Neutrino masses in RPV models with two pairs of Higgs doublets

We study the generation of neutrino masses and mixing in supersymmetric R-parity violating models containing two pairs of Higgs doublets. In these models, new RPV terms $\hat H_{D_1} \hat H_{D_2} \hat E$ arise in the superpotential, as well as new soft terms. Such terms give new contributions to neutrino masses. We identify the different parameters and suppression/enhancement factors that control each of these contributions. At tree level, just like in the MSSM, only one neutrino acquires a mass due to neutrino-neutralino mixing. There are no new one loop effects. We study the two loop contributions and find the conditions under which they can be important.


Introduction
Neutrinos have a non-zero mass matrix, as is indicated by neutrino oscillation experiments. This fact requires some extension of the Standrad Model (SM) that incorporates both their masses and their mixing angles [1,2,3]. The experimental data [4], ∆m 2 32 = 2.32 +0.12 −0.08 × 10 −3 eV 2 , ∆m 2 21 = (7.5 ± 0.20) × 10 −5 eV 2 , sin 2 (2θ 32 ) > 0.95, sin 2 (2θ 12 ) = 0.857 ± 0.024, sin 2 (2θ 13 ) = 0.095 ± 0.010, exhibit a mild mass hierarchy, two large mixing angles, and one mixing angle that is somewhat smaller. This structure poses a challenge for new physics where, generally, mass hierarchies come with small mixing angles. This is solved when different neutrinos obtain their masses from different sources. Then, cancellations in the determinant of the mass matrix can arise naturally, making its value smaller than the typical values of the elements of the matrix. Neutrinos in R-Parity Violating (RPV) supersymmetric models have been widely studied [5] and have been shown to be a framework in which this property is accomplished. In these models one neutrino acquires a mass at tree level through neutrino-neutralino mixing, while the other two acquire their masses from loop effects.
Models with extra Higgs doublets have been widely studied both in the context of the SM [6] and supersymmetry (SUSY) [7]. In the SUSY case, the simplest way to ensure anomaly cancellation is to add pairs of down-type and up-type Higgs fields. Lately, such models have been proposed as a way of naturally lifting the mass of the lightest Higgs boson, which in the Minimal Supersymmetric Model (MSSM) cannot be 125 GeV without some amount of fine tuning [8]. When R-parity is not imposed in these models, new renormalizable terms of the formĤ D iĤ D jÊ arise in the superpotential. Such new terms can substantially contribute to the neutrino mass matrix since their couplings are less constrained than the conventional leptonic RPV couplings.
In this work we study how neutrino masses arise in a general supersymmetric model with more than the minimal number of Higgs doublets. The large number of free parameters in the model does not allow to make predictions without any kind of further assumption. Nevertheless, we identify the suppression and enhancement factors in the various contributions to the neutrino mass matrix.
We find that, even with two pairs of Higgs doublets, only one neutrino acquires a mass at tree level, just like in the MSSM. We describe the loop diagrams generated by the new RPV terms in the superpotential, which arise at the two loop level, and in the appendix we give expressions for the relevant one loop diagrams within our model. We study which of these diagrams may give relevant contributions to the neutrino masses.
One major issue in models with several Higgs doublets is that generally they generate flavor changing neutral currents (FCNCs) that can cause severe phenomenological problems. There are several ways to avoid such bounds, for example by assuming a specific texture for the Yukawa couplings to the quark sector, or by assuming Minimal Flavor Violation (MFV), see, for example [6].
In this work we only concentrate on the leptonic sector and thus we do not elaborate on the quark sector, and just assume that one of the available solutions to the FCNCs bounds is in place.

The model
We work with a general RPV low-energy supersymmetric model with one extra pair of Higgs doublets, namely, we consider two up-type and two down-type Higgs doublets. We follow the notation of [9] where the model with just one pair of Higgs doublets is fully described. Neutrino masses arise from diagrams which violate lepton number by two units. In order to avoid the bounds from proton stability, we choose only terms which still preserve Z 3 baryon triality [10]. When Rparity is not imposed, the down-type Higgs supermultipletsĤ D 1 andĤ D 2 have the same quantum numbers as the lepton supermultipletsL i . We denote the five supermultiplets by one only symbol L I (I = 0, 1, 2, 3, 4) such thatL 0 ≡Ĥ D 1 ,L 1 ≡Ĥ D 2 andL 1+i ≡L i . Throughout this work we will use the following index notation: upper-case Latin letters for the extended five-dimensional lepton flavor space, Greek letters for four-dimensional flavor spaces and lower-case Latin letters for three-dimensional ones.
The relevant renormalizable superpotential for this model is whereĤ U i , i = 1, 2, are the two up-type Higgs supermultiplets,Q n are the quark doublet supermultiplets,Û m (D m ) are the up-type (down-type) quark supermultiplets, andÊ m are the singlet charged lepton supermultiplets. The n and m are flavor indices. The coefficients λ IJm are antisymmetric under the exchange of the indices I and J. The usual MSSM µ-term is now extended to two fivedimensional vectors, µ 1I and µ 2I . Note that, in comparison with the RPV models already studied in [11,12,13], a new type of trilinear λ-term arises for the two down-type Higgs supermultiplets, which is less constrained than the conventional leptonic RPV terms, whereλ m = λ 01m .
In order to compute all the contributions to the neutrino masses we need to consider the following soft supersymmetry breaking terms: which correspond to the A-terms and B-terms of the superpotential and the new scalar mass terms.
The usual MSSM B-term is now extended to a combination of five-dimensional vectors B 1I and B 2I , and the MSSM single mass term for the down-type Higgs boson together with the 3 × 3 lepton mass matrix are now extended to a 5 × 5 matrix, M 2 L IJ . We also define with The value of these vacuum expectation values can be determined by minimizing the potential.
Performing this minimization is beyond the scope of this work.

Tree level neutrino masses
The neutrino mass matrix receives contributions both from tree and loop level effects. In this section we study the mass matrix that arises at tree level.
The tree level masses arise from RPV mixing between the neutrinos and the neutralinos, as shown in Fig. 1. Below we will first study which are the alignment conditions of the five-dimensional expectation value ofL I and the couplings µ 1I and µ 2I , so that the mass which arises at tree level is within the experimental bounds. We will see how, even though we have doubled the number of Higgs-fields, still only one neutrino acquires a mass at tree level and we will give an explicit expression for that mass.
In our model we have a 9 × 9 mass matrix for the neutralinos. In the basis {B,W , where we neglect the effects of non-renormalizable operators, it is given by where M 1 is the Bino mass, M 2 is the Wino mass,v x = v x /v, c W ≡ cos θ W , s W ≡ sin θ W and θ W is the Weinberg angle. Note that none of the angles between the five-vectors v I , µ 1I and µ 2I is small.
The R-parity conservation limit corresponds to the case where the three vectors are coplanar. Small R-parity breaking manifests itself by the deviation of v I from the plain determined by µ 1I and µ 2I .
Such deviation can be parametrized by the angle ξ such that whereâ is a unit vector in the direction of the vector a and the angle measures the alignment of µ 1I and µ 2I . The cross product is defined on the three-dimensional space generated by the three five-vectors.
In order to find the masses, the first thing to note is that the mass matrix has rank seven and thus there are two massless states at tree-level. The product of the seven non-vanishing eigenvalues can be extracted from Eq. (8), and reads: where we have defined mγ = M 1 c 2 W + M 2 s 2 W . Note that when v I , µ 1I , and µ 2I are in the same plane, ξ = 0 and thus det M N = 0.
In order to get an estimate of the masses we consider the electroweak breaking and SUSY breaking scales to be roughly equal and we denote them bym. When we consider all the relevant masses to be of orderm the product of the seven non-vanishing masses should satisfy: det M N ≤ m 6 m 3 . where m 3 is the mass of the heaviest neutrino. In order for the neutrino masses to be within the experimental bounds we thus require where we used sin χ ∼ 1. We see that the expression we get is similar to the one for the case of the MSSM [11]. The small angle in the MSSM is the one between the µ and v vectors while here it is the angle between the plane generated by the two µ-like vectors and v. ν i ν j µ kiμkj m χ α In order to obtain the neutrino mass matrix we need to diagonalize the 9 × 9 matrix M N . This computation is simplified by considering the hierarchical structure of the matrix to diagonalize: where M µ and therefore we may integrate out the six neutralinos. From now on we work in the Note that a basis in which all the v I 's except one are zero could also be chosen, however, we prefer to keep our results in a more basis independent fashion. To integrate out the neutralinos we use the see-saw mechanism, where M is a Majorana mass and µ is a Dirac mass, and obtain the eigenvalues: Now, defining the following ratios, where β is the usual angle defined by the ratio v u /v d = tan β. We find the neutrino mass matrix: where and we have defined, In the last step of Eq. (17) we have taken all the relevant masses to bem.
The tree level neutrino masses are the eigenvalues of the rank one matrix in Eq. (16) and therefore there is just one massive neutrino: where µ i = µ ij . We define in the following m 3 > m 2 > m 1 . As expected, the tree level neutrino mass is quadratically proportional to the small parameter that measures the RPV.

Loop contributions to the neutrino mass matrix
The neutrino mass matrix receives contributions from loop diagrams with ∆L = 2. There are one loop contributions due to RPV couplings that are present also in models with one pair of Higgs doublets. They have already been thoroughly studied (see for example [13,17,18]), and we collect them in Appendix B for completeness.
Here we concentrate on the new diagrams that arise only once the second pair of Higgs doublets is introduced. Strictly speaking, the only new term that is introduced isλ. Yet, below we also consider effects that are due to the extended B-term, namelyB, which has been defined in (38).
We find that the new effects that are generated by the newλ term in the superpotential enter the neutrino masses only at two loops. Roughly speaking, this is because theλ term does not involve any neutrinos. Thus it only breaks lepton number by one unit in the charged lepton sector and the transformation of this breaking into the neutrinos appears at one loop. Since we need two of them, we end up with a two loop effect.
The effects of theB coupling on the neutrino mass matrix arise both at one and two loops.
The one loop effect is collected in Appendix B. Here we include some of the results for two loop diagrams in order to give an estimate of their possible importance. In general, we expect such two loop effects to be smaller than the one loop effects that the MSSM also presents. Yet, since the coefficientsλ k are less constrained than the usual RPV coefficients, these two loop diagrams could give important contributions to the neutrino mass matrix.
There are two types of effects that we call separable and non-separable two loop contributions to the neutrino matrix. We study them both below.

Separable contributions
For the separable contributions we study the Dirac-like neutrino-neutralino mixing (see Fig. 2). We define an effective coupling for this mixing at first order, The effective coupling µλ iα corresponds to the diagram in Fig. 2(a), and can be expressed as: where the Z's refer to the appropriate mixing matrices defined as in the MSSM [14,15,16] but enlarged so that they accommodate the extra particle states of our model. In the last step, we have  set ∆m 2 l i ≈ 2m l im and ml i ∼ m χm ∼m. The effective coupling µB iα is represented in Fig. 2(b) and can be expressed as where B ik is defined in Eq. (39) and I 3 is given in Eq. (58) (Eq. (59) for the equal masses case). In the final step we have taken all the masses to be at the supersymmetry breaking scale and we use C αnki The separable contribution to the neutrino mass matrix that is proportional to the couplingλλ where we used the approximation m χα ∼m. This contribution is suppressed by two loop factors, two RPV couplings and two leptonic Yukawa couplings. The latter makes this contribution irrelevant in most cases.
Moving to the one that depends onλ kB we get where in the last step we consider m χα ∼m. The suppression factors in this case are given by two loop factors, one Yukawa coupling and the two RPV couplingsλ andB.  Last we show the result for the loop that depends onBB. It is given by where in the last step m χα ≈m is considered. The suppression factors in this case are given by two loop factors and the two RPV couplingsλ,B. Since there is no leptonic Yukawa coupling in this case, this is the least suppressed of these contributions.

Non-separable contributions
We now move to discuss non-separable two loop diagrams. We have found that there are several of them. We include here three representative cases in order to have an insight of their possible importance. These diagrams are represented in Fig. 3, and we discuss them in turn below.
For theBλ-diagram in Fig. 3(a) we find where and I 6 is defined in Eq. (67). Note that C αmnki 2 has several subtractions of Z's and so it could undergo large cancellations. Taking all the masses to be at the electroweak scale, and using C αmnki where I 6 for the equal masses case has been computed in Eq. (68). This contribution to the neutrino mass matrix is suppressed by a lepton mass, the trilinear RPVλ-coupling, the bilinear supersymmetry-breaking RPVB-coupling, and two loop factors.
Moving to theλλ-diagram in Fig. 3 where, and I 5 is defined in Eq. (62). Taking all the masses to be at the electroweak scale, and considering C αmn 3 ∼ 0.5, we find: where I 5 for the equal mass case has been computed in Eq. (66). This contribution to the neutrino mass matrix is suppressed by two lepton masses, two trilinear RPVλ-couplings, and two loop factors.
Finally, for theBB-diagram in Fig. 3(c), the result reads where, and I 7 is defined in Eq. (69). Note that C αmnkij

4
, just as C αmnki 2 , has several subtractions of Z's and so it could also undergo large cancellations. Taking all the masses to be at the electroweak scale, and using C αmnkij 4 ∼ 0.5, we find: where I 7 for the equal masses case has been computed in Eq. (70). This contribution to the neutrino mass matrix is suppressed by two bilinear supersymmetry-breaking RPVB-couplings, and two loop factors. Note that there is no Yukawa suppression for this diagram.

Conclusions
We study new sources of neutrino masses in RPV supersymmetric models with an extra pair of Higgs doublets. In these models there is a new type of RPV term in the superpotential of the formλ kĤD 1Ĥ D 2 E k . Such a term is forbidden in the MSSM since λ is antisymmetric in its first two indices. There are also similar new soft SUSY breaking terms. These new terms violate lepton number by one unit and therefore two such terms can induce Majorana neutrino masses. We find that the tree level effects that arise due to neutrino-neutralino mixing, contribute to the mass of only one neutrino, just like it happens in the MSSM. The value of this mass is quadratically proportional to the small R-parity breaking parameter, which in this case is measured by the deviation of the vector v from planarity with respect to the two µ-like vectors.
At the loop level we find that the new term can contribute to the mass matrix only through two loop diagrams. Thus, in general we expect such terms not to be significant. The estimates of the different diagrams are given in Eqs. There is, however, one factor that tells them apart which is the amount of Yukawa suppressions. We see that the number of Yukawa factors is the same as the number ofλ couplings.
If we make the assumption that all RPV parameters are of the same order, that is,B/m 2 ∼ µ/m ∼λ, the Yukawa suppression governs the hierarchy. In that case the diagrams without anỹ λ couplings are the most important, that is, Eqs. (26) and (35) are expected to give the dominant effect. Nevertheless, the are one loop effects proportional to twoB's as in Eq. (55) and thus it is unlikely that the two loop effects will be important.
On the other hand, if we consider another plausible assumption, namely that the only coupling that is significant isλ, we find that its effect is always suppressed by one small Yukawa, and so it can be important only whenλ is very large. In this case, we could considerB/m ∼μ ∼λm l and so the leading contributions will be Eqs. (24), (25), (29), and (32).
Our results can be extended to other similar models. They include models where the extra Higgs states are not just simple duplication of the MSSM one. They may be relevant also to a case study in [20] where non-holomorphic terms like EH D H † U can appear. To conclude, neutrino masses can be used to put bounds on any model with lepton number violation. In the model we considered, due to the fact that the new term we study couples only to right handed charged leptons, its contribution to neutrino masses is somewhat suppressed. Thus, neutrino masses may not give severe bounds on such models.

Acknowledgments
We thank Daniele Alves, Jeff Dror, Javi Serra, and Tomer Volansky for helpful discussions.

Appendix A Feynman Rules
In this Appendix we give the set of Feynman rules in our model necessary for describing all the diagrams studied in this work. As a reference for notation we have followed the MSSM Feynman rules in [14]. For every rule described here, there is one with all arrows reversed and complex conjugated couplings (except for the explicit i). In all the cases, fermions are taken to be in their eigenstate basis and sfermions in a basis where they are their supersymmetric partners.
In this model there are RPV bilinear µ-like terms involving a neutrino which arise from Eq. (2), RPV bilinear terms involving neutral scalars and RPV bilinear terms involving charged scalars, both arising from Eq. (4). These vertices and their Feynman rules are represented below where we used The trilinear RPV vertices which include two Higgs fields arise from Eq. (3) and are represented below The triliniear R-parity conserving vertices involving a neutrino are There are other R-parity conserving vertices which we have also extended to include them in our diagrams. They are : ν i ν j l nL l kRlkL l nR λ ink λ jkn The soft supersymmetric breaking RPV terms combined inB ik andB i{h,H j ,A j } , defined in Eqs.
(39) and (38) respectively, also produce contributions to the neutrino masses at the loop level as represented in Fig. 5(a).
The one loop contribution is given by where I 4 is defined in Eq. (60).
Finally, we study theμB Loops represented in Fig. 5(b). These kind of loops contribute like: whereμ iα is defined in Eq. (36). Note that in this result we have neglected terms which are proportional to the tree-level masses.
ν i ν j χ α ν iν j h, H k , A k and where µ is the dimensional regularization scale, and T 3 has been evaluated in [19]. Note that even where in the last step the integral is computed numerically. Next we have .