV. However, this region is excluded by the µ → eγ search and so such a behavior is unimportant.
In figure
Unfortunately, the rate 10 −16 is much below the future sensitivity of a high-luminosity Z-factory proposed in Ref. [21].
We observe that for NH, we can hope that the h → eτ decay is detected at a rate Br(h → eτ )/Br(h → τ τ ) ∼ 10 −12 and that the h → µτ decay is detected at a rate Br(h → µτ )/Br(h → τ τ ) ∼ 10 −11 even when the model satisfies the current experimental bound on Br(µ → eγ). If IH is the correct mass hierarchy, both Br(h → eτ ) and Br(h → µτ ) roughly decrease by 1/10. Unfortunately, the predicted rate is too small to explain the hint of h → µτ decay reported by CMS [22].

Summary
We have investigated the neutrinophilic Higgs+seesaw model, in which right-handed neutrinos couple only with an extra Higgs field that develops a tiny VEV and they also have Majorana mass, and which realizes the low-scale seesaw naturally. We have concentrated on CFLV processes induced by loop diagrams of the charged scalar and heavy neutrinos. First, we have studied the current constraint on the model's parameter space from the search for µ → eγ. Second, we have predicted the branching ratios of other CLFV processes (µ → 3e, µ + Al → e + Al, µ + Ti → e + Ti, Z → eµ, Z → eτ , Z → µτ , h → eτ , h → µτ ), and discussed whether these processes can be detected in the future. An important finding is that considering the future sensitivities, the µ → 3e, µ + Al → e + Al and µ + Ti → e + Ti processes can be detected in a wide parameter region in the future, even when the model satisfies the current stringent bound on the µ → eγ branching ratio.