Soft See-Saw: Radiative Origin of Neutrino Masses in SUSY Theories

Radiative neutrino mass generation within supersymmetric (SUSY) construction is studied. The mechanism is considered where the lepton number violation is originating from the soft SUSY breaking terms. This requires MSSM extensions with states around the TeV scale. We present several explicit realizations based on extensions either by MSSM singlet or SU(2)_w triplet states. Besides some novelties of the proposed scenarios, various phenomenological implications are also discussed.


Introduction
One of the missing piece of the Standard Model (SM) is the consistent neutrino sector required for accommodation of the neutrino date [1]. Extensions based on type I [2], type II [3] and type III [4] see-saw mechanisms have been suggested, which at tree level induce effective dimension five (d = 5) ∆L = 2 lepton number violating operator [5] λ ij M * l i l j HH , where l i (i, j = 1, 2, 3 are family indices) and H are SM lepton and Higgs doublets respectively. The (1) type couplings, in turn, generate neutrino masses and mixings after EW symmetry breaking. It was shown [6][7][8][9] that, augmenting the SM by specific states and couplings, the operators of Eq. (1) can be generated radiatively at one (or higher) loop level. This possibility, referred as radiative neutrino mass generation mechanism, offers many interesting scenarios with rich phenomenological implications [6][7][8][9]. It is a curious fact that, besides some studies [10], within SUSY constructions such possibilities have not been pursued much. 3 In this work, aiming to fill this gap and address this issue within SUSY scenarios, we offer possibilities of loop induced neutrino masses, where lepton number violation occurs in soft SUSY breaking couplings and is transferred in the SM neutrino sector at the loop level. Referring to this possibility as 'soft see-saw' mechanism, we present several extensions which naturally realize this programme. The following three types of extensions are considered: i) extension with MSSM singlet (matter) superfields, ii) extension with a pair of SU(2) w triplet-antitriplet scalar superfields carrying U(1) Y hypercharges ±2, and iii) extension with matter SU(2) w triplet and U(1) Y neutral superfields. We call these three scenarios soft type I, soft type II and soft type III see-saw scenarios respectively. We work out details of each construction and discuss phenomenological implications.
The paper is organized as follows. In the next section we discuss lepton number violating operators of different dimensions, involving MSSM states and list some of them. Then, as a demonstration, we compute the d = 5, ∆L = 2 operator of type (1), induced via 1-loop due to presence of a quartic (d = 4) ∆L = 2 operator. In Sect. 3 we present models -extensions of the MSSM -where the lepton number violation takes place in soft SUSY breaking terms and show how integrating out the extra states give effective ∆L = 2 operators. We consider extensions with MSSM singlet matter (right handed neutrino) superfields and with SU(2) w triplet states. Based on these extensions different scenarios (e.g. type I, type II-A, type II-B and type III soft see-saw) emerge. In each case neutrino masses are induced via loops. Corresponding 1-loop and 2-loop results are presented. Sect. 4 contains discussion and outlook with some prospects for future studies. In appendix A we present supergravity formalism for calculation of soft SUSY breaking terms emerged from hidden sector superpotential and non-minimal Kähler potential couplings. We give the details of the SUSY breaking via Polonyi superpotential and specific non-minimal Kähler potential -the system insuring for SUSY breaking X superfield F X ∼ m 3/2 M P l (where m 3/2 is a gravitino mass) and adequately suppressed value for X . Both these are needed for our model building. In appendix B detailed derivations of effective ∆L = 2 quartic couplings, emerging by integration of MSSM singlet and SU(2) w triplet states, are given. In appendix C we present the calculation of the 2-loop contribution to the neutrino masses.

Some ∆L = Lepton Number Violating Operators with MSSM States
The lepton sector of the MSSM involves the following D and F -term couplings: l † e V l + e c † e V e c D →l * Ṽ l +ẽ c * Ṽ e c + h.c.
where schematically V indicates all appropriate gauge superfields multiplied by proper generator and gauge coupling. Without soft SUSY breaking terms, it is possible to define the lepton number (L-numbers for l, e c ) and also slepton number (L -numbers forl,ẽ c ) separately. 4 Thus, the couplings in (2) and (3) possess two independent symmetries. However, including soft SUSY breaking terms -the gaugino masses 1 2 MṼ aṼ aṼa -give the slepton numbers fixed equal to the corresponding lepton numbers:L = L. 5 Thus, all components from a given superfield have same lepton number. Therefore, if for instance, in the soft SUSY breaking sector the lepton number will be broken, it will be transferred in the fermion sector via radiative corrections.
In our consideration we assume that the whole Lagrangian respects the matter parity, under which states transform as Φ → (−1) 3(B−L)+2S Φ, where B and L are baryon and lepton numbers respectively and S indicates the spin of the Φ state. This symmetry insures that no baryon and lepton number violating couplings present at renormalizable level and LSP is a stable.
Considering higher dimensional operators, in SUSY (with matter parity) the lepton number violation starts from the F -term d = 5 operator which is the SUSY analog of the (1) coupling. At r.h.s of Eq. (4) l denotes SM fermionic lepton and h u is up type Higgs doublet. Symbols with tildes denote their superpartners respectively. The coefficients at r.h.s of Eq. (4) are related due to SUSY. However, in general they can be different especially if some of these couplings emerge via SUSY breaking terms. Therefore, we write each of them with independent coefficients: d = 5, ∆L = 2 operators : λ 5 -type couplings are directly constrained from the neutrino masses. The λ ′ 5 and λ ′′ 5 operators induce λ 5 -term via loops in which soft SUSY breaking couplings participate. Therefore, constraints on λ ′ 5 , λ ′′ 5 will depend on SUSY spectroscopy. The λ ′ 5 operator emerges in one of the model (named as type II-B soft see-saw model) we present in Sect. 3.2. Before going to the higher dimensional operators, note that with MSSM states one can write ∆L = 2 quartic (d = 4) coupling d = 4, ∆L = 2 operator : This term is non-supersymmetric operator and to discuss its origin one needs to have some UV completion. In Sect. 3 we present models, where this operator emerges and compute λ 4 coupling in terms of model parameters. Let us also give some d = 6, ∆L = 2 couplings involving states from l and h u superfields: d = 6, ∆L = 2 operators : where in λ ′ 6 -term under gṼ we assume g 1Ṽ1 or g 2Ṽ2 (with appropriate contraction of the SU(2) w gauge indices). This coupling emerges in type II-B soft see-saw model (discussed in Sect. 3.2).
Other ∆L = 2 couplings can be obtained from the operators in Eqs. (5)- (7) by the substitution h u → h † d . Also, the variety of ∆L = 2 operators involving e c andẽ c states can be constructed. The models with their generation would be interesting to consider, but in this work we pursue only specific constructions. ∆L = 2 operators, including those we gave above, by radiative corrections will be converted to the operator λ 5 (l h u ) 2 -responsible for the neutrino mass generation. In the next subsection we give details of the calculation of this λ 5 -term emerged from the λ 4 coupling of Eq. (6) via gaugino/higgsino loop dressings.

∆L = 2 Quartic Coupling and Neutrino Mass
The coupling (6) is invariant under SM gauge symmetry. Since we are interested in neutrino mass generation, we extract from it the neutral components: The combinationνν will be converted to νν by the neutralino dressing diagram(s) shown in Fig.  1. The relevant terms are also (given in 2-component notations of Ref. [11]) whereB andW a b are U(1) Y and SU(2) w gauginos respectively. The indices a, b = 1, 2 belong to the SU(2) w gauge group.W 's matrix is defined as We are looking for the operator: where κ ν ij will be computed in terms of couplings appearing in (8), (9), together with other model parameters given below.
By diagonalization of 4 × 4 neutralino mass matrix (of the states {−iB, −iW 0 ,h 0 d ,h 0 u }), theB andW 0 get transformed as where χ 0 a is a physical mass eigenstate neutralino with mass Mā χ0 . Using these in Eq. (9), the relevant couplings will be Interactions in Eqs. (8), (13) together with neutralino Majorana type mass terms, generate the operator of Eq. (11) via 1-loop diagram(s) shown in Fig. 1. Upon evaluation of the loop integral for the expression of κ ν we obtain: where tan θ W = g 1 g 2 and From these expressions we can see that in order to have adequately suppressed neutrino masses( < ∼ 0.1 eV), for the SUSY particle masses∼ TeV, we need to have λ 4 < ∼ 10 −8 . In the next section we will see that this suppression can be naturally realized within presented models.

Soft See-Saw
In this section we present models which generate some of the ∆L = 2 operators [in particular: λ 4 , λ ′ 5 and λ ′ 6 -type couplings of Eqs. (6), (5) and (7) resp.]. This happens through the soft SUSY breaking sector. Therefore, we refer to it as a soft see-saw mechanism. Scenarios we consider are based on extensions either with MSSM singlet matter superfields 6 , or with a pair of SU(2) w triplet-antitriplet scalar superfields, or on a model with matter SU(2) w triplet superfields. Each case is investigated separately.

Type I Soft See-Saw
In this case, the MSSM is extended with right handed neutrino (RHN) superfields N. Since within our scenario, the masses of these states are near TeV scale, in order to avoid unacceptably large neutrino masses, we will need to suppress the Y ν lNh u type Yukawa superpotential couplings. Instead demanding ad hoc condition |Y ν | < ∼ 10 −7 , we will forbid such superpotential coupling by the R-symmetry, under which the superfield φ i and the superpotential transform as: With the R charge assignment for the superpotential W and superfields l, h u , N given in Table 1, together with the couplings lNh u , also the NN-type superpotential terms are forbidden. However, as we will see shortly, the lepton number violating soft SUSY breaking terms will be induced. These type of terms will come from the Kähler potential, via the SUSY breaking. The latter will occur in a hidden sector with MSSM singlet superfield X having non-zero F -term F X ∼ m 3/2 M P l . The R charge of the X is selected to be R X = 1. In Table 1 we summarize R charges of all superfields to be considered. With this assignment, the following minimal and non-minimal Kähler (denoted below as K m and K nm respectively) potential couplings are allowed: (On the second term in K nm , with cut off scaleM < M Pl , we comment below.) With these, the generated soft SUSY breaking terms involvingÑ will be: For detailed discussion and derivations see Appendix A. In the same Appendix we discuss the generation of F X via Polonyi superpotential and specific Kähler potential for X.
From the couplings in Eq. (18), by integrating out theÑ states we get the operator of Eq. (6). In particular, by assumption κ < ∼ 1/3, κ A v u < ∼ 0.3m 3/2 , the expression for λ 4 can be approximated as: Corresponding diagram is given in Fig. 2. Derivation and more accurate expression for λ 4 is given in Appendix B by Eq. (B.7). From the structure of (19) and process given in Fig. 2 it is clear why it is fair to call this mechanism (for generating of λ 4 coupling) the type I soft see-saw. The quartic operator of (6) is forbidden in the SUSY limit, but is generated due to SUSY breaking ∆L = 2 terms via integrating out theÑ states. With these, as was shown in section 2.1, the neutrino masses are generated at 1-loop level. The needed suppression for the λ 4 is easily obtained. For instance, with the SUSY particle masses∼few·TeV and (19) we can get λ 4 ∼ 10 −8 -guaranteeing suppressed values of the neutrino masses(∼ 0.1 eV). The second operator in K nm of Eq. (17) can be obtained by integrating some heavy states. Here we give one example. By introducing additional MSSM singlet states N , N , N ′ and N ′ we can have the Kähler coupling K N = κ N X † M P l NN and the superpotential terms: Comparing this with Eq. (17), one can identify As shown in Appendix A, by the construction one can insure that the VEV of the X field can be adequately suppressed. With X < ∼ 10 −7 M P l the neutrino Dirac Yukawa couplings, generated via the Kähler potential will be Y ν < ∼ 10 −7 (see expression in (A.23) and related discussion before and after of this equation), i.e. not relevant for the neutrino masses. On the other hand, there is a low bound on the values of Y ν couplings. They should be sizable enough to insure decays of the fermionic RHN states N within < ∼ 0.3 sec. in order to not affect the standard Big Bang Nucleosynthesis. With M N > M l + M hu (Ml + Mh) we will have decays N → lh u (lh), with the lifetime given by Since Y ν ∼ X /M P l , we will have the low bound on the VEV X > ∼ 5.6 · 10 5 GeV. Therefore, summarizing all above, we will have the following range: The mass M N is generated from the first coupling of K nm of (17), with the value M N ∼ κF X /M Pl ∼ κm 3/2 . Note that, it is easy to satisfy M N > M l + M hu (Ml + Mh) insuring the decays described in (20). Closing this subsection, with the R-charge assignment given in Table 1, the MSSM Yukawa superpotential couplings are: Note that the direct h u h d superpotential term is forbidden. However, as shown first in Ref. [12], the last coupling in (17) Table 2: R charges in the Type II-A soft see-saw model, with R W = ω. By ω = m + 1/2 (m ∈ Z) the matter parity is automatic.
With the R-charge assignments given in Table 1 the lepton number violating superpotential couplings X n h u l (which break also matter parity) are all forbidden for ω = m + 1/2 (where m is an integer). Also the trilinear lepton number violating interactions qd c l and e c ll will be forbidden in the superpotential. The baryon number violating term u c d c d c can be easily avoided by specific selection of the phases α q and α. By similar selection, one can make sure that matter parity violating terms do not emerge also from the Kähler potential.

Type II Soft See-Saw
In this case we extend the MSSM by introducing the pair of SU(2) w triplet and antitriplet superfields Σ,Σ with U(1) Y hypercharges 2 and −2 respectively (in this normalization, the hypercharge of the lepton doublet l = (e − , ν) equals to −1). Σ andΣ are 2 × 2 matrices in SU(2) w space and their compositions are given by: where signs in superscripts indicate electric charges [e.g. (++) stands with the double charged state]. With this field content, two scenarios which somewhat differ from each other, can be considered. We refer to them as type II-A soft and type II-B soft see-saw models.
Type II-A soft see-saw model In this case, the R-charge assignment is given in Table 2 and relevant Kähler potential couplings are: The MSSM superpotential couplings will be same as given in (22). The couplings of Eq. (24) and SUSY breaking hidden sector, together with the µ and B-terms for the MSSM Higgs doublets, generate the following soft terms involving the scalar components of Σ andΣ: Here Σ andΣ denote bosonic components. Their fermionic partners will be denoted byΣ andΣ respectively. Integration of Σ,Σ states lead to the operator of Eq. (6)  for λ 4 given by: (B.10)]. The first diagram of Fig. 3 corresponds to the generation of this operator, which give loop induced neutrino masses as shown in Sect. 2.1. Also within this case, the needed suppression of λ 4 can be achieved, for example, by the selection Via exchange of Σ,Σ states, the four slepton interaction operator is also generated, which in a same approximation (as Eq. (26)) is: where a, b = 1, 2 are SU(2) w indices [see Eq. (B.11) for more accurate expression]. The corresponding diagram is the second one in Fig. 3. By 2-loop gaugino/higgsino dressing diagrams, the operator (27) will be converted to the four lepton operator. The latter in turn induce e i → e j e m e m rare decays, including processes such as τ → 3µ, µ → 3e etc. These will constrain the A Σ matrix elements via experimental limits [13] and open window to probe this mechanism of neutrino mass generation. Finally, the κ-term in Eq. (24) generates the mass term for the fermionicΣ,Σ components µ ΣΣΣ with µ Σ ∼ κm 3/2 .

Type II-B soft see-saw model
In this modified version, the neutrino masses are induced at 2-loop level. The R-charges of the states are given in Table 3 and the Kähler potential couplings are The X †Σ h u h u term is forbidden in the Kähler potential, but the following superpotential coupling, involvingΣ, is allowed: The MSSM superpotential terms are same as given in Eq. (22). The coupling (29), in general   would generate the trilinear soft term AΣΣh u h u . However, within the SUSY breaking scenario we are considering, this A-term is suppressed: (see Eq. (A.25) and discussion therein). Kähler potential terms of Eq. (28), besides higgsino µ and higgs B terms induce the potential terms: With the couplings in (30), (31), integration of Σ,Σ states lead to the operator of Eq. (6) with expression for λ 4 having the same form as given in Eq. (26), but with extra strong suppression factor < ∼ 10 −7 . Therefore, 1-loop contribution to the neutrino masses will be very suppressed and can be ignored. Neutrino masses will be induced at 2-loop level, which we discuss below.
The superpotential of Eq. (29) gives the following Yukawa interactions: These couplings, together with those given in (31), upon integration of scalar components from Σ, Σ superfields, induce the following ∆L = 2 dimension 5 operator (λ ′ 5 -type of Eq. (5)): The way of derivation of this d = 5 operator is similar to that given in Appendix B. We just need to make replacement AΣh u h u → λΣh uhu in Eq. (B.10). The operator (33), by the gaugino dressings at 2-loop give the d = 5 operator λ 5 (l h u ) 2 responsible for the neutrino mass. The relevant diagrams are given in Fig. 4. For consistency, one should also take into account the effective d = 6, ∆L = 2 operators (the λ ′ 6 -type coupling of Eq. (7)) which are induced by integration of the scalar and fermionic components fromΣ, Σ superfields [by using the mass term µ ΣΣΣ and couplings (31), (32)]. In (34), gṼ = {g 1B , g 2W } denote U(1) Y and/or SU(2) w gaugino contributions.
Care must be taken to treat properly divergences of the diagrams in Fig. 4. If the effective vortexes (33) and (34) are used without specifying the model (the operators are induced from), the loops need to be cut off by the characteristic scale. Since within type II soft see-saw, these operators are obtained by integrating Σ,Σ states, full calculation can be done. With 'opening' vortexes, the diagrams are shown in Fig. 5. It is obvious that log divergence of the first and third diagrams are coming from upper triangles corresponding to the parts of one loop renormalization of the soft AΣ term (vanishing at tree level). Indeed, where µ and µ Σ are the MSSM higgsino µ-term and the mass of the fermionicΣ,Σ states respectively, g a is the gauge coupling (with c (1,2) a being corresponding group theoretical factors and MṼ a the corresponding gaugino's soft mass). In the last line of (35), a = 1, 2, 3 labels U(1) Y , SU(2) w and SU(3) c gauge groups respectively (the coupling g 1 is taken in SU(5) normalization). The ln Λ terms coincide to those obtained via integration of the RG equation for the AΣ soft term. Expressions (35), after the renormalization, leave us with a non-divergent and finite parts. Assuming the Further, we compute the lower triangle diagrams of Fig. 5-(a),(c), which by the gaugino dressings convertll to ll. Then, integrate out the Σ,Σ states and obtain the contribution to the neutrino mass to be given by: where the function f is defined in Eq. (15) and m i , m j denote masses ofl i ,l j . There is no any divergence from the diagram of Fig. 5-(b). Detailed evaluation of this diagram is given in Appendix C. Contribution to the neutrino mass matrix from the 2-loop diagram of Fig.  5-(b) is given by: where J is given in Eq. (C.11) and γ αnm and other appearing factors are defined in Eqs. (C.9), (C.4) and (C.5) respectively. Concluding, note that additional operators where h u scalar is replaced by h † d will be also induced. The relevance of this kind of operators will depend on the value of the VEV h (0) d . The latter, on the other hand, depends on the parameter tan β. Thus, the details and numerical results will depend on SUSY spectroscopy, which should be taken such that LHC constraints are satisfied. The latter study is beyond the scope of this work.

Type III Soft See-Saw
In this case, the MSSM is extended with SU(2) w triplets superfields T with zero hypercharges. For a realistic neutrino sector, at least two such superfields should be introduced. The matrix representation of this superfield(s) has the form with superscripts indicating the electric charges of the fragments. In this case, the R charges are: Note that with ω = m + 1 2 (m ∈ Z) the matter parity is automatic. The superpotential couplings with the T states will be forbidden, while the Kähler potential couplings will be The R charge selection of Eq. (40) is consistent with MSSM superpotential couplings given in Eq. (22). From the coupling of (41), likewise of previous cases, together with the MSSM µ and B-terms, the following soft SUSY breaking terms are induced: where by T the scalar components are denoted. Their fermionic partnersT gain the mass 1 2 µ TTT with µ T ∼ κm 3/2 . (Once more, we refer the reader to Appendix A for details.) Now, one can easily verify that integration of T 's scalar components, via (42) couplings induce the ∆L = 2 operator of Eq. (6) with: This expression is approximate and works well with M 2 T > ∼ B T /3. The derivation of this expression is very similar to that corresponding to the case of Type I soft see-saw model, presented in Appendix B. 8 With the λ 4 coupling the neutrino mass will be generated (as discussed in Sect. 2.1) at Fig. 1. 1-loop contribution will dominate if Y T lT h u type Yukawa interactions, induced through the Kähler potential (with Y T ∼ X /M P l ) will be adequately suppressed. This requires X < ∼ 2 · 10 11 GeV. For this case [unlike the Eq. (21)] we will not have low bound on X , because the states from T can decay via MSSM gauge interactions.

1-loop level via diagram shown in
From the potential (42), upon integration of the T states, betweenl and T , the mixing of the order ∼ A T vu M 2 T is emerged. This, in turn, via T -gaugino (higgsino) loops will induce e i → e j γ decays. However, with M T ∼ few·TeV and A T /M T < ∼ 1/10, thel − T mixing will be < ∼ 10 −2 . This will be enough for adequate suppression of such rare processes. Much will also depend on a details of the flavor structure of A T and M 2 T matrices, which require separate investigation and is not pursued here.
Concluding, let us emphasize that the decays of the states from Σ, Σ and T (within type II and type III soft see-saw scenarios respectively) in SM and LSP states proceed via EW interactions and there is no low bound on the value of X (unlike to the type I soft see-saw scenario. See the discussion before and after Eq. (20)).

Discussion and Outlook
In this paper we have presented possibilities for radiative neutrino mass generation within SUSY theories. Suggested mechanisms open broad prospects for further investigations.
Since extensions, we have suggested, have the lepton number violation in the soft SUSY breaking sector with new states near the TeV scale, the models would have peculiar collider signatures. This gives possibilities for testing the origin of neutrino masses at accelerator experiments. This issue deserves separate investigation in a spirit of Refs. [14], [15]. It would be also interesting to exercise the lepton number violating higher dimensional operators (such as d ≥ 7, ∆L = 2 couplings studied in different scenarios earlier [16], [15]) and investigate how they may emerge from the soft SUSY breaking terms.
Since within presented models new states lie near the TeV scale and they have some family dependent interactions, one expects to have new contributions to the rare processes such as µ → eγ, τ → (e, µ)γ, µ → 3e, etc. These also could be the signatures of the presented scenarios and may serve for models' test. Similar concern to new possible contributions to the EW precision parameters (T, S and U).
Finally, would be challenging to embed considered models in SUSY Grand Unification (GUT) such as SU (5) and SO(10) GUTs. Because of the GUT symmetry, new relations and constraints would emerge, making models more predictive. These and related issues will be addressed elsewhere.

A SUSY Breaking. µ, B and A-type Couplings
In the superconformal formulation, the supergravity Lagrangian density is given by [17], [18] where K and W (Φ) are the Kähler potential and the superpotential respectively, while f IJ (Φ) is the gauge kinetic function (for the chiral superfield strength W αI obtained from the vector superfield V I ). φ is the compensator chiral superfield. Superspace integrals of (A.1) (and throughout this paper) should be understood as F and D-densities of (conformal) SUGRA as given in refs. [17], [18]. In general, the scalar potential consists of two parts. The D-term potential (A.3) and the F -term potential V F . The latter is obtained from (A.1) by integrating F -terms. Doing so, setting φ = M P l and going to the Einstein-frame one obtain: Few comments about the definitions are in order. In the 'covariant' derivative D I (with respect to the field Φ I ) we have: ∂ I ≡ ∂ ∂Φ I and K I ≡ ∂K ∂Φ I . The object KJ I is an inverse of the matrix K IJ build from the second derivatives of the Kähler potential K IJ = ∂ 2 K ∂Φ I ∂Φ † I . Thus, The DW can be defined as a column, while DW is its hermitian conjugate row: Thus, the first entry in the brackets of Eq. (A.4) can be written as DW (K ′′ ) −1 DW . We will be dealing with superfields X i of the visible sector and with superfield X through which the SUSY breaking takes place in a hidden sector. For all chiral superfields unified notation Φ I = {X, X i } is used. The Kähler potential K, together with minimal quadratic (kinetic) terms, will also include non-holomorphic higher order terms: Φ J Φ K Φ M + · · · + · · · + h.c. (A.5) The superpotential is the sum of the visible and hidden sector superpotentials, denoted by W v (X i ) and W h (X) respectively. The form of W h (X) should insure SUSY breaking, i.e. F X = 0. Within our consideration, the couplings in Eq. (A.5), in combination with the superpotential, will be responsible for generation of µ type terms as well as for B and A-type soft SUSY breaking terms. Namely, with the expression of the scalar potential we can calculate the mass 2 terms (corresponding to soft mass 2 and B-terms) and trilinear scalar couplings (corresponding to the soft SUSY breaking A-terms): More detailed discussion about these couplings will be given in the next subsection after discussing the details of the SUSY breaking.

A.1 SUSY Breaking via Polonyi Superpotential
For the SUSY breaking hidden sector superpotential W h (X) we consider Polonyi superpotential [19] of the form: The coupling J insures non zero F X , while the constant C is needed to cancel cosmological constant. The hidden sector superpotential (as usually) explicitly breaks the R-symmetry. 9 This may be considered as a soft breaking, because it does not makes troubles in a visible sector. With X's minimal Kähler potential K = X † X, the superpotential (A.8), by proper selection of J and C gives SUSY breaking Minkowski vacuum, but with X ∼ M P l . The reason for this is the following. Minimum condition in a Minkowski vacuum fixes J and C. With this, the potential for X has the linear term ∼ m 2 3/2 M P l X and the quadratic term ∼ m 2 3/2 |X| 2 (i.e. potential's curvature) and the VEV of X is basically set from the ratio of these two terms: X ∼ m 2 3/2 M P l /m 2 3/2 = M P l . The situation will change, i.e. one can get desirable value of F X and suppressed X , with higher order terms in the Kähler potential. For our purposes will be enough to include the coupling ∆K(X) = −(X † X) 2 /(4M * 2 ). As we will see, with M * ≪ M P l the scalar potential will develop high curvature(∼ m 2 ), which will suppress the VEV of X. Thus, we will consider X's Kähler potential: .9) and analyze this case in details. Since the gravitino mass is given by [11], [21]: and we are looking for a solution with suppressed X , we will use the parametrization J =mM P l , C = cmM 2 P l , α = M P l /M * , x = X/M P l , (A.11) wherem will turn out to be of the order of m 3/2 and c is a constant. With these, using in (A.4) forms of (A.8) and (A.9), we will get the potential: One can easily check that in the region |x| < 1/α, with α ≫ 1 (i.e. for M * ≪ M P l ), the potential has the minimum. The constant c can be selected in such a way that the cosmological constant is zero in this minimum. Doing this, upon the minimization of (A.12), we find: (A.14) In In the minimum, the masses of real and imaginary scalar components of the X field will be ≃ √ 3m 3/2 M P l /M * . The region |x| < 1/α has minimum with a SUSY breaking. Although, the VEV of the X field can be strongly suppressed, the F X -term and D X W in the minimum are of the order of ∼ m 3/2 M P l : 16) In the regions |x| > /α the potential is not bounded from below. However, the tunneling from the branch of |x| < 1/α to the either other branches are extremely suppressed. As far as the Kähler coupling −(X † X) 2 /(4M * 2 ) [the second term of Eq. (A.9)] is concerned, it can be generated by integration of states of mass ∼ M * which couple with the X field. For instance, having two chiral superfields Ω, Ω and the superpotential couplings λXΩ 2 + M Ω ΩΩ, at 1-loop level the Kähler potential receives the quartic correction 10 ∆K(X) = − |λ| 4 12π 2 M 2 Ω (X † X) 2 . This justifies analysis performed above.
Within considered framework, the visible sector superfields X i will couple with X via higher order terms in the Kähler potential. Within our consideration, the latter will have the form: These couplings, as shown in [12] will generate µ, B and A-type terms, as well as the Yukawa couplings. From the expression of the F -term potential (A.4), we have following relevant terms: Simple way for computing the loop corrections to the Kähler potential is to use formalism given in Ref. [22].
Therefore we have obtained the B-terms: (A.19) of the needed value. The A-terms will have desirable values withM = (3m 3/2 M 2 P l ) 1/3 ∼ 10 −5 M P l . With this scale, we will have For the fermionic states Ψ i (coming from X i superfield) the µ-terms 1 2 µ ij Ψ i Ψ j can be calculated from the mass formulae [21]: .21) Using this and the forms of (A.8), (A.9) and (A.17), we get: From the mass formulae (A.21), we can also extract The Yukawa couplings. With the definition of the Yukawa coupling 1 2 Since, X can be strongly suppressed, the Yukawa coupling also can have desirable suppression. Namely, with M * M P l < ∼ 3 · 10 −4 , if the Y ijk corresponds to the neutrino Dirac Yukawa couplings Y ν , then according to Eq. (A.23) they will be |Y ν | < ∼ 10 −7 , i.e. irrelevant for the neutrino masses. If the (visible) superpotential involves the trilinear Yukawa-type terms (A.24) then, using (A.4) and (A.7) we can see that the corresponding A-term is also induced due to λ ijk coupling, but is strongly suppressed by the M 2 * /M 2 P l factor: We close this appendix by commenting on the possibility of the soft gaugino mass generation. Since within our framework we are applying the R-symmetry, the corresponding operator should be consistent with it. For instance, if the superpotential's R charge is selected as ω = 4, 11 then we also have R(W α W α ) = ω = 4 and the D-term effective operator responsible for gaugino mass will be d 4 θ X † X 2 W α Wα M P l . From this, the gaugino mass will be MṼ ∼ 1 8M P l (M * /M P l ) 4 , which for M * /M P l ∼ 2 · 10 −4 , m 3/2 ∼ TeV, gives the desirable value MṼ ∼ m 3/2 . Different possibility, for geugino mass(∼ m 3/2 ) generation would be to include the operator d 2 θ X M P l W α W α . The latter, although explicitly breaks R-symmetry (similar to hidden sector superpotential), but do not spoil any phenomenology, also can be considered as a plausible option.

B Deriving Effective Couplings λ 4 Coupling induced byÑ states
Here we derive the effective operator (6) obtained by integrating out the statesÑ . The relevant terms are given in the potential of Eq. (18). Not taking into account the EW symmetry breaking effects, we set D-terms to zero and do not consider their effects.
Since we are deriving an effective operator, relevant at low energies, we ignore the kinetic terms (setting momenta to zero). Then the equations of motion forÑ,Ñ * are: where p, j indices numerate RHN states, while α(= 1, 2, 3) is a family index. We will be interested in the case when the mass scales M 2 N are larger than the terms A N h u . Then the mixings betweeñ N andν can be ignored at the leading order. Therefore,Ñ states can be integrated from Eqs. (B.1). The latter in a matrix form are: where the matrix at l.h.s of (B.2) contains the block sub-matrices. Having in general the block matrix of the form: it's inverse (in case DetA = 0 and DetD = 0 ) is given by [11]: Using this, from (B.2) we obtaiñ Plugging the solution (B.5) back in (18), after grouping various terms and some simplifications, we obtain the effective d = 4, ∆L = 2 interaction term i.e. that given in Eq. (6), with: The corresponding diagram is given in Fig. 2. In the case of relatively small B N -terms (e.g. is simplified to the form given in Eq. (19). In this case, we integrate out the scalar components from the superfields Σ andΣ. The latter will be denoted by same symbols as superfields they are coming from. The relevant potential terms are given in Eq. (25). As for the case withÑ states, here we also ignore kinetic terms, EW symmetry breaking effects and also mixings of components of h u , h d with corresponding states of Σ,Σ. Eqs. of motion for Σ * andΣ (only two of them are independent) are: From (B.8) we find solutions: (B.9) Plugging these solutions back into the potential (25), we get λ 4 -type [of Eq. (6) B.10) and also quartic term with respect tol: With scales M Σ MΣ > ∼ |B Σ |/3, the couplings of (B.10) and (B.11) reduce to those given in Eqs. (26) and (27)  The amplitude, corresponding to the diagram of Fig. 5-(b), is given by where C and P L,R are charge conjugation and projection matrices respectively. Using the properties / pP L,R = P R,L/ p , CP L,R = P L,R C , C −1 = −C , P 2 R,L = P R,L , P R P L = 0 = P L P R (C.2) the numerator of the integral in (C.1) gets simplified to µ 2 (pq)P L C −1 and we obtain: We can rewrite the fractions of (C.3) in a following way and . (C.5) Note the following properties of the coefficients: Moreover, using the identity , (C.8) the identities of Eq. (C.6) and 2 α=1 (−1) α+1 = 0, from (C.7) we get: with γ αnm ≡ (−1) α+1 (c 2 α − a 2 n − b 2 m )A n B m . (C.9) Using the Feynman parametrization (C. 10) in the integral of Eq. (C.9), allows to perform integration with p and q. Due to the properties in (C.6) and 2 α=1 (−1) α+1 = 0, the divergent parts disappear (as should be) and we remain with finite result given by: (C.11) Since the amplitude, defined in (C.3) (we have calculated), accounts for the operator 1 4 (l T Ml)h u h u , we can identify the contribution to the neutrino mass matrix given in Eq. (38) [for definitions' references see also the comment after Eq. (38)].