Systematic classification of two-loop realizations of the Weinberg operator

We systematically analyze the $d=5$ Weinberg operator at 2-loop order. Using a diagrammatic approach, we identify two different interesting categories of neutrino mass models: (i) Genuine 2-loop models for which both, tree-level and 1-loop contributions, are guaranteed to be absent. And (ii) finite 2-loop diagrams, which correspond to the 1-loop generation of some particular vertex appearing in a given 1-loop neutrino mass model, thus being effectively 2-loop. From the large list of all possible 2-loop diagrams, the vast majority are infinite corrections to lower order neutrino mass models and only a moderately small number of diagrams fall into these two interesting classes. Moreover, all diagrams in class (i) are just variations of three basic diagrams, with examples discussed in the literature before. Similarly, we also show that class (ii) diagrams consists of only variations of these three plus two more basic diagrams. Finally, we show how our results can be consistently and readily used in order to construct two-loop neutrino mass models.


Introduction
Neutrino masses observed in oscillation experiments [1][2][3] are so far the only signal for physics beyond the standard model (SM) measured in laboratories. However, while we do know now mass squared differences and mixing angles to a very high precision [4][5][6], there is no experimental data on whether neutrinos are Dirac or Majorana particles.
From the low energy point of view Majorana neutrino masses are described by a leptonnumber-breaking dimension five effective operator, known as the Weinberg operator [7]: The smallness of the observed light neutrino masses can then be explained from eq. (1) as being due to either a large scale Λ or a small coffecient c αβ (or both). However, disentangling these possibilities requires going beyond this effective operator picture. It is well-known that at tree-level only three UV completions for the Weinberg operator exist [8]: These are usually called type-I [9][10][11][12], type-II [13][14][15][16] and type-III [17] seesaw. All of them have in common that for c αβ O(10 −2 − 1), Λ 10 13−15 GeV is needed to produce sub-eV neutrino masses. Thus, while being an attractive possibility from the theoretical point of view, experimentally the classical seesaws do not offer any possible tests-apart from neutrino masses themselves and the fact that neutrinos are predicted to be Majorana particles, thus a finite rate for 0νββ decay should exist.
However, c αβ could easily be smaller than O (1). Essentially there are three possibilities to arrange this: 1. The neutrino mass is generated at tree level, but an additional suppression enters through a small lepton-number-violating (LNV) coupling. The so-called "inverse" seesaw [18] or "linear" seesaw [19,20] are examples for this approach.
2. The neutrino mass is generated radiatively. The additional suppression is guaranteed by a combination of loop integrals and sub-EW scale masses (for example SM charged lepton masses) entering the diagrams. At the one-and two-loop level, the Zee [21] 1 and the Cheng-Li-Babu-Zee models stand as benchmark references [16,26,27] and probably due to this reason they have been the subject of intensive phenomenological studies [25,[28][29][30][31][32][33][34].
3. The neutrino mass is forbidden at d = 5, but appears from effective operators of higher dimension [35,36]. Such an approach is not feasible in models with only the SM Higgs doublet, since (H † H) is a complete singlet and can not carry any charges. However, in a two-Higgs doublet world (or more complicated setups) forbidding the d = 5 while allowing d = 7 could be realized with, for example, the help of some flavor symmetry that prevents the direct Yukawa coupling of the SM Higgs doublet to the light fermions.
In this paper we will focus on the second possibility: Loop neutrino masses. In [37] the Weinberg operator was studied systematically at the one-loop level. Two topologies (for a total of four diagrams) were identified to give neutrino masses at the 1-loop level genuinely (i.e. without producing neutrino masses at tree-level), see fig. (1). Three more diagrams were found, that can be understood as 1-loop realizations of one of the known tree-level seesaws and the relation between tree-and 1-loop diagrams were discussed. In our current work, we extend this analysis [37] to the 2-loop level, following the same diagrammatic-based approach.
The systematic decomposition used in [37] allows one to identify all possible realizations of O 5 at a given loop level, in principle. However, while there are only 12 diagrams (out of which only seven turn out to be of any interest) at the 1-loop level, at the 2-loop level one can naively expect to find order O(100) diagrams, which need to be studied. We have followed therefore a sort of "algorithm" for O 5 at the two-loop order: (i) derive all possible two-loop topologies; exclude all 1-loop reducible topologies from this list. (ii) into the remaining topologies insert fermions and scalars, to create all possible diagram variations; exclude from further analysis all those diagrams which need non-renormalizable vertices. For these first two steps we have extensively used FeynArts [39], in order to ensure that no topology is missed. (iii) Identify in this list of diagrams (with renormalizable vertices only) all those, for which no 1-loop diagram (nor a tree-level neutrino mass) exist. We call these diagrams "genuine 2-loop diagrams" and classify them as class-I diagrams (or models). (iv) For all remaining diagrams one can then distinguish diagrams which lead to finite loop integrals from those with inifinite integrals. The former cases, which are our "class-II" diagrams, can present interesting models of neutrino The four diagrams leading to genuine 1-loop neutrino mass models. The notation of [37] is used to classify these diagrams. Just to mention two examples: Diagram T1-ii corresponds to the the classical Zee model [21], while an example for T-3 is the "scotogenic" model of [38].
mass, even though they are not genuinely 2-loop. The characterization of class-II diagrams (and their corresponding models) is similar to the discussion given in [37] for the 1-loop order: Class-II diagrams can give a theoretical motivation for the smallness of a particular vertex, generated at 1-loop order. This particular vertex then appears in one of the four genuine 1loop neutrino mass diagrams (see fig. 1), making the whole construction effectively 2-loop. Diagrams with infinite loop integrals, on the other hand, can never lead to interesting models and can therefore be discarded. Surprisingly, the result of the above exercise allows one to show that in the moderate number of diagrams of class-I all cases are variations of only two basic diagrams, known in the literature for a long time: The Cheng-Li-Babu-Zee [16,26,27] diagram (CLBZ in the following) and another similar diagram first considered in two independent papers by Petcov and Toshev [40] and by Babu and Ma [41] (PTBM in the following). Similarly, it can be shown that all diagrams in class-II can be described by variations of just five basic types of diagrams: we call them the non-genuine CLBZ and PTBM diagrams, the rainbow diagram and internal scalar correction diagrams (two categories, called ISC-i and ISC-ii).
Before entering into the details, let us mention that our study considers only scalar bosons, while, for example, the original papers on the PTBM diagram [40,41] use the SM W -boson. We decided to concentrate on scalars for essentially two reasons: (a) From a topological point of view, diagrams with scalar or vector bosons are equivalent. Thus, from our list of diagrams for scalars the corresponding diagrams for vectors can be easily derived. 2 And (b) apart from the few cases with SM W -bosons, new vector-mediated cases require that the vector should be a gauge boson under a new symmetry, and the mass should be given by the spontaneous breaking of that symmetry. This means that the scalar sector of the model needs to be discussed as well; see [42] for a recent example.
Our list of diagrams allows us to recover 2-loop models discussed previously in the literature. Apart from the standard diagrams CLBZ [16,26,27] and PTBM [40,41] in their original incarnations (enumerated as CLBZ-1 and PTBM-1 in the following) 3 , we have found a number of variations of these genuine diagrams and also several realizations of our class-II diagrams have been discussed in the literature. For example, a variant of CLBZ-1 with an additional neutral scalar vev to generate the lepton number violating triple scalar vertex h − h − k ++ has been discussed in [43,44]. A supersymmetric extension of CLBZ has been discussed in [45]. A new model with a scalar diquark and a scalar leptoquark has been discussed in [46], 2-loop neutrino masses are generated by the CLBZ-1 diagram. Ref. [47] considers a model with neutrino masses due to CLBZ-1 and a Z 3 symmetry to eliminate tree-level seesaw and also explain dark matter. There are also models in the literature based on other variants of CLBZ. CLBZ-3 appears in [48], CLBZ-9 in [49,50] and CLBZ-8 and CLBZ-10 appear within the 331-model of [51]. Then there are also models, based on CLBZ, using vectors instead of scalars [52][53][54]. In [54] an effective LNV vertex for W bosons with doubly charged scalar is considered, while in [52,53] an SU (2) L triplet scalar mixes with a charged singlet, the diagram thus being effectively CLBZ-9. This construction [52,53] requires that the tree-level coupling between the triplet and the leptons is absent. Then there are models based on variants of PTBM such as [55], which uses leptoquarks and a colour octet fermion. Also in R-parity violation PTBM diagrams appear [56] and can be used to constrain the RPV soft SUSY breaking parameters. Such RPV SUSY models have not only PTBM diagrams, but also rainbow type 2-loop diagrams [57]. Then there are leptoquark models [58] and extensions of the SM with vector-like quarks [59], with scalar and SM W-boson diagrams. In this case, both CLBZ and PTBM in various variants contribute to the neutrino mass. Rainbow diagrams (RB-6) where considered, for example, in [60] and [61]. The 1-loop diagram T-3, see fig. 1, contains a quartic scalar vertex, usually its parameter is called λ 5 . The radiative generation of λ 5 for this diagram, via diagrams of class ISC-i has been considered in [62]. Similarly ISC-i variant-5 was discussed in [63].
On top of these "pure" 2-loop models, also mixed situations, where one (or more) neutrinos have tree-level masses, while one neutrino mass is generated at 2-loop level have been considered. Ref. [64] considers such a situation, with some neutrinos getting a mass through CLBZ-1. Similarly, [65] assumes two neutrino masses to be tree-level and calculates the minimal mass for the remaining neutrino, generated through diagrams with Higgses of the form PTBM-1 in both SM and MSSM. Ref. [66] considers a variant with some neutrinos receiving 1-loop neutrino masses and others are 2-loop. Also, [67] consider models where neutrino mass appear at 1-loop level and also at 2-loop level with CLBZ-1, PTBM-4 and two variants of the rainbow diagram 4 .
We mention that there are also papers on two-loop models in the literature, which are not covered by our classification, because they are of higher dimension than the Weinberg operator. To quote two examples, the papers [68][69][70][71] have several variants of CLBZ at d = 7, while [72] has a variant of the rainbow at d = 7. Finally there is the paper [73], that discusses Dirac neutrino masses at 2-loop.
Our work is, of course, not the first attempt to organize neutrino mass models systematically. Apart from the above-mentioned paper [37], which treats the 1-loop case, a set of "rules and recipes" for neutrino masses at 1-loop, 2-loop and higher orders has been discussed in [74] and our approach has some overlap with this paper, too. Then, there is the interesting work of [75], which writes down all lepton number violating operators from d = 5 to d = 11. Decomposing these operators, one can find a list of tree-level, 1-loop, 2-loop etc. diagrams, which allow to specify neutrino mass models [75][76][77][78]. Our study is complementary to the analysis done in these papers in that it provides further insight for the specific two-loop case, exhaustively listing all possible diagrams. However, different from [75][76][77][78], we put quite some emphasis on our classification schemes, which allow us to distinguish "genuine" models, i.e. those for which the absence of 1-loop masses is guaranteed from the "non-genuine" (or class-II) models.
The rest of this paper is organized as follows. In section 2.1 we discuss the "strategy" followed in this paper and introduce our notation. Section 2.2 is devoted to the classification of relevant topologies. "Genuine" and "non-genuine" diagrams are discussed next and SM electroweak-sector quantum number assignments are given. In sec. 3 we exemplify the use of our results by constructing two specific examples of UV completions. In sec. 4 we summarize and present our conclusions. The bulk of the technical details of our calculation is collected in appendices A, B and C, where we list renormalizable topologies leading to non-genuine finite and infinite diagrams, non-renormalizable topologies, as well as non-genuine diagrams, tables with the different quantum number assignments and relevant formulas for the evaluation of two-loop integrals.
2 Two-loop 1PI topologies, diagrams, genuine models and quantum numbers 2.1 Generalities, strategy and notation A systematic classification of the two-loop order realizations of O 5 using the diagrammatic method does so far not exist. The underlying reason is probably related with the fact that tackling the problem via the diagram-based method turns out to be challenging, due to the large number of two-loop diagrams and unless precise guidelines are followed such study is not possible. Thus, in this section we discuss some generic guidelines that will allow us to deal with the two-loop classification of O 5 . At the two-loop order, the dimension-five effective operator consist of a set of topologies: . . , T 2 r }. We have identified 29 distinct topologies, see below, out of which only a subset turns out to be relevant. Once all topologies have been identified, the next step is then that of specifying the fermion and scalar internal lines (F and S) as well as the external lines (L and H) of each topology, i.e. "promoting" topologies to diagrams. Here, renormalizability fixes possible vertices to only dimension-four three and four point vertices (3-PVs and 4-PVs). Due to the freedom when placing the two external L and H lines, however, in general, a given two-loop order topology T 2 i can involve quite a few number of Feynman diagrams. At this point it is possible to discard 11 topologies, since these will always lead to non-renormalizable diagrams, see appendix.
From the remaining 18 topologoes only a subset leads to genuine two-loop diagrams. In order to identify non-genuine diagrams, one can assign arbitrary quantum numbers q i (q i , related with a new symmetry or in some cases with the gauge symmetry itself, e.g. hypercharge) to the external and internal fields, and then enforce conservation of these charges vertex by vertex. These conservation rules define a set of conditions (denoted by C 2 i ) that whenever satisfied guarantee the presence of the corresponding diagram. These conditions should be confronted with those arising from the 1-loop diagrams shown in fig. 1 (denoted by C 1 i ). Thus, if C 2 i ⊂ C 1 i , the corresponding 2-loop diagram will be necessarily accompanied by a 1-loop diagram, hence being non-genuine. Diagrams for which C 2 i ⊂ C 1 i is not satisfied are potentially genuine, but their particle content must satisfy further constraints which guarantee the absence of both tree and 1-loop level diagrams (see sec. 2.4 for more details). Once these constraints are assured, the full list of truly genuine diagrams is fixed Genuine diagrams define a set of renormalizable vertices, which will lead to a 2-loop UV completion (Lagrangian) once the gauge quantum numbers of the beyond SM fermion and scalar fields are specified. For that purpose the lepton and Higgs gauge quantum numbers can be used to constrain the possible quantum numbers of the internal fermion and scalar fields. This procedure, however, provides an unambiguous determination only in the case of trilinear couplings involving two SM fields. Let us discuss this in more detail. Yukawa (or pure scalar trilinear) couplings can involve two, one or none SM fields, schematically:F LH;F LS,F 1 F 2 H; F 1 F 2 S. In the first case, clearly F has to be-unambiguously-a vanishing hypercharge SU (2) singlet or triplet (type-I or type-II seesaws) while in the other cases SU (2) × U (1) Y invariance requires: where n X corresponds to the SU (2) representation under which the X field transforms. From (2), it can be seen that-in principle-an infinite number of SU (2) representations as well as hypercharge assignments are consistent with the constraints implied by the lepton and Higgs quantum numbers. 4-PVs allow even for more freedom. As the trilinear couplings, these vertices can involve two, one or none SM fields (Higgses), schematically: So, in this case gauge invariance implies: which shows that there is no coupling which allows for an unambiguous determination of the SM gauge quantum numbers of the new fields. Since trilinear couplings involving two SM fields are only possible in tree level realizations, the discussion above implies that once going beyond the tree level, a given genuine diagram leads to an infinite number of UV completions. From the practical point of view nevertheless one can impose an upper limit on the dimensions of the representations used. In our tables we list all combinations with singlets, doublets and triplets of SU (2) L . Tables for larger representations can be easily derived. We also mention that we do not give explicitly color quantum numbers Figure 2: 1PI box-based (upper row) and triangle-based (lower row) two-loop topologies, which can lead to genuine 2-loop models. All other topologies can lead only to class-II models, infinite renormlizations of 1-loop diagrams or contain non-renormalizable vertices. Those topologies are given in the appendix.
in our tables since, as pointed out in [37], the inclusion of color is straightforward, see also the discussion in section 2.5.
Finally, once the UV completions are identified the only step which remains to be done is the determination of the light neutrino mass matrix, which requires calculating 2-loop integrals. Although the list of genuine diagrams is large this does not means that the number of 2-loop integrals to be evaluated is large. Different diagrams, not necessarily arising from the same topology, can involve the same 2-loop integral, essentially because after electroweak symmetry breaking the couplings to Higgs legs are just couplings to a background field: if coupled to fermions (scalars) they imply chirality flips (scalar mixing). This observation allows to reduce the number of integrals to be evaluated to just combinations of a few basic integrals, which we list in the appendix.

Two-loop 1PI topologies
Following the strategy described in sec. 2.1, our starting point consist in determining the complete set of two-loop one-particle irreducible (1PI) inequivalent topologies. At the two-loop order this set is expected to be large, so in order to generate an exhaustive list we proceed with FeynArts [39]. To simplify the output we suppress from the calculation topologies involving external legs self-energies, tadpoles and one-particle-reducible 2-loop topologies. The complete set of topologies is displayed in fig. 2 and figs. 13 and 14 in the appendix, respectively. In total we have identified 29 topologies, but only topologies listed in fig. 2 lead to genuine diagrams. Topologies shown in fig. 13 will lead to class-II models, while fig.14 shows for completeness the non-renormalizable topologies.

Constructing diagrams
No matter the topology, any diagram can be constructed from the following schematic Lagrangian: In order to illustrate the method we have used for constructing diagrams we focus on the first two box-based topologies in fig 2 (T 2 B 1 and T 2 B 2 ), and base our discussion on the diagrams sketched in fig. 3.
In the (6,0) case, external vertices always involve Y L , Y H or µ H couplings. So, in order to find an exhaustive list of possible Feynman diagrams one can start by fixing any of these couplings at V 1 (see fig 3) and then inserting sequentially in clockwise direction all possible vertices combinations. Table 1 illustrates the procedure for topology T 2 B 1 , where we have fixed at V 1 the Y L coupling. It can be seen that out of the 15 diagrams, 3 are possible only for four fermion external legs and so have nothing to do with O 2−loop 5 . In addition two diagrams appear twice (CLBZ-2 and PTBM-1), so at the end the two-loop box-based topology T 2 B 1 involves 10 diagrams. For topology T 2 B 2 , determining the complete list requires considering at V 1 not only Y L but also Y H and µ H , due to its non-symmetric character. By doing so, the resulting list involves repeated diagrams (redundant diagrams) whose identification turns out to be complex. For that aim it is therefore useful to introduce the following sextuplet (n L , n 3 , n H , n Y , n S , n 4 ) , where the different entries label the number of Y L , µ H , Y H , Y , µ s and λ HH vertices defining a given diagram, and are such that depending on the topology obey certain constraints. For (6,0)-based diagrams these constraints read: thus implying that the sextuplet structure of any diagram arising from (6,0) topologies will necessarily belong to one of the following nine sextuplets: (2, 2, 0, 2, 0, 0) , (2, 2, 0, 0, 2, 0) , (2, 2, 0, 1, 1, 0) , (2, 0, 2, 2, 0, 0) , (2, 0, 2, 0, 2, 0) , (2, 0, 2, 1, 1, 0) , (2, 1, 1, 2, 0, 0) , (2, 1, 1, 0, 2, 0) , (2, 1, 1, 1, 1, 0) . For T 2 B 2 , the procedure outlined above yields 21 diagrams which can be grouped in five sets: one (2,2,0,0,2,0), six (2,2,0,1,1,0), five (2,0,2,2,0,0), five (2,1,1,2,0,0) and four (2,1,1,1,1,0). Possible redundant diagrams should belong to a specific set, however since not all diagrams belonging to a given set are redundant, the identification of superfluous diagrams requires labelling the fermion and scalar lines of all diagrams within each set and comparing the different couplings. If the couplings of a couple of diagrams match, those diagrams count as one. So, proceeding in that way we have found that the number of diagrams arising from the T 2 B 2 is nine.
Following this procedure for the remaining 16 topologies we have found the full set of diagrams for 1PI two-loop topologies.

Genuine diagrams
Having identified all possible diagrams, we are now in a position to build the full list of genuine diagrams. The procedure to be followed involves two steps. First, we assume that the lepton and Higgs SU (2) doublets as well as the heavy fields flowing in the loops carry arbitrary charges q i , and impose q i charge conservation vertex by vertex, as outlined in sec. 2.1 (and exemplified below). By doing so, we identify the non-genuine diagrams in our list. We are then left with diagrams which potentially lead to genuine models, see fig. 4. Their genuineness can then be guaranteed provided their particle content obeys the additional constraints discussed near the end of this section. Let us first illustrate the q i charge procedure we have employed to identify non-genuine diagrams. The example we discuss is based on the 1-and 2-loop diagrams displayed in fig. 5. For diagram T1-i, the equations for q i conservation can be written as For diagram T1-3 one has: The solution of these system of equations then leads to the following charge constraints: C T1-3 : where C T1-i is the solution for the diagram T 1 − i in fig. (5), whereas C T1-3 the solution for the one in eq. (10). The constraints in (11) and (12) are to be used to know whether the 2-loop box-based diagram ISC-i-3 in fig. 5 is non-genuine or not. For that aim the q i charge conservation equations for this diagram has to be written. Conservation of charges implies: and their solution is given by Comparing this solution with C T1-i R in (11), one can see that q 5 = 0 forbids the 1-loop boxbased diagram T1-i (right-hand side in fig. 5). However, when comparing with C T1-3 (trading q 2 → q 3 and q 3 → q 4 ) in (12) it is clear that constraints C ISC-i-3 allow the 1-loop triangle-based diagram T1-3, independent of the choice of charges. One can then conclude that ISC-i-3 is not a genuine diagram.
Following this procedure, we have identified all non-genuine diagrams. These emerge from the topologies in fig. 13 in appendix A. Moreover, we have found that the non-genuine but finite diagrams all belong to one of the following five different types, namely: (a) non-genuine CLBZ (NG-CLBZ), (b) non-genuine PTBM (NG-PTBM), (c) rainbow, (d) ISC-i, (e) ISC-ii. figs. 15-18 in appendix A show this complete list of non-genuine but finite diagrams.
Let us now turn to the remaining (potentially) genuine diagrams that can not be eliminated after this procedure has been applied to the full list of diagrams. These are given in fig. (4). All of these fall, as already stressed above, into only two classes, which we call CLBZ and PTBM. So far we have worked from the full set of (topologies and) diagrams, excluding one after the other the non-interesting cases. However, for those remaining 16 diagrams, there is one more subtlety to be discussed: One can write down Lagrangians, which produce, say, only one neutrino mass at tree-level (or 1-loop) level, while the other neutrino mass 5 (or masses) are generated radiatively. In this case, restrictions on the particle content of the model are determined by the requirements at that lower order. For example, a model with one righthanded neutrino will produce one non-zero neutrino mass at tree-level, while the other neutrino masses are then automatically generated by the genuine 2-loop diagram PTBM-1 (with SM W + gauge bosons).
The following additional (but rather trivial) conditions, which finally guarantee that Lagrangians producing the diagrams in fig. (4) are genuine 2-loop Lagrangians in our sense should therefore be understood as constraints per neutrino generation for which one wants to generate genuine 2-loop masses. Genuiness in this sense requires: i) Absence of hypercharge zero fermion electroweak singlets or triplets, or hypercharge 2 scalar SU (2) triplets is required, otherwise the neutrino mass will be determined by tree level type-(I,II,III) seesaw diagrams.
ii) Absence of hypercharge zero scalar SU (2) singlets or triplets. The presence of these fields allow constructing 1-loop diagrams by making a simple cut in the 2-loop diagram. The position where this cut appears depends (in a rather obvious way) on the 2-loop diagram under consideration.
iii) Absence of internal Higgses, i.e. all internal scalars must be beyond SM scalar fields. iv) And, lastly, in order to guarantee absence of 1-loop contributions from T1-3, if not already excluded by the previous three conditions, one needs to check the SU (2) quantum numbers of internal scalars. The quartic vertex in T1-3 can be generated [37] by attaching a pair of Higgses to S 1 S 2 with S 1 = S D and S 2 = S D or S 1 = S S and S 2 = S T or S 1 = S T and S 2 = S T , where S, D and T indicate singlet, doublet and triplet under SU (2). If any of these combinations appear, the difference in hypercharge of these states must be different from 2Y H in order to forbid T1-3. Different from all previous conditions, this rule has (exactly) one exception, see table (2).

SM gauge quantum numbers
Due to the large number of diagrams involved, it is desirable to apply a strategy where the quantum number assignments are done mostly at the topology level rather than at the diagrammatic level. Since both the leptons and the Higgs are SU (2) doublets, for these quantum PTBM-3 model Table 2: Quantum number assignment for the new particles appearing in the PTBM-3 model, which is the one possible exception to rule (iv). Naming conventions for particles as in fig. (9). Strict application of rule (iv), would forbid this model. However, for any q different from zero this model has no lower order neutrino mass diagram. number this turns out to be trivially possible. However, for hypercharge (Y = 2(Q − T 3 )) the procedure is more subtle due to the different hypercharges the lepton and Higgs doublets have. This implies that different external lepton-Higgs attachments lead to different hypercharges for the internal fields. So, when discussing hypercharge assignments, in order to avoid a diagrammatic approach we group the different diagrams according to the different external lepton-Higgs structures, which once fixed lead to a unique set of hypercharges for the internal fields.
For all the relevant topologies we will label the internal fields as X i (see fig. 6), where X i can be either a scalar or a fermion depending on the specific diagram ( fig. 4). For the field X i , no matter whether it is a fermion or a scalar, we will use the notation r for the SU (2) quantum numbers (with r labelling the the SU (2) representation r = 1, 2, 3: singlet, doublet and triplet). Hypercharge of a given field X i will be denoted by Y i . In what follows we discuss quantum number assignments for the double-box diagrams CLBZ-i and PTBM-i (i=1,2,3) in fig. 4. Results for the remaining diagrams are summarized in appendix A. For all possible genuine diagrams we display the possible quantum number assignments in tables. Quantum numbers for diagrams of type (a) fig 6: We start with T 2 B 1 -based diagrams, as shown in fig 6-(a). SU (2) invariance of the different a a a a a a  fig 6-(a). Symbols S i refer to allowed external lepton-Higgs structures according to fig 7. Hypercharge of field X i is denoted by Y i (see the text for further details). Since the lepton and Higgs doublets are color singlets, color charges can be trivially included.
vertices imply the following constraints: This means that fixing the representation of X 1,2 fixes X 5 . With X 1 fixed X 7 is determined too, and this in turn allows settling X 6 . With X 2 specified, X 3 can be determined as well and this finally fixes X 4 . The SU (2) assignment "chain" is then given by: The setup of constraints in (15), (16) and (17) are summarized in tab. 3, where in addition to the SU (2) possible quantum number assignments (upper table) we have as well added a  table with the different set of possible hypercharge assignments (lower table).
Some words are in order regarding tab. 3. The upper table is divided in three subtables delimited by the double vertical lines. The subtable in the left hand side shows the possible SU (2) charges of the internal fields for X 1 fixed to be a singlet and for any X 2 (singlet, doublet and triplet). The following subtables give the SU (2) charges for X 1 transforming as a doublet (middle subtable) or as a triplet (right hand side subtable) for any X 2 . Note that there is no relation between the choices for X 1 and X 2 , e.g. while X 1 can transform as a triplet X 2 can do so as a singlet. For fields which admit two SU (2) charge assignments within a certain row (see e.g. X 6 and X 4 for X 1,2 ∼ 1 or X 6 and X 3 for X 1 ∼ 1 and X 2 ∼ 2 in table 3), a horizontal internal line indicates that no crossed assignments are possible. For example, when X 1,2 ∼ 1, X 6 can transform as either a singlet or a triplet. If fixed to be a singlet (triplet), X 4 is fixed univocally to be a singlet (triplet) too. If instead the horizontal internal line is absent, crossing is possible. This is indeed the case for X 6 and X 3 when X 1 ∼ 1 and X 2 ∼ 2. Fixing X 6 to be a singlet (triplet) allows X 3 to be either a singlet or a triplet.
Finally, the lower table shows the different hypercharge assignments derived by taking hypercharge flow according to fig 6, and the possible lepton-Higgs external structures S 1 and S 2 , schematically represented in fig 7. Note that since the number of internal fields exceed the number of hypercharge conservation constraints (one per each vertex), hypercharge is not univocally fixed. The arbitrariness is encoded in the parameters α and β.
We do not give explicitly color quantum numbers in our tables. As mentioned above, the inclusion of color is straightforward, since Higgs and lepton doublets are color singlets. This implies that (pairs of) internal particles coupled to either L or H can come only in combinations of 1 ⊗ 1, 3 ⊗3, 6 ⊗6 etc. Moreover, once the colour quantum numbers for internal particles coupled to L or H are chosen, the color quantum numbers of the remaining inner particles are fixed by consistency conditions, derived from SU (3) rules such as3 ⊗3 = 3 a ⊕6 s and 3 ⊗3 = 1 ⊕ 8.

Assigning quantum numbers: some examples
We now exemplify the use of these results by constructing a couple of models. For that purpose we take diagrams CLBZ-1 and PTBM-1 (see fig 4): Starting with CLBZ-1, and comparing with the symbolic diagram in fig 6 it can be seen that: X 1,2,5 → S 1,2,5 and X 3,4,6,7 → F 3,4,6,7 . Whether the resulting model involves three (four) different scalar (fermion) fields should be determined by their transformation properties, for which tab. 6 should be used.
Sticking to the case X 1,2 ∼ 1, one is left with X 5 ∼ 1 and X 7 ∼ 2. For X 6 there are two possible choices, taking X 6 ∼ 1 one then has X 3 ∼ 2 and univocally X 4 ∼ 1. Diagram CLBZ-1 follows a S 1 lepton-Higgs structure (see fig 7), so for hypercharge assignments one has to focus on the S 1 row in tab. 6. Fixing α = −1 and β = 1, one gets Y 1 = −2, So, the resulting UV completion consist of: one hypercharge +2 scalar singlet and its complex conjugate (S 2 = S * 1 ), one hypercharge +4 scalar singlet (S 5 ), one hypercharge −1 fermion doublet and its conjugate (F 3 =F 7 ), and one hypercharge −2 fermion singlet and its conjugate (F 4 =F 6 ). Thus, the fermions can be identified with SM lepton doublets and singlets, and so the UV completion constructed in this way is nothing else but the CLBZ model [16,26,27]. Other quantum number choices, as dictated by tab. 3, will of course produce variants of the CLBZ model.
• A PTBM-1-based model: In this case comparing diagram PTBM-1 with that in fig. 6-(a) allows the identification: X 1,4 → S 1,4 and X 2,3,5,6,7 → F 2,3,5,6,7 . For the SU (2) charges we fix them as in the previous example. For hypercharge one has to bear in mind the lepton-Higgs structure, which for this diagram follows S 2 (see fig 7). Thus, fixing α = β = 1/3 one gets the following UV completion: a hypercharge +1/3 fermion doublet and its copy (F 3 = F 7 = F ), one hypercharge −2/3 fermion singlet and its copy (F 2 = F 6 = F ), one vanishing hypercharge fermion singlet (F 5 = f ) and one hypercharge −2/3 scalar singlet and its copy (S 1 = S 4 ). Assigning non-trivial color charges to these fields: f ∼ 8 c , F ∼ 3 c , F ∼ 3 c and S 1 = S 4 ∼3, one can then identify F with quark SU (2) doublets while F with quark SU (2) singlets. The resulting model in that case then matches the model of Angel et. al [55]. Using tab. 3, further variants can be constructed.  (1) and (2)) show the two cases corresponding to "genuine" or "true 2-loop" models, while diagrams in the lower row show the special cases of the rainbow diagram and the ISC-i and ISC-ii diagrams (diagrams (3), (4) and (5) respectively). We will treat the two classes separately and start with the genuine models.

Genuine 2-loop models
In genuine 2-loop model one encounters two types of integrals: Here, {k 2 , q 2 , (q + k) 2 } implies that the numerator could be any of k 2 , q 2 or (q + k) 2 , depending on the helicity structure of the underlying Lagrangian, see discussion below. We choose the convention of labelling the fermion masses as a, b and the scalar masses as α, β. X is the inner  (1) and (2) in the upper row correspond to "genuine" 2-loop models, whereas diagrams (3), (4) and (5) in the lower row to 2-loop models where a coupling entering in a 1-loop diagram is generated at the 1-loop level (non-genuine 2-loop models). particle that can be either a scalar (CLBZ-type) or fermion (PTBM-type). Note that integral in (18) is finite per se, while a finite result for integrals (19) requires summation over internal mass eigenstates. Integrals in (18) and (19) can be evaluated by rewriting them in terms of a "master integral" (see eq. (52) in appendix C). In order to illustrate the way in which this is done, we write down a specific PTBM-3-based model which arises from the diagram shown in fig. 9. For all other possible genuine 2-loop models, the procedure follows very closely the one outlined for this particlar example.
The diagram in fig. 9 is generated from the following Lagrangian (20) and scalar potential terms Table 4: Quantum number assignment for the new particles appearing in the diagram shown in fig. 9. For simplicity, all states are assumed to be color singlets.

PTBM-3 model
where the parenthesis indicate SU (2) index contractions. The particle content of the resulting model and its SM transformation properties are displayed in tab. 4. In addition, the fermions can have vectorlike mass terms, namely: Coupling µ 34 in (21) induces mixing between the Q = 3/2 scalars, while µ 12 mixing between the Q = 1/2 states. The mass matrices for these states then reads Assuming the mixing parameters to be real, these matrices are diagonalized by 2 × 2 rotation matrices: with the rotation angles given by: Rotating the interactions in (20) and (21) to the scalar mass eigenstate basis, one can then calculate the full neutrino mass matrix. The chiral structures appearing in the diagram are determined by the different chiral projectors (P L or P R ) entering in each of the Yukawa vertices involved. Since chirality of external vertices (those where the SM neutrinos enter) is fixed by the neutrino chirality, the number of possibilities is determined by the different chiral structures of internal Yukawa vertices. For PTBM models there are three chiral structures: internal vertices with P L − P L , P R − P L or P R − P R stuctures. The (internal) combination P L − P L leads to integrals of type eq. (18), while the other two possibilities project out integrals of type eq. (19). The full final result reads: with the different dimensionful functions F (A) ac,αβ,b , determined by With the aid of eqs. (53)- (56) in appendix C, one can then express these functions in terms of the "master" functionĝ(s, t) (see eq. (60)).  (26)). Note, that m (4) ν is numerically equal to m ν . In the plots there are some points for m F b , for which the different contributions can actually exactly cancel each other. Note also, that for (m F b → ∞) m ν goes to zero, as expected. Obviously, as these plots demonstrate, neutrino masses of the correct order of magnitude can be achieved for a wide range of input parameters.
We close this section with a brief discussion on neutrino flavour fits. Any model, aiming at explaining neutrino oscillation data, must of course not only reproduce the overall neutrino mass scale, but also have sufficient freedom to fit the two neutrino mass squared differences and the three neutrino angles. Our numerical examples have been done with only one non-zero neutrino mass, fits to all data can nevertheless be easily done. The actual form of the fit, however, depends on the number of copies of new fermions and scalars present in the model under consideration. For exactly one copy of new states, both fermions and scalar, eq. (26) has     (29)) and the common global factor in (26). The plots are for Yukawa couplings equal to 1 and thus show that neutrino masses of the correct order of magnitude can be obtained easily in this model.

Non-genuine but finite 2-loop models
As discussed at length in the previous sections, diagrams in the lower row in fig. 8 will not lead to genuine 2-loop models 6 . However, models generating such kind of diagrams might still be interesting constructions in the following sense: Consider, for example, a model with some new fermions in which, invoking a non-Abelian flavor symmetry, the direct coupling of the new fermions with the standard model Higgs is forbidden. The flavour symmetry is then broken at some large, unspecified scale and upon integrating out some heavy fields, an effective fermion-fermion-Higgs vertex is generated at 1-loop order. Such a construction would allow to understand, at least in principle, why that particular coupling is small compared to all others, simply due to the 1/(16π 2 ) suppression from the loop. An example of this approach is the Figure 11: Example for a "finite" rainbow diagram. The diagram corresponds to RB-1 in fig. 15 in appendix A. Table 5: Quantum number assignment for new particles appearing in the diagram shown in fig.  11. All states are assumed to be color singlets for simplicity. d = 7 rainbow model of [72], but the very same idea could, of course, be applied to any of our non-genuine d = 5 diagrams.

RB-1 model
In all such cases one can carry out the calculation either by solving the full 2-loop integral or by first calculating that particular vertex at 1-loop order and then doing a 1-loop calculation for the neutrino mass using this effective vertex in the second step. We will call the former the "full" or "2-loop" calculation, while we call the second approach "2-step" calculation in the following. The two calculations should, of course, lead to the same numerical result (only) in the limit where there is a certain hierarchy of masses for the particles in the loop. In this subsection, we will discuss one particular example of such a model, based on the rainbow diagram RB-1, see fig. 11, in some detail. The treatment of all other "finite" but non-genuine models is very similar. Here, we are mainly interested in demonstrating the numerical agreement between full and 2-step calculation and therefore will not work out the details of a suitable flavor symmetry model. We again refer to [72] for an example for the (d=7) rainbow diagram based on the discrete symmetry T 7 and to [63] for another example based on a variant of the ISC-i diagram using the symmetry Z 2 × Z 2 .
The diagram in fig. 11 can be generated from the following interaction Yukawa Lagrangian: This fixes the SM quantum numbers as shown in table 5. The scalars appearing in the inner and outer loops, denoted by S I and S O respectively, have the same SM quantum numbers and thus could be the same particles. For generality, however, and since in the ultra-violet completion they could transform differently under the flavour group, we will treat them as independent states.
The scalar potential of the model contains the following terms: We add then the following three fermion mass terms: Only F b can have a Majorana mass, as indicated by the charge conjugation "C" in eq. (33), but F a and F c can have vector-like fermion mass terms 7 . The mass matrices for the inner and outer scalars can be written as with k = I, O. This matrix can be diagonalized, as in the example in the previous section, by a simple rotation matrix with the angle given as: The expression for the neutrino mass is given as with the two-loop integral defined as in (19): Rescaling the momenta we rewrite this integral in terms of dimensionless mass squared ratios. Using the formulas in appendix C one finds: As before, Now consider the 2-step calculation. The inner loop can be evaluated as .
The solution to eq. (40) gives the well-known function ∆ m gives an entry to the mass matrix of the fermions F a and F c : Diagonalizing this mass matrix gives two eigenvalues m F 1 and m F 2 , which can be used in the calculation of the outer loop, which has the same form than the inner loop just calculated. This results in: where V ij is the matrix which diagonalizes eq. (42). Fig. 12 shows a comparison of the neutrino mass calculated with eq. (36) (full 2-loop result) and eq. (43) (2-step result) for different combinations of internal masses. We show the ratio of the two calculations, thus all coupling constants cancel and need not to be specified. The calculation is for one neutrino mass only and not meant to be a complete fit to all neutrino data. The plot on the left shows the result for negligibly small outer fermion masses, varying the (common) mass of the inner-loop particles, keeping the masses of the outer scalars constant.
The plot on the right shows the result for fixed values of the scalar masses, but varying the ratio of inner to outer fermion mass. In both cases, it is clear that if there is a hierarchy in the masses of the particles in the inner loop with respect to the masses in the outer loop, then the two calculations agree very well. Comparison of the plot on the right to the plot on the left demonstrates that especially the value of the ratio of the fermion masses is important: Fermion F b should be heavier than the outer fermions, otherwise the 2-step calculation starts to fail.
We close of this discussion with one more comment. Eq. (31) specifies that the vertex connecting F a and F b has a projector P R . However, we have given a vector-like mass term to these fermions and vector-like fermions can couple, in prinicple, with both chiralities. A model, in which the other projector P L also appears, however, will produce terms proportional to (m Fa + m Fc ) q.k and, different from the genuine 2-loop models discussed in the last subsection, the corresponding integral will be logarithmically divergent in q. In our definition, such a construction is not considered an interesting model and we will not discuss this further.

Conclusions
Using a diagrammatic approach we have systematically studied the d = 5 Weinberg operator at the 2-loop order. Out of the large number of possible diagrams the majority are just corrections to lower order diagrams. We have shown that the relevant 2-loop models can be classified as follows: (A) Class-I models, which only involve genuine diagrams, i.e. diagrams for which the absence of lower order diagrams is assured. Interestingly, we have found that class-I models implicate only variants of the CLBZ (Cheng-Li-Babu-Zee) [16,26,27] and PTBM (Petcov-Toshev-Babu-Ma) [40,41] models. (B) Class-II models, which involve non-genuine but finite 2-loop diagrams. Diagrams belonging to this class correspond to 1-loop diagrams that contain a 1-loop generated vertex, and are variations of just five different diagrams which we have dubbed: NG-CLBZ (non-genuine CLBZ), NG-PTBM (non-genuine PTBM), Rainbow, ISC-i and ISC-ii (internal scalar correction type i and ii).
We provided the full list of class-I diagrams in fig. 4. This list combined with our results for the internal fields SM quantum number assignments (summarized in tabs. 3, 6-9), allows the construction of novel 2-loop neutrino mass models, something that we have exemplified in sec. 2.5.1 and, in more details in section 3. We have given as well the full list of non-genuine but finite 2-loop diagrams in figs. 15 -18. This list enables the construction of novel 2-loop models where the smallness of certain coupling can be, in principle, understood as due to its 1-loop radiative origin. Also, the "tools" needed for numerical calculations have been collected in appendix C.
In summary, we have identified the possible 2-loop neutrino mass models arising from the d = 5 Weinberg operator. Our findings can be understood as a guide for the construction of 2-loop neutrino mass models, which arguably might serve for several purposes, e.g: systematic study of neutrino mass model signals at the LHC (testing the origin of neutrino masses at the LHC, as has been pointed out at the 1-loop level e.g. in [80][81][82], and at the 2-loop level in [83]) or systematic construction of common frameworks for neutrino masses and dark matter (in the same vein it has been done for the 1-loop case [84]

A Non-renormalizable topologies and finite non-genuine diagrams
In this appendix, we present in fig. 13 Figure 13: 1PI two-loop topologies leading to non-genuine finite or infinite diagrams. Topologies T 2 B 4,6 belong to set (6,0), topologies T B 5 and T T 9,10 to (2,2), while the remaining to the (4,1) set. Further details can be found in sec. 2.2.  1).

B Quantum numbers
In this appendix, we give tables from which the SM quantum numbers of genuine diagrams CLBZ-i and PTBM-i (i=4,5,6,. . . ) can be determined. The tables obey the same conventions as tab. 3, i.e. symbols S i refer to allowed external lepton-Higgs structures according to fig. 7, and hypercharge of field X i is denoted by Y i . Their utilization requires as well using fig. 6, as  1).
ISC-ii-3 ISC-ii-4 Figure 18: Non-genuine and finite two-loop diagrams which correspond to the one-loop generation of one of the couplings entering in the one-loop diagram T1-3 (see fig 1).
Here, n is the number of dimensions. In the case of infinite integrals one has to carefully evaluate all terms for n = 4 + before taking the limit → 0. Since we are interested only in models with finite integrals, we will not write out terms containing poles in 1/ .
Here, (2m|m i |m j ) is a short-hand for (m, m|m i |m j ). The "p 2 decomposition" is another relation which proves to be useful for calculating integrals with momentum dependent numerators, namely .
Using only eq. (45) results in expressions [55] which are more compact than those given in [85,86], which make repeated use of both, eq. (45) and eq. (46). Also, for numerical evaluation, it is useful to define the final expressions in terms of dimensionless quantities. By convention we scale all masses appearing in the integrals with respect to the "innermost" scalar/fermion mass. This implies rescaling the momenta, and for I ab,αβ,X factoring out this overall mass squared. We thus write I ab,αβ,X = 1 (2π) 8 withÎ ab,αβ = d 4 k d 4 q where {k 2 , q 2 , (q +k) 2 } stands for k 2 , q 2 or (k +q) 2 , r a = (m Fa /m X ) 2 and t α = (m Sα /m X ) 2 . The "strategy" then for calculating these integrals consists of reducing them to "master integral" form: I(s, t) = µ d n k d n q 1 (k 2 − s)(q 2 − t)[(k + q) 2 − 1] .
Below we repeat the solution forÎ ij,αβ in the convention of [85]. This solution rewrites eq. (52) into a less divergent expression using eq. (46). Introducing dimensionless arguments as before, one findŝ Here, Eq. (63) is more complicated than eq. (53), but leads to exactly the same numerical result. We have found it therefore a useful cross-check for our calculation.