Revisiting the R$\nu$MDM Models

Combining neutrino mass generation and a dark matter candidate in a unified model has always been intriguing. We revisit the class of R$\nu$MDM models, which incorporate minimal dark matter in radiative neutrino mass models based on the one-loop ultraviolet completions of the Weinberg operator. The possibility of an exact accidental $Z_2$ is completely ruled out in this scenario. We study the phenomenology of one of the models with an approximate $Z_2$ symmetry. In addition to the Standard Model particles, it contains two real scalar quintuplets, one vector- like quadruplet fermion and a fermionic quintuplet. The neutral component of the fermionic quintuplet serves as a good dark matter candidate which can be tested by the future direct and indirect detection experiments. The constraints from flavor physics and electroweak-scale naturalness are also discussed.


Introduction
The particle identity of dark matter is one of the most important problems in physics beyond the Standard Model (SM). Generally in dark matter models a discrete symmetry, Z 2 symmetry as the simplest example, has to be imposed by hand or show up as the remnant symmetry of a larger group to protect the dark matter particle from decaying. Alternatively, the SM gauge group can be used to stabilize dark matter as discussed in minimal dark matter (MDM) [1; 2] models. Higher representations of SU(2) L are introduced in MDM models as dark matter candidates. They couple to the SM sector only through gauge interactions, while other types of interactions such as Yukawa interactions or scalar interactions are all forbidden by SU(2) L due to their large dimensions 1 . As a result, an accidental (approximate) Z 2 symmetry is present. The symmetry might be exact or approximately realised if the lifetime is larger than the age of the Universe.
If we consider more than just a single SU(2) L multiplet, we might be able to write down an interaction between SM fermions or Higgs and a pair of dark sector particles. It is of great interests to see whether it is possible to simultaneously explain the origin of neutrino masses and mixings, another clear evidence for physics beyond the SM. There is a rich literature on dark matter in neutrino mass models (See Refs. [4][5][6][7][8][9] for example.). The usual seesaw mechanisms [10][11][12][13][14][15][16][17][18][19][20][21] obviously can not be incorporated in the framework of MDM, as the intermediate particles couple to a pair of SM particles and thus are even under the accidental Z 2 of MDM. Therefore we turn our eyes to the next-to-minimal solution, radiative neutrino mass generation [22]. We will focus on minimal ultraviolet (UV) completions of the Weinberg operator.
This idea of realizing radiative neutrino mass in a minimal dark matter model, coined as RνMDM, was proposed in Ref. [23], where a scalar sextet and a fermionic quintuplet are introduced. The neutrino mass would be generated at one-loop level. However, the accidental Z 2 symmetry is broken by a quartic coupling of three scalar multiplets with one Higgs [24] and it can be only approximately realised in the limit of a small quartic coupling. There are also attempts at higher loop order (See e.g. Ref. [25][26][27]) and based on the generalized Weinberg operator [28]. Now we would like to take a second look at the one-loop completions of the Weinberg operator and perform a systematic study and explore the possibility of RνMDM.
This work is organized as follows: In Sec. 2 we discuss the one-loop topologies and determine the possible RνMDM model realizations. In Sec. 3 we discuss one RνMDM model and obtain an expression for neutrino mass. The relevant constraints from Higgs and flavor physics are elaborated on in Sec. 4 and electroweak-scale naturalness in Sec. 5. In Sec. 6 we discuss the allowed parameter space as well as other phenomenological issues, like collider phenomenology. We conclude in Sec. 7. Technical details are collected in the appendices.

One-Loop RνMDM Models
Majorana neutrino masses can be conveniently expressed in the form of the Weinberg operator where Λ is the suppression scale and L and H denote the lepton and the Higgs doublets in the SM. The minimal UV completions of the Weinberg operator at tree level are the seesaw mechanisms. However the Weinberg operator can also be UV completed at loop level and neutrino mass is generated radiatively. A systematic study of the one-loop UV completions has been performed in Ref. [29]. Part of the one-loop completions also induce one of the seesaw mechanisms whose contribution is mostly dominant. These models are thus reducible and not genuine one-loop models. We show all the topologies that allow irreducible one-loop completions in Fig. 1. The well-known radiative seesaw model [6] is a realization of T3 shown in Fig. 1d. We want to first check the possibility to accommodate MDM in these topologies. A good RνMDM model should at least satisfy the following three conditions: 1. At least one of the exotic particles in the loop should be a plausible MDM candidate, which can be either a fermionic quintuplet or a scalar septuplet with zero hypercharge [2].
3. The Lagrangian should of course be invariant under SU(2) L × U(1) Y before electroweak symmetry breaking. So the existence of the Yukawa interaction Lχ i φ i for fermions χ i and scalars φ i implies where d φ i and d χ i denote the dimension of φ i and χ i , and n is a positive integer. Similarly a scalar trilinear coupling φ i φ j H will result in We discuss the four possible topologies in turn: For topology T1-i, any of the scalar field φ i being a dark matter candidate will assign another scalar field φ j with hypercharge ± 1 2 through trilinear scalar couplings φ i φ j H. So according to condition 1 and 2, both φ i and φ j are odddimensional, which violates condition 3. Alternatively χ i can play the role of dark matter. This will also assign φ i with hypercharge ± 1 2 through Yukawa interaction Lχ i φ i . Similarly both χ i and φ i are odd-dimensional according to condition 1 and 2, which contradicts with condition 3. So topology T1-i does not provide any valid UV completion.
We follow the same argument for topology T1-ii. It suffices to consider either φ 1 or χ 2 to be a dark matter candidate as the diagram is symmetric. So the only possible assignment of hypercharges is Y (φ 1 ) = 0, Y (φ 2 ) = − 1 2 , Y (χ 1 ) = 1 2 and Y (χ 2 ) = 0. We immediately notice that φ 2 has to be in an odd-dimensional representation according to condition 2. Therefore if φ 1 is the dark matter candidate, it must be an odd-dimensional representation of SU(2) L and does not satisfy condition 3. Alternatively if χ 2 is the dark matter candidate, it should be in an odd-dimensional representation and does not satisfy condition 3. So topology T1-ii is also ruled out.
For topology T1-iii, we find two minimal completions as shown in Tab. 1, one for scalar dark matter and one for fermion dark matter. Besides the exotic particles shown in the loop, another fermionψ has to be introduced to cancel the anomaly and to write down a mass term.
We turn to topology T3 in the end. If we take χ i to be dark matter candidate, φ i should have hypercharge 1 2 and thus be an odd-dimensional representation according to condition 2, which is contradicting with condition 3. This failed attempt is exactly the model considered in Ref. [23]. Another possibility is φ i being the dark matter candidate. Again we can just take φ 1 due to the symmetry of the diagram. The quantum numbers of the exotic particles of the minimal UV completion of T3 are listed in Tab. 2. Non-minimal completions of T3 contain larger SU (2) representations.
The last sanity check is to see if any renormalizable term invariant under the SM gauge group breaks the accidental Z 2 . For the completions shown in Tab. 1, the Yukawa coupling ψψφ spoils Z 2 [30] as well as the cubic coupling φ 3 [2; 31]. Similarly χ 1 χ 2 φ 1 and φ 1 φ † 2 φ 2 for the model in Tab. 2 do the same [30]. Similar arguments apply to higher representations. Hence, there is no one-loop UV completion of the Weinberg operator in the framework of minimal dark matter, which leads to an exact accidental Z 2 symmetry. However, in the limit of vanishing renormalizable Z 2 -breaking couplings, the symmetry of the Lagrangian is enlarged and thus it is technically natural to have small Z 2 -breaking couplings [32]. All quantum corrections to these couplings are proportional to Z 2 -breaking couplings. With regards to minimal dark matter, the fermionic dark matter candidate is preferred [2]. We will consider the fermionic dark matter model in Tab. 1 and perform a detailed phenomenological study under the assumption that the Z 2 -breaking couplings are sufficiently small, such that the lifetime of χ is longer than the age of the Universe and satisfies all indirect detection constraints. Indirect detection constraints from photon, ν, e + and p − searches are the strongest and generally require lifetimes τ DM 10 26 sec [33][34][35][36]. The maximum allowed size of the coupling can be estimated from the bound given in Eq. (13) in Ref. [30] where λ denotes the Z 2 -breaking Yukawa coupling and m the mass of the particles in the loop. Taking m ∼ m DM , the bound on τ DM can be translated to a bound on the Z 2 -breaking coupling, λ 3 × 10 −21 . Despite the coupling being required to be extremely small, it is technically natural and quantum corrections will not induce larger Z 2 -breaking couplings. Thus the neutral component of χ constitutes a viable dark matter candidate. Note that besides the three models in Tabs. 1 and 2 there are other one-loop radiative neutrino mass models with a viable dark matter candidate including the original RνMDM model [23].
Weyl notation is used.
We leave the details of the kinetic terms to Appendix B as well. From the discussion later in this section, we know that more than one copy of φ is needed to explain both, the solar and the atmospheric, mass splittings. Thus we attach a subscript as the family indices φ i . Then the Yukawa interactions are explicitly expressed in component fields as Without loss of generality we can choose Y H to be real and positive using a phase redefinition of ψ. Similarly three phases of Y L ij can be absorbed by a phase redefinition of L i . Apparently the Yukawa interaction in Eqn. (7) will induce mixing in the fermion sector. The mass matrix for the neutral sector in basis Ψ 0 = ψ 0 ,ψ 0 , χ 0 is where v = 246 GeV is the vev of the Higgs boson. Similarly the mass matrix for the singly charged fermions in the basis ψ + ,ψ + , χ + , ψ − ,ψ − , χ − reads The doubly-charged fermions also mix through the Yukawa interaction. In the basis of interaction eigenstates ψ ++ , χ ++ ,ψ −− , χ −− , the mass matrix reads The diagonalization of these mass matrices can be done using the Takagi factorization and singular value decomposition, respectively. The details of which are collected in Appendix A.
The neutral component of χ is the dark matter candidate of this model. To saturate the dark matter density observed by Planck [37], the dark matter mass should be [38] m χ = (9.4 ± 0.47) TeV, where Sommerfeld effects have been taken into account [2]. Other dark sector particles have to be heavier than m χ , which makes the mixing between χ and ψ, roughly given by Y H v/m χ,ψ , negligible.
The neutrino mass after electroweak symmetry breaking is defined by L = − 1 2 m ν νν and thus with the factor The calculation is done neglecting mass mixing among the fermions which could only give rise to a tiny correction proportional to Y H v/m χ,ψ . The neutrino mass matrix m ν is full rank if and only if n φ ≥ 3. The Yukawa couplings Y L can be expressed as using a Casas-Ibarra-type parametrisation [39], where O is a general complex orthogonal matrix and U diagonalises the neutrino mass matrix withm ν = V T ν m ν V ν . To keep minimality, we will only introduce two copies of φ, which suffices to explain the solar and atmospheric mass splittings. One of the neutrinos remains massless. Both Y L and O are 3 × 2 matrices. The Yukawa couplings Y L are entirely determined by the solar and atmospheric mass squared differences and the leptonic mixing parameters. The only relevant undetermined parameter is a complex angle θ parameterising the 3 × 2 orthogonal matrix O. The two elements of the orthogonal matrix O associated with the massless neutrino do not enter the Yukawa coupling Y L . Neutrino mass fixes the product of the Yukawa couplings Y L ij Y H . We will study the phenomenology induced by these Yukawa couplings and work out the experimental constraints on the model parameter space in the following sections.

Constraints from Higgs physics and flavor physics
The Yukawa interactions introduced in Eqn. (6) modify the SM in two aspects: The coupling between the SM Higgs and the exotic particles will change the decay width of the Higgs; Lepton family number is also violated and lead to lepton flavor-changing rare processes. Both impose constraints on the parameter space of the model. Since exotic particles introduced are not charged under SU (3) c , there will be no new contribution to processes such as meson mixings and b → s transition. Derivation of the constraints involves calculation of one-loop Feynman diagrams, which is assisted by the Mathematica packages FeynRules [40], FeynArts [41], FormCalc [42], and ANT [43]

Constraints from the Higgs
The Yukawa coupling Y H can modify the behavior of the Higgs boson. As all exotic particles are colorless, the production of the Higgs at the LHC is untouched. The decay from the Higgs to the exotic particles is also forbidden kinematically. However, the exotic particles can contribute to the decay of the Higgs at loop level such as h → γγ or h → Zγ as depicted in Fig. 2. LHC has measured the production of the Higgs boson in the diphoton channels. Among them the diphoton measurement is the most accurate one, which we will use to derive our constraint. The Higgs signal strength in the diphoton channel is defined as with the decay width of Higgs to diphoton in this model where the first two terms are the well-known results for the contributions from W boson and the SM fermions, and the last term is the new contribution from the Yukawa interaction H † ψχ. Γ SM h→γγ can be achieved by simply setting Y H to be zero in Eqn. (17). The dimensionless loop factors in Eqn. (17) are where f (τ ) = sin −1 ( 1/τ ) and I(η) has been obtained in the limit m χ , m ψ m h . We derive the limit on Y H for specific fermion masses from µ = 1.17 +0. 19 −0.17 [44]. In Fig. 3 we plot the contours of the maximal Y H allowed by the measurements at the LHC in solid lines. As the exotic particles have masses of 9.4 TeV to saturate the relic density, the current experiments impose literally no constraints on Y H . We also project the future constraints on Y H at 14 TeV LHC with 3000 f b −1 dataset. An optimistic estimate of 5% sensitivity [45] is used to extract the limits shown in the dashed contours also in Fig. 3. However, the heavy masses of the exotic particles make it impossible to place any meaningful constraints on Y H even at the high luminosity (HL) -LHC.

Lepton Flavor Violating Processes
The Yukawa interaction in Eqn. (8) will induce lepton flavor violating (LFV) processes. Several LFV processes, such as µ − → e − γ, µ → eee and µ-e conversion in nuclei, have been probed with extremely high sensitivity, but no signal has been found. The experimental results will surely place strong constraints on the model parameters. We will study the most well measured and thus most stringent ones in this section. The implications of these constraints will be discussed in Sec. 6.

l i → l j γ
We consider the process of the form l i → l j γ first and follow the convention in Ref. [46]. This type of decay is rare compared with the dominant tree-level decay via a virtual W boson. The amplitude of such process, depicted in Fig. 4, reads where e is the magnitude of electron charge and P L,R ≡ 1 2 (1 ∓ γ 5 ) are the projection operators. The coefficients σ Lij,Rij can be written as where F (η) is defined as The partial width then can be easily calculated with In the minimal variant of the model with two scalar quintuplets only, the relative rates are fixed and they are of similar order of magnitude. The current experimental limits are given by Br(µ → eγ) < 5.7 × 10 −13 [47], Br(τ → eγ) < 3.3 × 10 −8 and Br(τ → µγ) < 4.4 × 10 −8 [44]. Thus the most constraining limit is from µ → eγ, which we will discuss in Sec. 6. The proposed upgrade of MEG will improve the future sensitivity of µ → eγ down to Br(µ → eγ) ∼ 6 × 10 −14 [48].

Anomalous Magnetic Moment
The lepton anomalous magnetic moment can also be expressed with the terms in Eqn. (20) The anomalous magnetic moment of the muon has been measured to a very high precision ∆a µ = 0.0011659209 ± 0.0000000006 [49]. Given the heavy scalar quintuplet masses, the correction to the anomalous magnetic moment of the muon is well below the experimental precision and thus it does not impose any constraint.

µ → eee
Another well-measured LFV process is µ → eee, which this model will have several different contributions including γ-penguin, Z-penguin, Higgs-penguin, and box diagrams. The contribution from the Higgs-penguin is proportional to the electron mass and negligible. We show the relevant Feynman diagrams for this process in Fig. 5. The amplitude of the γ-penguin can be written in the form of where p is the momentum of the initial muon and p 1,2,3 are the momenta of the two electrons and the positron in the final state. The loop functions are given by where we have set the electron mass and the external momenta to zero. Similarly the contribution from the Z-penguin is where Z e L,R is the weak charge of the left-or right-handed electron. The weak charge of any fermion f is defined as being the isospin of f L,R and Q f the electrical charge of f . Finally the leading order contribution from the box diagram can be written as where we have neglected subdominant contributions further suppressed by the heavy mass scale and the loop function B L 1 is The analytic expression of the four-point function D 00 can be found in the Appendix of Ref. [43]. Among the various contributions, only the Z-penguin is suppressed by the Z boson mass, while the others are all suppressed by the masses of the exotic particles. With the amplitude we can easily write down the decay width [50][51][52], with the factors F LL,LR = F L Z e L,R /(e m Z ) 2 , where we have omitted all the vanishing or next-toleading order contributions. The heavy exotic particles in this model imply that the Z-penguin contribution dominates over the contributions of the γ-penguin and the box diagrams. The branching ratio for this rare decay channel can be approximately obtained by dividing Eqn. (32) by the decay width of µ − → e −ν e ν µ . The current experimental limit is given by Br(µ → eee) < 10 −12 [53] and there is a proposal for an experiment with an substantially increased sensitivity down to Br(µ → eee) < 10 −16 [54].

µ-e Conversion in Nuclei
The last LFV process we consider is µ-e conversion in nuclei. There are two types of contributions: the long-range interactions determined by the electromagnetic dipole, and the short-range interactions from the penguin diagrams as shown in Fig. 5 with the final electron pair replaced by a quark pair. For this model, there is no contribution from the box diagrams because no colored exotic states are introduced. Therefore the effective Lagrangian can be expressed as where the first and second lines denote the long-and short-range interactions. The Wilson coefficients g LV (q) only receive a contribution from the γ-penguin, while g LA(q) from both γ-and Z-penguins. They can be written as where Q f is the electrical charge of the quarks. Similar to µ → eee the Z-penguin contribution dominates over the γ-penguin. Coherent processes generally dominate over incoherent processes, if the final state nucleus is the same as the initial state nucleus. Thus we will focus on coherent contributions to µ-e conversion and neglect any incoherent contribution. So the conversion rate is [55] ω conv = 4 whereg (p,n) LV are the coefficients of the vector interactions with protons and neutrons defined as g (p) We use the values in Tab. 1 of Ref. [55] for the overlap integrals D, V (p) and V (n) . The branching ratio of µ-e conversion is defined as the ratio between the conversion rate and the capture rate where the rate ω capt takes the value 13.07 × 10 6 s −1 , 2.59 × 10 6 s −1 and 0.7054 × 10 6 s −1 in Au, Ti and Al [55]. The current bounds on the branching ratio are Br(µAu → eAu) < 7 × 10 −13 and Br(µTi → eTi) < 4.3 × 10 −12 [44]. There are good prospects to increase the future sensitivities for Al and Ti to Br(µAl → eAl) 10 −16 [56][57][58][59][60] and Br(µTi → eTi) 10 −18 [56; 57], respectively.

Naturalness
The newly introduced particles lead to corrections to the Higgs effective potential. Defining the tree-level Higgs potential as  In order to estimate the corrections of the new particles on electroweak scale naturalness, we calculate the corrections to the Higgs bilinear in the scalar potential. The correction to the quartic term is dimensionless and is not quadratically enhanced by the heavy mass scale compared to the electroweak scale. We use dimensional regularization with the modified minimal subtraction (M S) renormalization scheme to calculate the one-loop correction given in Fig. 6a. After canceling the divergent part with the counterterm, there is a finite contribution to the effective bilinear term of the Higgs µ 2 Thus it receives a quadratic correction from the new fermions in the loop. This poses a naturalness problem. The correction to the quartic coupling is of order (Y H ) 4 , but does not receive a quadratic enhancement by the large mass hierarchy. Similarly the Higgs couples to the scalar φ leading to a one-loop correction to the Higgs mass, which is shown in Fig. 6b. This quartic coupling, however, is unrelated to neutrino mass and could in principle be arbitrarily small at a given renormalization scale. Thus the two-loop contribution is effectively the leading order contribution related to φ, that we take into account.
In the unbroken phase its main contribution is given by diagrams of the type shown in Fig. 6c with W bosons and φ in the loop, which we estimate as follows The constant C is an order one factor, which we do not explicitly determine. We will naively set it to 1 in the following for simplicity, which is enough for an order of magnitude estimate. Evaluating the expression for the Higgs mass at the scale µ = m h The required fine-tuning to arrange for the correct Higgs mass by canceling the finite correction with the term tree-level term µ 2 H can be estimated by ∆ −1 with which quantifies the amount of tuning required to obtain the correct Higgs mass.

Discussion
Currently the LUX dark matter experiment [61] places the strongest limit on the dark matter direct detection cross section. Despite the mixing with ψ, there is no tree-level contribution to the spin-independent cross section from Z-boson exchange due to the Majorana nature of the dark matter candidate. Thus the dominant contribution arises from loop-level processes [2; 62]. The limit on dark matter direct detection cross section is given up to 1 TeV, which is roughly 10 −44 cm 2 . If we extrapolate the limit to the quintuplet mass, 9.4 TeV, the limit will be much weaker and thus will not put any constraint on this model at the moment. This model, however, will be probed by future direct detection experiments such as XENON 1T [63]. Dark matter annihilation in our galaxy produces high energy gamma rays. Experiments such as High Energy Stereoscopic System (H.E.S.S) [64] and the planned Cherenkov Telescope Array (CTA) [65] are searching for such signals and can place limits on the model. This limit on quintuplet has been discussed in Refs. [38; 66; 67] thoroughly. It is in tension with the current observation for a cuspy profile of dark matter halo, but allowed for a cored one. Nevertheless, this model is within the reach of the future CTA independent of the dark matter profile and will be tested soon.
The possibility to test MDM at a collider has been extensively studied for the LHC, the HL-LHC and even a 100 TeV proton-proton collider [68; 69]. Generally electroweak multiplets can be tested in events with mono-jet, mono-photon, vector-boson fusion and disappearing tracks. Among them, the test with disappearing tracks has the best sensitivity, with a reach of about 3 TeV for a 100 TeV proton-proton collider for an integrated luminosity of 30 ab −1 , which is still far below the mass of the dark matter candidate of this model.
Lepton flavor violating processes are governed by the Yukawa coupling Y L , while the Higgs to diphoton branching ratio and naturalness are controlled by Y H . Finally, neutrino mass depends on the product of Yukawa couplings Y H Y L and thus connects the phenomenology creating an interesting interplay. The smaller Y H , the larger Y L and consequently the larger the LFV branching ratios. In Fig. 7 we show the branching ratio for the lepton flavor violating processes µ → eγ (red) , µ → eee (green), and µ-e conversion 2 in gold (blue), aluminium (maroon), and titanium (magenta) as a function of Y H for fixed masses m χ = 9.4 TeV, m ψ = 15 TeV, m φ 1 = 16 TeV, and m φ 2 = 16.5 TeV as solid lines. The dotted lines indicate the change if the masses of ψ and φ i are doubled. The solar and atmospheric mass squared differences and the leptonic mixing parameters are fixed to the best-fit values of v2.0 of the nu-fit collaboration [70]. Leptonic CP phases are set to zero, as well as the complex angle θ parameterizing O. The horizontal solid lines show the current experimental bounds on Br(µ → eγ) < 5.7 × 10 −13 [47] (red), Br(µ → eee) < 10 −12 [53] (green), Br(µTi → eTi) < 4 × 10 −13 [71] (magenta), and Br(µAu → eAu) < 7 × 10 −13 [72] (blue), while the dashed lines indicate the future sensitivities: Br(µ → eγ) ∼ 6 × 10 −14 [48], Br(µ → eee) ∼ 10 −16 [54], Br(µAl → eAl) ∼ 10 −16 [56][57][58][59][60]) and Br(µTi → eTi) ∼ 10 −18 [56; 57] For fixed masses and Yukawa coupling Y H , it is possible to quantify how much the parameters of the model have to be tuned to obtain the electroweak scale. We show the required fine-tuning ∆ −1 , which is defined in Eqn. (43), as a black solid line for our benchmark point and dotted line for doubled particle masses. The values can be read off the y-axis on the right-hand side. The Yukawa coupling Y H becomes non-perturbative on the right-hand side of the figure, while the largest entry of the Yukawa coupling Y L becomes non-perturbative on the left-hand side of the figure. We plot the maximum value of the Yukawa couplings, max(Y L ij , Y H ), in gray. The value can be read off the y-axis on the right-hand side. Fig. 8 illustrates the uncertainty in the solar and atmospheric mass squared differences and leptonic mixing parameters. All these parameters are varied within their 3σ allowed ranges. The   different colors and line styles are chosen in the same way as in Fig. 7. Fixing everything but the leptonic mixing parameters and mass squared differences, there is an uncertainty of up to three orders of magnitude in the rates for the different processes. Fig. 9 shows a contour plot of Br(µAu → eAu) in the plane of the imaginary part of the complex angle θ in the complex orthogonal matrix O and the Yukawa coupling Y H . All LFV rates are relatively insensitive to the real part of θ and change at most at the percent level. A large imaginary part however leads to large Yukawa couplings Y L canceling among each other to accommodate the light neutrino mass.
Finally we want to comment on the renormalization group evolution of the couplings in this model. The large SU(2) representations lead to a strong running of the gauge couplings resulting in a Landau pole at a scale of about 10 9 GeV, where we have taken two-loop running into account. Similarly the quartic couplings of large scalar SU(2) representations suffer from the triviality bound [73][74][75], because gauge couplings will induce these couplings at one-loop order, which will be further amplified by the running of the respective quartic coupling itself [75]. In particular, the study in Ref. [75] finds that the quartic coupling of the real quintuplet suffers from a Landau pole below 10 15 GeV. A viable UV completion has to preserve the accidental Z 2 symmetry to prevent the minimal dark matter candidate from decaying. We are agnostic about the UV completion and mainly concentrate on phenomenology, since there are good prospects to ultimately test minimal dark matter models in the near future.

Conclusion
Embedding radiative neutrino mass models in the framework of MDM to make one theory work for two major fields of physics beyond the SM is aesthetically appealing. We systematically studied the possibility to realize this idea with radiative neutrino mass models as UV completions of the Weinberg operator. None of the minimal UV completions at one-loop leads to a stable minimal dark matter candidate. However we argued that it is feasible to obtain a cosmologically stable dark matter candidate, because the decay is controlled by a coupling unrelated to neutrino mass generation, which can naturally be arbitrarily small.
We studied the phenomenology of one model explicitly. The model contains two real quintuplet scalars and also a quintuplet fermion whose neutral component field plays the role of dark matter. Both fields have zero hypercharge. In addition, a vector-like quadruplet fermion are introduced with hypercharge ± 1 2 . We discussed the neutrino mass generation in this model and performed a detailed phenomenological study of lepton flavour violation and Higgs decay. There is a sizable allowed region of parameter space consistent with all current experimental constraints from Higgs physics and lepton flavor changing processes. The most stringent bound is placed by µ-e conversion in nuclei and will be further improved by future experiments. It places a lower bound on the Yukawa coupling Y H and thus increases the electroweak fine-tuning. Current bounds already require at least ∼ 10% tuning. In the near future, the remaining parameter space of this model can be tested by direct detection experiments like XENON 1T and indirect detection experiments such as CTA.

A Mass Matrix Diagonalization
The mass mixing in this model is proportional to Y H v/m ψ,χ and thus very suppressed for heavy masses of O(10) TeV. Although it can be safely neglected in most of the calculation as we did, we show the technical details for the diagonalization of the mass matrices for completeness.
The mass matrix for the neutral fermions can be diagonalised using a Takagi factorization, At leading order we find For the singly-charged fermions, the mass matrix M 1 can be diagonalised with a singular value decomposition V * 1 X 1 W † 1 = X D 1 with a diagonal matrix X D 1 and two unitary matrices V 1 , W 1 . To the leading order they are given by Similarly for the doubly-charged fermions, the mass matrix M 2 can be diagonalised with a singular value decomposition V * 2 X 2 W † 2 = X D 2 with a diagonal matrix X D 2 and two unitary matrices V 2 , W 2 . To leading order they are given by B SU(2) L generators and the kinetic terms All the odd-dimensional representations are real and even-dimensional representations pseudo-real. The generators of the four-dimensional representations can be explicitly written as and the generators of the five-dimensional representation are given by J 5 3 = diag (2, 1, 0, −1, −2) . The kinetic terms for the exotic field are expressed as where the covariant derivatives are D µ = ∂ µ − igJ a W a µ − ig Y B µ .