Loop neutrino masses from d=7 operator

TL;DR

This paper systematically analyzes all possible one-loop topologies for generating small neutrino masses via dimension-7 operators, identifying genuine models where the dimension-7 contribution is dominant and classifying them by particle content.

Contribution

It provides a comprehensive classification of all genuine d=7 one-loop neutrino mass models, including their particle content and minimal examples.

Findings

Only 8 of 48 topologies lead to genuine d=7 neutrino masses.

One diagram involves no representation larger than triplet, 22 involve quadruplets.

Three minimal example models are briefly discussed.

Abstract

We discuss the generation of small neutrino masses from d=7 1-loop diagrams. We first systematically analyze all possible d=7 1-loop topologies. There is a total of 48 topologies, but only 8 of these can lead to "genuine" d=7 neutrino masses. Here, we define genuine models to be models in which neither d=5 nor d=7 tree-level masses nor a d=5 1-loop mass appear, such that the d=7 1-loop is the leading order contribution to the neutrino masses. All genuine models can then be organized w.r.t. their particle content. We find there is only one diagram with no representation larger than triplet, while there are 22 diagrams with quadruplets. We briefly discuss three minimal example models of this kind.

Equations

$$m_{\nu}\propto\frac{v^{2}}{\Lambda}\times\Big{(}\frac{1}{16\pi^{2}}\Big{)}^{n}\times\epsilon\times\Big{(}\frac{v}{\Lambda}\Big{)}^{d-5}.$$

$${\mathcal{O}}^{d=7}\propto LLHHH{H}^{†}$$

$$\frac{1}{{\Lambda }^3}LLHHH{H}^{†}\to \frac{1}{16{\pi }^2}\frac{1}{\Lambda }LLHH$$

Full text

Neutrino mass, lepton number violation

††preprint: IFIC/17-21

I Introduction

If neutrinos are Majorana particles, one can roughly estimate their mass as:

[TABLE]

Here, $\Lambda$ is the energy scale of new physics, where lepton number violation (LNV) occurs and $v$ is the standard model vacuum expectation value. The different terms in eq. (1) can be understood easily. The first term corresponds to the famous Weinberg operator, ${\cal O}^{W}\equiv{\cal O}^{d=5}\propto LLHH$ Weinberg:1979sa . This operator can be generated either at tree-level or at loop level. The second term in eq. (1) takes into account this fact, with $n=0,1,2,\cdots$ being the number of loops. Then, there are models of neutrino mass, in which the Weinberg operator is suppressed by some small factor $\epsilon$, which could be either due to some small coupling in the corresponding model or due to some nearly conserved symmetry. R-parity violating supersymmetry is an example of the former Dreiner:1997uz ; Hirsch:2000ef , models such as the inverse Mohapatra:1986bd or the linear Akhmedov:1995ip ; Akhmedov:1995vm seesaw are examples of the latter. And, finally, neutrino masses could be due to higher dimensional operators. This is expressed by the last term in eq. (1), with $d=5,7,\cdots$ the dimension of the operator.

In this paper we will focus on $d=7$ neutrino mass models at 1-loop order. Our aim is to give a systematic analysis of such models, constructing first all possible 1-loop topologies and then identify those topologies, which allow to construct genuine models. Here, we define genuine models such models, where the 1-loop $d=7$ contribution to the neutrino mass gives the leading order contribution. This assumption implies, of course, that both the $d=5$ and the $d=7$ tree-level, as well as the $d=5$ 1-loop, contributions should be absent.

$d=5$ neutrino masses have been studied extensively in the literature. In Ma:1998dn it was shown that there are only three tree-level realizations of ${\cal O}^{W}$ at tree-level. A systematic analysis of the Weinberg operator at 1-loop level was presented in Bonnet:2012kz , at 2-loop level in Sierra:2014rxa , see also Farzan:2012ev for a general discussion of tree versus loop neutrino masses.

Disregarding derivative operators, the authors of Babu:2001ex have written down all $\Delta L=2$ operators up to $d=11$. Only one of these operators is important for us here:

[TABLE]

All other $d=7$ operators in the list of Babu:2001ex will lead to $d=5$ 1-loop neutrino mass models, while the $d=9$ and $d=11$ operators can lead only to $d=7$ neutrino masses, if the underlying model is 2-loop or higher. Note that $\Delta L=2$ operators with derivatives have been studied in delAguila:2012nu . Two operators with derivatives at $d=7$ exist, but neither can lead to a 1-loop $d=7$ model, see also the discussion in Helo:2016vsi .

Bonnet et al. Bonnet:2009ej analyzed the $d=7$ operator of eq. (2) at tree-level in detail. As noted in this work, the $d=7$ operator of eq. (2) will always also generate a higher order $d=5$ neutrino mass:

[TABLE]

One can straightforwardly estimate that this loop contribution will become more important than the tree-level one if (\Lambda/v)\raise 1.29167pt\hbox{;>\kern-7.5pt\raise-4.73611pt\hbox{\sim;}}4\pi. This means \Lambda\raise 1.29167pt\hbox{;<\kern-7.5pt\raise-4.73611pt\hbox{\sim;}}2 TeV is required for the $d=7$ contribution to dominate. Since this is unavoidable in the standard model (SM), the authors of Bonnet:2009ej considered a two Higgs doublet extension of the SM in their discussion of the $d=7$ tree-level neutrino mass. 111$HH^{\dagger}$ is a singlet under any discrete symmetry. With more than one Higgs it is possible to introduce an additional discrete symmetry, under which the two Higgses transform differently. We instead will stick to only the SM Higgs and take eq. (3) as a motivation that any $d=7$ model of neutrino mass must have new particles below 2 TeV, otherwise it will not give the leading contribution to the neutrino mass matrix.

As mentioned above, both $d=5$ and $d=7$ tree-level contributions should be forbidden, otherwise the $d=7$ 1-loop contribution might be just some minor correction to the neutrino mass matrix. Absence of these lower order contributions could be attributed to either: (i) the existence of some symmetry; or (ii) absence of fields which generate neutrino masses at lower order. An example of the former at $d=5$ is the scotogenic model Ma:2006km . In this model, a right-handed neutrino (plus an extra doublet scalar) exists, but due to a $Z_{2}$ symmetry, there is no tree-level $d=5$ neutrino mass.222The well-known bonus of the $Z_{2}$ symmetry is that it allows to “stabilize” the lightest $Z_{2}$ odd particle, thus relating the stability of the dark matter to the generation of neutrino masses. Instead neutrinos have mass at ($d=5$) 1-loop order. The classic example for (ii) is the Zee model Zee:1980ai . In the Zee model, none of the particles necessary for a tree-level seesaw exist. Instead an additional charged singlet scalar (plus an additional doublet scalar) generate neutrino masses radiatively.

We will not discuss in detail additional discrete (or gauge) symmetries here, since models of $d=7$ neutrino masses have been discussed in this context already in a number of references, see for example Bonnet:2009ej ; Chen:2006hn ; Kanemura:2010bq ; Krauss:2011ur ; Krauss:2013gy . Also we will not consider $d=7$ operators with additional singlets, see for example Gogoladze:2008wz ; Gu:2009hu . Instead, we will follow the second route mentioned above: After constructing all 1-loop $d=7$ topologies, we will classify the underlying models according to their particle content.

The rest of this paper is organized as follows. In the next section, we will first give a short summary of neutrino masses at tree-level at $d=5$ and $d=7$, as well as $d=5$ at 1-loop order. This provides the basis for the discussion in section III. Section III then provides the core of our present work. It discusses all possible topologies and classifies them into different groups. The section then also introduces the three most minimal example models that one can construct at $d=7$ 1-loop order. We then close with a short summary. More complete lists of topologies and diagrams are relegated to the appendix.

II Preliminaries

Since we are interested in identifying models, in which a 1-loop $d=7$ diagram gives the leading order contribution to the neutrino mass matrix, we first need to discuss briefly neutrino masses at lower order. When discussing possible models will use $S$ (and $\phi$) for scalars and $\psi$ ($\chi$) for fermions. For a more compact notation we will also use a notation which gives the $SU(2)_{L}$ representation and hypercharge in the form ${\bf R}_{Y}$ with a superscript $S$ or $F$, where necessary, i.e. for example ${\bf 5}_{1}^{S}$ is a scalar 5-plet with $Y=1$. Note that, for some of the fields, particular symbols are common in the literature, such as $\nu_{R}$, $\Delta$ and $\Sigma$ for the type-I, type-II and type-III seesaw.

II.1 Tree-level $d=5$

As noted in Ma:1998dn , there are only three possibilities to de-construct the Weinberg operator at tree-level. A seesaw type-I is generated via the introduction of a right-handed neutrino, $\nu_{R}\equiv{\bf 1}^{F}_{0}$ Minkowski:1977sc ; Yanagida:1979as ; Mohapatra:1979ia . It generates Dirac mass terms for the active neutrinos via ${\bar{\nu}_{R}}HL$. For the singlet a Majorana mass term is allowed, $M_{M}{\bar{\nu}_{R}^{c}}\nu_{R}$, which implies $\Delta L=2$. The type-II seesaw requires a scalar triplet $\Delta\equiv{\bf 3}^{S}_{1}$ Schechter:1980gr ; Mohapatra:1980yp ; Cheng:1980qt . Here, the simultaneous presence of the couplings $L\Delta L$ and $H\Delta^{\dagger}H$ violates lepton number $\Delta L=2$. And, finally, a type-III seesaw Foot:1988aq can be generated with a fermionic triplet $\Sigma\equiv{\bf 3}^{F}_{0}$. Here, the vector-like mass term $M_{\Sigma}{\bar{\Sigma}^{c}}\Sigma$ is the source of the lepton number violation. Note, that type-I and type-III seesaw are generated by the same topology, reducing the total number of $d=5$ topologies at tree-level to two.

II.2 Tree-level $d=7$

At $d=7$ level one can construct five different topologies, one of which however can not lead to any renormalizable model. The remaining four topologies have been discussed in Bonnet:2009ej . Only one of these topologies can generate a genuine $d=7$ neutrino mass model in our sense, see figure (1). All other models will require additional symmetries to avoid $d=5$ tree-level neutrino masses.

This genuine $d=7$ tree-level model, BNT model in the following, was first discussed in Babu:2009aq . For a discussion of lepton flavour violation in the BNT model see Liao:2010rx . The model requires two new particles beyond the SM field content: (i) A (vector-like) triplet fermion, $\Psi={\bf 3}^{F}_{1}$. 333Its vector partner ${\bf 3}^{F}_{-1}$ is needed for a (effectively lepton number violating) mass term. And (ii) a scalar quadruplet $S\equiv{\bf 4}^{S}_{3/2}$. Note that the quadruplet is the smallest representation which allows a contraction $S^{0}(H^{0})^{3}$.

II.3 1-loop $d=5$

In addition to the tree-level, there are many 1-loop $d=5$ models. The classical example is the Zee model Zee:1980ai . A systematic analysis of all 1-loop $d=5$ topologies has been given in Bonnet:2012kz . In total, 6 topologies where found, but only two of them (denoted as T-1 and T-3) can give genuine models in our sense. All other topologies lead either to non-renormalizable models or diagrams with infinite loop integrals (thus representing loop corrections to tree-level quantities) 444Topology T-4 has two divergent and two finite diagrams. In Fraser:2015mhb the diagram T-4-2-ii was used to generate a coupling $L\Delta L$ at one-loop. This diagram is classified as divergent in Bonnet:2012kz . However, in Fraser:2015mhb two diagrams of this type appear, with the infinity cancelled between diagrams. This can not be justified in terms of symmetry, instead it is due to the fact that lepton number is broken softly in the model of Fraser:2015mhb . or can be understood as finite 1-loop realizations of some particular vertex of one of the tree-level $d=5$ seesaws. Topologies T-1 and T-3 lead to a total of four diagrams shown in fig. (2). The Zee model Zee:1980ai falls within category T-1-ii, the scotogenic model of Ma Ma:2006km is an example of T-3.

While at tree-level the size of the representations as well as the hypercharge of the new fields is fixed, at loop level there always exists a “tower” of possible models. This is easily understood. Consider, for example, the diagram T-3. The outside leptons couple to a scalar and a fermion. Since $L$ is a $SU(2)$ doublet, the representation of the scalar and the fermion can be: ${\bf 1}\otimes{\bf 2}$, ${\bf 2}\otimes{\bf 3}$, ${\bf 3}\otimes{\bf 4}$, etc. Similarly at the four scalar vertex the smallest possibility is ${\bf 2}\otimes{\bf 2}$, but larger representations can be inserted, with the only constraint that the product of the two scalars can build a triplet. In the same way, the hypercharge of the internal particles is fixed only up to an additive constant $x$, that runs in the loop.555Of course, not all choices of $x$ will lead to phenomenologically acceptable models.

The minimal possibility to build a model for T-3 is then that the fermion is a $\nu_{R}={\bf 1}^{F}_{0}$, while the scalar is a doublet ${\bf 2}^{S}_{1/2}$, i.e. the well-known scotogenic model. This is minimal in the sense that it uses the smallest representations and the smallest value of the hypercharge possible, i.e. $x=0$. However, ${\bf 2}^{S}_{1/2}$ can not be the SM Higgs, it must be an additional inert doublet.

Note that in our sense, strictly speaking, the scotogenic model is not a “genuine” model, since it requires an additional symmetry (in the minimal case a simple $Z_{2}$) to avoid the tree-level $d=5$ type-I seesaw. This does not mean, however, that topology T-3 is non-genuine: This topology has a finite loop integral and thus, no tree-level counter term is needed in models that generate this topology. Rather, in order to avoid the type-I seesaw contribution without the use of an additional symmetry, requires us to introduce fields with larger hypercharges. The smallest possible choice is: ${\bf 1}^{F}_{1}$ (together with its vector partner) and two scalars ${\bf 2}^{S}_{1/2}$ and ${\bf 2}^{S}_{3/2}$.

III Classification

In this section, we will discuss the classification of the different $d=7$ 1-loop diagrams. We first construct all possible 1-loop topologies with six external legs and then discard in different steps those topologies that can not lead to genuine models. For the remaining topologies we order all possible diagrams into different classes, depending on the minimum size of the largest required $SU(2)_{L}$ representation appearing in the corresponding diagram.

We note in passing that we will not discuss colour in detail, because colour assignments can be trivially added: All particles outside loops must be necessarily colour singlets, while pairs of particles in loops can always be assigned colour in combinations ${\bf X}+{\bf\bar{X}}$, for ${\bf X}={\bf 1}$, ${\bf 3}$, $\cdots$, which then couple to “outside” colour singlet particles.

III.1 Topologies

We construct all possible 1-loop topologies with six external legs, discarding from the start all self-energy corrections. This construction can be done in different ways and we used two different procedures to assure that all possible topologies were found. In total there are 48 possible topologies. The complete list is shown in the appendix.

We will briefly describe the methods we used to find the topologies. The first procedure consists in taking the five $d=7$ tree-level topologies and to generate loops in all possible combinations, connecting either lines to lines or lines to vertices or vertices to vertices, using only 3-point and 4-point vertices. From this list one has to discard in the end all duplicates.

The second procedure starts from the most simple realization of a 1-loop topology adding six external lines to the loop using only 3-point vertices, as shown in fig. (3). From this topology, all other toplogies can be found by systematically removing lines attached to the loop and adding them to an outside particle generating a new 3-point vertex, as shown in the figure for the examples of T2, T3 etc. Once all possible topologies with only 3-point vertices are found, all remaining topologies can be generated from the earlier ones by shrinking one line connecting two 3-point vertices to one new 4-point vertex, see the example fig. (4). Again, this procedure produces duplicates, which have to be identified and discarded.

This second procedure provides a systematic construction of all possible topologies in a more intuitive way than the first one described above. It allows to classify topologies according to the number of lines entering the loop, creating subgroups with the same number of 4-leg vertices. This procedure is the one we use for ordering the complete list of topologies, given in the appendix.

We then proceed to order topologies into different groups. We can discard immediately the six topologies shown in fig. 17, because none of them can lead to a renormalizable model. The next step is to generate all possible diagrams and check if any field which generates neutrino masses at lower order is required.

Two examples of topologies, which always necessarily will be accompanied by a tree-level $d=5$ seesaw, are shown in fig. 5. These diagrams can be easily understood. Every topology with at least two 3-leg vertices on two external lines will always generate a vertex $LH{\bar{\nu}_{R}}$ (or $LH{\bar{\Sigma}}$ or $H\Delta^{\dagger}H$) and, thus a seesaw at tree-level, as in the example $T4$ fig. 5 on the left. Topologies which contain one 3-leg vertex with two external lines isolated by a 4-leg vertex will always have a coupling of the type $L\Delta L$, as for example the topology T33 in fig. 5, to the right. The 27 topologies, for which all diagrams can be excluded due to this argument, are given in fig. 14. The topologies T7, T22, T23 and T24 in this figure are somewhat particular examples. For these a $\Delta$ always has to exist. One might think to bypass the $\Delta$ and introduce a quintuplet instead. However, the coupling of two doublets to a 5-plet is zero due to $SU(2)_{L}$.

Before moving on a brief comment might be in order. Integrals for the diagrams in topologies T4 and T33 are finite. One might therefore wonder, whether it is possible to forbid one of the “ingredients” of the tree-level $d=5$ seesaw, say one particular vertex, via a discrete symmetry, only to generate it at 1-loop order. This was discussed at length for $d=5$ 1-loop diagrams in Bonnet:2012kz . At the $d=7$ level, however, this will not be possible, since $H^{\dagger}H$ is a singlet under any discrete symmetry.

Next we turn to identifying topologies which generate diagrams reducible to 1-loop $d=5$ models. This is not as straightforward as the tree-level case. In particular, in this class of topologies many diagrams lead to $d=5$ tree-level models, while only the remaining diagrams can lead to $d=5$ 1-loop models. However, when the topology is highly symmetric, as for example in $T1$, one can always find a coupling between two internal fields and an external field which bypasses the $H^{\dagger}$, giving one of the diagrams of fig. 2. In addition, any diagram containing the structure given in fig. 6 can be reduced to the well-known diagram T-3 in fig. 2. We list all topologies excluded due to these arguments in fig. 15.

Then, there are diagrams which always contain the fields $\textbf{4}^{S}_{3/2}$ and $\textbf{3}^{F}_{1}$, responsible of generating neutrino mass at tree-level $d=7$ Babu:2009aq . Examples are diagrams of the topologies $T25$, $T29$ and $T35$ (fig. 16), which are excluded as genuine ones due to this reason.

Finally, in the remaining 8 topologies that are not completely excluded by one of the above arguments, many but not all the diagrams do not lead to genuine models. For instance, from the 10 different diagrams that one can generate from topology $T10$ (fig. 7 left), only one is not reducible to 1-loop $d=5$ (fig. 7 right).

In summary, from the initial 48 topologies only 8 have at least one genuine 1-loop $d=7$ diagram. The excluded topologies are listed in the appendix in figs 14 - 17. The next step is to classify the surviving topologies in terms of the minimal $SU(2)_{L}$ representation needed to realize a genuine model.

III.2 Diagrams: Minimal $SU(2)_{L}$ representations

In all $d=7$ diagrams, in order to avoid neutrino masses at lower order, a minimal size for the $SU(2)_{L}$ representations of the model is required. We order the possible models according to the largest representation present in a given model. The “smallest” or minimal model that one can construct is then a model in which no representation larger than $SU(2)_{L}$ triplets is needed. The next smallest possibility is models with quadruplets. Here, one can distinguish three different subgroups: (i) diagrams in which one quadruplet is needed inside the loop; (ii) diagrams in which one quadruplet appears outside the loop and internal particles need not be larger than triplets; and (iii) models in which at least two quadruplets are needed. We will discuss these three possibilities in reverse order and then proceed to briefly discuss the triplet diagram.

For external fields, finding the minimal representation is straightforward. A recurrent example in most of the diagrams of the appendix is that of the vertex $HH^{\dagger}$-scalar. Since $2\otimes 2=3+1$, the scalar could be a trivial singlet or the triplet $\phi\equiv{\bf 3}^{S}_{0}$. The former case is directly reducible to a $d=5$ diagram, the latter is the one we are interested in. The same principle applies to the diagrams given in fig. 20, case (iii), for which the largest necessary representation is a quadruplet. In order to avoid lower order contributions, one needs the quadruplet $\textbf{4}^{S}_{1/2}$ ($\textbf{4}^{F}_{-1/2}$) outside the loop. Moreover, one should be able to distinguish between these quadruplets and a Higgs or a lepton doublet. For this reason, the external quadruplet must couple to a singlet and another quadruplet running inside the loop. All the diagrams of this type are depicted in fig. 20 and they always contain two quadruplets, one outside the loop and another inside.

As we are dealing with the operator $LLHHHH^{\dagger}$, the maximum hypercharges of an external quadruplet is $3/2$, i.e. $S$. These diagrams corresponds to group (ii) defined above. All the diagrams given in fig. 19 contains this scalar entering the loop and they belong to the same topology $T16$ (fig. 13). Note that the hypercharge $3/2$ of $S$ prevents the possibility to reduce these models to $d=5$ 1-loop.

The rest of the diagrams do not contain a external quadruplet. In the minimal case, all the diagrams given in fig. 18 just need one quadruplet running in the loop. Diagrams generated from the topologies $T2$, $T12$ and $T13$ with a triplet entering the loop have all similar structures to those given in fig. 8. The minimum representations for the fields ($\chi$, $\eta$) or ($\eta_{1}$, $\eta_{2}$) in fig. 8, are then a singlet and a quadruplet, in oder to prevent a coupling of these fields with a lepton doublet $L$ or the Higgs $H$, respectively.

The remaining diagram T10-i of fig. 18 is a rather singular case of this group (i), with only one internal quadruplet. Given the isolated $H^{\dagger}$ and the upper asymmetric structure with three Higgses, the representation between the Higgs vertices needs to be (at least) a quadruplet, otherwise a $d=5$ 1-loop contribution is possible.

After giving all the diagrams that can be construct with quadruplets as the highest representation (figs 18-20), the last case depicted in fig. 9 shows the only genuine diagram that can be constructed with no representation bigger than triplet. Despite its similarity to the structure given in fig. 8 (left), the 4-leg vertex prevents lower order neutrino masses already with triplets. The corresponding diagram fig. 9 (right) with a 4-legs vertex followed by a triplet $\Psi$ cannot be bridged to construct a 1-loop $d=5$ contribution given the relation between the hypercharges of the fields running inside the loop.

To summarize, from the 8 genuine topologies, one can generate 23 diagrams. Among them, only one can be realized with no representation larger than triplets as the maximum $SU(2)_{L}$ representation, fig. 9. The other 22 diagrams generated from the 7 topologies given in fig. 13 can generate models with representations up to quadruplets. This whole set can be divided depending if the diagrams require one quadruplet running in the loop (fig. 18), outside the loop (fig. 19) or two quadruplets both inside and outside the loop (fig. 20). Of course, models with larger representations can be constructed and we will give one example in the next subsection.

III.3 Example models

The complete list of diagrams, from which genuine $d=7$ 1-loop models can be built is given in the appendix. Here, we will briefly discuss three example models, which are among the most simple models one can built from these diagrams. These models are: (i) The simplest $d=7$ model, which requires no representation larger than a triplet; (ii) one example model with an external quadruplet $S$; and (iii) an example model with an $SU(2)_{L}$ quintuplet. The latter serves to show, how models with larger representations can easily be constructed from our list of diagrams.

III.3.1 Triplet model

As discussed above, there is only one possible diagram that has a triplet as the largest $SU(2)_{L}$ representation, see fig. 9. The model requires the fermionic triplet $\Psi=\textbf{3}^{F}_{1}$, that also appears in the BNT model. A priori, for the particles inside the loop hypercharge is not fixed. However, not all choices of hypercharge will lead to genuine models, since lower order contributions might appear. If we use only doublets and triplets inside the loop, the smallest hypercharge assignments that lead to a genuine model are:

[TABLE]

[TABLE]

The model generates neutrino masses via the diagram depicted in fig. 10. $\eta_{1}$ has the smallest hypercharge of the particles in the loop. For colorless particles it is not possible to find a smaller hypercharge assignment that leads to a genuine model. For example, choosing $\eta_{1}={\bf 2}_{1/2}^{S}$ instead would result in a model, which also has the diagram T-3 at $d=5$ level.

One interesting aspect of this model is that the scalar triplet inside the loop has a component which carries 4 units of electric charge.

III.3.2 Quadruplet model

While for triplets as the maximal representation there is only one diagram, for quadruplets three distinct groups of model exist, as discussed above. We choose an example based on an external quadruplet $S={\bf 4}^{S}_{3/2}$. The example model we choose is based on diagram T16-ii.

As in the triplet case, hypercharge and $SU(2)_{L}$ representation are not uniquely fixed. The minimal model, again in the sense of using the smallest possible hypercharge assignment for colourless internal fields, has the following particle content:

[TABLE]

[TABLE]

This model generates neutrino masses via the diagram of fig. 11. In this example, lower order contributions can be avoided due to the hypercharge of $S$. A model with smaller hypercharges, for example $\eta_{1}$ chosen to be $\eta_{1}={\bf 1}_{1}^{S}$, would again not be genuine, since it would necessarily have a $d=5$ 1-loop contribution, i.e. the classical Zee model Zee:1980ai , see diagram T-1-ii in fig. 2. Note that, while the triplet model contains a scalar with 4 units of electric charge, in the quadruplet model it is an internal fermion that has such a large electric charge.

III.3.3 Quintuplet model

Finally, our last example is a model based on diagram T13-i in fig. 18. It contains the field $\phi={\bf 3}_{0}^{S}$ and a quintuplet in the loop. The diagram for the generation of the neutrino masses is shown in fig. 12. The minimal particle content containing a 5-plet is given by:

[TABLE]

[TABLE]

The maximum representation in this model is a quintuplet, $\eta_{2}$. Since it couples to the Higgs, $\phi$ and $\eta_{1}$, $\eta_{1}$ could be either a ${\bf 2}$, ${\bf 4}$, ${\bf 6}$ or a ${\bf 8}$. However, only for the case of $\eta_{1}$ being a doublet, a genuine model results. This is because, once the coupling $\eta_{1}H\eta_{2}^{\dagger}$ is allowed, one can construct again a $d=5$ 1-loop diagram with the particle content of the model.

It is worth noting that from three example models we have discussed, the quintuplet model is the only one, in which the representations in the loop contain a neutral component. For this model, one can thus follow the idea of the scotogenic model Ma:2006km : Add a discrete $Z_{2}$ symmetry to the model, under which the internal particles are odd, and the lightest neutral particle can be a cold dark matter candidate.

IV Summary

We have discussed neutrino masses at 1-loop $d=7$ order. We have identified all possible topologies that can lead to genuine models, i.e. models that are not accompanied by either a $d=5$ or $d=7$ tree-level mass term nor by a $d=5$ 1-loop neutrino mass. We have found that only 8 out of a total of 48 topologies can lead to genuine models.

We then ordered the remaining, possibly genuine, diagrams into different groups, depending on the minimal field content necessary to construct a model. There is only one possible diagram for which the largest necessary representation is a triplet. The remaining 7 topologies yield 22 diagrams, with the largest representation being at least a quadruplet. We then briefly discussed three example models, starting from the triplet model, with one additional example for a quadruplet and one for a quintuplet each.

To avoid lower order neutrino masses, the “genuine” models we discussed always have to introduce five new multiplets, usually with quite a large hypercharge for at least one of them. Thus, these $d=7$ models are necessarily more complicated constructions than the classical seesaw. From a theoretical point of view this might make these models less attractive. However, in particular due to the large electrical charges in these models, one can expect interesting signatures for them at colliders. We reiterate that the $d=7$ 1-loop contribution can only be dominant, if at least some of the new particles have masses below roughly 2 TeV.

Acknowledgements

We thank R. Fonseca for discussion. This work was supported by the Spanish MICINN grants FPA2014-58183-P and Multidark CSD2009-00064 (MINECO), FPU15/03158 (MECD) and PROMETEOII/2014/084 (Generalitat Valenciana). J.C.H. is supported by Chile grants Fondecyt No. 1161463, Conicyt ACT 1406 and Basal FB0821.

V Appendix

In this appendix we present the list of all $d=7$ 1-loop topologies, classified into genuine and non-genuine topologies, as discussed in the main text. We also give the complete list of diagrams that can lead to ”genuine” $d=7$ neutrino mass models with $SU(2)_{L}$ quadruplet representations.

V.1 Topologies

Fig. 13 shows the 8 topologies that can lead to genuine $d=7$ 1-loop models. We stress again that not all diagrams, derived from these topologies, are necessarily genuine, as discussed in the main text. Note that only T11 can give a model in which the largest representation can be as small as a $SU(2)_{L}$ triplet. All other 7 topologies require at least one quadruplet for genuine models.

In fig. 14 we list all topologies for which all diagrams are excluded, since they contain either a singlet fermion $\nu_{R}$ ($\textbf{1}_{0}^{F}$) or a triplet scalar $\Delta$ ($\textbf{3}_{-1}^{S}$) or a triplet fermion $\Sigma$ ($\textbf{3}_{0}^{F}$). All diagrams from these topologies thus will also generate a tree-level $d=5$ seesaw contribution to the neutrino mass matrix.

In fig. 15 we list the topologies for which many but not all diagrams are excluded by a $d=5$ tree-level seesaw. For these topologies all remaining diagrams are excluded because a 1-loop $d=5$ contribution to the neutrino mass necessarily exists.

In fig. 16 we list topologies, which lead to a $d=7$ tree level neutrino mass. For each of these topologies one can construct diagrams, which have a $d=5$ tree-level mass. All remaining diagrams, contain the scalar $S$ ($\textbf{4}^{S}_{3/2}$) along with the fermion $\Psi$ ($\textbf{3}^{F}_{1}$) and thus generate the $d=7$ tree-level BNT model Babu:2009aq . We note in passing that one can, in principle, use these diagrams to radiatively generate one of the vertices in the BNT model. This is very similar to the discussion for the radiative generation of a seesaw coupling given in Bonnet:2012kz at $d=5$ level.

In fig. 17 for completeness we show the topologies which are excluded, since they can never lead to a renormalizable model.

V.2 Genuine diagrams

In this section we list diagrams with quadruplets. All diagrams are given in figs 18, 19 and 20. We have divided these diagrams into three groups, depending on whether there is a quadruplet in the loop (fig. 18), the scalar $S$ on the outside of the loop (fig.19) or models with at least two different quadruplets (fig. 20).