Lepton Flavor Violation in the singlet-triplet scotogenic model

We investigate lepton flavor violation (LFV) in the the singlet-triplet scotogenic model in which neutrinos acquire non-zero masses at the 1-loop level. In contrast to the most popular variant of this setup, the singlet scotogenic model, this version includes a triplet fermion as well as a triplet scalar, leading to a scenario with a richer dark matter phenomenology. Taking into account results from neutrino oscillation experiments, we explore some aspects of the LFV phenomenology of the model. In particular, we study the relative weight of the dipole operators with respect to other contributions to the LFV amplitudes and determine the most constraining observables. We show that in large portions of the parameter space, the most promising experimental perspectives are found for LFV 3-body decays and for coherent $\mu-e$ conversion in nuclei.

In this work we will concentrate on a simple extension 1 of the minimal setup introduced in [1]: the singlet-triplet scotogenic model [34]. In this variant of the scotogenic model, the fermion sector includes the SU (2) L triplet Σ, which can mix with the singlet fermions via the vacuum expectation value (VEV) of a real scalar, Ω, also triplet under SU (2) L . The most relevant features of the minimal model, radiative neutrino masses and a stable dark matter candidate, are kept in this variant, while the singlet-triplet mixing allows one to interpolate between pure singlet DM [1] and pure triplet DM [35], when the dark matter candidate is fermionic. This leads to a richer phenomenology, in particular to better prospects in direct DM detection experiments [34].
Lepton flavor violation (LFV) is one of the most important probes of models with extended lepton sectors. In fact, precision high-intensity experiments are sensitive to the existence of new physics at very high energies, which makes flavor physics a powerful discovery tool, as demonstrated by its central role in the making of the Standard Model. Furthermore, very promising experimental projects in the search for LFV will begin their operation in the near future. In addition to the planned upgrade for the MEG experiment, which will improve its sensitivity to µ → eγ branching ratios as low as 6 · 10 −14 [36], other new experiments will also join the effort. Among them, one can highlight the Mu3e experiment [37], which will look for the 3-body decay µ → 3 e, as well as a plethora of experiments looking for µ − e conversion in nuclei, like Mu2e [38][39][40], DeeMe [41], COMET [42,43] and PRISM/PRIME [44], in all cases with spectacular sensitivity improvements compared to previous experiments. This remarkable multi-channel experimental effort in the search for LFV encourages detailed LFV studies in specific neutrino mass models.
We study LFV in the singlet-triplet scotogenic model, in the spirit of previous works for the singlet [15] and triplet [45] models 2 . We will show that the model contains large regions of parameter space with observable LFV rates and hence will be probed in the near round of LFV experiments. Furthermore, we will explore some aspects of the LFV phenomenology of the model, such as the relative weight of the dipole operators with respect to other contributions to the LFV amplitudes, and determine that the most promising experimental perspectives are found for the LFV 3-body decays µ → 3 e and for coherent µ − e conversion in nuclei.
The rest of the paper is organized as follows: in Sec. II we introduce the model whereas in Sec.

II. THE MODEL
We consider the singlet-triplet scotogenic model introduced in [34]. The matter content of the model, as well as the charge assignment under SU (2) L , U (1) Y and Z 2 , is shown in Table I. The quark sector, not included in this table, is SM-like and has Z 2 = +1. The new fields beyond the SM particle content include two fermions: the singlet N and the triplet Σ, both with vanishing hypercharge and odd under the discrete Z 2 . Regarding the new scalars, these are the doublet η, also odd under Z 2 , and the real triplet Ω. The SU (2) L doublets φ and η can be decomposed as and can be identified with the usual SM Higgs doublet and a new inert doublet. Regarding the SU (2) L triplets, Σ and Ω, they are decomposed using the standard 2 × 2 matrix notation as With the charge assignment in Table I, the most general SU(3) c ⊗ SU(2) L ⊗ U(1) Y , Lorentz and Z 2 invariant Yukawa Lagrangian is given by Here we indicate the flavor indices α, β = 1, 2, 3 explicitly and denoteη = iσ 2 η * , as usual. Gauge contractions are omitted for the sake of clarity. The Σ and N fermions have Majorana mass terms, Finally, the scalar potential can be written as 3 A. Symmetry breaking and scalar sector We will assume that the scalar potential in Eq. (5) is such that with v φ , v Ω = 0. These vacuum expectation values (VEVs) are determined by means of the minimization conditions t where t i ≡ ∂V ∂v i is the tadpole of v i . The VEVs v φ and v Ω break the electroweak symmetry and induce masses for the gauge bosons, We note that the triplet VEV v Ω contributes to the W boson mass, thus receiving constraints from electroweak precision tests. We estimate that this VEV cannot be larger than about 4.5 GeV (at 3σ).
The scalar spectrum of the model contains the Z 2 -even scalars φ 0 , Ω 0 , φ ± and Ω ± , and the Z 2odd scalars η 0 and η ± . In the basis Re φ 0 , Ω 0 , the mass matrix for the Z 2 -even neutral scalars is given by The lightest of the S mass eigenstates, S 1 ≡ h, can be identified with the SM Higgs boson with a mass m h 126 GeV recently discovered at the LHC, whereas the heaviest mass eigenstate, S 2 , is a new heavy Higgs boson not present in the SM. Regarding the Z 2 -even charged scalars, their mass matrix in the basis (φ ± , Ω ± ) takes the form One of the H ± mass eigenstates is the Goldstone boson that becomes the longitudinal component of the W boson, whereas the other is a physical charged scalar. In what concerns the Z 2 -odd scalars η 0,± , we first express the neutral η 0 field in terms of its CP-even and CP-odd components The conservation of the Z 2 symmetry implies that the η R,I,± fields do not mix with the rest of scalars. Their masses are given by 4 We point out that the mass difference between the neutral η scalars is controlled by the λ 5 coupling, , and thus vanishes if λ 5 = 0. This will be relevant for the generation of neutrino masses, as shown in Sec. II B.
Finally, we emphasize that the vacuum in Eq. (6) breaks SU (2) L ⊗ U (1) Y → U (1) Q but preserves the Z 2 discrete symmetry. As we will discuss below, this gives rise to the existence of a stable neutral particle which may play the role of the dark matter of the universe.

B. Neutrino masses
Before discussing neutrino masses we must comment on the Z 2 -odd neutral fermions. The Z 2odd fields Σ 0 and N get mixed by the Yukawa coupling Y Ω and the non-zero VEV v Ω . In the basis Σ 0 , N , their Majorana mass matrix takes the form The mass eigenstates χ 1,2 are determined by the 2 × 2 orthogonal matrix V (α),   χ 1 such that The singlet-triplet scotogenic model generates Majorana neutrino masses at the 1-loop level. This is shown in Fig. 1, which actually includes four loop diagrams, since η 0 ≡ η R , η I and χ ≡ (χ 1 , χ 2 ). The resulting neutrino mass matrix can be written as 5 where h is a 3 × 2 matrix defined as and I(m 2 1 , m 2 2 ) is a Passarino-Veltman function evaluated in the limit of zero external momentum. We note that m 2 η R = m 2 η I leads to vanishing neutrino masses due to an exact cancellation between the η R and η I loops. This was indeed expected, since m 2 η R = m 2 η I implies λ 5 = 0 and a definition of a conserved lepton number would be possible in this case. Furthermore, this justifies the choice λ 5 1, which is natural in the sense of 't Hooft [46], given that the limit λ 5 → 0 increases the symmetry of the model.
It proves convenient to write the neutrino mass matrix in Eq. (20) as where A neutrino mass matrix as the one in Eq. (22) formally resembles that obtained in the standard type-I seesaw with two generations of right-handed neutrinos. In this case we can make use of an adapted Casas-Ibarra parameterization [47,48] to obtain an expression for the Yukawa matrix h, Here R is a 3 × 2 complex matrix such that RR T = I 3 , where I 3 is the 3 × 3 unit matrix, and the neutrino mass matrix is diagonalized as is the PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix. Here c ij = cos θ ij , s ij = sin θ ij and δ is the CP-violating Dirac phase 6 . Similarly to the type-I seesaw with two right-handed neutrinos, the singlet-triplet scotogenic model predicts a vanishing mass for the lightest neutrino. It has, however, enough freedom to accommodate both neutrino spectra, Normal Hierarchy (NH) and Inverted Hierarchy (IH), and the form of the complex R matrix introduced in Eq. (24) depends on this choice [48], We can finally make use of the previous expressions and write the Yukawa couplings h in terms of the PMNS matrix U , the eigenvalues m i and the complex angle γ. In case of NH, one obtains whereas for IH one finds

C. Dark matter
The lightest state charged under the conserved Z 2 parity is stable and hence, if electrically neutral, it constitutes a standard weakly-interacting dark matter candidate. Therefore, in what concerns dark matter, the singlet-triplet scotogenic model contains two distinct scenarios: (i) scalar dark matter, when the candidate is the lightest neutral η state, η R or η I , and (ii) fermion dark matter, when the candidate is χ 1 , the lightest χ state. Even though we will not be concerned about dark matter in this paper, we find it worth summarizing the main features of these two scenarios: • Scalar dark matter: In this case the dark matter phenomenology resembles that of the inert doublet model [49] (see also [50][51][52] for some recent works on dark matter in the inert doublet model). Since in this scenario dark matter production in the early universe is driven by gauge interactions, there is no direct relation with LFV (driven by Yukawa interactions).
• Fermion dark matter: This scenario presents some of the most interesting features of the singlet-triplet scotogenic model [34]. The phenomenology dramatically depends on the nature of the dark matter candidate. In the two extreme cases this can be a pure SU (2) L singlet (when χ 1 ≡ N ) or a pure SU (2) L triplet (when χ 1 ≡ Σ), while in general it will be an admixture of these two gauge eigenstates. When χ 1 is mostly singlet, the dark matter phenomenology is determined by Yukawa interactions and one expects a direct link between dark matter and LFV, as in the minimal scotogenic model [20]. In contrast, the DM phenomenology of a mostly triplet dark matter candidate is driven by the known gauge interactions.
This case has little impact on LFV and predicts a dark matter candidate with a mass of about ∼ 2.7 TeV in order to reproduce the observed dark matter relic density. The parameter Y Ω , which determines the N − Σ mixing, interpolates between these two cases, in a way completely analogous to DM in R-parity conserving supersymmetry.

III. LFV OBSERVABLES
A. Current experimental situation and future projects No observation of a flavor violating process involving charged leptons has ever been made. This has been used by many experiments to set strong limits on the most relevant LFV observables, usually translated into stringent bounds on the parameter space of many new physics models.
In what concerns the radiative decay α → β γ, the experimental search is led by the MEG collaboration. This experiment searches for the process µ → eγ and recently announced the limit BR(µ → eγ) < 5.7 · 10 −13 [53], about four times more stringent than the previous bound obtained by the same collaboration. The 3-body LFV decay µ → 3 e was also searched for long ago by the SINDRUM experiment [54], which obtained the limit BR(µ → 3 e) < 1.0 · 10 −12 , still not improved by any experiment after almost 30 years. Another µ − e LFV process of interest due to the existing bounds is µ − e conversion in nuclei. Among the experiments involved in this search we may mention SINDRUM II, which searched for µ − e conversion in muonic gold and obtained the impressive limit CR(µ − e, Au) < 7 × 10 −13 [55]. Finally, the current experimental limits for τ lepton observables are less stringent, with branching ratios bounded to be below ∼ 10 −8 .
In addition to the active LFV searches, some of them with planned upgrades, several promising upcoming experiments will join the effort in the next few years 7 . The MEG collaboration has announced plans for upgrades which will allow this experiment to reach a sensitivity to branching ratios as low as 6 · 10 −14 [36]. Significant improvements are also expected for τ observables from searches in B factories [59,60], although the expected sensitivities are still less spectacular than those for µ observables. Regarding the new projects, the most promising ones are expected in searches for µ → 3 e and µ − e conversion in nuclei. The Mu3e experiment, which plans to start data taking soon, announces a sensitivity for µ → 3 e branching ratios of the order of ∼ 10 −16 [37].
In case no discovery is made, this would imply an impressive improvement of the current bound by 4 orders of magnitude. Regarding µ − e conversion in nuclei, the competition will be shared by several experiments, with expected sensitivities for the conversion rate ranging from 10 −14 to an impressive 10 −18 . These include Mu2e [38][39][40], DeeMe [41], COMET [42,43] and, in the long term, the future PRISM/PRIME [44].
Finally, even though in this paper we concentrate on low-energy processes, we emphasize that colliders can also play a relevant role in the search for LFV. For instance, there is currently an intriguing hint at CMS for Higgs boson LFV decays into τ µ [61]. This anomaly seems to require an explanation based on an extended scalar sector (see e.g. [62,63]   for the observables of interest. Our numerical analysis reveals that the LFV phenomenology is mainly driven by two Wilson coefficients, both generated by photon penguin diagrams: the monopole K L 1 and the dipole K R 2 . Box diagrams also lead to sizable contributions, mainly to the Wilson coefficients A V LL , B V LL and C V LL , but we have found them to be always subdominant compared to the photonic monopole and dipole contributions. Therefore, we can obtain simple approximate expressions for the LFV observables in terms of only K L 1 and K R 2 . The most relevant photon penguin diagrams in the singlet-triplet scotogenic model are shown in Fig. 2. The diagram with the neutral fermions χ ≡ (χ 1 , χ 2 ) running in the loop is common to the scotogenic model [15], whereas the diagram with the charged Σ − state is only present in the singlet-triplet variant. This difference has an impact on the phenomenology, as we will see below.
Let us first consider the dipole coefficient K R 2 , which induces the radiative LFV decay α → β γ. It can be written as where the contributions from the two diagrams in Fig. 2 are approximately given by Similarly, the monopole coefficient K L 1 can be split as and the two contributions from the penguin diagrams in Fig. 2 are given by Here we have defined and used m 2 η R m 2 η I ≡ m 2 η 0 . Finally, the loop functions appearing in these expressions are given by We find that in the limit M Σ → ∞ our analytical results are in good agreement with those obtained in the scotogenic model [15] 8 . Finally, we emphasize that the numerical results discussed in the next Section are based on the full 1-loop evaluation of the LFV observables and not on these approximate expressions, only presented to gain insight.

IV. PHENOMENOLOGICAL ANALYSIS
Our phenomenological analysis uses a SARAH-generated SPheno [75,76] module for the numerical evaluation of the LFV observables. We solve the tadpole equations for the squared mass terms m 2 H and m 2 Ω and use an adapted Casas-Ibarra parameterization for neutrino masses to compute the Yukawa couplings Y N and Y Σ . For this purpose, the results of the global fit to neutrino oscillation data [77] will be used. Furthermore, given the little impact on the LFV phenomenology, we fix the following parameters in the scalar potential, We have explicitly checked that these parameters only affect the LFV observables indirectly, due to their influence on the scalar spectrum 9 . The large value chosen for the trilinear coupling µ 2 ensures the conservation of the Z 2 symmetry up to high energy scales [78]. We also fix v Ω = 1 GeV. This choice leads to a negligible deviation from ρ = 1, thus respecting limits from electroweak precision data. Finally, the doublet VEV v φ is fixed so that m W is correctly obtained, see Eq.     (44) and (45), use γ = 0, best-fit values for the neutrino oscillation parameters, as obtained in [77], normal hierarchy for the light neutrino spectrum and δ = 0.
regions of parameter space lead to cancellations among diagrams that strongly reduce some of the Wilson coefficients (see below for details), and (ii) the fit to neutrino oscillation data that leads to an increase in the Yukawa couplings when m η or M N,Σ increase.
The BR(µ → 3 e)/BR(µ → eγ) ratio We also observe in Fig. 3 that for most points in the selected m η -M N plane, one obtains BR(µ → eγ) BR(µ → 3 e). However, this is not a general prediction of the model, as we proceed to discuss now. Let us consider the benchmark points in Table III. The results for the LFV observables have been obtained making the same choices as for Fig. 3, but using specific values for m 2 η , M N and M Σ . First, we observe that the ratio show that for α 0, the fermion triplet loops lead to K R 2 ∝ F 2 (ξ 1 ) − 2 G 2 (ρ), both loop functions being positive. Therefore, one naturally expects to find parameter points where this cancellation in the dipole coefficient is effective, leading to a reduction in the µ → eγ rate. This is explicitly shown in Fig. 4, where we plot our numerical results for BR(µ → eγ) as a function of M N for the fixed values Y Ω = 0.1, m 2 η = 2.5 · 10 5 GeV 2 and M Σ = 500 GeV. The purple dots display the total branching ratio, whereas the pink and blue dots show partial results obtained with only the D 0 and D − contributions, respectively. This figure has been obtained by allowing the neutrino oscillation parameters to vary randomly within the preferred 3 σ ranges found by the global fit of [77], which explains the spread of the points. We observe that the D 0 and D − contributions approach a common value for large M N values, whereas the total branching ratio drops. This is due to the abovementioned cancellation in the Σ 0 -Σ − loops. For low M N values the singlet contributions to D 0 dominate and the cancellation in the triplet contributions is not relevant. However, as M N increases and the N contribution to D 0 gets smaller, the cancellation in the triplet contributions becomes visible. We point out that a similar cancellation in the monopole coefficient takes place, again due to the relative sign between M 0 and M − , see Eqs. (37) and (38). However, typically this cancellation has little impact on the LFV observables which receive contributions from the monopole operator due to the interplay with the other contributions (e.g. dipole).

LFV τ decays
So far we have concentrated on µ − e violating processes. Now we turn our attention towards LFV processes involving the τ lepton. Given the worse experimental limits, these can only be phenomenologically relevant when they have rates much larger than those for the µ lepton. For example, in the benchmark points 1 and 2 presented above one finds branching ratios for the radiative decays τ → α γ, with α = e, µ, in the ∼ 10 −13 − 10 −12 ballpark, clearly below the expected experimental sensitivity in the near future.
The results shown in Tab. III for points 1 and 2 were obtained with a vanishing R matrix angle γ. This parameter has a direct impact on the Yukawa couplings Y N and Y Σ , see Eqs. (29) - (32), and can lead to cancellations in the amplitudes of specific flavor violating transitions. This is illustrated in Fig. 5, where we show our numerical results for BR( α → β γ) as a function of the R matrix angle γ (assumed to be real for simplicity) for M Σ = 300 GeV (on the left) and M Σ = 800 GeV (on the right). The rest of the parameters are fixed to the same values as in points 1 and 2, with the exception of a smaller λ 5 coupling (λ 5 = 10 −10 ) in order to increase the resulting Yukawa couplings and get larger LFV rates. We see in these figures that even though most points are experimentally excluded due to a µ → eγ rate above the MEG bound, for certain γ values a strong cancellation takes place, leading to a tiny BR(µ → eγ) and BR(τ → eγ) ∼ 10 −9 − 10 −8 within reach of B factories.
Therefore, we conclude that the singlet-triplet scotogenic model can also be probed via τ observables. However, the scenarios that would be experimentally explored in this way are not generic and require a certain level of tuning in the Yukawa parameters in order to suppress the µ → e rates.

V. SUMMARY AND CONCLUSIONS
We have investigated the lepton flavor violating phenomenology of the singlet-triplet scotogenic model, a well-motivated scotogenic neutrino mass model in which neutrinos acquire their masses at the 1-loop level. The same symmetry that forbids the tree-level generation of neutrino masses stabilizes a weakly-interacting dark matter candidate, thus providing a natural solution for another fundamental problem of current physics.
Our main findings can be summarized as follows: • The model will be probed in the next generation of LFV experiments. In fact, we have found that parts of the parameter space are already ruled out by µ → eγ searches. This of course depends on the value of the λ 5 parameter, which sets the global size of the Yukawa parameters and is expected to be naturally small due to its crucial role in the violation of lepton number.
• Currently, the most stringent LFV bound on the model is the one set by the MEG experiment on BR(µ → eγ). However, this will soon change due to the impressive expected sensitivity in the incoming experiments. Experiments such as Mu3e (searching for µ → 3 e) and Mu2e or COMET (searching for µ − e conversion in nuclei) will soon probe larger portions of the parameter space of the model. • One naturally finds points of the parameter space with BR(µ → 3 e), CR(µ − e, Nucleus) BR(µ → eγ). This is caused by cancellations in the dipole coefficient which take place when the dominant contributions are generated by Σ 0 -Σ − loops. When this happens, MEG is usually unable to constrain the model. In case of the Σ fermions, their subsequent decays lead to final states including DM particles, hence to signatures with missing energy, in a way analogous to the standard R-parity conserving supersimmetric signals [79]. These interesting possibilities are left for future work.
Here e is the electric charge, q is the photon momentum, P L,R = 1 2 (1 ∓ γ 5 ) are the usual chirality projectors and the lepton flavors are denoted by α,β . We omit flavor indices in the Wilson coefficients for the sake of clarity. The first and second terms in Eq. (A2) are usually called monopole and dipole operators, respectively. Notice that we have singled out the photonic contributions, not included in other vector operators. On the contrary, Z-and Higgs boson contributions have been included whenever possible.
Finally, the last piece of Eq. (A1) is the general 2 2q 4-fermion interaction Lagrangian, given by and we have used d γ to denote the d-quark flavor.
Appendix B: Generic expressions for the LFV observables The radiative decays α → β γ only receive contributions from the dipole operators. The decay width is given by [80] Γ ( α → β γ) = αm 5 where α is the electromagnetic fine structure constant.

α → 3 β
In this case, in addition to the standard dipole contributions, the decay width receives contributions from the monopole operators in Eq. (A2) and from the 4-lepton operators in Eq. (A3).
The resulting decay width can be written as [69] Γ ( α → 3 β ) = m 5 α 512π 3 e 4 K L  They induce the effective µeqq couplings g RV (q) = g LV (q) L→R , (B5) g RS(q) = g LS(q) L→R , where Q q is the quark electric charge (Q d = −1/3, Q u = 2/3) and C IXK qq = B K XY C K XY for d-quarks (u-quarks), with X = L, R and K = S, V . These couplings at the quark level must be where the G (q,p) K and G (q,n) K numerical coefficients were computed in [81] and given in [69]. For an improved calculation of the scalar coefficients we refer to [82]. Finally, the conversion rate, normalized to the standard muon capture rate Γ capt , is given by [83] CR(µ − e, Nucleus) = p e E e m 3 µ G 2 F α 3 Z 4 eff F 2 p 8 π 2 Z Γ capt × (Z + N ) g (B10) Z and N are the number of protons and neutrons in the nucleus under consideration and Z eff is its effective atomic charge [84]. Furthermore, G F is the Fermi constant, F p is the nuclear matrix element and p e and E e ( m µ ) are the momentum and energy of the electron.