Radiative accidental matter

Accidental matter models are scenarios where the beyond-the-standard model physics preserves all the standard model accidental and approximate symmetries up to a cutoff scale related with lepton number violation. We study such scenarios assuming that the new physics plays an active role in neutrino mass generation, and show that this unavoidably leads to radiatively induced neutrino masses. We systematically classify all possible models and determine their viability by studying electroweak precision data, big bang nucleosynthesis and electroweak perturbativity, finding that the latter places the most stringent constraints on the mass spectra. These results allow the identification of minimal radiative accidental matter models for which perturbativity is lost at high scales. We calculate radiative charged-lepton flavor violating processes in these setups, and show that $\mu\to e \gamma$ has a rate well within MEG sensitivity provided the lepton-number violating scale is at or below $10^6$ GeV, a value (naturally) assured by the radiative suppression mechanism. Sizeable $\tau\to \mu \gamma$ branching fractions within SuperKEKB sensitivity are possible for lower lepton-number breaking scales. We thus point out that these scenarios can be tested not only in direct searches but also in lepton-flavor violating experiments.


Introduction 2 Effective scales in accidental matter scenarios
Accidental matter models are weak-scale extensions of the SM in which the beyond-SM (BSM) degrees of freedom (R) preserve the accidental and approximate symmetries of the SM at the renormalizable level. Thus, this means that even if present at the renormalizable level their effects will not be accessible in indirect searches. One might wonder whether departures from their standard formulation could change that picture. For that aim one can consider the SM as the renormalizable part of a larger Lagrangian: where L Ren SM are the renormalizable SM interactions, whereas the second term are effective operators where the only dynamical degrees of freedom are SM fields. They do break the SM accidental and approximate symmetries and so their effects include CP and flavor violation in the quark and lepton sectors as well as baryon and lepton number breaking.
In the standard approach the coefficients of the effective expansion C N are assumed to be O(1). When combined with the assumption that neutrinos are Majorana particles, this fixes the effective scale, Λ Eff ∼ 10 15 GeV. Automatically then, all possible signatures related with departures from SM accidental and approximate symmetries are suppressed, thus explaining their absence in indirect searches. Testability of these scenarios is possible only if the new states can be directly produced and detected in collider experiments. Otherwise the new physics, although present, could be hard-if not ℓ ℓ H H R Figure 1: Dimension five LNV effective operator in the limit SM + R. The presence of R allows for a naturally small expansion coefficient C and therefore for a lower effective scale.
impossible-to reveal. A possible departure from the standard formulation consists then in allowing for lower cutoff scales that in turn allow for other observables to have sizeable values, thus increasing the testability of these scenarios. The effective operators in (1) are subject to different phenomenological constraints, with the most stringent limits enforced by neutrino masses on the LNV ones. Without any further assumption the leading-order LNV operator is given by the dimension five Weinberg operator that after EW symmetry breaking leads to m ν ∼ C v 2 /Λ. A lower cutoff scale, say O(TeV), is possible provided C ∼ 10 −12 −10 −11 . There are two generic mechanisms through which such a small coefficient can be naturally obtained: (i) the operator is related with a slightly broken symmetry (for example slightly broken lepton number) [25][26][27][28], (ii) the operator is radiatively induced. Though mechanisms of type (i) can be envisaged, in the presence of SM + R the natural option relies on (ii). Let us discuss this in more detail. Below Λ the only degrees of freedom are the SM fields and R, and so lepton number violation should be determined by effective operators (as required by the accidental matter scenario). Since R ⊗ R ⊃ 1 ⊕ 3 ⊕ 5 ⊕ · · · ⊕ (2R − 1), the effective operator in (2) can be endowed with the new degree of freedom as shown in fig. 1. Therefore, in this case the effective expansion coefficient is suppressed by the loop factor and by extra couplings that originate in the UV completed theory. Roughly it can be written as C ∼ Y 4 ν /16π 2 , which means that Λ can be as small as ∼ 10 5 GeV for Y ν ∼ h τ (τ Yukawa coupling). These values mean that the degrees of freedom of the UV theory, although not reachable at the LHC can manifest in indirect searches, e.g. in µ → eγ, µ → 3e or in µ − e conversion in nuclei, processes that are/will be searched for in MEG [12], M u3e [29,30] and PRISM/PRIME [31,32] respectively (see e.g. refs. [27,33] for phenomenological studies).
Some words are in order regarding the effective scale for the operator in (2). This scale determines the cutoff scale where different UV completions-involving new states-enable writing down the operator in fig. 1 through renormalizable couplings. This scale differs from that where perturbativity is lost (Λ Landau-pole ), in contrast to standard accidental matter scenarios where these two scales match (see fig. 2). At scales above Λ new states kick in (generically denoted by R ), and their renormalizable interactions break lepton number and lepton flavor, but quark flavor and baryon number are still symmetries of the renormalizable Lagrangian (at that scale). Since the new states : Energy scales in standard (left-hand side) and radiative (right-hand side) accidental matter models (benchmark scenario). Note that the LNV, LFV, and QFV scales in the latter are smaller, thus allowing for potential observability of the corresponding processes. The energy scales on the right side represent the benchmark scenario we will use for our discussion.
contribute to α 1 and α 2 (α i = g 2 i /4π) running, perturbativity is lost more rapidly than in the case SM + R. The exact value where this happens depends upon the number and dimensionality of the new representations. Assuming that perturbativity is restored by a larger gauge theory, one could expect quark flavor and/or baryon number to be broken at that scale. If only quark flavor is broken, Λ Landau-pole ∼ 10 8 GeV suffices to satisfy current constraints on QFV processes [34], otherwise Λ Landau-pole 10 15 GeV is required to ensure proton stability. Here we will select viable accidental matter scenarios by the condition Λ Landau-pole 10 8 GeV, which implicitly assumes that the new dynamics does not involve any baryon number violation.

Accidental matter representations
The quantum numbers of the new representation are determined by whether R is a fermion or a scalar. Since R should preserve the SM symmetries, as pointed out in [17] fermionic representations should be such that operators of the form should not be possible writing.
3) e -breaking sources will be introduced. For scalar representations instead renormalizable operators are possible without affecting G F , provided their quantum numbers do not enable couplings with SM fermionic bilinears. Bearing in mind this discussion and that the first relevant fermionic representation is R F = 4 Y F , with hypercharge Y = 1/2 or 3/2 determined by whether it couples to H H † or H † H † (here we will use the notation R Y F,S where R labels the representation, F and S its fermionic or scalar character and Y its hypercharge. Hypercharge is normalized according to Q = T 3 + Y ). The remaining fermionic representations follow from the rule in (4) with hypercharge fixed by the SM operator to which they couple. For scalars, at the renormalizable level, the first representation is indeed a SM singlet, R S = 1 0 S , with the remaining representations given by R S = 3 0 S , 4 1/2 S , 4 3/2 S and 6 1/2 S , with the latter being loop-induced (see sec. 3) 3 . For non-renormalizable operators the first scalar representation is R S = 2 Y S with Y = 3/2, 5/2. The remaining scalar representations are: S (Y = 1/2, 3/2, 5/2, 7/2) and so on. From this list, viable representations are selected from cosmological constraints and the condition of perturbativity.
The neutral component of higher-order SU (2) representations is cosmologically stable: For scalars Direct DM searches constraints, however, rule out all those representations for which Y = 0 [19][20][21]. Perturbativity criteria as well places constraints on viable representations. Using a two-loop RGE analysis, ref. [17] has shown that for R 0 S > 8 0 S and R 0 F > 6 0 F a Landau pole is obtained for scales below ∼ 10 8 GeV. Thus, in a model defined by SM + R the accidental matter representations are: R Y S ≤ 7 0 S and R Y F < 5 0 F . A major difference between the standard accidental matter models and the setups we are considering here is that in the latter there are new degrees of freedom that enter at relatively low scales (10 6 GeV). The presence of these states induces fast decays of those representations that otherwise would be cosmologically stable [22]. Thus they no longer involve a DM particle, and therefore direct DM constraints no longer hold. This enables Y = 0 representations, something that is particularly important for the scalar sextet (see sec. 3) 4 .
In summary, the setups we will consider henceforth are defined by the accidental matter representations listed in tab. 1. This list differs from that found in standard accidental matter scenarios in that it contains the scalar sextet representations, which are enabled due to the instability of higherorder representations induced by the presence of additional representations. The UV completed models we will construct are therefore defined by these representations and subject to the condition of the full UV model satisfying Λ Landau-pole > 10 8 GeV (see sec. 4.2).
3 Higher-order SU (2) representations and their cosmological instability As we have already pointed out, higher-order SU (2) representations in SM + R models are cosmologically stable. The neutral component of the hypercharge-zero fermion quintet and scalar septet are-in principle-WIMP DM particles. For these representations tree level effective operators of 3 Note that SU (2) ⊗ U (1) Y invariance allows for R S = 3 1 S . This representation however couples to the fermion bilinear c and so introduces G F -breaking sources. 4 This argument applies as well to 5 1,2 F , however for these representations alone a Landau pole is reached at scales below ∼ 10 8 GeV. The presence of additional representations at 10 6 GeV reduces that scale to values well below those that define our perturbativity criteria (see fig. 2), and so we do not consider them.

Radiative accidental matter representations
with O SM an operator entirely consisting of only SM fields, are dim=6 and dim=7, respectively. Thus, lifetimes amounting to 10 26 seconds (as required by the non-observation of γ-ray, ν, e + or p − signals in DM indirect detection experiments [35][36][37][38]) are found for Λ 10 15 GeV for 5 0 F and Λ 10 10 GeV for 7 0 S , provided m DM ⊂ [5, 10] TeV [22]. However, if one considers one-loop induced effective operators one finds that scalar septet decays are instead driven by dim=5 operators [17,39], for which not even Λ = M Planck leads to sufficiently large DM decay lifetimes. This observation then singles out the hypercharge-zero fermion quintet as the only representation containing a viable DM particle.
It is worth pointing out that this representation is however subject to stringent constraints coming from indirect DM searches. Particularly relevant are limits derived from γ-ray line searches from the galactic center, for which it has been found that if the Milky Way exhibits a Navarro-Frenk-White or Einasto DM profile this representation is not viable either [40,41]. It can be however consistently considered in the context of cored profiles such as Burket or Isothermal. Or by relaxing the hypothesis of WIMP DM, allowing for Y = 1 and so leading to millicharged DM scenarios [39]. In scenarios defined by UV completions of the operator in fig. 1, slow decays of higher-order EW representations do not hold anymore. The point is that at Λ = 10 6 GeV new states kick in, introducing new renormalizable couplings that allow writing down operators of type (5) with cutoff scales fixed by neutrino data. Roughly one can write m ν ∼ v 2 Y 4 ν /16π 2 Λ, which means that for m ν = m Atm = 50 meV [42][43][44] and Y ν = 1 the cutoff scale should be below ∼ 10 13 GeV. This scale is below the value required for cosmological stability of the fermion quintet, thus showing that these setups are not reconcilable with slow DM decays (for a more detailed discussion see [22,23]). This can be put more precisely in the context of explicitly broken symmetries: DM slow decays can be understood as due to an accidental Z 2 symmetry under which R → −R and X SM → X SM , and which results as a consequence of the SM gauge symmetry. For R = 5 0 F , UV completions of the operator in fig. 2 always allow for couplings that break Z 2 , hence DM instability 5 .
These arguments can apply to Y = 0 representations, depending on the value of Y and on the specific UV completion. Indeed, they are responsible for the scalar sextet as a viable accidental matter representation as we now discuss. For 6 1/2 S , gauge invariance allows the following scalar coupling: that explicitly breaks the accidental Z 2 symmetry. The presence of this coupling enables the operator in fig. 3-(a), which induces fast decay processes of the neutral component of the multiplet, Since slow decays are-in general-not possible, its density is rapidly depleted and thus should be consistently included in the list of possible accidental matter representations. Note that this operator can be written even in the absence of additional representations, and so this conclusion proves to be true even in standard accidental matter scenarios.
Operators for 6

3/2 S
can also be written, but in contrast to the 6 1/2 S case they require additional representations. The different decay operators are shown in fig. 3, diagrams (b) − (d). As can be seen, they all involve Z 2 -breaking couplings and therefore lead to fast decay processes of the lightest component of the multiplet, in this case ϕ 0 ⊂ 6 3/2 S . Among those processes one can identify e.g.
These couplings are of three kinds: independent of 6 3/2 S and bilinear and linear in 6 3/2 S . Explicitly, for each operator, they are given by As we will show in sec. 4.2, these couplings cover all possible UV completions associated with this representation, apart from one which involves the following representations: 5 2 S , 4 3/2 F and 5 2 F and for which we did not find a coupling enabling fast decay (Z 2 -breaking coupling). This UV completion therefore has not been included in our analysis. Finally, for 6 5/2 S several decay operators can be written too. Here, however, we present just a single one (see fig. 3-(e)). The reason is that for this representation only few UV completions are consistent with our perturbativity criteria (see sec. 4.2), and this operator covers all of them. Being Z 2 -breaking it induces fast decays of the neutral component of the multiplet and so allows for a viable accidental Y = 5/2 sextet.
The dots indicate Z 2 -breaking couplings, which ensure fast ϕ 0 ⊂ 6 Y S decays thus making them suitable accidental matter representations.

Radiative accidental matter
We now turn to the discussion of UV completions of the operator in fig. 1. For certain representations, in particular for higher-order ones, a certain UV completion can as well generate a dim=7 or dim=9 lepton-number-violating operator. In these cases one can find therefore regions in parameter space where the effective neutrino mass matrix receives contributions from several operators, or even where the neutrino mass matrix is entirely determined by the higher-order operator. Our assumption here is that m ν is solely determined by the operator in fig. 1, and this imposes a condition on the scale of the beyond-the-SM (BSM) degrees of freedom. The contribution to neutrino masses from the tree level dim=7 lepton-number breaking operator is m dim=7 ν v 4 /Λ 3 . Thus, when compared with the one-loop contribution from the Weinberg operator it can be seen that Λ 3 TeV guarantees m dim=7 ν < m dim=5 ν , a condition satisfied by the BSM spectrum that defines the radiative accidental matter scenarios we are discussing (see fig. 2).

Additional constraints: ρ parameter and BBN
Beyond the constraints we have already mentioned, there are other constraints one needs to bear in mind. Of particular relevance are those related with the breaking of the custodial symmetry, which place bounds on the mass of the accidental matter representations. A detailed analysis of these constraints has been presented in [17] and therefore here we discuss only those aspects that directly apply to radiative scenarios. In the limit sin θ W → 0 (g → 0), the weak gauge bosons W ± and Z transform as a triplet of an SU (2) L+R global symmetry, which implies m W ± = m Z . In that limit the SM ρ-parameter, defined as ρ = m 2 W /m 2 Z cos 2 θ W , is one. Departures from this limit removes the gauge bosons mass degeneracy through cos θ W , but still one finds ρ tree = 1, with small deviations induced by radiative corrections, that remain under control due to the SU (2) L+R custodial symmetry. This value is consistent with its experimental value, ρ Exp = 1.0004 +0.0003 −0.0004 [13]. Contributions to the ρ-parameter from BSM scalar fields that develop vevs can produce sizeable deviations from such value. For a set of scalars {ϕ T,Y } that acquire a vev, ϕ T,Y , and whose total weak isospins are T and their hypercharges are Y , the tree level ρ-parameter reads [46] where c T,Y = 1 (c T,Y = 1/2) for complex (real) fields and Q = T 3 + Y . There is an infinite set of scalar fields that satisfy ρ tree = 1, determined by the condition The list of the viable scalar representations (that contain a neutral component) are determined by the following quantum numbers: (T, Y ) = {(0, 0), (1/2, ±1/2), (3, ±2), (25/2, ±15/2), · · · } 6 . Which shows that apart from the singlet, none of the other accidental matter scalar representation satisfies such condition and thus their vevs are subject to constraints. With only the Higgs and a single extra scalar field, expression (8) at order ϕ T,Y 2 /v 2 can be cast according to from which using the experimental upper limit for ρ Exp one finds ϕ T,Y /v 1% [17]. This constraint is particularly important for accidental matter representations which develop an induced vev, namely the triplet and the quartet (the singlet does as well but its vev does not contribute to ρ tree , as we have mentioned). For these representations this restriction translates into a lower bound on their masses, which can be (roughly) estimated from the minimization condition of the corresponding scalar potentials: As can be seen, these values are consistent with radiative accidental matter models and in particular with the benchmark scenario we have chosen. It is worth emphasizing that in models where several of these states are found, these bounds will be more stringent with the values estimated to increase multiplicatively with the number of states. Non-vanishing contributions to the ρ-parameter arise as well from radiative corrections to gauge boson masses. Mass splittings between the different components of a representation R lead to large radiative contributions, provided the splittings are large [48]. These splittings can arise from oneloop corrections (for fermions and scalars) and from off-diagonal terms in the tree level scalar mass matrices. The former are of order MeV [19] and so lead to negligible corrections to the ρ-parameter. The latter instead can involve large splittings and so can induce in turn sizeable deviations on ρ. However, when used to derive limits on scalar masses, the values found are less competitive than those in (12) or those coming from direct accelerator searches [17].
We now turn to the discussion of the constraints arising from BBN. Long-lived particles with lifetimes larger than ∼ 0.1 seconds can significantly affect light-elements abundances through their electromagnetic and/or hadronic activity. Thus, consistency with observed light-elements abundances translates into constraints which lead e.g. to lower bounds on their masses/couplings [49,50]. Whether such constraints hold for the scenarios we are considering here depends on the lifetime of the different decay processes. For the representations in tab. 1 there are two types of decays. Intermultiplet decay processes in which heavier components of a multiplet undergo decays into lighter components, and decays of the lightest state (LS) into SM particles. The former are fast decay processes such as e.g. R + → R 0 + π + , and so they take place at early times well before BBN. The latter can be fast too, depending on whether the lightest particle can decay via renormalizable couplings or, in case it does not, on the cutoff scale. As it has been discussed in sec. 2.1, effective decay processes are driven by either dim=5 or dim=6 operators, for which the decay lifetimes for the LS can be estimated to be τ dim=5 9.8 Λ 10 6 GeV Note that this result assumes that the LS can directly decay via the non-renormalizable operator. However, this is not the case for 2 by off-shell heavier components of the representation. These processes have been studied in ref. [17] assuming Λ = 10 15 GeV and resulting in lifetimes amounting to ∼ 10 3 seconds, but the rescaling of these results according to our cutoff scale lead to lifetimes comparable to those in (13). All in all, the decay processes of the accidental matter representations in radiative scenarios are fast, with the largest lifetimes amounting to at most microseconds, and so BBN constraints are of no relevance.

UV completions and perturbativity
Our assumption is that at or above 10 6 GeV new degrees of freedom defining different UV completions for the operator in fig. 1 become available. For a given representation the full set of UV completed models can be derived by considering the diagrams in fig. 4, which correspond to all possible irreducible one-loop realizations of the operator in fig. 1 [51] 7 . We systematically fix the accidental matter representation within the loop in each of the diagrams and then fix the SU (2)×U (1) Y quantum numbers of the remaining fields according to R ⊗ 2 = R ± 1. UV completions involving hypercharge-zero fermion singlets or triplets or hypercharge-one scalar triplets are discarded, since they lead to seesaw neutrino masses ("seesaw filtering criterion"). These cases are found for 1 0 S , 3 0 S and 4 Y F (Y = 1/2, 3/2). Rather than explicitly listing the resulting models, which amount to hundreds and that can be straightforwardly derived, we list in tab. 2 the representations which are needed in each case. These results are then used to identify those models that lead to the highest Landau pole scales.
For 1 0 S all viable models are forbidden by the "seesaw filtering criterion". For 3 0 S some models are found, but still the filtering criterion removes most of them leaving just few. For 4 Y F , instead, this criterion removes only few models. Instead, in this case, a fairly large number of such models are found to be non-viable because they lead to non-perturbative effects below 10 8 GeV, something found as well for other higher-order SU (2) representations.

A.M.
Fermion sector Scalar sector # of models S 12 24 17 10  fig. 4 and selected according to the condition that the particle content of the model does not enable type-I, type-II or type-III seesaw and that α 1,2 remain perturbative at least up to 10 8 GeV.
Models that become non-perturbative below 10 8 GeV, as defined by our benchmark scenario, are identified by using two-loop RGEs subject to the following energy thresholds (see appendix A): • From M Z and up to m R (R being the accidental matter representation), that we take to be 1 TeV, the particle content is entirely given by the SM.
• From m R and up to 10 6 GeV, where according to our benchmark scenario the UV completions for the operator in fig. 1 kick in, the particle content is determined by the SM + R.
• Above 10 6 GeV, RGE running takes into account the SM + R + R , with R referring to representations that define the UV completions.
For low-order SU (2) representations up to the triplet, Landau poles are found at rather high scales, ranging from 10 11 GeV up to 10 19 GeV, with the exception being few models for 2 5/2 S for which α 1 = g 2 1 /4π develops a Landau pole at ∼ 10 8 GeV, due to the large hypercharges of the extra representations. Thus, apart from this representation, all low-order radiative accidental matter models are consistent with perturbativity up to 10 8 GeV.
For higher-order accidental matter representations this behavior remains like that for the 4 1/2 S and also for 4 3/2 S , but for 4 3/2 S some models fail to pass the perturbativity condition. For 5 0 S all models are consistent with perturbativity and Λ highest LP remains at 10 13 GeV, depending on the model category (defined by diagrams D1-D2). For 5 Y S (Y = 1, 2), the largest Landau pole scales are somehow degraded with values even as low as 10 9 GeV for 5 2 S , again depending on the model category. In these cases many models reach Landau poles well below 10 8 GeV, and so fail to pass the pertubativity cut. This trend persists for the remaining accidental matter representations, including the 4 Y F , being Among all the viable models we select those for which the Landau pole scale is the largest. These radiative accidental matter models are arguably the most compelling ones. For models involving 3 0 S and 2 3/2 S several setups for which Λ LP 10 19 GeV are found. In these cases the selection criterion is that of the model involving the least number of representations. In almost all cases the corresponding models are D1-based, something somehow expected since these UV completions involve the least number of fermions and so gauge couplings run slower. In all cases as well we have found that α 2 reaches the Landau pole before α 1 does, with a single exception given by 2

Acc. Matter Extra representations Model
> 10 19 GeV 3.41 × 10 17 GeV  Table 3: Radiative accidental matter models for which the Landau pole is reached at the highest possible scale. Apart from 5 0 S and 5 1 S , all models are D1-based. This list therefore defines the most compelling radiative accidental matter models. Note that the relatively low Landau pole scale for 5 2 S , 6 3/2 S and 6

Lepton flavor violation: generic approach
In this section we quantify the expected size of SM charged-lepton flavor violating (CLFV) radiative processes in the models depicted in tab. 3 8 . Three body CLFV decay processes, in particular µ + → e + e − e + , and µ − e conversion in nuclei are relevant as well due to the large sensitivity of near-future experimental facilities: M u3e at PSI aims at measuring µ + → e + e − e + to a precision of 10 −16 [29,30], while PRISM/PRIME at J-PARC µ − e conversion in nuclei down to 10 −18 [31,32]. Results for these processes will be presented elsewhere [55].
Rather than sticking to a particular realization or analyzing them all we make use of the fact that the problem follows a generic treatment. Since most of the compelling models are determined by D1 diagrams, we will focus on such models for which a schematic Lagrangian can be used for the discussion: where F (S) refer to vectorlike fermions (scalars) in any of the representations displayed in tab. 3, and so SU (2) contractions are assumed. Sum over lepton flavor and fermion generation indices is understood, while a and b just label different scalars. Note that in addition to the terms in (15), there are also pure scalar and fermion mass terms (that can be of Majorana type if Y (F ) = 0) which we are not writing, but are essential since they determine scalar mixing and eventually fermion mixing too. Furthermore, we are assuming for simplicity that Yukawa couplings are the same regardless of the scalar to which the fermion bilinear is coupled. We take two generations of vectorlike fermions, which is the minimal number required to generate two non-zero light neutrino masses in this simplified setup we have assumed. This is a direct consequence of assuming same Yukawa couplings, for different Yukawa couplings a single fermion will suffice. Note that in models where more than a single fermion is needed the Landau pole scale will be below the numbers quoted in tab. 3. Since in this section we aim just at showing that CLFV effects are within reach and we are not specifying quantum numbers we stick to the values in tab. 3. Components of the scalar multiplets (that we assume there is only one copy per representation) with the same electric charge Q mix through the scalar coupling in (15). Depending on Q, their mass eigenstates will contribute to the neutrino mass operator. One can distinguish for example: (i) one pair of scalar mass eigenstates with electric charge Q determine the neutrino mass matrix, (ii) two pairs with electric charges Q and Q are responsible for the operator. Just to mention a couple of cases, in the model for 2 1/2 S mix and define the eigenstates (S + 1 , S + 2 ) and (S ++ 1 , S ++ 2 ) that in turn lead to the neutrino mass matrix. Therefore, after rotation to the scalar mass eigenstate bases and regardless of the radiative accidental matter model, the neutrino mass matrix consist of a series of diagrams that guarantee cancellation of the divergent piece of the Passarino-Veltman (PV) function B 0 (0, m 2 Sa , m F ) [56]. The finite piece defines in turn the neutrino mass matrix that reads: where Θ 2 a parameterizes scalar mixing and satisfies a Θ 2 a = 0. In the simplest case a = 1, 2, Θ 2 a = (−) a sin θ cos θ (θ the mixing angle) and so the finite piece of the PV function under the sum over a can be written according to Note that the neutrino mass matrix is rank-two and so there is a massless light neutrino, as already anticipated. To determine the Yukawa structure required by neutrino oscillation data [42][43][44], that combined with the mass scale for the fermions and scalars determines the LFV rates, we start by recasting the neutrino mass matrix according to where Y is a 3 × 2 Yukawa coupling matrix whose entries are given by Y iα and F is a dimensionful 2 × 2 diagonal matrix whose non-vanishing entries read: With the mass matrix written as in (18) the Yukawas can be then parameterizedà la Casas-Ibarra [57], namely Here U is the lepton mixing matrix and R is a 3 × 2 orthogonal complex matrix that can be written as [58] R = 0 cos z sin z 0 − sin z cos z , where z is a complex angle. With the aid of eq. (20), radiative CLFV processes can be calculated by fixing the scalar mass spectrum and using neutrino data. The branching fractions for these processes can be written according to [59] Br with t αa = m 2 Fα /m 2 Sa , Γ l i Tot the total decay width of l i and the loop functions given by With these results at hand we calculated the muon and tau decay branching ratios as a function of the heaviest fermion mass. For that aim we fixed fermion masses according to m F 2 −m F 1 = 500 GeV, while randomly varying m F 2 in the range [10 4 , 10 9 ] GeV. The scalar spectrum according to m S 2 = 10 3 GeV and m S 1 = 500 GeV, neutrino low-energy observables to their best-fit point values [43] and scalar mixing in the range Θ 2 = [10 −12 , 10 −8 ], randomly varied too. For simplicity we have taken z real and equal to π/10. We have checked that the result is pretty insensitive to the value of z as long as z is real. The result as well is not very sensitive to the value of Q F provided Q F < 2. For all points in the scan we have checked max(Y ) < √ 4π. Note that this parameter choice has nothing special and has been taken just to exemplify the typical CLFV behavior one expects in these scenarios. Fig. 6 shows the result for µ → eγ as a function of the heavy fermion mass. It shows that for a reasonable mass range this process falls within the region determined by MEG near-future sensitivities [60]. The result is representative of what is expected for the radiative accidental matter models listed in tab. 3. Since it has been done without sticking to a particular realization, it is of course subject to numerical variations determined by the details of the model and its particular behavior with relevant parameters. For radiative tau decay processes we have found that τ → eγ lies one order of magnitude below SuperKEKB sensitivities [61], while τ → µγ can be within the range of observability if the mass gap between R (the accidental matter representation) and the radiative dim=5 operator UV completion is not large, O(m UV ) ∼ 10 m R . This, however, would lead to a lower Λ LP scale and so according to our approach is disfavored.  [12], whereas the lower line MEG future sensitivity [60]. This result has been derived by assuming that the process is mediated by a pair of charged fermions (scalars) with Q F = +1 (Q S = +2). The result is representative of what is expected from the radiative accidental matter models listed in tab. 3 (compelling radiative accidental matter models), with possible numerical variations determined by the specific model and/or parameter space region.

Conclusions
Accidental matter models involve BSM degrees of freedom that automatically preserve the SM accidental and approximate symmetries up to a certain cutoff scale. Assuming that this scale is determined by lepton number violation and in the absence of a "non-conventional" suppression mechanism, this scale is fixed at ∼ 10 15 GeV. In this paper, we studied accidental matter models assuming that the accidental matter representations play an active role in neutrino mass generation (radiative accidental matter models), something that we have shown necessary leads to radiatively induced neutrino masses and therefore to a suppressed LNV scale that can naturally be as low as 10 6 GeV.
By defining a benchmark radiative accidental matter scenario (see fig. 2), we have shown that in this new context higher-order accidental matter representations are no longer cosmologically stable. In particular, we have shown that this observation combined with a lower perturbative scale enables Y = 1/2, 3/2, 5/2 scalar sextets. By considering EW and BBN constraints, we have as well derived lower bounds on the masses of various accidental matter representations, showing that masses of about 1 TeV always imply a consistent picture.
We have identified the different UV completions of the radiative dimension five operator in fig.  1. We have systematically studied their perturbative behavior and filtered viable models according to whether α 1 and α 2 remain perturbative up to at least 10 8 GeV, our results are summarized in fig. 5. From this classification we have determined the most compelling radiative accidental matter models by the condition that the Landau pole scale is the largest possible. Our results are listed in tab. 3. For the resulting models we have studied in a fairly model-independent way the typical size of radiative CLFV effects. According to our findings, µ → eγ is within reach for all models provided the UV completion of the operator in fig. 1 does not exceed ∼ 10 6 GeV. In contrast, for τ → µγ observability requires a somewhat BSM compressed spectrum that in turn leads to a lower Landau pole scale, something disfavored by our approach but not excluded. Thus, these setups offer a rich LFV phenomenology that increases their experimental testability and which motivates further phenomenological studies of the resulting setups [55].

Acknowledgments
We would like to thank Luca di Luzio for useful comments. This work was supported by the "Fonds de la Recherche Scientifique-FNRS" under grant number 4.4501.15. The work of C.S. is supported by the "Université de Liège" and the EU in the context of the MSCA-COFUND-BeIPD project.

A Two-loop Renormalization group equations
The evolution of the gauge coupling constants, α i (i = 1, 2, 3), with the energy scale µ, at two-loop level, is given by the Renormalization Group Equations (RGEs) with α 1 = 5/3α y , α i = g 2 i /4π and b i and b ij the one-and two-loop beta functions, respectively. For a generic multiplet with G SM = SU (3) C × SU (2) L × U (1) Y quantum numbers (d 3 , d 2 , y), b i and b ij are given by [62,63] for gauge bosons for scalars (26) with s = 1/2 (s = 1) for real (complex) scalars, while f = 1/2 (f = 1) for Weyl (Dirac) fermions. d k and d m are the dimensions of the multiplet with respect to the remaining subgroups, G k and G m , and m, k = i. The quantities C i and T i are respectively the Casimir invariant and the Dynkin index for the multiplet under consideration with respect to the subgroup G i ⊂ G SM . The Casimir for the adjoint representation of G i is denoted by C i (Adj). The Casimir invariants and Dynkin indices (C 2 and T 2 ) for SU (2) representations up to octets are given in tab. 4. For U (1) Y one has instead T 1 = y 2 . Note that since all the multiplets we consider are color singlets they do not contribute to SU (3) running