Neutrino masses from SUSY breaking in radiative seesaw models
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Abstract
Radiatively generated neutrino masses (\(m_\nu \)) are proportional to supersymmetry (SUSY) breaking, as a result of the SUSY nonrenormalisation theorem. In this work, we investigate the space of SUSY radiative seesaw models with regard to their dependence on SUSY breaking ( Open image in new window ). In addition to contributions from sources of Open image in new window that are involved in electroweak symmetry breaking ( Open image in new window contributions), and which are manifest from \(\langle F^\dagger _H \rangle = \mu \langle \bar{H} \rangle \ne 0\) and \(\langle D \rangle = g \sum _H \langle H^\dagger \otimes _H H \rangle \ne 0\), radiatively generated \(m_\nu \) can also receive contributions from Open image in new window sources that are unrelated to EWSB ( Open image in new window contributions). We point out that recent literature overlooks pure Open image in new window contributions (\(\propto \mu / M\)) that can arise at the same order of perturbation theory as the leading order contribution from Open image in new window . We show that there exist realistic radiative seesaw models in which the leading order contribution to \(m_\nu \) is proportional to Open image in new window . 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 Open image in new window . We show that in oneloop realisations \(L L H H\) operators are suppressed by at least \(\mu \, {m_\text {soft}}/ M^3\) or \(m_{\mathrm{soft}}^2/M^3\). We construct a model example based on a oneloop typeII seesaw. An interesting aspect of these models lies in the fact that the scale of soft Open image in new window effects generating the leading order \(m_\nu \) can be quite small without conflicting with lower limits on the mass of new particles.
1 Introduction
The large hierarchy between neutrino masses (\(m_\nu \)) and the electroweak (EW) scale may be regarded a symptom of an hierarchy between the latter and a new mass scale (\(M\)) that holds lepton number (\(L\)number) breaking. The simplest extensions to the Standard Model (SM) that implement this hypothesis (typeI seesaws [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]) generate \(L L H H\) [12] with the naively expected dimensionful suppression factor of \(1/M\). Both direct [13] and indirect [14, 15, 16] bounds on \(m_\nu \) suggest \(M\) as heavy as \(10^{15}~\text {GeV}\) if the underlying parameters are of order one and obey no special relations.^{1}
 1.
the additional scale is the EW scale (\({\sim }v\)). In this case \(LLHH\) is not generated in perturbation theory, but higher dimensional operators are. This replaces the \(1/M\) dimensionful suppression by \(v^n / M^{n+1}\), where \(5+n\) is the dimension of the leading order (LO) operator. See for example [18] for a model in which the LO contribution to \(m_\nu \) comes from the dimension7 operator \(L L H H H^\dagger H\). See also [19] and references therein.
 2.
the additional scale (\(m\)) is an intermediate scale between \(m_\nu \) and \(M\). In this case \(L L H H\) is suppressed by some power of \(m / M\). For example, in the inverse seesaw [20, 21, 22] \(m\) is connected to some small (\({\ll }M\)) \(L\)number breaking scale that is transmitted to the actual leptons by dynamics at the scale \(M\). In the typeII seesaw [7, 8, 9, 10, 11] \(m\) could be the coupling scale of the scalar triplet to the Higgses. Both examples lead to a \(m / M^2\) dimensionful suppression.
Two new scales are introduced by supersymmetric (SUSY) extensions to the SM: the soft SUSY breaking ( Open image in new window ) scale, \({m_\text {soft}}\); and the scale at which Open image in new window takes place, \(M_X\). Naive dimensional analysis gives us grounds to speculate that \(M_X\) is much heavier than \({m_\text {soft}}\), since the strengths of hard and soft Open image in new window are related by powers of \({m_\text {soft}}/ M_X\) (see for e.g. [36]). The minimal SUSY SM (MSSM) introduces yet another scale: the Higgs bilinear, \(\mu \). Though, in general, correct EW symmetry breaking (EWSB) requires \(\mu \sim {m_\text {soft}}\). Do any of these scales play any role in neutrino mass generation?
It has been contemplated in [37, 38, 39, 40] that hard Open image in new window is the source of \(L\)number violation, so that \({m_\text {soft}}/ M_X \ll 1\) might be the reason for \(m_\nu / v \ll 1\). For example, if Open image in new window generates \(\tilde{L} \tilde{L} H_u H_u\), then \(L L H_u H_u\) arises at oneloop level via an EWino–slepton loop and is suppressed by \({m_\text {soft}}/ M_X\) [37]. Another possible connection to Open image in new window is in identifying the seesaw mediators with the mediators of Open image in new window to the visible sector [41, 42, 43, 44].
Our analysis will be carried out using perturbation theory in superspace (supergraph techniques,^{4}) as it renders the SUSY nonrenormalisation theorem a very simple statement and its implications in terms of component fields easier to identify. Points of contact with results in terms of component fields will be established throughout. Another advantage is that perturbation theory in superspace is much simpler than the ordinary QFT treatment. For instance, aside from the algebra of the SUSY covariant derivatives (\(D_{\alpha }\) and \(\bar{D}_{\dot{\alpha }}\)), supergraph calculations in a renormalisable SUSY model made of chiral scalar superfields resemble the Feynman diagrammatic approach to an ordinary QFT made of scalars with trilinear interactions. Open image in new window can be parameterised in a manifestly supersymmetric manner by introducing superfields with constant \(\theta \)dependent values ( Open image in new window spurions). Thus, Open image in new window effects will be conveniently taken into account in supergraph calculations by means of considering couplings to external Open image in new window spurions [54]. This allows one to see the Open image in new window contributions to neutrino masses as small Open image in new window effects upon a fundamentally SUSY topology.
2 Radiative seesaws in SUSY
2.1 Pure Open image in new window contributions
We then set up to ask a different question. Do Higgs bilinears imply the existence of a pure Open image in new window contribution to \(m_\nu \)? Or are there models in which this implication does not hold? We show that there is always a pure Open image in new window contribution (Sect. 3.1), however, models exist in which the LO contribution to \(m_\nu \) is proportional to Open image in new window (Sect. 4), as we exemplify in Sect. 5.
2.2 Models in the literature
A thorough evaluation of soft Open image in new window contributions to \(L L H H\) up to order \(2\) and in the simplifying limit \(M_{N_i} = \mu _{s3} = \mu _{L2} = M_N\) is given in Appendix D.
3 Open image in new window contributions
3.1 Are there models in which the pure Open image in new window subset of \(\text {OP}_\nu \) is empty?
 (a)
at least one external Higgs \(\hat{H}\) (or \(\hat{H}^\dagger \)) is locally connected to loop superfields, i.e. at least one external Higgs is 1PI;
 (b)
all external Higgses are connected to the loop(s) by means of 1PR propagators, i.e. all external Higgses are 1PR.
The procedures described above can be applied to each classa or b supergraph of the set contributing to \(\widehat{\text {OP}}\) up to any given order of perturbation theory. Hence, if classa or b supergraphs for superoperator \(\widehat{\text {OP}}\) do not add up to zero, the transformed ones do not add up to zero for \(\hat{H}^\dagger \hat{H} \, \widehat{\text {OP}}\) either. Now, if there exists a Higgs bilinear, \(\hat{H}^\dagger \hat{H} \, \widehat{\text {OP}}\) yields a pure Open image in new window \(\text {OP}\in \text {OP}_\nu \) regardless of \(\widehat{\text {OP}}\in \widehat{\text {OP}}_\nu \). We will illustrate this for a particular model in Sect. 5.
4 Models in which the leading order subset of \(\text {OP}_\nu \) is proportional to Open image in new window
A possible strategy to construct models of this kind is the following. Pick a set of superoperators that cannot yield a pure Open image in new window [cf. Eqs. (33) and (34)]. Choose the LO topologies at which these operators appear. Write the necessary superfields and couplings. As a final step, pick an internal symmetry group that precludes, at least up to the same order of perturbation theory, all superoperators that yield a pure Open image in new window . In particular, it is essential that the “wrong” Higgs does not communicate (at least up to the same order as the “right” Higgs) to the sector that holds \(L\)number breaking. To illustrate this, consider for example the oneloop realisation of 1PI \(\hat{L} \hat{L} \hat{H}_u \hat{H}_u\). \(\hat{H}_u\) couples to, say, \(\hat{X}_1 \hat{X}_2\), where \(\hat{X}_{1,2}\) have mass terms. Without loss of generality let the mass terms be \(\hat{\overline{X}}_i \hat{X}_i\). Hence, \(\hat{\overline{X}}_1 \hat{\overline{X}}_2 \hat{H}_d\) is invariant under non\(R\)symmetries in this phase. If such a term exists in the superpotential, this same model generates the supergraph topology shown in the middle panel of Fig. 3, leading to \(\hat{L} \hat{L} \hat{H}_u \hat{H}^\dagger _d\) which yields a pure Open image in new window .
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 oneloop typeII seesaw, also showing that this kind of models need not be complicated.

\(D^2 ( \hat{L} \hat{L} ) \hat{H}_u \hat{H}_u\), \(\hat{L} \hat{L} D^2 ( \hat{H}_u \hat{H}_u )\), \(D^2 ( \hat{L} \hat{L} ) \hat{H}^\dagger _d \hat{H}^\dagger _d\) and \(\hat{L} \hat{L} \bar{D}^2 ( \hat{H}^\dagger _d \hat{H}^\dagger _d )\) – typeII without a chirality flip;

\(\hat{L} \hat{L} \hat{H}_u \hat{H}_u\) (1PR) – typeII with a chirality flip, typeI and III;

\(\hat{L} \hat{L} \hat{H}_u \hat{H}_u\) (1PI).

\(\mu \, {m_\text {soft}}/ M^3\) or \(m_{\mathrm{soft}}^2 / M^3\) – \(D^2 ( \hat{L} \hat{L} ) \hat{H}_u \hat{H}_u\) and \(\hat{L} \hat{L} D^2 ( \hat{H}_u \hat{H}_u )\);

\(\mu \, m_{\mathrm{soft}}^2 / M^4\) or \({m_\text {soft}}^3 / M^4\) – \(\hat{L} \hat{L} \bar{D}^2 ( \hat{H}^\dagger _d \hat{H}^\dagger _d )\);

\(m_{\mathrm{soft}}^2 / M^3\) – \(D^2 ( \hat{L} \hat{L} ) \hat{H}^\dagger _d \hat{H}^\dagger _d\) and \(\hat{L} \hat{L} \hat{H}_u \hat{H}_u\) (both 1PR and 1PI).
In Appendix C, where we conduct a similar analysis for oneloop realisations with selfenergies, we find that these too have leading dimensionful suppression factors that range from \(\mu \, {m_\text {soft}}/ M^3\) or \(m_{\mathrm{soft}}^2 / M^3\) to \(\mu \, m_{\mathrm{soft}}^2 / M^4\) or \({m_\text {soft}}^3 / M^4\).
If we take \(\mu \sim {m_\text {soft}}\), we can conclude that in oneloop models of this kind \(L L H H\) operators have a dimensionful suppression of at least \(m_{\mathrm{soft}}^2 / M^3\). This result is naively expected for typeII seesaws without a chirality flip, since \(\int d^4\theta D^2 ( \hat{L} \hat{L} ) \hat{H} \hat{H}\) has mass dimension \(7\). For other realisations this dependence is not trivial, since for an underlying superoperator \(\hat{L} \hat{L} \hat{H} \hat{H}\) one in general expects a \({m_\text {soft}}/ M^2\) dependence, as was indeed found in Sect. 2.2.
5 A model example
Looking at the oneloop topology for \(D^2 ( \hat{L} \hat{L} ) \hat{H}_u \hat{H}_u\) (cf. Fig. 8) we see that the most general set of scalar superfields and superpotential terms involved is seven and five (four trilinears and one bilinear), respectively. The subset of \(U(1)^7\) (acting independently on each scalar superfield) under which the five terms are invariant consists of the hypercharge and a new \(U(1)_X\) charge carried by the superfields in the loop (say \(\hat{X}\)s). These are responsible for communicating \(L\)number breaking to the SM leptons via the exchange of a typeII seesaw mediator, \(\hat{\Delta }\).
Extension of the MSSM in the model example. We omitted the conjugates of \(\hat{\Delta }\) and \(\hat{X}_{1,2}\). \(U(1)_R\) stands for an \(R\)symmetry that is acquired as \(\bar{\lambda }_X \rightarrow 0\)
\(SU(2)_L \otimes U(1)_Y\)  \(U(1)_X\)  \(U(1)_L\)  \(U(1)_R\)  

\(\hat{\Delta }\)  \((\mathbf {3},1)\)  \(0\)  \(2\)  \(4\) 
\(\hat{\rho }\)  \((\mathbf {1},0)\)  \(0\)  \(2\)  \(0\) 
\(\hat{X}_1\)  \((\mathbf {2},1/2)\)  \(1\)  \(1\)  \(2\) 
\(\hat{X}_2\)  \((\mathbf {2},1/2)\)  \(1\)  \(1\)  \(0\) 
\(\hat{X}_3\)  \((\mathbf {1},0)\)  \(1\)  \(1\)  \(0\) 
\(\hat{\overline{X}}_3\)  \((\mathbf {1},0)\)  \(1\)  \(1\)  \(2\) 
It is instructive to illustrate in terms of component fields why there is no pure Open image in new window contribution to \(L L H H\). In order to yield \(L L H H\), the first supergraph of Fig. 10 necessitates the threescalar coupling \(\Delta ^\dagger H_u H_u\). There are three topologies contributing to this coupling at LO: two with scalars in the loop and the other with spinors (see Fig. 11). In the \(p_\text {ext}\rightarrow 0\) limit the latter cancels the former exactly. Another way to look at this result is the following. If one draws diagrams for \(\Delta ^\dagger H_u H_u\) using auxiliary fields – so that holomorphy becomes more transparent – one concludes that there does not exist a single diagram that is simultaneously holomorphy compliant and has at least an external \(F^\dagger \)–\( F\) pair. Moreover, all such diagrams that are holomorphy compliant can be paired in sets in such a way that a set with scalar loops is matched to a set with spinor loops and an exact cancellation in the \(p_\text {ext}\rightarrow 0\) limit is operative. Regarding the second supergraph, it necessitates \(F_{\overline{\Delta }} H_u H_u\) but no holomorphy compliant diagram for \(F_{\overline{\Delta }} H_u H_u\) can be drawn.
To understand, in terms of component fields, how these insertions are enablers of contributions to \(\text {OP}_\nu \) consider the following. As the insertion of an external auxiliary component of a gauge vector superfield (\(D\)) into a scalar line preserves chirality (or, diagrammatically, the arrowhead’s direction), any holomorphy compliant diagram with a \(D\) attached has a corresponding (underlying) holomorphy compliant diagram without that \(D\). Since in our example we are considering a single \(D\) insertion, the LO underlying diagrams are the ones depicted in Fig. 11, and no others. Once an external \(D\) is attached to an internal scalar line, the spinor loop diagram does not contribute and the sum of the others does not need to vanish anymore to respect the SUSY nonrenormalisation theorem. Regarding the \(\hat{H}^\dagger _u \hat{H}_u\) insertion, one can see that it allows for holomorphy compliant diagrams with an external \(F^\dagger \)–\( F\) pair by means of attaching \(F^\dagger _{H_u}\) and \(F_{H_u}\) to the scalar loop.
6 Conclusions
While the smallness of \(m_\nu \) points towards an high seesaw scale \(M\), the resolution of the hierarchy problem suggests that the scale of soft Open image in new window should lie close to the TeV scale. It is then tempting to conceive that \({m_\text {soft}}/ M\) is partially responsible for \(m_\nu \ll v\). Since in the SUSY limit there are no radiative corrections to the superpotential, models in which neutrino masses arise at the loop level provide a scenario in which such a connection is natural. How \(m_\nu \) is proportional to Open image in new window depends on the particular radiative seesaw model or, more specifically, on the form of the leading \(L\)number breaking superoperators.
By classifying the dependence on Open image in new window according to their involvement in EWSB, we identified a subset of model topologies in which the leading contributions to \(m_\nu \) depend on Open image in new window sources that are not involved in EWSB. In a first stage, we argued in favour of this by showing that, of all superoperators that can possibly contribute to neutrino masses, there is a subset which does it only by means of insertions of Open image in new window spurions. Then, in a second stage, we gave a complete description of the simplest model topologies in which all leading superoperators were of this type, and we calculated their dependence on soft Open image in new window up to order \(3\). We found that all oneloop realisations generated \(L L H H\) operators with a leading dimensionful dependence that ranged from \(\mu \, {m_\text {soft}}/ M^3\) or \(m_{\mathrm{soft}}^2 / M^3\) to \(\mu \, m_{\mathrm{soft}}^2 / M^4\) or \(m_{\mathrm{soft}}^3 / M^4\).
Even though the majority of all conceivable model topologies do in fact generate contributions to \(m_\nu \) proportional to Open image in new window , we pointed out that all models in the literature^{15} that we are aware of generate at least one leading topology that gives a contribution in which all Open image in new window sources are involved in EWSB. To serve as a proof of the existence of models in which \(m_\nu \) is proportional to Open image in new window at leading order, we built a model in which the leading neutrino mass operators were of dimension 5 and came from Open image in new window , whereas the pure Open image in new window ones had dimension 7.
One phenomenologically interesting aspect of these models is that soft Open image in new window effects generating the leading order \(m_\nu \) can be quite small without conflicting with lower limits on the mass of new particles. This is due to the fact that these effects involve states that can possess superpotential mass terms in the EWS phase, as we have seen in the model example. This is in contrast with models that contain pure Open image in new window contributions to \(m_\nu \) at leading order, because \(\mu \) and the soft Open image in new window effects driving EWSB provide the dominant contribution to the mass of the corresponding states, and are therefore severely constrained by present lower limits on sparticle masses.

\(\mu ^2 / M^2\) (and \(\mu A^*_\ell / M^2\)), due to superpotential terms involving the “wrong” Higgs. To be specific, \(\hat{L} \hat{L} \hat{H}_u \hat{H}^\dagger _d\) is generated by a 1PI twoloop topology that is constructed from the oneloop topology in the lefthand side of Fig. 10 by means of the coupling \(Y_\ell \hat{L} \hat{e}^c \hat{H}_d \subset \mathcal {W}\);

\(m_{\widetilde{\text {EW}}} / M\), due to topologies with internal EW gauge vector superfields in which a EWino mass term (\(m_{\widetilde{\text { EW}}}\)) is inserted.
The parameter space of these models is quite rich as there are many couplings and masses involved in the generation \(m_\nu \). From a qualitative point of view, one can identify two overlapping regions of parameter space of potential phenomenological interest. An interesting region is the one in which both \(\mu \) and \(m\) are particularly small w.r.t. \(M\), while higherorder contributions to \(m_\nu \) that are independent of both \(\mu \) and \(m\) remain subleading. In this region a small \(m_\nu / v\) can be generated with even larger couplings and/or lighter seesaw mediators. Since \(m_\nu \) is sensitive to at least the fourth power of couplings involved in \(L\)number breaking, another possibly interesting region comprises a lighter \(M\) at the expense of slightly weaker couplings. For instance, in the model of Sect. 5, decreasing all the couplings by a factor of \(1/2\) allows one to decrease \(M_X\) by a factor of \(1/10\) while keeping \(m_\nu \) fixed. A detailed phenomenological analysis of this model will be presented in a future publication.
To summarise, we have shown that there exist radiative seesaw models in which \(m_\nu / v \ll 1\) can be explained by \({m_\text {soft}}/ M \ll 1\) with \(M\) not very far above the EW scale. Under the assumption of \(L\)number breaking at the superpotential level and low \(M\), this explanation can be regarded to be more natural than that of treelevel seesaws in the sense that it does not require very small superpotential couplings (as canonical seesaws do) nor does it require two very different superpotential mass scales (as inverse seesaws do).
Footnotes
 1.
Some special textures in the seesaw parameters allow for relatively large couplings with a smaller \(M\), as discussed for e.g. in [17] and references therein.
 2.
 3.
Since SUSY and \(L\)number are very different symmetries, that the two are broken separately seems to be a plausible assumption.
 4.
Extensive details concerning supergraph calculations can be found in chapter 6 of [53].
 5.
Here and throughout the text, “mod X” means modulo insertions of X. For instance, suppose that \(\hat{V}~(\text {mod}\, \hat{H}^\dagger , \hat{H})\) is equal to \(\hat{U}\). This means that the general form of \(\hat{V}\) is \(\hat{V} = \hat{U} \hat{H}^{\dagger k} \hat{H}^{k'}\), where \(k,k' = 0,1,\ldots \).
 6.
We note that \(D := D \bar{D}^2 D \hat{V} \) is equal to \(\bar{D}^2 D^2 \hat{V} \) in the Wess–Zumino and Landau gauge, since in this gauge we have \(\hat{V}  = 0\) and \(\partial _\mu V^\mu = 0\).
 7.
Here, and throughout the text, a field (or a scalar chiral superfield) with a bar, say \(\overline{X}\) (\(\hat{\overline{X}}\)), transforms (under non\(R\)symmetries) in the conjugate representation of \(X\) (\(\hat{X}\)), so that \(X \overline{X}\) (\(\hat{X} \hat{\overline{X}}\)) is symmetric (i.e. invariant under the symmetries of the model). Moreover, the \(R\)charges satisfy \(Q_R(\hat{\overline{X}}) + Q_R(\hat{X}) = 2\) so that \(\int d^2\theta \hat{X} \hat{\overline{X}}\) is symmetric.
 8.
To simplify the discussion, from now on any \(\widehat{\text {OP}}\in \widehat{\text {OP}}_\nu \) is defined modulo Open image in new window insertions.
 9.
For example, \(D^2 (\hat{L} \hat{L}) \hat{H}_u \hat{H}_u \hat{V}_{U(1)_Y}\) is a term in the \(\hat{V}\)expansion of \(D^2 (\hat{L} \hat{L} e^{2 g' Y_L \hat{V}_{U(1)_Y}}) \hat{H}_u \hat{H}_u e^{2 g' Y_{H_u} \hat{V}_{U(1)_Y}}\).
 10.
A systematic method to derive this list is the following. The class of oneloop 4point supergraph topologies with a oneloop vertex can be partitioned w.r.t. the four possible types of 1PR propagators: \(\hat{\Phi }\hat{\Phi }^\dagger \), its \(\text {H.c.}\), \(\hat{\Phi }\hat{\overline{\Phi }}\) and its \(\text {H.c.}\). Of these topologies, only \(3+1+3+1\) (partitioned as mentioned) can underlie an \(\widehat{\text {OP}}\in \widehat{\text {OP}}_\nu \) as a consequence of requiring at least two external chiral lines that will be identified as a pair of \(\hat{L}\)s. Of these, only \(2+1+1+0\) can underlie a superoperator listed in Eq. (35). These \(2+1+1+0\) topologies can be identified by the superoperators \(D^2 ( \hat{A} \hat{B} ) \hat{C} \hat{D}\), \(D^2 ( \hat{A} \hat{B} ) \hat{C}^\dagger \hat{D}^\dagger \), \(\bar{D}^2 ( \hat{A}^\dagger \hat{B}^\dagger ) \hat{C} \hat{D}\) and \(\hat{A} \hat{B} \hat{C} \hat{D}\), respectively. Regarding irreducible topologies: only three have at least two external chiral lines and, of these, only one can underlie a superoperator listed in Eq. (35).
 11.
We disregard nonholomorphic soft Open image in new window trilinears as naive dimensional analysis indicates that they are suppressed by \({m_\text {soft}}/ M_X\) w.r.t. \(A\), \(\sqrt{B}\) and \({m_\text {soft}}\).
 12.
In spite of this, one could still be suspicious on whether our parameterisation for holomorphic soft Open image in new window is actually soft, since the \(A\)term vertex gives three factors of \(\bar{D}^2\), whereas only a maximum of four \(D_{\alpha }\) or \(\bar{D}_{\dot{\alpha }}\) is compatible with the renormalisability criterion for softness. To see that it is, notice that any subgraph in which one of these \(\bar{D}^2\) is not absorbed by \(\hat{X}^\dagger \) vanishes identically as there is a \(\bar{D}^2\) factor on every internal line attached to the vertex. Similarly, nonvanishing subgraphs with a \(B\)term are those in which the \(B\) is seen to introduce only a factor of \(\bar{D}^2\).
 13.
Although not relevant to our analysis, for definiteness we assume that the \(\hat{u}^c \hat{d}^c \hat{d}^c\) term is forbidden by, for instance, Rparity or baryon number conservation.
 14.
It does not appear in the expression above due to a fortuitous cancellation in the simplifying limit we have taken, cf. Eq. (93).
 15.
Barring those in which \(L\)number is a symmetry of the superpotential that is broken by the Open image in new window sector.
 16.
Recall that in those tables we suppressed insertions that were redundant due to some symmetry of the supergraph topology. In here, we are counting them provided they involve a distinct set of superfields.
Notes
Acknowledgments
This work has been partially funded by Fundação para a Ciência e a Tecnologia (FCT) through the fellowship SFRH/BD/64666/2009. We also acknowledge the partial support from the projects EXPL/FISNUC/0460/2013 and PESTOE/FIS/UI0777/2013 financed by FCT.
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