How many 1-loop neutrino mass models are there?

JHEP08(2022)023 Published for SISSA by Springer Received:June 3, 2022 Accepted:June 21, 2022 Published:August 2, 2022 How many 1-loop neutrino mass models are there? Carolina Arbeláez,aRicardo Cepedello,bJuan Carlos Helo,c,d Martin Hirsche and Sergey Kovalenkod,f aDepartment of Physics, Universidad Técnica Federico Santa María and Centro Científico Tecnológico de Valparaíso CCTVal, Avenida España 1680, Valparaíso, Chile bInstitut für Theoretische Physik und Astrophysik, University of Würzburg, Campus Hubland Nord, Würzburg D-97074, Germany cDepartamento de Física, Facultad de Ciencias, Universidad de la Serena, Avenida Cisternas 1200, La Serena, Chile dMillennium Institute for Subatomic Physics at the High Energy Frontier (SAPHIR), Fernández Concha 700, Santiago, Chile eInstituto de Física Corpuscular (CSIC-Universitat de València), C/ Catedrático José Beltrán 2, Paterna E-46980, València, Spain fDepartamento de Ciencias Físicas, Universidad Andrés Bello, Sazie 2212, Piso 7, Santiago, Chile E-mail: [email protected], [email protected],[email protected], [email protected],[email protected] Abstract: It is well-known that at tree-level the d= 5 Weinberg operator can be generated in exactly three different ways, the famous seesaw models. In this paper we study the related question of how many phenomenologically consistent 1-loop models one can construct at d=5. First, we discuss that there are two possible classes of 1-loop neutrino mass models, that allow avoiding stable charged relics: (i) models with dark matter candidates and (ii) models with “exits”. Here, we define “exits” as particles that can decay into standard model fields. Considering 1-loop models with new scalars and fermions, we find in the dark matter class a total of (115+203) models, while in the exit class we find (38+368) models. Here, 115 is the number of DM models, which require a stabilizing symmetry, while 203 is the number of models which contain a dark matter candidate, which maybe accidentally stable. In the exit class the 38 refers to models, for which one (or two) of the internal particles in the loop is a SM field, while the 368 models contain only fields beyond the SM (BSM) in the neutrino mass diagram. We then study the RGE evolution of the gauge couplings in all our 1-loop models. Many of the models in our list lead to Landau poles in some gauge coupling at rather low energies and there is exactly one model which unifies the gauge couplings at energies above 1015 GeV in a numerically acceptable way. Keywords: Other Weak Scale BSM Models, Models for Dark Matter, Neutrino Interactions ArXiv ePrint: 2205.13063 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP08(2022)023 JHEP08(2022)023 Contents 1 Introduction 1 2 Setup and models 3 2.1 Exit models 4 2.2 Dark matter models 7 3 Renormalization group running 11 4 Discussion 14 A Complete lists of 1-loop neutrino mass models 18 1 Introduction It is well-known that the Weinberg operator, OW, [1] can be generated at tree-level in exactly three different ways [2], the famous seesaw mechanisms [3–9]. The smallness of the observed neutrino masses has also motivated many papers on radiative neutrino mass models, starting with the classical papers [10–13]. For a recent review on loop models for neutrino mass, see [14]. One interesting question to ask then naturally is: how many possibilities actually exist to generate OWat 1-loop level? We will explore this question in the current paper. Partial answers to our question already exist in the literature, starting with [2]. Ref. [15] worked within a diagrammatic approach: construct all possible topologies and from there derive all possible diagrams that can lead to OW. Ref. [15] found a total of only four diagrams, descending from two topologies, which can yield “genuine” neutrino mass models, see figure 1.1Here, “genuine” models are defined as models for which at n-loop level all contributions to OWwith n-1 loops or less (can) vanish, i.e. the list of diagrams in these constructions do not contain self-energies and other loop diagrams that are guaranteed to be only corrections or subdominant to lower order contributions. Using these criteria ref. [15] then provides lists of all 1-loop models up to electro-weak triplets. Note also that [19] lists 1-loop neutrino mass models with dark matter candidates up to electro-weak triplets. An alternative approach to the problem is based on effective field theory. Here, one first constructs all lepton number violating operators, starting with OWand up to the desired dimension, allowed by the SM field content and symmetries. Ref. [20] lists all ∆(L) = 2 operators up to dimension d= 11, see also [21]. “Opening up” or “exploding” the operators in all possible ways, together with adding the appropriate numbers of Higgses 1A similar approach was followed in [16,17] for 2-loop models and in [18] for 3-loop models. – 1 – JHEP08(2022)023 L L HH L L H H L L HH L L HH Figure 1. The four different 1-loop diagrams that can lead to genuine neutrino mass models [15]. Top line: T-I-1 (left) and T-I-2 (right), bottom T-I-3 (left) and T-3 (right). to close loops, then, in principle, also allows a systematic construction of neutrino masses models [22]. None of the above papers, however, gives a “complete” list of 1-loop models. While ref. [15] provided a partial list of 1-loop neutrino mass models, here we aim at giving the complete list of phenomenologically consistent models. Of course, all models for neutrino mass should be able to reproduce experimental neutrino data, see for example [23], while at the same time obey upper limits on charged lepton flavour violation searches [24]. However, there are also other criteria that a successful model should fulfill and our main concern here is to avoid problems with cosmology. Consider the following, very basic observation: in loop models of OWthe particles internal to the n-loop diagram always couple in pairs to some external SM field, compare figure 1. If all particles in the loop are fields beyond the SM (BSM) and there are no other interactions in the model for the BSM fields than those appearing in the loop diagram, the corresponding model will have some accidental symmetry (in the simplest case a Z2). The lightest of the loop particles (LLP) will then be absolutely stable. Stable BSM fields, however, might lead to drastic changes in cosmology, in particular if they are electrically charged (and/or coloured). Experimental searches for stable charged relics put severe bounds on their abundance in the mass range M∼[1,105]GeV, see for example [24–27]. To avoid problems with cosmology, (at least one of) the BSM particles in 1-loop models should therefore be able to decay to SM fields — unless the lightest of them is electrically neutral. This simple consideration limits the number of allowed, electrically – 2 – JHEP08(2022)023 charged BSM particles, see section 2.1, and thus one can have only a finite set of 1-loop neutrino mass models. Electrically neutral BSM particles, on the other hand, can be candidates for dark matter in the form of WIMPs. Thus, one can construct 1-loop models of neutrino mass in which the lightest loop particles is a WIMP candidate, instead of decaying to SM fields. One can write down an infinite number of electro-weak multiplets, that contain one neutral state. However, the list of phenomenologically acceptable multiplets for WIMP candidates is rather short. To the best of our knowledge, this was first discussed in [28] and the subject has very recently been reconsidered in [29,30], see also [31] and [32]. Note that the recent papers [29,30] allows a larger list of acceptable multiplets than [28]. The possible connection between dark matter in SU(2) representations larger than triplets and loop models of neutrino mass has been discussed before in [33–37]. We will discuss more details in section 2.2. We thus will discuss two types of loop models not disfavoured by cosmology:2(i) “exit” models, i.e. models in which there are no stable particles in the loop and (ii) dark matter models, i.e. models in which one of the loop particles can be a good WIMP candidate. For both cases we construct the complete list of models. We find that there are (38+368) exit models, while in the dark matter class there are a total of (115+203) models. In the exit class the 38 refers to models, for which one (or more) of the internal particles in the loop is a SM field, while the 368 models contain only fields beyond the SM. The 115 is the number of DM models, which require an additional symmetry to give an acceptable WIMP candidate, while 203 is the number of models which contain a dark matter candidate, which maybe accidentally stable. The rest of the paper is organized as follows. In the next section we discuss the concrete criteria applied in the construction of our models. Subsection 2.1 deals with models in the exit class, while 2.2 discusses the dark matter class. In section 3we then turn to RGE evolution of the SM gauge couplings. Since many of our models contain large SU(2) and/or coloured multiplets, Landau poles at rather low energies appear in many of these constructions. Interestingly in our long list of models there is exactly one variant, which leads to a near-perfect unification of the gauge couplings at a scale of roughly mG≃1017 GeV. In section 4, we summarise briefly our results and discuss how our lists would change, modifying or dropping some of the assumptions that went into their construction. The complete lists of models are relegated to the appendix. 2 Setup and models Following [15] in our discussion we will concentrate on models with scalars and fermions. For models with new gauge vectors, the standard model gauge symmetry has to be extended and that symmetry needs to be broken to the SM. This implies that the scalar sector of the model needs to be discussed as well. This is beyond the scope of our present work. We will, however, briefly discuss loops with vectors in section 4. Note that models with new vectors will not require any additional diagram, beyond those shown in figure 1. This 2Somewhat fuzzily we call these “phenomenologically consistent” models, see discussion in section 4. – 3 – JHEP08(2022)023 Name S,aS1S2ϕΞ Ξ1/∆,a,bΘ1,cΘ3,c Irrep (1,1,0) (1,1,1) (1,1,2) 1,2,1 2(1,3,0) (1,3,1) 1,4,1 2 1,4,3 2 Name ω1ω2ω4Π1Π7ζ Irrep 3,1,−1 3 3,1,2 3 3,1,−4 3 3,2,1 6 3,2,7 6 3,3,−1 3 Name Ω1Ω2Ω4Υ Φ Irrep 6,1,1 3 6,1,−2 3 6,1,4 3 6,3,1 3 8,2,1 2 aThe field does not appear in the list of valid 1-loop decompositions of OW. bRef. [38] uses the symbol Ξ1. In neutrino physics this field is usually denoted as ∆(seesaw type-II). cThree Higgses. Table 1. Scalar “exits”: scalar bosons that can couple to a pair of standard model fields. section is divided into two parts. We will examine first models with exits, before turning to dark matter models. 2.1 Exit models In this subsection we will discuss the construction of models that contain no new stable particle, i.e. “exit” models. In this case, all particles in the neutrino mass loop can be charged and/or coloured. No new symmetry, beyond those of the SM, is needed to make these models genuine, if we also demand that the particle content of a given model does not generate OWat tree-level. Thus, from our list of models in this class we delete all possible constructions containing either F1,1,0=NR,F1,3,0= Σ or S1,3,1= ∆.3Here and elsewhere in this paper, we use the notation For Sto denote fermions or scalars, with the subscript showing the transformation properties and quantum numbers for the SM gauge group, SU(3)C×SU(2)L×U(1)Y. Alternatively, for compactness, we use the notation introduced in [38], see also tables 1and 2. We can divide the exit class of models into two sub-classes: (i) the model contains at least one SM field in the loop. In this case the model by construction does not contain any stable BSM particles. And (ii) all particles appearing in the loop are BSM fields. In that case, there must be at least one particle among the BSM fields, which can decay to SM fields. A list of all BSM scalars that can decay to SM fields at tree-level are given in table 1. This table coincides with table 1 in reference [38]. Ref. [38] arrived at this table from a completely different consideration, namely, from the construction of all treelevel completions for the d= 6 SM effective field theory (SMEFT). The lists coincide simply because in both cases a BSM field must appear linearly in at least one term of the 3One can avoid the tree-level generation of OWalso using an additional (discrete) symmetry. Models with additional symmetries can contain NR,Σand ∆, they are discussed in the next subsection. Here, we only mention that one can construct, in principle, an additional 78 1-loop models in the exit class using these fields. We disregard all of them in the following as “non-genuine”. – 4 – JHEP08(2022)023 Name N,aE∆1∆3Σ,aΣ1 Irrep (1,1,0) (1,1,−1) 1,2,−1 2 1,2,−3 2(1,3,0) (1,3,−1) Name U D Q1Q5Q7T1T2 Irrep 3,1,2 3 3,1,−1 3 3,2,1 6 3,2,−5 6 3,2,7 6 3,3,−1 3 3,3,2 3 aField does not appear in the list of ordinary genuine 1-loop decompositions of OW, since it mediates tree-level seesaw type-I/III. Symbols are again taken from [38]. Table 2. Fermion exits: new vector-like fermions that can couple to standard model fields. Lagrangian. That term will allow to generate a d= 6 operator in SMEFT at tree-level and, at the same time, is responsible for the decay of the BSM field. In the table we give the quantum numbers of the scalars in the order SU(3)C×SU(2)L×U(1)Yand also the symbols proposed in [38]. A few comments are in order. First of all, Sdoes not appear in the list of genuine 1-loop exit models, we include it in the table only for completeness.4Second, the field Ξ1/∆does also not appear in our list of “genuine” exit models, since ∆is the mediator of the tree-level seesaw type-II. This field has been denoted as Ξ1in [38], but in the neutrino physics community it is more commonly known as ∆.5Also note that all scalars in the table can decay to SM fermion pairs, except S,S1and Ξ, which decay to pair of Higgses. Finally, Θ1and Θ3will decay to three Higgses. Particularly interesting, from the point of view of model building, is Θ3, since this quadruplet scalar appears in the only genuine tree-level model [40,41] for the operator O7,W = (H†H)·OW. Table 2contains all BSM fermion fields that can couple to a SM fermion plus a Higgs. In the mass eigenstate basis, in addition to decays to the physical Higgs and a SM fermion, these fields will also decay to a SM fermion plus a gauge boson. Again, we include F1,1,0= Nand F1,3,0= Σ for completeness, although model constructions involving these fields are not included in our lists of genuine exit models, since they generate tree-level seesaws. There are five fields which have quantum numbers coinciding with some SM fermion. However, all fields in table 2should be understood as vector-like fields (or self-conjugate Majorana fields, in case of Nand Σ). We do not write the vector-like partners explicitly. Given these lists of all possible BSM fields, that can decay directly to standard model particles via renormalisable interactions at tree-level, we can construct all possible 1-loop neutrino mass exit model variants, using the diagrams in figure 1. The task is in principle straight-forward, albeit tedious. We use our own code written in Mathematica to automatise the systematic generation of neutrino mass models. The diagrams in figure 1can be represented as adjacency matrices by giving numbers to all the vertices. Each entry of the matrices will then correspond to a field in the diagram, i.e. entry (i, j)will be the field 4All 1-loop diagrams with Swill either contain Nor Ξ1/∆and thus are eliminated as non-genuine. 5We also note that the fields ω1,ω4,Π1,Π7and ζare known in the literature as scalar “leptoquarks”. In the notation of the classic paper [39], these are called S0,˜ S0,˜ S1/2,S1/2and S1, respectively. – 5 – JHEP08(2022)023 # Fields # Fields # Fields 1LF1,1,1F1,2,3/2S1,1,12LeRF1,2,3/2S1,1,13LF1,3,1F1,2,3/2S1,1,1 4LF1,1,1S1,1,1S1,2,1/25LeRS1,1,1S1,2,1/26LF1,3,1S1,1,1S1,2,1/2 7eRF1,2,1/2F1,2,3/2S1,1,18eRF1,2,1/2S1,1,1S1,2,1/29QF3,1,−1/3F3,2,−5/6S3,1,−1/3 10 QF3,1,−1/3F3,2,−5/6S3,3,−1/311 QdRF3,2,−5/6S3,1,−1/312 QdRF3,2,−5/6S3,3,−1/3 13 QF3,3,−1/3F3,2,−5/6S3,1,−1/314 QF3,3,−1/3F3,2,−5/6S3,3,−1/315 QF3,3,−1/3F3,4,−5/6S3,3,−1/3 16 QF3,1,−1/3F3,3,2/3S3,2,1/617 QdRF3,3,2/3S3,2,1/618 QF3,3,−1/3F3,1,2/3S3,2,1/6 19 QF3,3,−1/3F3,3,2/3S3,2,1/620 QF3,3,−1/3F3,3,2/3S3,4,1/621 QF3,1,−1/3S3,2,1/6S3,1,−1/3 22 QF3,1,−1/3S3,2,1/6S3,3,−1/323 QdRS3,2,1/6S3,1,−1/324 QdRS3,2,1/6S3,3,−1/3 25 QF3,3,−1/3S3,2,1/6S3,1,−1/326 QF3,3,−1/3S3,2,1/6S3,3,−1/327 QF3,3,−1/3S3,4,1/6S3,3,−1/3 28 uRF3,2,7/6F3,2,1/6S3,1,2/329 uRF3,2,7/6F3,2,1/6S3,3,2/330 uRF3,2,7/6F3,3,5/3S3,2,7/6 31 uRF3,2,7/6S3,2,7/6S3,1,2/332 uRF3,2,7/6S3,2,7/6S3,3,2/333 dRF3,2,1/6F3,2,−5/6S3,1,−1/3 34 dRF3,2,1/6F3,2,−5/6S3,3,−1/335 dRF3,2,1/6F3,3,2/3S3,2,1/636 dRF3,2,1/6S3,2,1/6S3,1,−1/3 37 dRF3,2,1/6S3,2,1/6S3,3,−1/338 HF1,3,1S1,4,3/2 Table 3. 1-loop neutrino mass models for which some internal field can be a SM fermion. For discussion, see text. connecting vertices iand j. The power of this approach is twofold: (i) the contraction of all fields along a row or column should contain always a singlet, and (ii) with adjacency matrices one can then use tools from graph theory, for instance, to delete isomorphic diagrams. The external fields are already known, so once given the quantum number for one of the fields in the loop, the rest can be computed. This can be actually done for a general set of quantum numbers for the starting field (seed). As the external particles are colour blind, all the particles in the loop will have the same SU(3)Crepresentation as the seed, while SU(2)Land hypercharge can be obtained by systematically solving the set of equations for each vertex, i.e. for each row/column of the adjacency matrix. Note that one should keep track of the several possibilities for the products of SU(2) representations, for example, a representation rtimes a doublet gives two possible representations r±1, where only those representations larger than 1are possible. Numbers can then be systematically given to the free charges of the seed to get a complete list of models, to which we apply our genuineness criteria and further classify them, as explained in the text. Chirality is also being tracked along the fermion line to afterwards check whether any of the internal fermions may be a SM fermion. It is worth noticing a slight subtlety in this approach first shown in [18]: the antisymmetric contractions of SU(2) implies that some couplings with identical particles vanishes exactly, for example, the coupling of two identical SU(2) doublets (like two Higgses) to a singlet. Diagrams with such couplings should be removed.6 The resulting lists are given in table 3and in the appendix. The tables in the appendix are divided first into the four diagrams of figure 1. The models are then ordered first with respect to the scalar exits in table 1, then w.r.t. fermions as in table 2and then sub-divided again into increasing number of exits that occur in each diagram. 6Even if this is a local feature of SU(2), diagrams with such non-local (effective) couplings may also vanish if, for example, the identical particles get a VEV, which is the case of the SM Higgs. – 6 – JHEP08(2022)023 We will discuss now a few, particular cases found in those tables. A subset of the models appearing in the diagrams T-I-2 and T-I-3 use fermions, with quantum numbers and couplings identical to one of the standard model fermion. In these cases, either one or two of the internal particles can be identified with SM quarks or leptons. These are particularly simple models, in the sense that fewer BSM fields are needed than in all other cases. We have identified a total of 38 possibilities in this special sub-class and list all of them in table 3. Note that in case there are two SM fields in the diagram, as for example in model #5, there are two more models (in this example #4 and #8) in which one of the two SM fields could also be a new, vector-like particles. These are counted in this table as extra models, since they contain a different number of degrees of freedom, see also section 3. A number of the models listed in table 3have appeared in the literature before. For example, model #5 is the famous Zee-model [10]. Models #23 and #24 are leptoquark models [42]. These are based on the idea to break lepton number in LQ models via LQHiggs interactions [43]. Note that supersymmetry with R-parity violation generates the same diagrams [44] with scalars that have the same quantum numbers as in the Zee model and the LQ model #23 of table 3. The particle content of models #11,#12 and #17, appeared first in tables 6 and 7 of [45], 1-loop neutrino masses in this setup were discussed then in [46]. Model #38 is special, first because it is the only model in this class based on T-3. Also, while this model is technically a “genuine” 1-loop model in the sense, that there is no tree-level d= 5 neutrino mass, this model generates actually a tree-level d= 7 mass. The model was first discussed in [41]. Whether the tree-level d= 7 or the 1-loop d= 5 contribution is numerically more important, depends essentially on the mass scale of F1,3,1 and S1,4,3/2. If these particles are heavier than, roughly Λ≃2TeV, the loop tends to dominate, while for lighter masses the tree-level contribution is more important. All models with only BSM particles in the loop are given in the appendix. We note, that all models in the diagram class T-I-1 will also have a contribution to the neutrino mass matrix via diagram T-3. In principle, one can find all T-3 models from the models in T-I-1, eliminating simply the “middle” scalar, compare with figure 1. Since the number of degrees of freedom in T-3 and T-I-1 models are different, however, we count these models as different. Even though the size (and number) of the representations is limited in the exit class, very exotic states appear in our lists. For example, a model with Θ3allows SU(2) representations up to 6-plets. As one can see from the tables, there are many models that have more than one exit particle in the diagram. In fact, there are several models in which all particles in the loop are one of the particle in the exit lists and this is possible within any of the four diagrams. Since none of the fields in these tables are singlets, one can expect interesting phenomenology at the LHC, if the mass scale of the BSM particles is around the electro-weak scale. A complete study of possible LHC signals is, however, beyond the scope of our present work. 2.2 Dark matter models In this subsection we will discuss 1-loop neutrino mass models containing a WIMP dark matter candidate. The classical proto-type for this class of models is the scotogenic – 7 – JHEP08(2022)023 F1,1,0≡Nc R × S1,2,1/2S1,2,1/2 L L H H F1,4,1/2 × S1,5,0S1,5,1 L L H H Figure 2. Two examples of dark matter models. To the left the original scotogenic model [47]; to the right an accidentally stable DM model, see text. model [47], see figure 2(left). Models in the DM class can again be sub-divided into two sub-classes: (i) models that need a stabilizing (discrete) symmetry and (ii) models in which the DM candidate maybe accidentally stable [28], for an example see figure 2 (right). We will call these two classes (i) DM-E (since at least one of the particles in these models has quantum numbers coinciding with one of the exit particles) and (ii) DM-A (for accidental). The division into these two classes can be easily understood. Consider the scotogenic model. The particle content of this model is such, that without any additional symmetry, beyond the gauge symmetries of the SM a tree-level type-I seesaw would exist and the loop particles would be unstable, i.e. they will decay to SM fields (a right-handed neutrino can decay to SM Higgses or gauge bosons plus SM leptons, for example). Ref. [47] solves these “problems” with the simple assumption that all particles in the loop transform odd under a new Z2symmetry. The lightest of the NRiand S1,2,1/2is then absolutely stable. On the other hand, for larger SU(2)Lmultiplets as DM candidates, the model might have an accidental Z2symmetry, like in the example model shown in figure 2(right). For this model it is easy to see that SU(2)Ldictates that, at the renormalisable level, the particles in the loop always couple in pairs to SM fields. Thus, the model has an accidental Z2and the LLP is stable automatically. We could call this an “accidental dark matter candidate” [48], in [28] this was named “minimal dark matter”. Note, that this reasoning assumes that there are no other BSM fields present in the model beyond those appearing in the 1-loop diagram. Consider again the example model figure 2(right). If we add to this model a S1,3,0, for example, then the vector-like fermion can decay to a SM Lplus a S1,3,0, while the latter decays to two Higgses and this extended model will have no DM candidate — unless we postulate an additional symmetry, which would put this model back into the first subclass. Thus, for the DM candidate to be accidental DM, there should be no particle from the exit class in the diagram and we always have to assume in our model constructions implicitly that there are no other BSM particles in the model, beyond those appearing in the 1-loop diagram. Otherwise, the model will belong to DM class (i) DM-E. However, at this point we would like to stress that for us DM-E and DM-A are just a convenient classification scheme, dividing the model lists into those models with “small” – 8 – JHEP08(2022)023 T-I-1.Max .Min mNP =1 TeV 14 2 13 24 35 46 50 42 31 20 9, , , , 3 5 7 9 11 13 15 10 3 10 5 10 7 10 9 DM n -plet ΛLP [GeV ] T-I-2.Max .Min mNP =1 TeV 39 54 60 66 72 78 77 53 59 65 71 , , ,, 3 5 7 9 11 13 15 10 3 10 5 10 7 10 9 DM n -plet ΛLP [GeV ] T-I-3.Max .Min mNP =1 TeV 60 89 100 111 122 130 126 71 83 93 104 115 , , , ,, 3 5 7 9 11 13 15 10 3 10 5 10 7 10 9 DM n -plet ΛLP [GeV ] .Max .Min mNP =1 TeV 135 146 157 168 177 136 147 158 169 , , , , T-3 3 5 7 9 11 13 15 10 3 10 5 10 7 10 9 DM n -plet ΛLP [GeV ] Figure 5. For each topology and electro-weak DM representation, we show the models with the minimal (magenta points) and maximal (blue points) SU(2)-Landau pole ΛLP2and its specific values. In the running of the gauge couplings all the additional fields beyond the SM are added at the scale mNP = 1 TeV. The numbers above the points refer to the specific model as listed in the DM tables in the appendix. All model numbers refer to DM-A in table 9, except the ones associated to 3-plet DM (and also model 71 of T-I-3) which belong to DM-E in table 8. DM n-plet MχMin ΛLP2 3 2.5 TeV 3×106TeV 5 15.4 TeV 9×102TeV 7 54.2 TeV 3×102TeV 9 117.8 TeV 3×102TeV 11 199 TeV 4×102TeV 13 338 TeV 5×102TeV Table 4.SU(2) n-plet WIMP thermal masses, for which the relic abundance of dark matter would be correctly reproduced and energy scale of the Landau pole, ΛLP2. In the running of the gauge couplings, all the extra fields up to the SM are added at the Mχscale. – 15 – JHEP08(2022)023 detail in section 2, we have found a total of 724 1-loop neutrino mass models: 406 in the exit class and 318 in the dark matter class. While these are certainly uncomfortably big numbers, especially compared to the fact that there are only three tree-level seesaws, many of these models could actually be excluded in the future. For the dark matter class, future DD experiments, such as DARWIN [50], will either finally detect WIMP dark matter or exclude most of the larger SU(2) multiplets [29,30] as DM candidates. From our 318 DM models only 109 would survive non-observation of DM in DARWIN. Also, there are theoretical considerations, such as perturbativity up to some large energy scales, that we have discussed in section 3. Conservatively, we have listed all possible models in our tables. However, if we require our new physics scale, at which the 1loop neutrino mass is generated, to be around the electro-weak scale and add the condition that all gauge couplings remain perturbative up to the GUT scale, only 57 models (out of 406) in our exit class survive. Similarly, in the DM class only 59 out of the whole 318 would survive this constraint, eliminating in particular all models with representations larger than 5-plets. Two important assumptions on model building were used in all our constructions: (i) use only scalars and fermions as BSM fields; and (ii) avoid stable charged relics. Both of these assumptions can be questioned. Let us discuss 1-loop models with vectors first. Note that very few 1-loop models with gauge vectors do exist in the literature, a few examples are [53–55]. The two main problems with gauge vectors are that: (a) for many of the vectors, which appear in the 1-loop diagrams, it is not even possible to find a phenomenologically consistent or interesting gauge group [56]; and (b) complete gauge models in many cases also contain the ingredients for a tree-level seesaw, thus loops are most likely only a sub-dominant contribution to the neutrino mass in these constructions. Disregarding these problems in the construction of valid gauge models, however, our automated diagram-based approach allows us, of course, to search also for valid 1-loop neutrino mass diagrams with vectors instead of scalars. From the list of valid “exit” vectors, see table 3 of [38], one can show that there are a total of 499 vector models in the exit class, out of which 34 models contain either one or two SM fermions. Two examples are shown in figure 6. The example on the left is from the diagram class T-I-2, while the one on the right is from T-I-3. Both diagrams have vector LQs as internal particles. Let’s have a closer look to the diagram on the left first. The quantum numbers of the vectors are the same as in the scalar LQ model of table 3, model #23, except the hypercharge Y= 2/3of the second vector in the diagram. Thus, the diagram contains uRinstead of dR, but is otherwise very similar to the corresponding scalar LQ model. Many, but not all of our scalar models can be “vectorized” by such simple replacements. The two examples shown in figure 6can also serve to discuss the main problems one encounters in the construction of neutrino mass models based on extended gauge theories. Consider the model shown in the figure on the right. The vector in this diagram, V3,1,2/3, can be generated from the adjoint of SU(4), when the Pati-Salam (PS) group [57], is broken to the standard model. Both, Qand uRare of course present in Pati-Salam as members of the 4and 4. However, the F3,2,7/6is not part of a minimal PS model and thus another – 16 – JHEP08(2022)023 QuR V3,1,2/3V3,2,1/6 L L H H V3,1,2/3 QF3,2,7/6 uR L L HH Figure 6. Two example 1-loop diagrams with vectors instead of scalars. For a discussion see text. multiplet containing this fermion has to be added to complete the particle content of this 1-loop model. On the other hand, the 4necessarily contains a F1,1,0, i.e. a right-handed neutrino. Since the 1-loop diagram necessarily violates lepton number, it seems reasonable that the model also generates a Majorana mass term for NR. This could, for example, occur if the PS is first broken to the left-right group, which is then broken by a right-triplet to the SM group. The model then could generate the 1-loop diagram shown, but also has a tree-level seesaw to which the loop diagram would be only a minor correction in large parts of the parameter space. In other words, according to our criteria, in this setup the 1-loop diagram would not be considered “genuine”. This problem — the presence of NR or also other tree-level seesaws — occurs in many of the popular gauge groups, in which the SM group could be embedded. While it seems possible to construct a full model along the lines just discussed, in which the tree-level seesaw is sub-dominant (or absent entirely at tree-level), the model building required clearly is beyond our minimalistic approach to neutrino masses. A second problem with vector diagrams is demonstrated by the diagram on the left of figure 6. Here, no BSM fermion appears, but two different vectors are needed to complete the diagram. Again, V3,1,2/3appears in the adjoint of SU(4). The other vector, V3,2,1/6, appears, for example, in flipped SU(5) [58]. Flipped SU(5) has no NR, so no tree-level seesaw type-I, but it induces proton decay and the “vector leptoquark” V3,2,1/6of the 1loop diagram has also diquark couplings in this setup. Thus, when V3,2,1/6is interpreted as the gauge vector of flipped SU(5), its mass must lie at the grand unification scale. This mass scale is too large to generate the atmospheric neutrino mass scale with perturbative couplings from a 1-loop diagram. The problem is exacerbated by the fact, that the diagram needs two vectors. Thus, one would need to identify a group — or semisimple group [59] — which contains both, the Pati-Salam group and the SU(5), plus suitable model building to avoid proton decay and many other constraints in this extended theory. Finally, we would like to stress again, that for most of the exit vectors [38] it was shown in [56], that no suitable gauge group can be constructed at all, since they can not lead to models which contain the SM particle content. Our second main assumption is to avoid stable charged relics. For the models in the exit class, one can actually question the validity of this criterion. From experimental – 17 – JHEP08(2022)023 data the absence of stable charged particles is established only for a certain mass window, roughly M∼[1,105]GeV [24,25]. 1-loop models for neutrino mass, however, can fit the observed data even for considerably more massive BSM states in the loop, roughly up to 10(12−13) GeV for perturbative couplings. Such ultra-heavy particles would decouple very early in the history of the universe and therefore not be produced in any measurable quantities.9Thus, there is a window of parameter space for 1-loop neutrino mass models, where this criterion is not supported by experimental data. Clearly, we have disregarded this possibility. We note in passing, that such models would use, of course, even larger multiplets than what we have considered and thus Landau poles would exist in these constructions always not far above the mass scale of the BSM states. For the dark matter models, on the other hand, the two most important constraints for valid WIMP candidates are (i) unitarity bounds on the annihilation cross section in the early universe and (ii) limits by direct detection experiments. Here we followed [29] and [30]. While unitarity bounds put a definitive upper limit on the size of the SU(2) multiplet, that can be a good WIMP candidate, the argument (ii) is slightly more fragile. We have considered models with Y= 0 DM candidates, as well as inelastic DM candidates. However, one could think about cooking up other ways to avoid the DD constraints and we have simply disregarded this possibility. In summary, we provide “complete” lists of possible 1-loop models for neutrino masses. We have considered two possible classes of models, which can be consistent with cosmology: “Exit” models, with no stable particles in the loop and dark matter models, which assume that the lightest particle in the loop is neutral, stable and can be in agreement with known constraints. In the appendix we give the lists of all possible models, consistent with these assumptions. It would be interesting to study, whether some of these models can lead to phenomenology at colliders, say the LHC or FCC, that has not already been covered in previous work, see for example [14,42,60–68]. A Complete lists of 1-loop neutrino mass models Here we give tables containing the 1-loop neutrino mass models as discussed in the previous sections. The models are divided in tables for each of the four 1-loop neutrino mass diagrams: T-I-1, T-I-2, T-I-3 and T-3, see figure 1. For each diagram, the models are classified into two large classes: “exit” (tables 5,6,7) and dark matter (tables 8,9,10) models. The exit models have been ordered in the tables from top to bottom by models with 1 exit to models with 4 exits. To identify the exits particles we have used the notation of ref. [38], shown in tables 1and 2. The DM models have been separated in four class of models: models with exits that need a stabilizing symmetry to give an acceptable DM candidate (DM-E: table 8), models in which the DM is stable due to an accidental symmetry (DM-A: table 9) and another two cases for exceptional candidates with Y= 1 which are separated again in exit DM models (DM-E exceptional: table 10) and accidental DM models (DM-A exceptional: table 10). See section 2for discussion. 9It is even conceivable such states are not produced at all, if the reheat temperature of the universe is sufficiently below the mass of these BSM states. – 18 – JHEP08(2022)023 T-I-1 Exit 1 (S1,2,5/2, S1,3,3,S2, F1,2,5/2) 2 - (S2, S1,2,5/2, S1,2,3/2, F1,1,2) 3 (S2, S1,2,5/2, S1,2,3/2, F1,3,2) 4 (Θ1, S1,5,1, S1,5,0, F1,4,1/2)5 (Θ1, S1,5,1, S1,5,0, F1,6,1/2) 6 (S1,5,1, S1,6,3/2,Θ1, F1,5,1) 7 (S1,5,0,Θ1,Θ1, F1,5,0)8 (S1,5,0,Θ1, S1,6,1/2, F1,5,0) 9 (S1,3,2, S1,2,5/2,Θ3, F1,3,2) 10 (S1,3,2, S1,4,5/2,Θ3, F1,3,2)11 (S1,3,2, S1,4,5/2,Θ3, F1,5,2) 12 (Θ3, S1,3,2, S1,5,1, F1,4,3/2) 13 (Θ3, S1,5,2, S1,5,1, F1,4,3/2)14 (Θ3, S1,5,2, S1,5,1, F1,6,3/2) 15 (S1,5,2, S1,4,5/2,Θ3, F1,3,2) 16 (S1,5,2, S1,4,5/2,Θ3, F1,5,2)17 (S1,5,2, S1,6,5/2,Θ3, F1,5,2) 18 (S1,5,1,Θ3, S1,6,1/2, F1,5,1) 19 (S3,2,−11/6, S3,3,−7/3, ω4, F3,2,−11/6)20 (ω4, S3,2,−11/6, S3,2,−5/6, F3,1,−4/3) 21 (ω4, S3,2,−11/6, S3,2,−5/6, F3,3,−4/3) 22 (S3,1,5/3, S3,2,13/6,Π7, F3,1,5/3)23 (S3,1,5/3, S3,2,13/6,Π7, F3,3,5/3) 24 (Π7, S3,3,5/3, S3,3,2/3, F3,4,7/6) 25 (S3,3,5/3, S3,2,13/6,Π7, F3,1,5/3)26 (S3,3,5/3, S3,2,13/6,Π7, F3,3,5/3) 27 (S3,3,5/3, S3,4,13/6,Π7, F3,3,5/3) 28 (S3,4,1/6, S3,3,2/3, ζ, F3,4,1/6)29 (S3,4,1/6, S3,5,2/3, ζ, F3,4,1/6) 30 (S3,2,−5/6, S3,3,−4/3, ζ, F3,4,−5/6) 31 (S3,4,−5/6, S3,3,−4/3, ζ, F3,4,−5/6)32 (S3,4,−5/6, S3,5,−4/3, ζ, F3,4,−5/6) 33 *(ζ, S3,4,1/6, S3,4,−5/6, F3,5,−1/3) 34 (S6,2,5/6, S6,3,4/3,Ω1, F6,2,5/6)35 - (Ω1, S6,2,5/6, S6,2,−1/6, F6,1,1/3) 36 (Ω1, S6,2,5/6, S6,2,−1/6, F6,3,1/3) 37 (S6,2,−1/6,Ω1, S6,3,−2/3, F6,2,−1/6)38 (S6,2,−7/6, S6,3,−5/3,Ω2, F6,2,−7/6) 39 - (Ω2, S6,2,−1/6, S6,2,−7/6, F6,1,−2/3) 40 (Ω2, S6,2,−1/6, S6,2,−7/6, F6,3,−2/3)41 (S6,2,11/6, S6,3,7/3,Ω4, F6,2,11/6) 42 (Ω4, S6,2,11/6, S6,2,5/6, F6,1,4/3) 43 (Ω4, S6,2,11/6, S6,2,5/6, F6,3,4/3)44 (S6,2,5/6, S6,3,4/3,Υ, F6,2,5/6) 45 *(S6,2,5/6, S6,3,4/3,Υ, F6,4,5/6) 46 (S6,4,5/6, S6,3,4/3,Υ, F6,2,5/6)47 *(S6,4,5/6, S6,3,4/3,Υ, F6,4,5/6) 48 *(S6,4,5/6, S6,5,4/3,Υ, F6,4,5/6) 49 (S6,2,−1/6,Υ, S6,3,−2/3, F6,2,−1/6)50 *(S6,2,−1/6,Υ, S6,3,−2/3, F6,4,−1/6) 51 - (Υ, S6,2,5/6, S6,2,−1/6, F6,1,1/3) 52 (Υ, S6,2,5/6, S6,2,−1/6, F6,3,1/3)53 (Υ, S6,2,5/6, S6,4,−1/6, F6,3,1/3) 54 (Υ, S6,4,5/6, S6,2,−1/6, F6,3,1/3) 55 (Υ, S6,4,5/6, S6,4,−1/6, F6,3,1/3)56 *(Υ, S6,4,5/6, S6,4,−1/6, F6,5,1/3) 57 (S6,4,−1/6,Υ, S6,3,−2/3, F6,2,−1/6) 58 *(S6,4,−1/6,Υ, S6,3,−2/3, F6,4,−1/6)59 *(S6,4,−1/6,Υ, S6,5,−2/3, F6,4,−1/6) 60 (S8,1,1, S8,2,3/2,Φ, F8,1,1) 61 (S8,1,1, S8,2,3/2,Φ, F8,3,1)62 (Φ, S8,1,1, S8,3,0, F8,2,1/2) 63 (Φ, S8,3,1, S8,1,0, F8,2,1/2) 64 (Φ, S8,3,1, S8,3,0, F8,2,1/2)65 *(Φ, S8,3,1, S8,3,0, F8,4,1/2) 66 (S8,3,1, S8,2,3/2,Φ, F8,1,1) 67 (S8,3,1, S8,2,3/2,Φ, F8,3,1)68 (S8,3,1, S8,4,3/2,Φ, F8,3,1) 69 - (S8,1,0,Φ,Φ, F8,1,0) 70 (S8,1,0,Φ,Φ, F8,3,0)71 (S8,3,0,Φ,Φ, F8,1,0) 72 (S8,3,0,Φ,Φ, F8,3,0) 73 (S8,3,0,Φ, S8,4,1/2, F8,3,0)74 (S3,4,7/6, S3,3,5/3, S3,3,2/3, Q7) 75 (S3,5,−1/3, S3,4,1/6, S3,4,−5/6, T1) 76 (S3,3,2/3, S3,4,7/6, S3,4,1/6, T2)77 (S3,5,2/3, S3,4,7/6, S3,4,1/6, T2) 78 - (S1,2,3/2, S1,3,2,S1,∆3) 79 (Θ1, S1,5,1,Ξ, F1,4,1/2) 80 (Ξ,Θ1,Θ1, F1,5,0) 81 (S1,5,1,Θ3,Θ1, F1,5,1)82 - (S3,2,−5/6, S3,3,−4/3, ω1, Q5) 83 (Π1, S3,3,2/3, ζ, F3,4,1/6) 84 (S3,3,2/3, S3,4,7/6,Π1, T2)85 (Π7, S3,1,5/3, S3,3,2/3, Q7) 86 (Π7, S3,3,5/3, S3,3,2/3, Q7) 87 (S3,3,2/3,Π7, S3,4,1/6, T2)88 (S3,4,1/6, S3,3,2/3, ζ, Q1) 89 - (S3,2,−5/6, S3,3,−4/3, ζ, Q5) 90 (S3,4,−5/6, S3,3,−4/3, ζ, Q5)91 (ζ, S3,4,1/6, S3,2,−5/6, T1) 92 (ζ, S3,4,1/6, S3,4,−5/6, T1) 93 (S6,2,−1/6,Υ,Ω2, F6,2,−1/6)94 (S6,2,5/6,Ω4,Υ, F6,2,5/6) 95 - (S1, S1,2,3/2, ϕ, E) 96 - (S1, S1,2,3/2, ϕ, Σ1) 97 - (S1,5,1,Θ3,Θ1,Σ1) 98 - (Π1, S3,3,2/3, ω1, Q1)99 - (ω1,Π1, S3,2,−5/6, D) 100 - (ω1,Π1, S3,2,−5/6, T1) 101 (Π7, S3,3,5/3, ω2, Q7) 102 - (S3,2,−5/6, ω4, ζ, Q5) 103 - (Π1, S3,3,2/3, ζ, Q1) 104 - (S3,3,2/3,Π7,Π1, U) 105 (S3,3,2/3,Π7,Π1, T2) 106 - (ζ, Π1, S3,2,−5/6, D) 107 (ζ, Π1, S3,2,−5/6, T1) 108 (ζ, Π1, S3,4,−5/6, T1) 109 - (ϕ, S1,Ξ,∆1) 110 - (ω2,Π7,Π1, U) 111 - (ω2,Π7,Π1, T2) 112 - (Π1, ω2, ζ, Q1) Table 1: TI-1 models with scalar and fermionic exits. The four main cells represent the models with 1,2,3,4 exits. The * (−) represents the models where: One of the Landau pole scales is very low, i.e Λ1or λ2or Λ3<100 TeV (All Landau pole scales, larger than mNP , are very large Λ1,2,3>1015 GeV) Table 5. T-I-1 models with scalar and fermionic exits. The four horizontal divisions represent the models with 1, 2, 3 and 4 exit fields. The *and “−” represent the models where one of the Landau pole scales is very low, i.e. Λ1,2,3<100 TeV, and where there is no Landau pole up to 1015 GeV. – 19 – JHEP08(2022)023 T-I-2 Exit 113 (F1,1,2, F1,2,5/2, S1,2,5/2,S2) 114 (F1,3,2, F1,2,5/2, S1,2,5/2,S2) 115 (F1,4,1/2, F1,5,1, S1,5,1,Θ1) 116 (F1,6,1/2, F1,5,1, S1,5,1,Θ1)117 (F1,4,1/2, F1,5,0, S1,5,0,Θ1) 118 (F1,5,0, F1,6,1/2,Θ1, S1,5,0) 119 (F1,4,3/2, F1,3,2, S1,3,2,Θ3)120 (F1,4,3/2, F1,3,2, S1,5,2,Θ3) 121 (F1,4,3/2, F1,5,2, S1,3,2,Θ3) 122 (F1,4,3/2, F1,5,2, S1,5,2,Θ3)123 (F1,6,3/2, F1,5,2, S1,5,2,Θ3) 124 (F1,5,1, F1,4,3/2,Θ3, S1,5,1) 125 (F1,5,1, F1,6,3/2,Θ3, S1,5,1)126 (F3,1,−4/3, F3,2,−11/6, S3,2,−11/6, ω4) 127 (F3,3,−4/3, F3,2,−11/6, S3,2,−11/6, ω4) 128 (F3,4,7/6, F3,3,5/3, S3,3,5/3,Π7)129 *(F3,4,−5/6, F3,5,−1/3, ζ, S3,4,−5/6) 130 *(F3,5,−1/3, F3,4,1/6, S3,4,1/6, ζ) 131 (F6,1,1/3, F6,2,5/6, S6,2,5/6,Ω1)132 (F6,2,−1/6, F6,1,1/3,Ω1, S6,2,−1/6) 133 (F6,2,−1/6, F6,3,1/3,Ω1, S6,2,−1/6) 134 (F6,3,1/3, F6,2,5/6, S6,2,5/6,Ω1)135 (F6,1,−2/3, F6,2,−1/6, S6,2,−1/6,Ω2) 136 (F6,2,−7/6, F6,1,−2/3,Ω2, S6,2,−7/6) 137 (F6,2,−7/6, F6,3,−2/3,Ω2, S6,2,−7/6)138 (F6,3,−2/3, F6,2,−1/6, S6,2,−1/6,Ω2) 139 (F6,1,4/3, F6,2,11/6, S6,2,11/6,Ω4) 140 (F6,2,5/6, F6,1,4/3,Ω4, S6,2,5/6)141 (F6,2,5/6, F6,3,4/3,Ω4, S6,2,5/6) 142 (F6,3,4/3, F6,2,11/6, S6,2,11/6,Ω4) 143 (F6,1,1/3, F6,2,5/6, S6,2,5/6,Υ) 144 (F6,2,−1/6, F6,1,1/3,Υ, S6,2,−1/6) 145 (F6,2,−1/6, F6,3,1/3,Υ, S6,2,−1/6) 146 *(F6,2,−1/6, F6,3,1/3,Υ, S6,4,−1/6)147 (F6,3,1/3, F6,2,5/6, S6,2,5/6,Υ) 148 *(F6,3,1/3, F6,2,5/6, S6,4,5/6,Υ) 149 *(F6,3,1/3, F6,4,5/6, S6,2,5/6,Υ) 150 *(F6,3,1/3, F6,4,5/6, S6,4,5/6,Υ) 151 *(F6,4,−1/6, F6,3,1/3,Υ, S6,2,−1/6) 152 *(F6,4,−1/6, F6,3,1/3,Υ, S6,4,−1/6)153 *(F6,4,−1/6, F6,5,1/3,Υ, S6,4,−1/6) 154 *(F6,5,1/3, F6,4,5/6, S6,4,5/6,Υ) 155 (F8,2,1/2, F8,1,1, S8,1,1,Φ) 156 (F8,2,1/2, F8,1,1, S8,3,1,Φ) 157 *(F8,2,1/2, F8,3,1, S8,1,1,Φ) 158 *(F8,2,1/2, F8,3,1, S8,3,1,Φ) 159 *(F8,4,1/2, F8,3,1, S8,3,1,Φ) 160 (F8,1,0, F8,2,1/2,Φ, S8,1,0) 161 (F8,1,0, F8,2,1/2,Φ, S8,3,0)162 *(F8,2,1/2, F8,3,0, S8,1,0,Φ) 163 *(F8,2,1/2, F8,3,0, S8,3,0,Φ) 164 *(F8,3,0, F8,4,1/2,Φ, S8,3,0)165 (∆3, F1,1,2, S1,3,2, S1,2,3/2) 166 (∆3, F1,3,2, S1,3,2, S1,2,3/2) 167 (Q5, F3,1,−4/3, S3,3,−4/3, S3,2,−5/6)168 (Q5, F3,3,−4/3, S3,3,−4/3, S3,2,−5/6) 169 (Q5, F3,3,−4/3, S3,3,−4/3, S3,4,−5/6) 170 (Q7, F3,3,5/3, S3,3,5/3, S3,4,7/6)171 (T1, F3,4,1/6, S3,4,1/6, S3,5,−1/3) 172 (F3,4,−5/6, T1, S3,5,−1/3, S3,4,−5/6) 173 (T2, F3,4,7/6, S3,4,7/6, S3,3,2/3)174 (T2, F3,4,7/6, S3,4,7/6, S3,5,2/3) 175 (F3,4,1/6, T2, S3,3,2/3, S3,4,1/6) 176 (F3,4,1/6, T2, S3,5,2/3, S3,4,1/6) 177 - (∆3, F1,1,2,S2, S1,2,3/2) 178 (∆3, F1,3,2,S2, S1,2,3/2) 179 (F1,4,1/2, F1,5,0,Ξ,Θ1) 180 (F1,4,1/2,Σ1, S1,5,1,Θ1)181 (∆3, F1,3,2, S1,3,2,Θ3) 182 (Σ1, F1,4,3/2,Θ3, S1,5,1) 183 (Q5, F3,1,−4/3, ω4, S3,2,−5/6)184 (Q5, F3,3,−4/3, ω4, S3,2,−5/6) 185 (F3,4,1/6, T2, S3,3,2/3,Π1) 186 (Q7, F3,1,5/3, S3,1,5/3,Π7)187 (Q7, F3,1,5/3, S3,3,5/3,Π7) 188 (Q7, F3,3,5/3, S3,1,5/3,Π7) 189 (Q7, F3,3,5/3, S3,3,5/3,Π7)190 (T2, F3,4,7/6,Π7, S3,3,2/3) 191 (T1, F3,4,1/6, S3,4,1/6, ζ) 192 (F3,4,−5/6, T1, ζ, S3,2,−5/6)193 (F3,4,−5/6, T1, ζ, S3,4,−5/6) 194 (Q1, T2, S3,3,2/3, S3,4,1/6) 195 (T2, Q7, S3,4,7/6, S3,3,2/3) 196 - (E, ∆3, S1,2,3/2,S1) 197 - (Σ1,∆3, S1,2,3/2,S1) 198 - (Q5, D, ω1, S3,2,−5/6) 199 (Q5, T1, ω1, S3,2,−5/6)200 - (Q1, U, S3,3,2/3,Π1) 201 (Q1, T2, S3,3,2/3,Π1) 202 (T1, F3,4,1/6,Π1, ζ)203 (U, Q7,Π7, S3,3,2/3) 204 (T2, Q7,Π7, S3,3,2/3) 205 - (Q5, D, ζ, S3,2,−5/6)206 (Q5, T1, ζ, S3,2,−5/6) 207 (Q5, T1, ζ, S3,4,−5/6) 208 (T1, Q1, S3,4,1/6, ζ) 209 - (∆1, E, S1, ϕ) 210 - (∆1,Σ1,S1, ϕ) 211 †(D, Q1,Π1, ω1) 212 (T1, Q1,Π1, ω1) 213 (U, Q7,Π7, ω2) 214 - (Q1, U, ω2,Π1) 215 (Q1, T2, ω2,Π1) 216 (T2, Q7,Π7, ω2) 217 - (D, Q1,Π1, ζ) 218 (T1, Q1,Π1, ζ) Table 2: TI-2 models with scalar and fermionic exits: The model †unifies at a scale of mG= 1017 GeV . Table 6. T-I-2 models with scalar and fermionic exits: the model †unifies at a scale of mG≃ 1017 GeV. – 20 – JHEP08(2022)023 T-I-3 Exit 219 (F1,5,0, F1,4,1/2, F1,4,1/2,Ξ) 220 (F1,4,1/2, F1,5,1, F1,5,0,Θ1) 221 *(F1,6,1/2, F1,5,1, F1,5,0,Θ1) 222 (F1,4,3/2, F1,3,2, F1,5,1,Θ3)223 (F1,4,3/2, F1,5,2, F1,5,1,Θ3) 224 *(F1,6,3/2, F1,5,2, F1,5,1,Θ3) 225 *(F3,5,−1/3, F3,4,1/6, F3,4,−5/6, ζ)226 (F6,1,1/3, F6,2,5/6, F6,2,−1/6,Ω1) 227 *(F6,3,1/3, F6,2,5/6, F6,2,−1/6,Ω1) 228 (F6,1,−2/3, F6,2,−1/6, F6,2,−7/6,Ω2)229 *(F6,3,−2/3, F6,2,−1/6, F6,2,−7/6,Ω2) 230 (F6,1,4/3, F6,2,11/6, F6,2,5/6,Ω4) 231 *(F6,3,4/3, F6,2,11/6, F6,2,5/6,Ω4)232 (F6,1,1/3, F6,2,5/6, F6,2,−1/6,Υ) 233 *(F6,3,1/3, F6,2,5/6, F6,2,−1/6,Υ) 234 *(F6,3,1/3, F6,2,5/6, F6,4,−1/6,Υ) 235 *(F6,3,1/3, F6,4,5/6, F6,2,−1/6,Υ) 236 *(F6,3,1/3, F6,4,5/6, F6,4,−1/6,Υ) 237 *(F6,5,1/3, F6,4,5/6, F6,4,−1/6,Υ) 238 *(F8,2,1/2, F8,1,1, F8,3,0,Φ) 239 *(F8,2,1/2, F8,3,1, F8,1,0,Φ) 240 *(F8,2,1/2, F8,3,1, F8,3,0,Φ) 241 *(F8,4,1/2, F8,3,1, F8,3,0,Φ) 242 (F1,1,2, F1,2,5/2,∆3, S1,3,2) 243 (F1,3,2, F1,2,5/2,∆3, S1,3,2)244 (F1,3,2, F1,4,5/2,∆3, S1,3,2) 245 (F1,4,3/2, F1,3,2,Σ1, S1,2,3/2) 246 (Σ1, F1,4,3/2, F1,4,1/2, S1,5,1)247 (F3,1,−4/3, F3,2,−11/6, Q5, S3,3,−4/3) 248 (F3,3,−4/3, F3,2,−11/6, Q5, S3,3,−4/3) 249 (F3,3,−4/3, F3,4,−11/6, Q5, S3,3,−4/3)250 (F3,1,5/3, F3,2,13/6, Q7, S3,1,5/3) 251 (F3,1,5/3, F3,2,13/6, Q7, S3,3,5/3) 252 (F3,3,5/3, F3,2,13/6, Q7, S3,1,5/3)253 (F3,3,5/3, F3,2,13/6, Q7, S3,3,5/3) 254 *(F3,3,5/3, F3,4,13/6, Q7, S3,3,5/3) 255 *(F3,4,1/6, F3,5,2/3, T1, S3,4,1/6)256 (F3,4,−5/6, F3,3,−4/3, T1, S3,2,−5/6) 257 (F3,4,−5/6, F3,3,−4/3, T1, S3,4,−5/6) 258 *(F3,4,−5/6, F3,5,−4/3, T1, S3,4,−5/6)259 *(T1, F3,4,1/6, F3,4,−5/6, S3,5,−1/3) 260 (F3,4,7/6, F3,3,5/3, T2, S3,4,7/6) 261 *(F3,4,7/6, F3,5,5/3, T2, S3,4,7/6)262 *(T2, F3,4,7/6, F3,4,1/6, S3,3,2/3) 263 *(T2, F3,4,7/6, F3,4,1/6, S3,5,2/3) 264 *(F3,4,1/6, T2, F3,5,−1/3, S3,4,1/6) 265 (F1,1,2, F1,2,5/2,∆3,S2) 266 (F1,3,2, F1,2,5/2,∆3,S2) 267 (F1,4,1/2,Σ1, F1,5,0,Θ1) 268 (F1,4,3/2, F1,3,2,Σ1,Θ3)269 (F1,4,3/2, F1,5,2,Σ1,Θ3) 270 (F3,1,−4/3, F3,2,−11/6, Q5, ω4) 271 (F3,3,−4/3, F3,2,−11/6, Q5, ω4)272 (F3,4,7/6, F3,3,5/3, T2,Π7) 273 *(T1, F3,4,1/6, F3,4,−5/6, ζ) 274 (∆3, F1,3,2, E, S1,2,3/2)275 (∆3, F1,1,2,Σ1, S1,2,3/2) 276 (∆3, F1,3,2,Σ1, S1,2,3/2) 277 (Q5, F3,3,−4/3, D, S3,2,−5/6)278 (T2, F3,4,7/6, Q1, S3,3,2/3) 279 (Q5, F3,1,−4/3, T1, S3,2,−5/6) 280 (Q5, F3,3,−4/3, T1, S3,2,−5/6)281 (Q5, F3,3,−4/3, T1, S3,4,−5/6) 282 (Q7, F3,3,5/3, T2, S3,4,7/6) 283 (T2, Q7, F3,4,1/6, S3,3,2/3)284 (F3,4,1/6, T2, T1, S3,4,1/6) 285 (∆3, F1,3,2,Σ1,Θ3) 286 (F3,4,1/6, T2, T1,Π1) 287 (Q7, F3,1,5/3, T2,Π7) 288 (Q7, F3,3,5/3, U, Π7)289 (Q7, F3,3,5/3, T2,Π7) 290 (T1, Q1, F3,4,−5/6, ζ) 291 (T1, F3,4,1/6, Q5, ζ)292 - (U, Q7, Q1, S3,3,2/3) 293 (Q1, T2, T1, S3,4,1/6) 294 (T2, Q7, Q1, S3,3,2/3) 295 - (E, ∆3,∆1,S1) 296 - (Σ1,∆3,∆1,S1) 297 - (D, Q1, Q5, ω1) 298 (T1, Q1, Q5, ω1)299 - (U, Q7, Q1, ω2) 300 (T2, Q7, Q1, ω2) 301 (Q1, U, T1,Π1)302 (Q1, T2, D, Π1) 303 (Q1, T2, T1,Π1) 304 - (D, Q1, Q5, ζ)305 (T1, Q1, Q5, ζ) T-3 Exit 306 (F1,2,5/2,S2, S1,3,3) 307 (F1,4,1/2,Ξ, S1,5,1) 308 (F1,5,1,Θ1, S1,6,3/2) 309 (F1,5,0,Θ1,Θ1)310 (F1,5,0,Θ1, S1,6,1/2) 311 (F1,3,2,Θ3, S1,2,5/2) 312 (F1,3,2,Θ3, S1,4,5/2)313 (F1,5,2,Θ3, S1,4,5/2) 314 (F1,5,2,Θ3, S1,6,5/2) 315 (F1,5,1, S1,6,1/2,Θ3)316 (F3,2,−11/6, ω4, S3,3,−7/3) 317 (F3,1,5/3,Π7, S3,2,13/6) 318 (F3,3,5/3,Π7, S3,2,13/6)319 (F3,3,5/3,Π7, S3,4,13/6) 320 (F3,4,1/6, ζ, S3,3,2/3) 321 (F3,4,1/6, ζ, S3,5,2/3)322 (F3,4,−5/6, ζ, S3,3,−4/3) 323 (F3,4,−5/6, ζ, S3,5,−4/3) 324 (F6,2,5/6,Ω1, S6,3,4/3)325 (F6,2,−1/6, S6,3,−2/3,Ω1) 326 (F6,2,−7/6,Ω2, S6,3,−5/3) 327 (F6,2,11/6,Ω4, S6,3,7/3)328 (F6,2,5/6,Υ, S6,3,4/3) 329 *(F6,4,5/6,Υ, S6,3,4/3) 330 *(F6,4,5/6,Υ, S6,5,4/3)331 (F6,2,−1/6, S6,3,−2/3,Υ) 332 *(F6,4,−1/6, S6,3,−2/3,Υ) 333 *(F6,4,−1/6, S6,5,−2/3,Υ) 334 (F8,1,1,Φ, S8,2,3/2) 335 (F8,3,1,Φ, S8,2,3/2) 336 (F8,3,1,Φ, S8,4,3/2)337 - (F8,1,0,Φ,Φ) 338 (F8,3,0,Φ,Φ) 339 (F8,3,0,Φ, S8,4,1/2)340 (Q7, S3,3,2/3, S3,1,5/3) 341 (Q7, S3,3,2/3, S3,3,5/3) 342 (T1, S3,2,−5/6, S3,4,1/6) 343 (T1, S3,4,−5/6, S3,4,1/6) 344 (T2, S3,4,1/6, S3,4,7/6) 345 - (∆3,S1, S1,3,2) 346 - (E, ϕ, S1,2,3/2) 347 - (Σ1, ϕ, S1,2,3/2) 348 - (Σ1,Θ1, S1,2,3/2) 349 (F1,5,1,Θ1,Θ3) 350 - (Q1, ω1, S3,3,2/3) 351 - (Q5, ω1, S3,3,−4/3) 352 (Q7, ω2, S3,3,5/3) 353 (T2,Π1, S3,4,7/6) 354 - (D, S3,2,−5/6,Π1) 355 - (T1, S3,2,−5/6,Π1) 356 (T1, S3,4,−5/6,Π1) 357 (T2, S3,4,1/6,Π7) 358 - (Q1, ζ, S3,3,2/3) 359 - (Q5, ζ, S3,3,−4/3) 360 (F6,2,−1/6,Ω2,Υ) 361 (F6,2,5/6,Υ,Ω4) 362 - (∆1,Ξ,S1) 363 - (Σ1, ϕ, Θ3) 364 - (Σ1,Θ1,Θ3) 365 - (Q1, ζ, ω2) 366 - (Q5, ζ, ω4) 367 - (U, Π1,Π7) 368 - (T2,Π1,Π7) Table 3: TI-3 and T-3 models with scalar and fermionic exits. Table 7. T-I-3 and T-3 models with scalar and fermionic exits. – 21 – JHEP08(2022)023 T-I-1 DM-E 1 (S1,2,1/2, S1,3,1, S1,1,0, F1,2,1/2) 2 (S1,1,0, S1,2,1/2, S1,2,1/2, F1,1,0) 3 (S1,1,0, S1,2,1/2, S1,2,1/2, F1,3,0) 4 (S1,2,1/2, S1,1,1, S1,3,0, F1,2,1/2)5 (S1,3,0, S1,2,1/2, S1,2,1/2, F1,1,0) 6 (S1,3,0, S1,2,1/2, S1,2,1/2, F1,3,0) 7 (S1,3,0, S1,2,1/2, S1,4,1/2, F1,3,0)8 (S1,2,1/2, S1,3,1, S1,3,0, F1,2,1/2) 9 (S1,2,1/2, S1,3,1, S1,3,0, F1,4,1/2) 10 (S1,4,1/2, S1,3,1, S1,3,0, F1,2,1/2)11 (S1,4,1/2, S1,3,1, S1,3,0, F1,4,1/2) 12 (S1,4,1/2, S1,5,1, S1,3,0, F1,4,1/2) 13 (S1,3,0, S1,4,1/2, S1,4,1/2, F1,3,0)14 (S1,3,0, S1,4,1/2, S1,4,1/2, F1,5,0) 15 (S1,4,1/2, S1,3,1, S1,5,0, F1,4,1/2) 16 (S1,5,0, S1,4,1/2, S1,4,1/2, F1,3,0)17 (S1,5,0, S1,4,1/2, S1,4,1/2, F1,5,0) 18 (S1,5,0, S1,4,1/2, S1,6,1/2, F1,5,0) 19 (S1,3,1, S1,4,3/2, S1,2,1/2, F1,3,1)20 (S1,3,1, S1,4,3/2, S1,4,1/2, F1,3,1) 21 (S1,3,1, S1,4,3/2, S1,4,1/2, F1,5,1) 22 (S1,5,1, S1,4,3/2, S1,4,1/2, F1,3,1)23 (S1,5,1, S1,4,3/2, S1,4,1/2, F1,5,1) 24 (S1,1,1, S1,2,3/2, S1,2,1/2, F1,1,1) 25 (S1,1,1, S1,2,3/2, S1,2,1/2, F1,3,1)26 (S1,3,1, S1,2,3/2, S1,2,1/2, F1,1,1) 27 (S1,3,1, S1,2,3/2, S1,2,1/2, F1,3,1) 28 (S1,3,1, S1,2,3/2, S1,4,1/2, F1,3,1)29 (S1,4,1/2, S1,5,1, S1,5,0, F1,4,1/2) 30 (S1,4,1/2, S1,5,1, S1,5,0, F1,6,1/2) 31 (S1,5,1, S1,6,3/2, S1,4,1/2, F1,5,1) T-I-2 32 (F1,1,0, F1,2,1/2, S1,2,1/2, S1,1,0) 33 (F1,2,1/2, F1,3,0, S1,1,0, S1,2,1/2) 34 (F1,2,1/2, F1,3,0, S1,3,0, S1,2,1/2) 35 (F1,1,0, F1,2,1/2, S1,2,1/2, S1,3,0)36 (F1,3,0, F1,4,1/2, S1,2,1/2, S1,3,0) 37 (F1,2,1/2, F1,3,0, S1,3,0, S1,4,1/2) 38 (F1,3,0, F1,4,1/2, S1,4,1/2, S1,3,0)39 (F1,4,1/2, F1,5,0, S1,3,0, S1,4,1/2) 40 (F1,4,1/2, F1,5,0, S1,5,0, S1,4,1/2) 41 (F1,2,1/2, F1,1,1, S1,1,1, S1,2,1/2)42 (F1,2,1/2, F1,1,1, S1,3,1, S1,2,1/2) 43 (F1,2,1/2, F1,3,1, S1,1,1, S1,2,1/2) 44 (F1,2,1/2, F1,3,1, S1,3,1, S1,2,1/2)45 (F1,4,1/2, F1,3,1, S1,3,1, S1,2,1/2) 46 (F1,2,1/2, F1,3,1, S1,3,1, S1,4,1/2) 47 (F1,4,1/2, F1,3,1, S1,3,1, S1,4,1/2)48 (F1,4,1/2, F1,5,1, S1,3,1, S1,4,1/2) 49 (F1,4,1/2, F1,3,1, S1,5,1, S1,4,1/2) 50 (F1,4,1/2, F1,5,1, S1,5,1, S1,4,1/2)51 (F1,6,1/2, F1,5,1, S1,5,1, S1,4,1/2) 52 (F1,3,0, F1,4,1/2, S1,4,1/2, S1,5,0) 53 (F1,5,0, F1,6,1/2, S1,4,1/2, S1,5,0) T-I-3 54 (F1,1,0, F1,2,1/2, F1,2,1/2, S1,1,0) 55 (F1,3,0, F1,2,1/2, F1,2,1/2, S1,1,0) 56 (F1,1,0, F1,2,1/2, F1,2,1/2, S1,3,0) 57 (F1,3,0, F1,2,1/2, F1,2,1/2, S1,3,0)58 (F1,3,0, F1,2,1/2, F1,4,1/2, S1,3,0) 59 (F1,3,0, F1,4,1/2, F1,4,1/2, S1,3,0) 60 (F1,5,0, F1,4,1/2, F1,4,1/2, S1,3,0)61 (F1,2,1/2, F1,1,1, F1,3,0, S1,2,1/2) 62 (F1,2,1/2, F1,3,1, F1,1,0, S1,2,1/2) 63 (F1,2,1/2, F1,3,1, F1,3,0, S1,2,1/2)64 (F1,4,1/2, F1,3,1, F1,3,0, S1,2,1/2) 65 (F1,2,1/2, F1,3,1, F1,3,0, S1,4,1/2) 66 (F1,4,1/2, F1,3,1, F1,3,0, S1,4,1/2)67 (F1,4,1/2, F1,3,1, F1,5,0, S1,4,1/2) 68 (F1,4,1/2, F1,5,1, F1,3,0, S1,4,1/2) 69 (F1,4,1/2, F1,5,1, F1,5,0, S1,4,1/2)70 (F1,6,1/2, F1,5,1, F1,5,0, S1,4,1/2) 71 (F1,3,0, F1,4,1/2, F1,4,1/2, S1,5,0) T-3 72 (F1,2,1/2, S1,1,0, S1,3,1) 73 (F1,1,1, S1,2,1/2, S1,2,3/2) 74 (F1,3,1, S1,2,1/2, S1,2,3/2) 75 (F1,3,1, S1,2,1/2, S1,4,3/2)76 (F1,2,1/2, S1,3,0, S1,1,1) 77 (F1,2,1/2, S1,3,0, S1,3,1) 78 (F1,4,1/2, S1,3,0, S1,3,1)79 (F1,4,1/2, S1,3,0, S1,5,1) 80 (F1,3,1, S1,4,1/2, S1,2,3/2) 81 (F1,3,1, S1,4,1/2, S1,4,3/2) 82 (F1,5,1, S1,4,1/2, S1,4,3/2) 83 (F1,5,1, S1,4,1/2, S1,6,3/2) 84 (F1,1,0, S1,2,1/2, S1,2,1/2) 85 (F1,3,0, S1,2,1/2, S1,2,1/2) 86 (F1,3,0, S1,2,1/2, S1,4,1/2) 87 (F1,4,1/2, S1,5,0, S1,3,1) 88 (F1,3,0, S1,4,1/2, S1,4,1/2) 89 (F1,5,0, S1,4,1/2, S1,4,1/2) 90 (F1,5,0, S1,4,1/2, S1,6,1/2) 91 (F1,5,1, S1,6,1/2, S1,4,3/2) Table 8. DM models with exits (DM-E) which need a stabilizing symmetry to give an acceptable WIMP candidate. – 22 – JHEP08(2022)023 T-I-1 DM-A 1 (S1,5,0, S1,6,1/2, S1,6,1/2, F1,5,0) 2 (S1,5,0, S1,6,1/2, S1,6,1/2, F1,7,0) 3 (S1,6,1/2, S1,5,1, S1,5,0, F1,4,1/2) 4 (S1,6,1/2, S1,5,1, S1,5,0, F1,6,1/2)5 (S1,6,1/2, S1,5,1, S1,7,0, F1,6,1/2) 6 (S1,6,1/2, S1,7,1, S1,5,0, F1,6,1/2) 7 (S1,6,1/2, S1,7,1, S1,7,0, F1,6,1/2)8 (S1,6,1/2, S1,7,1, S1,7,0, F1,8,1/2) 9 (S1,7,0, S1,6,1/2, S1,6,1/2, F1,5,0) 10 (S1,7,0, S1,6,1/2, S1,6,1/2, F1,7,0)11 (S1,7,0, S1,6,1/2, S1,8,1/2, F1,7,0) 12 (S1,7,0, S1,8,1/2, S1,8,1/2, F1,7,0) 13 (S1,7,0, S1,8,1/2, S1,8,1/2, F1,9,0)14 (S1,8,1/2, S1,7,1, S1,7,0, F1,6,1/2) 15 (S1,8,1/2, S1,7,1, S1,7,0, F1,8,1/2) 16 (S1,8,1/2, S1,7,1, S1,9,0, F1,8,1/2)17 (S1,8,1/2, S1,9,1, S1,7,0, F1,8,1/2) 18 (S1,8,1/2, S1,9,1, S1,9,0, F1,8,1/2) 19 (S1,8,1/2, S1,9,1, S1,9,0, F1,10,1/2)20 (S1,9,0, S1,8,1/2, S1,8,1/2, F1,7,0) 21 (S1,9,0, S1,8,1/2, S1,8,1/2, F1,9,0) 22 (S1,9,0, S1,8,1/2, S1,10,1/2, F1,9,0)23 (S1,9,0, S1,10,1/2, S1,10,1/2, F1,9,0) 24 (S1,9,0, S1,10,1/2, S1,10,1/2, F1,11,0) 25 (S1,10,1/2, S1,9,1, S1,9,0, F1,8,1/2)26 (S1,10,1/2, S1,9,1, S1,9,0, F1,10,1/2) 27 (S1,10,1/2, S1,9,1, S1,11,0, F1,10,1/2) 28 (S1,10,1/2, S1,11,1, S1,9,0, F1,10,1/2)29 (S1,10,1/2, S1,11,1, S1,11,0, F1,10,1/2) 30 (S1,10,1/2, S1,11,1, S1,11,0, F1,12,1/2) 31 (S1,11,0, S1,10,1/2, S1,10,1/2, F1,9,0)32 (S1,11,0, S1,10,1/2, S1,10,1/2, F1,11,0) 33 (S1,11,0, S1,10,1/2, S1,12,1/2, F1,11,0) 34 (S1,11,0, S1,12,1/2, S1,12,1/2, F1,11,0)35 (S1,11,0, S1,12,1/2, S1,12,1/2, F1,13,0) 36 (S1,12,1/2, S1,11,1, S1,11,0, F1,10,1/2) 37 (S1,12,1/2, S1,11,1, S1,11,0, F1,12,1/2)38 (S1,12,1/2, S1,11,1, S1,13,0, F1,12,1/2) 39 (S1,12,1/2, S1,13,1, S1,11,0, F1,12,1/2) 40 (S1,12,1/2, S1,13,1, S1,13,0, F1,12,1/2)41 (S1,12,1/2, S1,13,1, S1,13,0, F1,14,1/2) 42 (S1,13,0, S1,12,1/2, S1,12,1/2, F1,11,0) 43 (S1,13,0, S1,12,1/2, S1,12,1/2, F1,13,0)44 (S1,13,0, S1,12,1/2, S1,14,1/2, F1,13,0) 45 (S1,13,0, S1,14,1/2, S1,14,1/2, F1,13,0) 46 (S1,13,0, S1,14,1/2, S1,14,1/2, F1,15,0)47 (S1,14,1/2, S1,13,1, S1,13,0, F1,12,1/2) 48 (S1,14,1/2, S1,13,1, S1,13,0, F1,14,1/2) 49 (S1,14,1/2, S1,15,1, S1,13,0, F1,14,1/2)50 (S1,15,0, S1,14,1/2, S1,14,1/2, F1,13,0) T-I-2 51 (F1,4,1/2, F1,5,0, S1,5,0, S1,6,1/2) 52 (F1,5,0, F1,6,1/2, S1,6,1/2, S1,5,0) 53 (F1,5,0, F1,6,1/2, S1,6,1/2, S1,7,0) 54 (F1,6,1/2, F1,7,0, S1,5,0, S1,6,1/2)55 (F1,6,1/2, F1,7,0, S1,7,0, S1,6,1/2) 56 (F1,6,1/2, F1,7,0, S1,7,0, S1,8,1/2) 57 (F1,7,0, F1,8,1/2, S1,6,1/2, S1,7,0)58 (F1,7,0, F1,8,1/2, S1,8,1/2, S1,7,0) 59 (F1,7,0, F1,8,1/2, S1,8,1/2, S1,9,0) 60 (F1,8,1/2, F1,9,0, S1,7,0, S1,8,1/2)61 (F1,8,1/2, F1,9,0, S1,9,0, S1,8,1/2) 62 (F1,8,1/2, F1,9,0, S1,9,0, S1,10,1/2) 63 (F1,9,0, F1,10,1/2, S1,8,1/2, S1,9,0)64 (F1,9,0, F1,10,1/2, S1,10,1/2, S1,9,0) 65 (F1,9,0, F1,10,1/2, S1,10,1/2, S1,11,0) 66 (F1,10,1/2, F1,11,0, S1,9,0, S1,10,1/2)67 (F1,10,1/2, F1,11,0, S1,11,0, S1,10,1/2) 68 (F1,10,1/2, F1,11,0, S1,11,0, S1,12,1/2) 69 (F1,11,0, F1,12,1/2, S1,10,1/2, S1,11,0)70 (F1,11,0, F1,12,1/2, S1,12,1/2, S1,11,0) 71 (F1,11,0, F1,12,1/2, S1,12,1/2, S1,13,0) 72 (F1,12,1/2, F1,13,0, S1,11,0, S1,12,1/2)73 (F1,12,1/2, F1,13,0, S1,13,0, S1,12,1/2) 74 (F1,12,1/2, F1,13,0, S1,13,0, S1,14,1/2) 75 (F1,13,0, F1,14,1/2, S1,12,1/2, S1,13,0)76 (F1,13,0, F1,14,1/2, S1,14,1/2, S1,13,0) 77 (F1,13,0, F1,14,1/2, S1,14,1/2, S1,15,0) 78 (F1,14,1/2, F1,15,0, S1,13,0, S1,14,1/2) T-I-3 79 (F1,4,1/2, F1,5,1, F1,5,0, S1,6,1/2) 80 (F1,5,0, F1,4,1/2, F1,4,1/2, S1,5,0) 81 (F1,5,0, F1,4,1/2, F1,6,1/2, S1,5,0) 82 (F1,5,0, F1,6,1/2, F1,6,1/2, S1,5,0)83 (F1,5,0, F1,6,1/2, F1,6,1/2, S1,7,0) 84 (F1,6,1/2, F1,5,1, F1,5,0, S1,6,1/2) 85 (F1,6,1/2, F1,5,1, F1,7,0, S1,6,1/2)86 (F1,6,1/2, F1,7,1, F1,5,0, S1,6,1/2) 87 (F1,6,1/2, F1,7,1, F1,7,0, S1,6,1/2) 88 (F1,6,1/2, F1,7,1, F1,7,0, S1,8,1/2)89 (F1,7,0, F1,6,1/2, F1,6,1/2, S1,5,0) 90 (F1,7,0, F1,6,1/2, F1,6,1/2, S1,7,0) 91 (F1,7,0, F1,6,1/2, F1,8,1/2, S1,7,0)92 (F1,7,0, F1,8,1/2, F1,8,1/2, S1,7,0) 93 (F1,7,0, F1,8,1/2, F1,8,1/2, S1,9,0) 94 (F1,8,1/2, F1,7,1, F1,7,0, S1,6,1/2)95 (F1,8,1/2, F1,7,1, F1,7,0, S1,8,1/2) 96 (F1,8,1/2, F1,7,1, F1,9,0, S1,8,1/2) 97 (F1,8,1/2, F1,9,1, F1,7,0, S1,8,1/2)98 (F1,8,1/2, F1,9,1, F1,9,0, S1,8,1/2) 99 (F1,8,1/2, F1,9,1, F1,9,0, S1,10,1/2) 100 (F1,9,0, F1,8,1/2, F1,8,1/2, S1,7,0)101 (F1,9,0, F1,8,1/2, F1,8,1/2, S1,9,0) 102 (F1,9,0, F1,8,1/2, F1,10,1/2, S1,9,0) 103 (F1,9,0, F1,10,1/2, F1,10,1/2, S1,9,0)104 (F1,9,0, F1,10,1/2, F1,10,1/2, S1,11,0) 105 (F1,10,1/2, F1,9,1, F1,9,0, S1,8,1/2) 106 (F1,10,1/2, F1,9,1, F1,9,0, S1,10,1/2)107 (F1,10,1/2, F1,9,1, F1,11,0, S1,10,1/2) 108 (F1,10,1/2, F1,11,1, F1,9,0, S1,10,1/2) 109 (F1,10,1/2, F1,11,1, F1,11,0, S1,10,1/2)110 (F1,10,1/2, F1,11,1, F1,11,0, S1,12,1/2) 111 (F1,11,0, F1,10,1/2, F1,10,1/2, S1,9,0) 112 (F1,11,0, F1,10,1/2, F1,10,1/2, S1,11,0)113 (F1,11,0, F1,10,1/2, F1,12,1/2, S1,11,0) 114 (F1,11,0, F1,12,1/2, F1,12,1/2, S1,11,0) 115 (F1,11,0, F1,12,1/2, F1,12,1/2, S1,13,0)116 (F1,12,1/2, F1,11,1, F1,11,0, S1,10,1/2) 117 (F1,12,1/2, F1,11,1, F1,11,0, S1,12,1/2) 118 (F1,12,1/2, F1,11,1, F1,13,0, S1,12,1/2)119 (F1,12,1/2, F1,13,1, F1,11,0, S1,12,1/2) 120 (F1,12,1/2, F1,13,1, F1,13,0, S1,12,1/2) 121 (F1,12,1/2, F1,13,1, F1,13,0, S1,14,1/2)122 (F1,13,0, F1,12,1/2, F1,12,1/2, S1,11,0) 123 (F1,13,0, F1,12,1/2, F1,12,1/2, S1,13,0) 124 (F1,13,0, F1,12,1/2, F1,14,1/2, S1,13,0) 125 (F1,13,0, F1,14,1/2, F1,14,1/2, S1,13,0) 126 (F1,13,0, F1,14,1/2, F1,14,1/2, S1,15,0) 127 (F1,14,1/2, F1,13,1, F1,13,0, S1,12,1/2) 128 (F1,14,1/2, F1,13,1, F1,13,0, S1,14,1/2) 129 (F1,14,1/2, F1,15,1, F1,13,0, S1,14,1/2) 130 (F1,15,0, F1,14,1/2, F1,14,1/2, S1,13,0) T-3 131 (F1,4,1/2, S1,5,0, S1,5,1) 132 (F1,5,1, S1,6,1/2, S1,6,3/2) 133 (F1,5,0, S1,6,1/2, S1,6,1/2) 134 (F1,6,1/2, S1,5,0, S1,5,1) 135 (F1,6,1/2, S1,5,0, S1,7,1) 136 (F1,6,1/2, S1,7,0, S1,5,1) 137 (F1,6,1/2, S1,7,0, S1,7,1) 138 (F1,7,1, S1,6,1/2, S1,6,3/2) 139 (F1,7,1, S1,6,1/2, S1,8,3/2) 140 (F1,7,1, S1,8,1/2, S1,6,3/2) 141 (F1,7,1, S1,8,1/2, S1,8,3/2) 142 (F1,7,0, S1,6,1/2, S1,6,1/2) 143 (F1,7,0, S1,6,1/2, S1,8,1/2)144 (F1,7,0, S1,8,1/2, S1,8,1/2) 145 (F1,8,1/2, S1,7,0, S1,7,1) 146 (F1,8,1/2, S1,7,0, S1,9,1) 147 (F1,8,1/2, S1,9,0, S1,7,1) 148 (F1,8,1/2, S1,9,0, S1,9,1) 149 (F1,9,1, S1,8,1/2, S1,8,3/2) 150 (F1,9,1, S1,8,1/2, S1,10,3/2) 151 (F1,9,1, S1,10,1/2, S1,8,3/2) 152 (F1,9,1, S1,10,1/2, S1,10,3/2) 153 (F1,9,0, S1,8,1/2, S1,8,1/2) 154 (F1,9,0, S1,8,1/2, S1,10,1/2) 155 (F1,9,0, S1,10,1/2, S1,10,1/2) 156 (F1,10,1/2, S1,9,0, S1,9,1) 157 (F1,10,1/2, S1,9,0, S1,11,1) 158 (F1,10,1/2, S1,11,0, S1,9,1) 159 (F1,10,1/2, S1,11,0, S1,11,1) 160 (F1,11,1, S1,10,1/2, S1,10,3/2) 161 (F1,11,1, S1,10,1/2, S1,12,3/2) 162 (F1,11,1, S1,12,1/2, S1,10,3/2) 163 (F1,11,1, S1,12,1/2, S1,12,3/2) 164 (F1,11,0, S1,10,1/2, S1,10,1/2) 165 (F1,11,0, S1,10,1/2, S1,12,1/2) 166 (F1,11,0, S1,12,1/2, S1,12,1/2) 167 (F1,12,1/2, S1,11,0, S1,11,1) 168 (F1,12,1/2, S1,11,0, S1,13,1) 169 (F1,12,1/2, S1,13,0, S1,11,1) 170 (F1,12,1/2, S1,13,0, S1,13,1) 171 (F1,13,1, S1,12,1/2, S1,12,3/2) 172 (F1,13,1, S1,12,1/2, S1,14,3/2) 173 (F1,13,0, S1,12,1/2, S1,12,1/2)174 (F1,13,0, S1,12,1/2, S1,14,1/2) 175 (F1,13,0, S1,14,1/2, S1,14,1/2) 176 (F1,14,1/2, S1,13,0, S1,13,1) 177 (F1,14,1/2, S1,13,0, S1,15,1) Table 9. Models in which the DM could be stable due to accidental symmetry (DM-A). – 23 – JHEP08(2022)023 T-I-1 DM-A (exceptional) 1 (S1,5,1, S1,4,3/2, S1,6,1/2, F1,5,1) 2 (S1,5,1, S1,6,3/2, S1,6,1/2, F1,5,1) 3 (S1,7,1, S1,6,3/2, S1,6,1/2, F1,5,1) T-I-2 4 (F1,4,1/2, F1,5,1, S1,5,1, S1,6,1/2) 5 (F1,5,1, F1,4,3/2, S1,4,3/2, S1,5,1) 6 (F1,5,1, F1,4,3/2, S1,6,3/2, S1,5,1) 7 (F1,5,1, F1,6,3/2, S1,4,3/2, S1,5,1)8 (F1,5,1, F1,6,3/2, S1,6,3/2, S1,5,1) 9 (F1,5,1, F1,6,3/2, S1,6,3/2, S1,7,1) 10 (F1,6,1/2, F1,5,1, S1,5,1, S1,6,1/2)11 (F1,6,1/2, F1,5,1, S1,7,1, S1,6,1/2) T-I-3 12 (F1,4,3/2, F1,3,2, F1,5,1, S1,4,3/2) 13 (F1,4,3/2, F1,5,2, F1,5,1, S1,4,3/2) 14 (F1,4,3/2, F1,5,2, F1,5,1, S1,6,3/2) 15 (F1,5,1, F1,4,3/2, F1,4,1/2, S1,5,1)16 (F1,5,1, F1,4,3/2, F1,6,1/2, S1,5,1) 17 (F1,5,1, F1,6,3/2, F1,4,1/2, S1,5,1) 18 (F1,5,1, F1,6,3/2, F1,6,1/2, S1,5,1)19 (F1,5,1, F1,6,3/2, F1,6,1/2, S1,7,1) 20 (F1,6,3/2, F1,5,2, F1,5,1, S1,4,3/2) 21 (F1,6,3/2, F1,5,2, F1,5,1, S1,6,3/2)22 (F1,6,3/2, F1,7,2, F1,5,1, S1,6,3/2) T-3 23 (F1,4,3/2, S1,5,1, S1,3,2) 24 (F1,4,3/2, S1,5,1, S1,5,2) 25 (F1,6,3/2, S1,5,1, S1,5,2) 26 (F1,6,3/2, S1,5,1, S1,7,2) T-I-2 DM-E (exceptional) 1 (F1,3,1, F1,2,3/2, S1,2,3/2, S1,1,1) 2 (F1,3,1, F1,2,3/2, S1,2,3/2, S1,3,1) 3 (F1,3,1, F1,2,3/2, S1,4,3/2, S1,3,1) 4 (F1,3,1, F1,4,3/2, S1,2,3/2, S1,3,1)5 (F1,3,1, F1,4,3/2, S1,4,3/2, S1,3,1) 6 (F1,3,1, F1,4,3/2, S1,4,3/2, S1,5,1) 7 (F1,5,1, F1,4,3/2, S1,4,3/2, S1,3,1) T-I-3 8 (F1,2,3/2, F1,1,2, F1,3,1, S1,2,3/2) 9 (F1,2,3/2, F1,3,2, F1,3,1, S1,2,3/2) 10 (F1,2,3/2, F1,3,2, F1,3,1, S1,4,3/2) 11 (F1,3,1, F1,2,3/2, F1,2,1/2, S1,1,1) 12 (F1,3,1, F1,2,3/2, F1,2,1/2, S1,3,1) 13 (F1,3,1, F1,2,3/2, F1,4,1/2, S1,3,1) 14 (F1,3,1, F1,4,3/2, F1,2,1/2, S1,3,1) 15 (F1,3,1, F1,4,3/2, F1,4,1/2, S1,3,1) 16 (F1,3,1, F1,4,3/2, F1,4,1/2, S1,5,1) 17 (F1,4,3/2, F1,3,2, F1,3,1, S1,2,3/2)18 (F1,4,3/2, F1,3,2, F1,3,1, S1,4,3/2) 19 (F1,4,3/2, F1,5,2, F1,3,1, S1,4,3/2) 20 (F1,5,1, F1,4,3/2, F1,4,1/2, S1,3,1) T-3 21 (F1,2,3/2, S1,3,1, S1,1,2) 22 (F1,2,3/2, S1,3,1, S1,3,2) 23 (F1,4,3/2, S1,3,1, S1,3,2) 24 (F1,4,3/2, S1,3,1, S1,5,2) T-I-1 DM-A (exceptional) 1 (S1,5,1, S1,4,3/2, S1,6,1/2, F1,5,1) 2 (S1,5,1, S1,6,3/2, S1,6,1/2, F1,5,1) 3 (S1,7,1, S1,6,3/2, S1,6,1/2, F1,5,1) T-I-2 4 (F1,4,1/2, F1,5,1, S1,5,1, S1,6,1/2) 5 (F1,5,1, F1,4,3/2, S1,4,3/2, S1,5,1) 6 (F1,5,1, F1,4,3/2, S1,6,3/2, S1,5,1) 7 (F1,5,1, F1,6,3/2, S1,4,3/2, S1,5,1)8 (F1,5,1, F1,6,3/2, S1,6,3/2, S1,5,1) 9 (F1,5,1, F1,6,3/2, S1,6,3/2, S1,7,1) 10 (F1,6,1/2, F1,5,1, S1,5,1, S1,6,1/2)11 (F1,6,1/2, F1,5,1, S1,7,1, S1,6,1/2) T-I-3 12 (F1,4,3/2, F1,3,2, F1,5,1, S1,4,3/2) 13 (F1,4,3/2, F1,5,2, F1,5,1, S1,4,3/2) 14 (F1,4,3/2, F1,5,2, F1,5,1, S1,6,3/2) 15 (F1,5,1, F1,4,3/2, F1,4,1/2, S1,5,1)16 (F1,5,1, F1,4,3/2, F1,6,1/2, S1,5,1) 17 (F1,5,1, F1,6,3/2, F1,4,1/2, S1,5,1) 18 (F1,5,1, F1,6,3/2, F1,6,1/2, S1,5,1)19 (F1,5,1, F1,6,3/2, F1,6,1/2, S1,7,1) 20 (F1,6,3/2, F1,5,2, F1,5,1, S1,4,3/2) 21 (F1,6,3/2, F1,5,2, F1,5,1, S1,6,3/2)22 (F1,6,3/2, F1,7,2, F1,5,1, S1,6,3/2) T-3 23 (F1,4,3/2, S1,5,1, S1,3,2) 24 (F1,4,3/2, S1,5,1, S1,5,2) 25 (F1,6,3/2, S1,5,1, S1,5,2) 26 (F1,6,3/2, S1,5,1, S1,7,2) T-I-2 DM-E (exceptional) 1 (F1,3,1, F1,2,3/2, S1,2,3/2, S1,1,1) 2 (F1,3,1, F1,2,3/2, S1,2,3/2, S1,3,1) 3 (F1,3,1, F1,2,3/2, S1,4,3/2, S1,3,1) 4 (F1,3,1, F1,4,3/2, S1,2,3/2, S1,3,1)5 (F1,3,1, F1,4,3/2, S1,4,3/2, S1,3,1) 6 (F1,3,1, F1,4,3/2, S1,4,3/2, S1,5,1) 7 (F1,5,1, F1,4,3/2, S1,4,3/2, S1,3,1) T-I-3 8 (F1,2,3/2, F1,1,2, F1,3,1, S1,2,3/2) 9 (F1,2,3/2, F1,3,2, F1,3,1, S1,2,3/2) 10 (F1,2,3/2, F1,3,2, F1,3,1, S1,4,3/2) 11 (F1,3,1, F1,2,3/2, F1,2,1/2, S1,1,1) 12 (F1,3,1, F1,2,3/2, F1,2,1/2, S1,3,1) 13 (F1,3,1, F1,2,3/2, F1,4,1/2, S1,3,1) 14 (F1,3,1, F1,4,3/2, F1,2,1/2, S1,3,1) 15 (F1,3,1, F1,4,3/2, F1,4,1/2, S1,3,1) 16 (F1,3,1, F1,4,3/2, F1,4,1/2, S1,5,1) 17 (F1,4,3/2, F1,3,2, F1,3,1, S1,2,3/2)18 (F1,4,3/2, F1,3,2, F1,3,1, S1,4,3/2) 19 (F1,4,3/2, F1,5,2, F1,3,1, S1,4,3/2) 20 (F1,5,1, F1,4,3/2, F1,4,1/2, S1,3,1) T-3 21 (F1,2,3/2, S1,3,1, S1,1,2) 22 (F1,2,3/2, S1,3,1, S1,3,2) 23 (F1,4,3/2, S1,3,1, S1,3,2) 24 (F1,4,3/2, S1,3,1, S1,5,2) Table 10. Exceptional DM candidates with Y= 1. The table at the top corresponds to DM models with exits (DM-E) which need a stabilizing symmetry and the table at the bottom to accidental DM models (DM-A). The symbol *has been placed next to each model where one of the Landau pole scales is very low, i.e Λ1,2,3<100 TeV. On the other hand the symbol “−”, placed next to a model, represents models where all Landau pole scales, larger than mNP, are very large Λ1,2,3>1015 GeV. The symbol †is marked next to the only model that unifies at a scale of mG≃1017 GeV. Acknowledgments This work is supported by the Spanish grants PID2020-113775GB-I00 (AEI/10.13039/ 501100011033) and CIPROM/2021/054 (Generalitat Valenciana). J.C.H. acknowledge support from grant ANID FONDECYT-Chile No. 1201673. S.K. is supported by ANID PIA/APOYO AFB180002 (Chile) and by ANID FONDECYT (Chile) No. 1190845. J.C.H. and S.K. acknowledge support from ANID — Programa Milenio — code ICN2019_044. R.C. is supported by the Alexander von Humboldt Foundation Fellowship. C.A. is supported by FONDECYT-Chile grant No. 11180722 and ANID-Chile PIA/APOYO AFB 180002. – 24 –