Neutrino masses from SUSY breaking in radiative seesaw models

Radiatively generated neutrino masses ($m_\nu$) are proportional to supersymmetry (SUSY) breaking, as a result of the SUSY non-renormalisation theorem. In this work, we investigate the space of SUSY radiative seesaw models with regard to their dependence on SUSY breaking ($\require{cancel}\cancel{\text{SUSY}}$). In addition to contributions from sources of $\cancel{\text{SUSY}}$ that are involved in electroweak symmetry breaking ($\cancel{\text{SUSY}}_\text{EWSB}$ contributions), and which are manifest from $\langle F^\dagger_H \rangle = \mu \langle \bar H \rangle \neq 0$ and $\langle D \rangle = g \sum_H \langle H^\dagger \otimes_H H \rangle \neq 0$, radiatively generated $m_\nu$ can also receive contributions from $\cancel{\text{SUSY}}$ sources that are unrelated to EWSB ($\cancel{\text{SUSY}}_\text{EWS}$ contributions). We point out that recent literature overlooks pure-$\cancel{\text{SUSY}}_\text{EWSB}$ contributions ($\propto \mu / M$) that can arise at the same order of perturbation theory as the leading order contribution from $\cancel{\text{SUSY}}_\text{EWS}$. We show that there exist realistic radiative seesaw models in which the leading order contribution to $m_\nu$ is proportional to $\cancel{\text{SUSY}}_\text{EWS}$. To our knowledge no model with such a feature exists in the literature. We give a complete description of the simplest model-topologies and their leading dependence on $\cancel{\text{SUSY}}$. We show that in one-loop realisations $L L H H$ operators are suppressed by at least $\mu \, m_\text{soft} / M^3$ or $m_\text{soft}^2 / M^3$. We construct a model example based on a one-loop type-II seesaw. An interesting aspect of these models lies in the fact that the scale of soft-$\cancel{\text{SUSY}}$ effects generating the leading order $m_\nu$ can be quite small without conflicting with lower limits on the mass of new particles.


Introduction
The idea that neutrino masses (m ν ) may owe part of their smallness to small supersymmetry (SUSY) breaking effects was put forward in [2], where it relied on the observation that the Rparity conserving minimal SUSY Standard Model (MSSM) can generate non-vanishing neutrino masses if one takes into account hard SUSY breaking ( SUSY) terms involving slepton doublets and Higgses, as for instance where M X is the underlying scale mediating SUSY ( X ∼ F X θ 2 ) to the visible sector, and m soft ∼ F X /M X is the soft-SUSY scale. Although non-standard SUSY terms are typically neglected in phenomenological analysis due to their expected smallness, this is a feature that could conceivably be responsible for m ν /v 1. For example, if SUSY generates (LH u ) 2 , then LLH u H u arises at the one-loop level via a gaugino-slepton loop and is suppressed by m soft /M X .
A shortcoming of the idea presented in the previous paragraph is that it does not explain why the more fundamental model is able to generate the (s)lepton number violating superfield operator (superoperator)X(LĤ u ) 2 while being unable to generate SUSY conserving superoperators that, if generated, would expectedly provide dominant contributions to m ν . For example, givenX(LĤ u ) 2 ⊂ W eff one would arguably expect that a generic model would also generate 1 A possible strategy to fix this shortcoming is to consider thatX carriesL-number, so that X(LĤ u ) 2 is symmetric, and is thus allowed, while (LĤ u ) 2 is forbidden. This kind of strategy in which the SUSY sector is postulated to share a visible sector symmetry is followed in [3]. A different strategy is to postulate a visible sector symmetry that forbids chiral superoperators while allowing non-chiral. Since non-chiral superoperators contribute to the effective action as D-terms, their contribution to neutrino masses is necessarily proportional to SUSY. For example, the authors of [4] consider an extra U (1) symmetry that allows but forbids the non-suppressed LH u N ⊂ d 2 θLĤ uN . In the aforementioned works the connection between SUSY and m ν results from making SUSY the unique provider of: i) L-number violating couplings suitable for a radiative seesaw with just the superfield content of the MSSM [2] or for a canonical type-I seesaw [3]; or ii) couplings for Dirac neutrino masses [3,4].
An alternative arises when one considers models in which neutrino masses are radiatively generated. In such models, the connection between SUSY and m ν is a consequence of the SUSY non-renormalisation theorem, so that one may have all the required couplings at the SUSY level while the radiative structure of a SUSY QFT precludes the generation of neutrino masses. Then, one may have L-number breaking at the superpotential level, i.e. independently of SUSY, and neutrino masses will still be proportional to SUSY. This dependence on SUSY is not entirely obvious in component fields calculations. To illustrate what we mean, consider the trilinear R-parity violation contribution to neutrino masses [5], which is generated at the one-loop level. It takes the schematic form where M is the loop mass scale, µ c ξ is the effective µ-term and A stands for holomorphic soft-SUSY trilinears. From this expression, one may erroneously think that the ∝ µ/M 2 contribution is a SUSY conserving one, since µ is a superpotential mass term. However, a closer inspection reveals that the µ-term contribution comes from the vacuum expectation value (VEV) of an F -term, specifically F † H d = µ H u , and is thus intrinsically SUSY. This is not that surprising since in the SUSY limit one has H = 0 and therefore H u,d = v u,d are proportional to SUSY. This suggests that SUSY contributions can be classified with respect to (w.r.t.) their involvement in electroweak symmetry breaking (EWSB) as follows: SUSY EWSB contributions are those which involve SUSY VEVs of the form where Hs are fields whose VEVs break the electroweak symmetry (EWS); while SUSY EWS contributions correspond to those in which at least one SUSY VEV is unrelated to EWSB.
In this work, we will start by analysing the space of radiative seesaw models w.r.t. their dependence on SUSY. In order to do this in the most efficient manner, we will use supergraph techniques 1 and incorporate the SUSY effects by means of considering couplings to external 1 A detailed introduction to supergraph techniques can be found in chapter 6 of [6].  [7]. This will allow one to see the SUSY EWS contributions to neutrino masses as small SUSY effects upon a fundamentally SUSY topology. Although L-number violation can possibly arise from SUSY, i.e. from the VEV of an auxiliary rather than scalar field, in here we assume that they are broken separately so that the nonrenormalisation theorem is the only bridge between m ν and SUSY. We thus assume that the radiative seesaw models are realised in the superpotential at a L-number breaking scale M that is higher than the scale of soft-SUSY effects involving the seesaw mediators. We will furthermore restrict ourselves to models with the low energy Higgs sector of the MSSM.
As we will see in section 2, contributions to neutrino mass operators whose dependence on SUSY arises entirely by means of SUSY sources involved in EWSB are expected to be suppressed by some power of µ/M or be of dimension higher than 5 and involve gauge couplings. Thus, in a generic radiative seesaw model one expects m ν to have the schematic dependence where m soft /M 2 corresponds to conceivable SUSY EWS contributions. Exploiting the power of the SUSY non-renormalisation in the space of radiative seesaw models, we then investigate if models exist in which the pure-SUSY EWSB contribution either vanishes or is subleading w.r.t. the contribution from SUSY EWS . Since in such a case where m soft is the scale of soft-SUSY effects involving the mediators of the radiative seesaw, models in which the leading contribution comes from soft-SUSY EWS offer the possibility of explaining the smallness of m ν with M in the vicinity of the TeV scale provided m soft /M 1. It should be noted that the smallness of m soft is not constrained by lower-limits on the mass of new particles because the fields upon which these m soft -effects are felt possess superpotential masses of order M . We catalogue one-loop topologies for such models in section 3. An explicit model example is presented in section 4 and consists of a one-loop type-II seesaw whose leading pure-SUSY EWSB contribution is of dimension-7 -comprising contributions ∝ µ/M and ∝ g 2 -, whereas the leading contribution from SUSY EWS is of dimension-5 and has the dimensionful dependence µ m soft /M 3 or m 2 soft /M 3 , the latter corresponding to pure-SUSY EWS contributions.
2. Radiative seesaws in SUSY: understanding the different SUSY contributions In order to understand the SUSY structure of radiatively generated neutrino mass operators (OP ν ), it is first convenient to notice that for every OP ∈ OP ν there exists a superoperator OP ∈ OP ν such that OP ⊂ d 4 θ OP. Then, since any OP ∈ OP ν is of the form OP = LL ⊗ Higgses, one arrives at the conclusion that OP ∈ OP ν belongs to one of two classes: with and where n = 0, 1, ... stands for conceivable insertions of superfields that yield Higgses at θ = 0 (a limit hereafter denoted by |). Now, if we assume that only scalar and gauge vector superfields exist, we can writeÂ where n = 0, 1, ... stands for arbitrary insertions of superfields within the given set (denoted by curly braces), though constrained by internal symmetries.â andV (modĤ † ,Ĥ) (b † and Z † (modĤ † )) are real (anti-chiral) scalar superfields whose D (F ) component is a constant or a product of Higgses 2 . We show in figure 1 characteristic examples of class A and B superoperators. Figure 1. Characteristic examples of supergraph topologies for radiative seesaws: type-II without a chirality flip (class A), type-I and -III, type-II with a chirality flip and 1PI seesaw, respectively.
2.1. Pure-SUSY EWSB contributions Pure-SUSY EWSB contributions are generated by superoperators containing superfields whose auxiliary fields yield Higgses upon being integrated out. We have two possibilities: whereV is the gauge vector superfield of any symmetry under which Higgses are charged; and whereẐ † is any anti-chiral scalar superfield that has a bilinear with an Higgs or a trilinear with two Higgses. For class B superoperators (cf. Eq. (9)),b † is thusẐ † or the anti-chiral projection ofV (D 2V ), while, for class A,â isV or the real product ofb (b †b ). Now, under the phenomenologically reasonable assumption of a superpotential mass term for Z, the contribution of a trilinear with two Higgses adds up to an overall derivative term of the form (HH ). Moreover, up to SUSY effects ∝ (m 2 soft ) Z /|µ Z | 2 . Therefore, the contribution that arises from a trilinear with two Higgses is more appropriately classified as a SUSY EWS contribution. (For a detailed discussion see the original work [1].) Neutrino mass operators that come from a class A or B superoperator via a gauge vector superfield must have mass dimension higher than 5, sinceV is an hypercharge singlet. The least is a dimension-6 operator 2 Here and throughout the text, "mod X" means modulo insertions of X. Since realistic SUSY models have Higgs bilinears, be them dynamically generated or otherwise, it is conceivable that in general models there are pure-SUSY EWSB contributions to LLHH. While pure-SUSY EWSB contributions are indeed guaranteed to exist in any model (see section 2.3), there are models in which they appear only at an higher order of perturbation theory, while the leading order (LO) contribution is proportional to SUSY EWS .

SUSY EWS contributions
SUSY EWS contributions are most conveniently understood by means of employing spurionic superfields: objects that are superfields at the formal level but that have constant θ-dependent values. The underlying idea is to perform perturbation theory in superspace while considering possible couplings to spurions (thus parameterising SUSY) in order to obtain the effective superoperators and see whether the θ-integration projects the spurions in a way that d 4 θ OP ⊃ OP ∈ OP ν . We can thus identify three cases: modulo D 2X ,D 2X † and D 2D2Ŷ insertions, and whereX andŶ are F -and D-term SUSY spurions, respectively.
In this work we are strictly interested in the case in which the SUSY spurions are neutral under any symmetry of visible sector superfields. In such a framework, one expects that both {X,X † ,Ŷ } OP (cases a, b and c, respectively) and OP are generated up to some order in perturbation theory. For example, if a model generates then it would most certainly also generate We can now ask ourselves which instances of OP ∈ OP ν do not yield an OP ∈ OP ν in the absence of SUSY spurions 3 . (This is the operational definition of superoperators that only give OP ∈ OP ν from SUSY EWS .) The general answer is: 1. OP = D 2 (LLĤ n ) ⊗ a superoperator whose D-term is zero at p ext = 0 ; 2. OP =LL ⊗ a superoperator whose F † -term is zero at p ext = 0 .
where n, n , k = 0, 1, ... stand for any number of insertions, though constrained by internal symmetries.
2.3. Are there models in which the pure-SUSY EWSB subset of OP ν is empty? First, let us analyse possible contributions of gauge vector superfields. As there are U (1) Y and SU (2) L charges flowing in internal lines of any supergraph contributing to OP ∈ OP ν , it seems natural to think that insertions of externalV U (1) Y andV α SU (2) L into internal lines can map any superoperator to one which yields a pure-SUSY EWSB OP ∈ OP ν . An example is as we will see in the context of the model example presented in section 4. Although supergraphs with any given number of externalV s can be constructed from any underlying OP ∈ OP ν , the so obtained OP ∈ OP ν may vanish as the supergraphs add up to zero. More generally,V s insertions can be seen to correspond to terms in theV -expansion of gauge completed superoperators. For example, D 2 (LL)Ĥ uĤuVU (1) Y is a term in theV -expansion of Next, we will see that for models in which there exists a Higgs bilinear, there is always a pure-SUSY EWSB contribution proportional to µ. First, pick a OP ∈ OP ν . Now, each supergraph contributing to OP has either at least one external Higgs locally connected to loop superfields (say, class-a) or none (class-b). Without loss of generality, say that for a particular supergraph belonging to class-a the vertex isĤX 1X2 , whereXs are loop superfields. One can then see (cf. figure 2) that an insertion ofĤ † (Ĥ) followed by an insertion ofĤ (Ĥ † ) leads to a supergraph for the superoperatorĤ †Ĥ OP .  By means of the procedure described in figure 3, each class-b supergraph can also be transformed into a supergraph forĤ †Ĥ OP.
Hence, if class-a or -b supergraphs for superoperator OP do not add up to zero, the transformed ones do not add up to zero forĤ †Ĥ OP either. Now, if there exists a Higgs bilinear,Ĥ †Ĥ OP yields a pure-SUSY EWSB OP ∈ OP ν regardless of OP ∈ OP ν . One expects such contributions to m ν to be proportional to  In order to describe all conceivable assignments of chiralities to external and internal superfields, the chiralities ofĤ,Ĥ ,Φ andΦ are left unspecified. However,Ĥ,Φ andΦ have the same chirality, as is implied by the vertex. Moreover, and so that all conceivable propagators are described, we also do not specify howΦ is connected to the loop(s) (depicted by the circle), nor howΦ is connected toĤ .

−→Φ
for class A or B superoperators, respectively, and where λ is the coupling strength ofĤs to the loop(s). We will see an example of this in the model example of section 4, where an insertion of H †Ĥ into the LO topologies leads to dimension-7 contributions proportional µ/M 4 and µ 2 /M 5 (cf. Eq. (34)).

Models in which m LO
ν is proportional to soft-SUSY involving seesaw mediators To construct models of this kind one may first choose a set of superoperators from those identified in Eq. (19), and then postulate the existence of superfields that generate the topologies for these superoperators. Finally, one should pick an internal symmetry group that forbids all couplings that could generate a pure-SUSY EWSB OP ∈ OP ν up to the same order of perturbation theory. We cannot think of any serious obstruction that would compromise this procedure for constructing general models of this kind. In fact, in the next section we give a proof of existence based on a one-loop type-II seesaw, also showing that this kind of models need not be complicated.
In here, we shall identify the simplest of such model-topologies. To be precise, we consider one-loop topologies that generate the lowest dimension superoperators that can only give OP ∈ OP ν from SUSY EWS . From Eq. (19) one can see that the candidate superoperators are (24) Then, from D-algebra considerations, and discarding topologies with self-energies 4 , one obtains the following list of possibilities: -type-II with a chirality flip, type-I and -III; The corresponding supergraph topologies are depicted in figure 4.
We calculated the contribution of these topologies to OP ν up to order 3 in m soft , finding three kinds of leading dimensionful suppression factors: 1PR and 1PI). The absence of a contribution linear in m soft for some topologies is most easily seen to stem from the fact that one-loop topologies forLLĤ uĤu , as well as the one-loop 1PI parts of , use vertices of a single chirality. Moreover, and in regard toLLD We also found that one-loop realisations with self-energies had leading dimensionful suppression factors that ranged from µ m soft /M 3 or m 2 soft /M 3 to µ m 2 soft /M 4 or m 3 soft /M 4 . If we take µ ∼ m soft , we may say that in one-loop models of this kind LLHH operators have a dimensionful suppression of at least m 2 soft /M 3 . This result is naively expected for type-II seesaws without a chirality flip, since d 4 θD 2 (LL)ĤĤ has mass dimension 7. For other realisations this dependence is not trivial, since for an underlying superoperatorLLĤĤ one in general expects a m soft /M 2 dependence. It is worthy to mention that at higher loops there are topologies that lead to the naively expected dimensionful suppression of m soft /M 2 . Indeed, the one-loop exception can be seen to be a consequence of the fact that the holomorphy of the superpotential restricts the non-trivial 1PI parts of one-loop topologies in a way that all couplings and masses involved must have the same chirality.

A model example
To construct a model example we follow the recipe outlined at the beginning of the previous section and choose a model based on an one-loop topology for the superoperator D 2 (LL)Ĥ uĤu (cf. figure 4). As the most general set of scalar superfields and superpotential terms in this topology has size 7 and 5, respectively, we can choose to supplement the model with a new U (1) symmetry, say U (1) X , that is carried by the superfields in the loop (sayXs). These will be responsible for communicating L-number breaking to the actual leptons via the exchange of a type-II seesaw mediator, say∆.
In order to make the coupling∆ †Ĥ uĤu genuinely radiative, while allowing for a mass term for∆, we have to link it to the VEV of a superoperator of at least dimension 4 in superfields. One simple example isρ †∆ †Ĥ To achieve this we will postulate a U (1) L-number symmetry that gets broken by the VEV of the scalar component ofρ. We leave unspecified the actual mechanism driving ρ = 0. Now, being L-number breaking communicated by Xs, the simplest choice is then to coupleXs directly tô ρ by making aρ † insertion in the loop line where chirality flips. This leads to the the left-hand side diagram of figure 5. We will furthermore takeX 1 andX 2 to possess superpotential mass terms already at the L-number symmetric phase, even though it is not required by the topology. Figure 5. Leading order subset of OP ν in the model example.
A model consistent with this description is summarised in table 1 and its most general renormalisable superpotential is given by 5 In the absence of the last term the model acquires the R-symmetry shown in the last column of table 1. This term allows for a chirality flipped type-II seesaw of superoperatorLLĤ uĤu , as shown in the right-hand side supergraph of figure 5. The broken L-number phase corresponds to λρX 3X 3 → M X 3X 3X 3 + λρX 3X 3 , M X 3 := λ ρ . Table 1. Extension of the MSSM in the model example. We omitted the conjugates of∆ and X 1,2 . U (1) R stands for an R-symmetry that is acquired asλ X → 0.
The set of LO superoperators that break L-number proceed from the two supergraphs of figure 5 and are To see that there is indeed no other contributions up to the same order of perturbation theory, notice that a suitably defined L-number symmetry is recovered as any coupling in {λ 1 , λ 2 , λ L }, or both λ X and any in {λ X , M ∆ , M X 1 , M X 2 }, goes to zero. Thus, any superoperator that breaks L-number is proportional to In the p ext → 0 limit the LO coefficients for these operators read respectively, and where C 0 and D 0 are abbreviations of scalar one-loop 3-and 4-point integrals. Then, in the SUSY limit we have while d 4 θLLĤ uĤu = 0. Hence, there is no pure-SUSY EWSB contribution to neutrino masses at LO. To arrive at this same conclusion by a different route, consider the following. First, we note that only the first supergraph has a non-vanishing 1PI part: Then, by adding these operators to the classical Lagrangian, one concludes that ∆ acquires a VEV, since F Hu = µ * H † d = 0 gives a (tadpole) contribution to F † ∆ ⊃ M ∆ ∆. However, such a VEV has no effect upon m ν in the absence of ∆ − ∆ mixing. This VEV is no Due to its instructive value, we will now analyse in terms of component fields the reason behind the absence of pure-SUSY EWSB contributions to LLHH. In order for the left-hand side supergraph of figure 5 to contribute to LLHH, the three-scalar coupling ∆ † H u H u would have to be generated. At LO there are in fact three topologies contributing to this coupling as we show in figure 6. However, in the p ext → 0 limit, the spinor loop topology cancels exactly the scalar loop topologies. This result can also be understood to be a consequence of the fact that one cannot draw diagrams for ∆ † H u H u that are both holomorphy compliant and have at least an external F † − F pair. In the case of the second supergraph, one can see that it would require the generation of the coupling F ∆ H u H u , whereas the holomorphy of the superpotential makes it impossible to to draw any component field diagram for such a coupling. Figure 6. Leading order diagrams for ∆ † H u H u in the model example.
The pure-SUSY EWSB subset of OP ν can be most easily identified by reminding the analysis carried out in section 2.3. One may then see that it comprises the superoperators Figure 7. Example of soft-SUSY insertions (grey blobs) that generate the LO subset of OP ν in the model example. For a detailed list up to order 3 in m soft see [1].

Conclusions
In this work we studied the connection between m ν and SUSY that arises when neutrino masses are radiatively generated. We started by distinguishing the different SUSY contributions w.r.t. their involvement in EWSB. This allowed us to identify a subset of superoperators that could only contribute to m ν through SUSY effects unrelated to EWSB and which involved the mediators of the radiative seesaw. An interesting aspects of these effects is that they can be quite small without being in conflict with experimental data. One may then conjecture that they are (at least partially) responsible for the smallness of neutrino masses, thus allowing for a seesaw scale within foreseeable experimental reach.
We analysed the simplest (one-loop) topologies that may underlie models with this interesting feature and found that they all generated LLHH operators with a leading dimensionful dependence that ranged from µ m soft /M 3 or m 2 soft /M 3 to µ m 2 soft /M 4 or m 3 soft /M 4 . As an application of this more abstract analysis, we constructed a model example based on a one-loop type-II seesaw, which also served as a proof of existence of such interesting models. A phenomenological analysis of this model will be presented elsewhere.