Minimal Radiative Neutrino Masses

We conduct a systematic search for neutrino mass models which only radiatively produce the dimension-5 Weinberg operator. We thereby do not allow for additional symmetries beyond the Standard Model gauge symmetry and we restrict ourselves to minimal models. We also include stable fractionally charged and coloured particles in our search. Additionally, we proof that there is a unique model with three new fermionic representations where no new scalars are required to generate neutrino masses at loop level. This model further has a potential dark matter candidate and introduces a general mechanism for loop-suppression of the neutrino mass via a fermionic ladder


I. INTRODUCTION
Since the discovery of neutrino oscillations and, therefore, non-vanishing neutrino masses, there have been numerous proposals explaining these small but non-zero masses. All of them add at least one new representation to the Standard Model of particle physics (SM). The Standard Model is a gauge theory based on the symmetry group  Here, hypercharge is normalised such that the electric charge Q em is given by Q em = Y /2 + I 3 with I 3 the third component of the weak isospin.
The most minimalistic extensions of the Standard Model which include neutrino masses are the wellstudied seesaw mechanisms. They each add one new representation to the Standard Model.
Type I seesaw adds a fermionic singlet with respect to the Standard Model gauge symmetry, ν R ∼ (1, 1, 0) [1][2][3][4]. This introduces the following interactions L ⊃ y ℓlLH ν R + M ν R ν c R ν R , (I.2) withH = iσ 2 H * . The type II seesaw enlarges the Standard Model scalar content by a scalar SU (2) Ltriplet, ∆ ∼ (1, 3, 2) [5][6][7][8], yielding the following interactions In type III seesaw, a fermionic SU (2) L -triplet, ρ R ∼ (1, 3, 0), is added to the Standard Model [9], which leads to the interactions The essence of the seesaw mechanisms thereby is the suppression of the neutrino mass due to the very heavy new representations. However, besides being difficult to experimentally test, these new heavy representation can contribute to the quantum corrections of the Higgs mass. A tuning would be necessary to explain the measured Higgs mass, see Ref. [10][11][12][13][14][15][16][17][18][19][20][21] for recent discussions. Moreover, none of the type I, II, and III seesaw fields has a clear theoretical motivation. Therefore, it is appealing to think about the possibility that the neutrino masses are pure quantum effects, or, in other words, radiatively generated by loop effects. An additional loop suppression of the neutrino mass could allow for smaller masses of the new particles due to the additional loop suppression. This could avoid the neutrino hierarchy problem, while, at the same time, allowing for better testability.
The first higher order operator generating neutrino masses is the dimension-5 Weinberg operator Whereas, the discussed seesaw mechanisms are tree-level realisations of the dimension-5 Weinberg operator, we will focus on models with additional suppression. This includes generation of the dimension-5 Weinberg operator at loop-level or suppression by a higher dimension of the operators inducing neutrino masses at tree level. Since the latter also induce the Weinberg operator at some loop level, we will refer to these models as radiative neutrino mass models. Known radiative neutrino mass models are for example the Zee-model [22], the Zee-Babu model [23,24], and the colored seesaw models [25]. Moreover, models like the scotogenic model [26] do not only introduce radiative neutrino masses but also provide a candidate for particle dark matter.
An effort has been made to find and understand radiative neutrino mass models systematically. This task was thereby approached from different angles. For example in Ref. [27,28] an effective field theory approach to organize neutrino mass mechanisms is used. Whereas, Ref. [29][30][31][32][33] categorise neutrino mass mechanisms by the topology of the diagram which generates non-vanishing masses. Additional systematic studies of neutrino mass generation can be found in Ref. [34][35][36].
In this paper, we conduct a systematic scan to answer the question: What are the neutrino mass mech- We will always explicitly state which combination of fermion fields we add.
In addition to pure radiative models, we will also find models introducing higher-dimensional versions of the Weinberg operator of the form However, these operators always also induce the dimension-5 Weinberg operator, by connecting the H † H-pairs via loops. A rough estimate shows that the loop-suppressed dimension-5 operator will give a larger contribution than the higher dimensional tree-level operator if where v is the Higgs vacuum expectation value and the scale Λ is associated with the mass of the new particles. If there are different couplings contributing to the different operators, their ratio will also 1 Note that if multiple fermionic representations are added to the Standard Model, this no longer has to be the case.
Apart from the seesaw mechanisms, there is exactly one model that produces neutrino masses at one-loop level with only one new beyond the Standard Model representation and a second copy of the Standard Model Higgs. This is the known Zee-model [22]. Going beyond this minimal model, we systematically search for radiative neutrino mass mechanisms with two new fields. A similar approach was discussed in Ref. [38]. However, in this paper we will focus on the number of representations and will allow for higher SU (2) L -representations and new coloured fields.
Additionally, we will proof that radiative neutrino masses cannot be generated with only two new fermionic representations and we will further present the unique mechanism to generate neutrino masses at loop-level with just three new fermionic representations without any new scalar fields. We will also comment on the possibility of dark matter in this scenario and on the possibility to generate large loopsuppression for the neutrino mass.
The paper is organized as follows. In section II, we will describe the systematic search for all possible radiative neutrino mass models with two new beyond the Standard Model representations. The discussion is thereby split into new fields transforming trivially with respect to SU (3) C and fields carrying colour.
We continue in section III with a formal proof why there is no model generating neutrino masses at the quantum level with just two new fermions and will then introduce the unique model with three new fermionic representations. Finally, we conclude in section IV.

II. SEARCH FOR RADIATIVE NEUTRINO MASS MODELS
In this section, we describe our systematic search for radiative neutrino mass models. We employ the following set of assumptions: • The Weinberg operator (I.5) appears only at loop-level.
• There are no new symmetries beyond the Standard Model symmetry group (I.1).
• There is only a minimal number of new G SM multiplets involved in the mass generation for a single neutrino generation.
The first point leads to the exclusion of the type I, II, and III seesaw fields, as discussed in the introduction. The last two constraints are implemented to allow for a bottom-up search for minimal viable models.
To identify viable models we pursue the following line of arguments. If we do not allow for new symmetries, the accidental global symmetry of lepton number is broken by the neutrino mass mechanism.
The new fields must therefore induce lepton number violation (LNV) by two units 2 . However, LNV is only introduced if at least one of the new fields has a coupling to Standard Model leptons. Hence, we should be able to identify all possible candidates by systematically scanning all possible couplings to the Standard Model leptons.
After identifying a set of candidates, we check all possible interactions of the candidates to Standard Model fields and to each other. If it is not possible to choose unique and non-trivial lepton numbers for the new fields and we find LNV by two units, we have identified a possible model. This is a generalisation of the LNV argument in [38].
When conducting a systematic model scan where minimality is an important criteria, it is essential to define minimality properly. We consider models as minimal, if the number of new representations required to generate radiative neutrino masses for a single generation is minimal. We thereby differentiate three types of new fields: • new scalar representations, • new Dirac fermions, • new Majorana fermions.
When counting the number of new representations, we do not consider multiple copies of the same representation and we count vector-like Dirac fermion pairs as a single new representation. The minimal number of new representations which we have to add to the Standard Model to generate neutrino masses at the quantum level is two, except for the already mentioned Zee-model.
In the following, we differentiate type A models with two new scalars, type B models with one new scalar and one new vector-like Dirac fermion, and type C models with one new scalar and one new Majorana fermion. We summarize our results in tables where we use the following terminology: • Within any type, the models are ordered by the dimension of the highest SU X , where a is the SU (2) Lrepresentation, and b is the hypercharge, and X can either be L or R for a left-or right-handed fermion. If a vector-like Dirac fermion is required both chiralities are noted separately for clarity.
If the new fermions carry colour, they are labelled ψ (c,a,b) X , respectively.
A. Minimal radiative neutrino mass models without colour We start our systematic search by considering new representations without colour. Hence, we have to consider the leptonic fermion bilinears of the Standard Model and the possible couplings of a Standard Model lepton to a scalar field. The complete lists of minimal models of type A, B, and C without new coloured representations found in scanning these interactions are presented in the next two subsections.

Minimal models with two new scalars without colour
We present all models with up to two new uncoloured scalar representations added to the Standard Model to generate neutrino masses at the quantum level in Table II. The first model (A0) on the list is the Zee-model [22], which was already mentioned. In addition to the new scalar singlet it requires a second Higgs field. Antisymmetry of the mass matrix in the flavour indices can be avoided if both Higgs doublets develop non-zero vacuum expectation values. This can also lead to flavour-violating effects [38].
The first model with two new representations (A1) is the Zee-Babu model [23,24]. It leads to a symmetric neutrino mass matrix with respect to the family indices with one copy of each of the new fields.
The model (A2) was discussed in Ref. [38] as a simplification of the Zee-model. Compared to the Zee-model it is more restrictive, since there are less new couplings of the Higgs boson and the new SU (2) L -doublet in the scalar potential. As a result, the mass matrix is traceless in flavour space and does not produce the correct mixing. This minimal model is therefore ruled out.
Similar to (A0), the model (A3) only works in the presence of two Higgs doublets. Otherwise the term H † σ a φ (2,3) H † σ aH vanishes identically and no neutrino mass is generated. This model was also discussed in Ref. [38].
The models (A1)-(A3) are all two-loop realisations of the dimension-5 Weinberg operator. In contrast, the model (A4) is a one-loop realisation of the dimension-7 operator (ℓ L HHℓ L )(H † H)/Λ 3 L . However, the mass matrix is antisymmetric in flavour space and no proper mixing is generated [38]. This minimal model is therefore also ruled out.
The final model with two new colour-neutral scalars produces the dimension-9 operator [38]. This models requires two copies of the quadruplet scalar since the couplingφ † (4,1) φ (4,1)φ(1,2) vanishes otherwise, and therefore also the neutrino mass. Moreover, a variation of the model where only one of the copies of φ (4,1) acquires a vacuum expectation value can be excluded, since it produces only a traceless neutrino mass matrix.

Minimal models with one new scalar and one new fermion without colour
In Table III, we list all minimal models generating neutrino masses at loop-level with a new scalar and a new fermionic field transforming trivially with respect to SU (3) C . In general, for models with one new fermion and one new scalar, one needs either two copies of the fermion or of the scalar to produce at least two independent non-zero neutrino masses.
The first model with one new scalar and a new vector-like fermion, (B1), was found in Ref. [31] in a study on dimension-7 effective operators. It is a tree-level realisation of the dimension-7 operator ℓ L ℓ L ℓ L e R H/Λ 3 L , closed off at two-loop to the dimension-5 Weinberg operator by connecting e R and ℓ L . The new fermion enables the coupling µφ † (1,2)H † H via a fermion loop. As a result, the neutrino mass matrix will be traceless, as in the case of (A2), and no proper mixing will be generated. This model is therefore ruled out.
The models (B2) and (B3) have not been discussed previously to the best of our knowledge. Both of them require two Higgs doublets. With only one Higgs the coupling H † φ (2,3) H †H vanishes identically, leading to a vanishing neutrino mass. While (B2) realises neutrino masses at one-loop level, (B3) generates them at two-loop level.
The model (B4) is the triplet-analogue to (B2). Hence, it also requires two copies of the Standard Model Higgs and induces radiative neutrino masses at one-loop. This model has not been discussed previously.
The model (B5) was introduced and studied in Ref. [43] in a version with two vector-like copies of the fermion field. The second copy of the fermion is necessary to produce the correct phenomenology. Depending on the masses of the fields, the dominant contribution can be either the dimension-5 operator generated at one-loop level or the tree-level realisation of the dimension-7 operator The model (C1) was introduced in Ref. [44] and studied in Ref. [49] as the starting point of a chain of models introducing higher dimensional operators at tree-level as their leading term. It realises the The dimension-5 Weinberg operator appears at one-loop.
(C2) is a class of models that all generate neutrino mass via the same mechanism. It contains the new scalar field φ (n,1) ∼ (1, n, 1) and the Majorana fermion where n > 4, and n even. The relevant interactions are then given by The resulting mass diagram is shown in Fig. 1. The case with φ (3,1) ∼ (1, 3, 1) and ψ (2,0) R ∼ (1, 2, 0) was discussed in Ref. [25]. However, it seems that in this case no neutrino mass is generated. The singlet from two fields in an even SU (2) representation is always a completely antisymmetric combination of the fields. So at least two flavours of the ψ (2,0) R are needed. Taking into account the anticommutiation of fermion fields, the Majorana mass matrix of these new particles is antisymmetric in flavour space. Since the rest of the interaction is totally symmetric, the resulting mass matrix for the neutrinos will be antisymmetric, leading to a vanishing Majorana mass.
It was discussed as a model generating neutrino masses radiatively and containing a viable dark matter candidate due to an accidental Z 2 symmetry.

B. Minimal radiative neutrino mass models with colour
In this section, we consider minimal models where one or both new representations transform nontrivially with respect to SU (3) C . We define the representations of SU (3) C such that 3 ⊗ 3 ∼6 ⊕3.
Note that now the conjugation of the field does not only change the sign of the hypercharges, but also the In order to break lepton number with new coloured scalar fields, we have to look at fermion bilinears of Standard Model quarks and leptons and all possible quark couplings which can inverse the fermion number flow. For new coloured fermions, we take all couplings of Standard Model quarks and the Higgs field into account. We find several coloured models of type A, called type cA, as well as coloured B-type (cB) and C-type (cC) models which are given in the next two subsections. Note that, apart from lepton number violation, some of the models also introduce baryon number violation. To avoid fast proton decays in these cases, the new particles have to be sufficiently heavy.
Hence, such models can be interesting to discuss in the context of supersymmetry. However, a realistic ultraviolet complete model is beyond the scope of this paper. Moreover, (cC3) is the triplet version of (cC1).
The model (cC6) is the generalisation of the model class (C2) to coloured models. In contrast to (C2), this class of models does not contain fundamental dark matter candidates since the new fields carry colour charge. For recent work on coloured dark matter see Ref. [72,73].

III. MINIMAL RADIATIVE NEUTRINO MASS MODEL WITH ONLY NEW FERMIONS
In this section, we will first proof that there does not exist a Standard Model extension with two new fermionic representations which generate the dimension-5 Weinberg operator only at the quantum level without employing new symmetries. We will then present the unique model with three new fermions which generates the Weinberg operator at the two-loop level.
Let us assume there exist two fermions ψ a and ψ b that fulfil the requirements for inducing LNV. In other words, there exist three interactions involving those fermions that do not allow for a well-defined non-trivial lepton number assignment for ψ a and ψ b . We can then differentiate three cases. This can always be solved by setting L(ψ a ) = L(ψ b ) = 0. There is no LNV in this case.
3. One of the fields, take ψ a w.l.o.g., couples directly to the Standard Model leptons, the other one, ψ b , does not. Since we excluded ν R ∼ (1, 1, 0) and ρ R ∼ (1, 3, 0) from these considerations as they lead to tree-level neutrino masses, it follows from the list (III.1) that ψ a has non-vanishing hypercharge. As a result either ψ a or ψ c a can couple to the Standard Model leptons, but not both. This fixes the lepton number of ψ a to L(ψ a ) = ±1.
Next, we list all possible representations of ψ b , that allow for a coupling to ψ a , but not to the Standard Model. Among all representations coupling to the candidates for ψ a from (III.1) via a Yukawa coupling, there are four such representations, Now we have to find two interaction terms for ψ b that lead to a well-defined hypercharge compatible with the list of candidates, but an ill-defined lepton-number. The first possibility would be to consider the Majorana mass term ψ c b ψ b . It requires Y (ψ b ) = 0. However, there is no candidate with Y (ψ b ) = 0, so we exclude this case. The remaining possibility is having both, the coupling of ψ b and ψ c b to ψ a . From them, we find the following conditions for the hypercharge of ψ a and ψ b Therefore, there is no model with two new fermions. This concludes the proof.
However, by going to three new fermionic representations, we find a unique model. Some aspects of this model were discussed in Ref. [37]. A proof of its uniqueness is given in the Appendix A. The field content is given by two vector-like Dirac fermions We thus find the relevant interactions Note that the dimension-5 operator is generated at two-loop level as can be inferred from Fig. 2.

IV. CONCLUSIONS
In this paper, we systematically studied radiative neutrino mass models, where the dimension-5 Weinberg operator is only generated at loop level. We add just two new beyond the Standard Model representations without employing new symmetries. The complete lists of known and new models can be found in the Tables II, III, IV, V, and VI. Thereby, two new representations is the minimal number of new fields which have to be added to the Standard Model to only generate neutrino masses at the quantum levelwith the only exception being the Zee-model [22] which just requires one new scalar representation and an additional Standard Model Higgs copy. This is a bottom-up approach to systematically study possible neutrino mass mechanisms. When considering ultraviolet completions, models which seem to be minimal from the low energy perspective can actually be non-minimal and vice versa. The study of minimal radiative neutrino masses from an ultraviolet perspective is therefore a complementary approach that might reveal a different list of minimal models.
The models C1 and C2 also contain stable neutral particles. These particles are potential dark matter candidates. The study of the interplay of neutrino and dark matter phenomenology in these models was partly done in Ref. [50] but a more complete study would be desirable.
In section III, we gave a formal proof that neutrino masses cannot be generated solely via quantum effects with just two new fermionic representations. We then introduced the minimal and unique model with three new fermionic representations ψ Since L a is determined by eq. (A.5), we find for ψ b It follows that ψ b needs to couple to both ψ a and ψ c to produce lepton number violation. We conclude that there is exaclty one model with three new fermionic multiplets and LNV namely ψ a ∼ (3, −2) ψ b ∼ (4, −1) ψ c ∼ (5, 0) , (A.14) where the signs of all hypercharges may also be inverted.