Radiative corrections to $M_h$ from three generations of Majorana neutrinos and sneutrinos

In this work we study the radiative corrections to the mass of the lightest Higgs boson of the MSSM from three generations of Majorana neutrinos and sneutrinos. The spectrum of the MSSM is augmented by three right handed neutrinos and their supersymmetric partners. A seesaw mechanism of type I is used to generate the physical neutrino masses and oscillations that we require to be in agreement with present neutrino data. We present a full one-loop computation of these Higgs mass corrections, and analyze in full detail their numerical size in terms of both the MSSM and the new (s)neutrino parameters. A critical discussion on the different possible renormalization schemes and their implications is included.


Introduction
In order to account for the impressive experimental data on neutrino mass differences and neutrino mixing angles [1] physics beyond the Standard Model (SM) is needed. On the other hand, after the discovery of a Higgs boson at the Large Hadron Collider (LHC) [2,3], the problem of stabilizing the Higgs mass at the electroweak scale within the SM became even more relevant. Similarly, the existence of Cold Dark Matter (CDM) [4] has to be accounted for by an extension of the SM. Consequently, in order to incorporate neutrino masses into the SM, to stabilize the Higgs-boson mass scale and to provide a viable CDM we choose here one of the most popular extensions of the SM: the simplest version of a supersymmetric extension of the SM, the Minimal Supersymmetric Standard Model (MSSM) [5], with the addition of heavy right-handed Majorana neutrinos, and where the well known seesaw mechanism of type I [6] is implemented to generate the observed small neutrino masses. From now on we will denote this model by "MSSM-seesaw". The lightest Higgs boson in this model can be interpreted as the Higgs particle discovered at the LHC [7].
In this MSSM-seesaw context, the smallness of the light neutrino masses, m ν ∼ m 2 D /m M , appears naturally due to the induced large suppression by the ratio of the two very distant mass scales. Namely, the Majorana neutrino mass m M , that represents the new physics scale, and the Dirac neutrino mass m D , which is related to the electroweak scale via the neutrino Yukawa couplings Y ν , by m D = Y ν v sin β. The Higgs sector content in the MSSMseesaw is as in the MSSM, i.e. composed of two Higgs doublets. tan β is the ratio of the two vacuum expectation values, tan β = v 2 /v 1 , and v 2 = v 2 1 + v 2 2 = (174 GeV) 2 . Small neutrino masses of the order of m ν ∼ O(0.1) eV can be easily accommodated with large Yukawa couplings, Y ν ∼ O(1), if the new physics scale is very large, within the range m M ∼ 10 13 − 10 15 GeV. This is to be compared with the Dirac neutrino case where, in order to get similar small neutrino masses, extremely tiny, hence irrelevant, Yukawa couplings of the order of Y ν ∼ 10 −12 − 10 −13 are required.
As for all SM fermions, the neutrinos in the MSSM are accompanied by their respective super partners, the scalar neutrinos. The hypothesis of Majorana massive (s)neutrinos is very appealing for various reasons, including the interesting possibility of generating satisfactorily baryogenesis via leptogenesis [8]. Furthermore, they can produce an interesting phenomenology due to their potentially large Yukawa couplings to the Higgs sector of the MSSM, such as corrections to the light CP-even Higgs-boson mass, M h [9,10] (see also [11][12][13] for previous evaluations). Further striking phenomenological implications [14] of the MSSM-seesaw scenario are the prediction of sizeable rates for lepton flavor violating processes (within the present experimental reach for specific areas of the model parameters [15,16]), non-negligible contributions to electric dipole moments of charged leptons [17], and also the occurrence of sneutrino-antisneutrino oscillations [18] as well as sneutrino flavor-oscillations [19].
While the initial evaluations and analyses of corrections to M h concentrated on the onegeneration case to analyze the general analytic behavior of this type of contributions, in this paper we investigate the Majorana neutrino and sneutrino sectors with three generations which can accommodate the present neutrino data. Specifically, we are interested here in the indirect effects of Majorana neutrinos and sneutrinos via their radiative corrections to the MSSM Higgs boson masses within the MSSM-seesaw framework. We will focus here on the corrections to the lightest M h and will present the full one-loop contributions from the 1 complete three generations of neutrinos and sneutrinos and without using any approximation. Another issue of interest in relation with the present computation is the question of decoupling of the heavy Majorana mass scales. While it was shown for the one generation case [9,10] that this crucially depends on the choice of the renormalization scheme, no such scheme could be identified being superior to the other in all respects. Consequently, we will also comment comparatively the advantages and disadvantages of the various renormalization schemes in the present case of three generations where there are several mass scales involved. On the one hand it may not be possible to obtain information from a precise M h measurement on the Majorana mass scale, on the other hand, however, the precise prediction of M h in the presence of Majorana (s)neutrinos and the understanding of these corrections in the different schemes used in the M h calculations, is desirable.
For the estimates of the total corrections to M h in the MSSM-seesaw, obviously, the oneloop corrections from the neutrino/sneutrino sector that we are interested here have to be added to the existing MSSM corrections. The status of radiative corrections to M h in the nonν/ν sector, i.e. in the MSSM without massive neutrinos, can be summarized as follows. Full one-loop calculations [20] have been supplemented by the leading and subleading two-loop corrections, see [21] and references therein. Together with leading three-loop corrections [22] and the recently added resummation of logarithmic contributions [23], the current precision in M h is estimated to be ∼ 2 − 3 GeV [21,23,24].
A summary and discussion of the previous estimates of neutrino/sneutrino radiative corrections to the Higgs mass parameters can be found in [9], where (as discussed above) the one-generation case was calculated and analyzed. In this work, we will consider the more general three generation MSSM-seesaw scenarios with no universality conditions imposed, and explore the full parameter space, without restricting ourselves just to large or small values of any of the relevant neutrino/sneutrino parameters. In principle, since the right handed Majorana neutrinos and their SUSY partners are SU (2) × U (1) singlets, there is no a priori reason why the size of their associated parameters should be related to the size of the other sector parameters. In the numerical estimates, we will therefore explore a wide interval for all the involved neutrino/sneutrino relevant input parameters.
The paper is organized as follows. In Sect. 2, we summarize the most important ingredients of the MSSM-seesaw scenario that are needed for the present computation of the Higgs mass loop corrections. These include, the setting of the model parameters and the complete list of the Lagrangian relevant terms. A complete set of the corresponding relevant Feynman rules in the physical basis is also provided here. They are collected in App. A (to our knowledge, they are not available in the previous literature). In Sect. 3 we discuss the renormalization procedure and emphasize the differences between the selected renormalization schemes. The corresponding analytic analysis can be found in Sect. 4. The numerical evaluation and in particular the dependence on the (hierarchical) Majorana mass scales is given in Sect. 5. Finally, our conclusions can be found in Sect. 6.

The MSSM-seesaw model
In order to include the proper neutrino masses and oscillations in agreement with present neutrino data (see, for instance, [25][26][27]), we employ an extended version of the MSSM, where three right handed neutrinos and their supersymmetric partners are included, in addition to the usual MSSM spectra. A seesaw mechanism of type I [6] is implemented which requires in addition to the Dirac neutrino mass matrix, m D , the introduction of a new 3 × 3 so-called Majorana mass matrix, m M . This matrix m M is the responsible for the Majorana character of the physical neutrinos in this MSSM-seesaw model.
The terms of the superpotential within the MSSM-seesaw that are relevant for neutrino and Higgs related physics are described by [9,18,19]: The indices i, j represent generations (with i, j = 1, 2, 3), the indices a, b refer to SU (2) doublets components, and 12 = −1. Omitting the generation indexes, for brevity, the involved superfields are as follows:N = {ν * R , (ν R ) c } is the new superfield that contains the right-handed neutrinos ν R i and their partnersν R i , while the other superfields are as in the MSSM, i.e.,L contains the SU (2) lepton doublet, (ν L , l L ) and its superpartner (ν L ,l L ),R contains the SU (2) sfermion and fermion singlets {l R , (l R ) c }, and theĤ 1 andĤ 2 are the Higgs superfields that give masses to the down and up-type (s)fermions, respectively. Here and in the following, f c refers to the particle-antiparticle conjugate of a fermion f defined as follows:Ĉ : whereĈ and C are the particle-antiparticle conjugation and charge conjugation respectively.
The superfieldsL,N andR can be chosen such that Y l and m M are real and non-negative diagonal 3 × 3 matrices, whereas Y ν , in contrast, is a general complex 3 × 3 matrix.
The additional sneutrinosν R i induce new relevant terms in the soft SUSY-breaking potential. Following [9,18,19] it can be written as: where m 2 L , m 2 R are 3 × 3 hermitian matrices in the flavor space, A ν is a 3 × 3 generic complex matrix and m 2 B is a complex symmetric matrix. After the Higgs fields develop a vacuum expectation value, the charged lepton and Dirac neutrino mass matrix elements can be written as: where v i are the vacuum expectation values (vev) of the M W and M Z denote the masses of the W and Z boson, respectively.
Finally, starting with the superpotential of (1), the Yukawa couplings of the neutrinos and their corresponding mass terms can be derived: where the ψ i are the two component fermion field superpartners of the corresponding scalar component φ i of the super fields.

Neutrino mass and interaction Lagrangians
After the Higgs field develops a vacuum expectation value, the mass Lagrangian of neutrinos in the MSSM-seesaw model with three generations of ν L and ν R is given by: where we have used again the notation i, j for generation indexes, and m D and m M are the Dirac and Majorana mass matrices, respectively, which have been introduced in the previous section (4). Notice that the particle-antiparticle conjugation operatorĈ flips the chirality of a particle and changes all the quantum numbers of it. Then, it changes a left handed neutrino by a right handed antineutrino and a right handed neutrino by a left handed antineutrino. Following (2):Ĉ If a neutrino is a Majorana fermion it is invariant underĈ. As a result, ν c = ν.
The L ν mass of (6) can be rewritten in a more compact form: where is a 6 × 6 complex symmetric matrix which can be diagonalized by an unitary matrix U : Here, the diagonal elements ofM ν , m n i , are the non negative square roots of the eigenvalues of (M ν ) † M ν .
The interaction eigenstates are the left and right handed components of the neutrino fields, ν L i and ν R i (with i = 1, 2, 3), and are related to the mass eigenstates n j (with j = 1, ..., 6) in the following way: 4 where here and from now on, we shorten the notation to U ij ≡ U i,j . Similarly for thê C-conjugate relations: In the seesaw limit, i.e. if ||m D || ||m M || 1 , an analytic perturbative diagonalization in blocks can be performed by expanding in powers of the dimensionless parameter matrix ξ = m D m −1 M . This allows us to separate the light sector from the heavy sector by the introduction of a 6 × 6 matrix: Two independent blocks of 3×3 neutrino mass matrices are obtained once thisÛ ν matrix is inserted in (10): The matrix m N of (15) is already diagonal and its diagonal elements m N 1 , m N 2 and m N 3 are approximately the three respective Majorana masses, m M 1 , m M 2 , m M 3 . The diagonalization of the matrix m ν of (14) is performed as usual by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary matrix [28], U PMNS given by: where and the notation c ij ≡ cos θ ij , s ij ≡ sin θ ij has been used. Here, θ ij are the mixing angles of the light neutrinos, δ is the Dirac phase and φ 1,2 are the two Majorana phases. As a result, the mass eigenvalues m n j , corresponding to light Majorana neutrinos (ν) and heavy Majorana neutrinos (N ) are given respectively by: In this work, in order to make contact with the experimental data, we have used the Casas-Ibarra parametrization [29], which provides a simple way to reconstruct the Dirac mass matrix by using as inputs the physical light m ν i and heavy m N i neutrino masses, the U PMNS matrix, and a general complex and orthogonal matrix R: 1 The euclidean matrix norm is defined by ||A|| = tr A † A 1/2 = Σ ij |a ij | 2 1/2 for a matrix A whose elements are given by a ij where R T R = RR T = 1 and where we have considered the following parametrization: where c i ≡ cos θ i , s i ≡ sin θ i and θ 1 , θ 2 and θ 3 are arbitrary complex angles. Thus, our set of input values consist of m M 1 , m M 2 , m M 3 and θ i , and for m ν 1 , m ν 2 , m ν 3 and U PMNS we use their suggested values from the experimental data are used. For the numerical estimates in this work we will use the following input values for the light neutrino mass squared differences and the angles in the U PMNS matrix: Notice that ∆m 2 32 > 0 for light neutrinos with a normal hierarchy and ∆m 2 32 < 0 for an inverted light neutrino hierarchy. These values are compatible with the present experimental data. Specifically, the recent global fit NuFIT 1.3 (2014) [26] sets: where NH and IH refer to the normal hierarchy and inverted hierarchy cases for the light neutrinos, respectively. The interaction Lagrangian of the MSSM neutral Higgs bosons with the three ν L and three ν R neutrinos is given, in compact form, by: Here α is the angle that diagonalizes the CP-even Higgs sector at the tree-level. By using (11) and (12) the interaction Lagrangian in (24) can be expressed in terms of the neutrino mass eigenstates n i = (n 1 , . . . , n 6 ): where j and i indexes run from 1 to 6 and l and m indexes run from 1 to 3.

6
The gauge interactions of ν L (the ν R have no interactions since they are singlets) with the neutral gauge boson Z are given, in compact form, by: When expressed in terms of the physical neutrino basis it gives: where the indexes i and j run from 1 to 6 and m runs from 1 to 3.

Sneutrino mass and interaction Lagrangians
Following [19], we will express the sneutrino mass terms in a compact 6 × 6 matrix form by defining two six-dimensional vectors φ L = (ν Lν * L ) T and φ N = (ÑÑ * ) T = (ν * Rν R ) T . In this new basis, the mass Lagrangian of the sneutrinos has the form: here M 2 LL and M 2 N N are 6 × 6 hermitian matrices while M 2 LN is a 6 × 6 complex matrix; and the three of them can be expressed in blocks of 3 × 3 matrices as follows: where the subscripts A, B stand for L and/or N . The matrices M 2 A † B and M 2 A T B for A = B are general complex matrices with no restrictions, but the M 2 A † A and M 2 A T A , for A = L, N , are 3 × 3 hermitian matrices and complex symmetric matrices, respectively.
The expressions of the different blocks of matrices that enter in the complete 12 × 12 sneutrino mass matrix M 2 ν are the following: 7 where we have assumed: and with the convention of: We have to diagonalize the sneutrino mass matrix in (28) in order to obtain the twelve mass eigenstates. This matrix is hermitian, so it can be diagonalized by an 12 × 12 unitary matrixŨ as follows: The relations between the interaction eigenstates and the mass eigenstates are then given by:ν where i runs from 1 to 3 and j from 1 to 12. Again we shorten the notation toŨ ij ≡Ũ i,j .
Finally, the contributions from the F -terms, the D-terms and the soft SUSY breaking terms to the interactions of the sneutrinos with the MSSM neutral Higgs bosons are given by: By using the rotations given in (37), the previous Lagrangians of (38) to (40) can be expressed in terms of the physical sneutrino basisñ j , (j = 1, . . . , 12). We have omitted to write them here for brevity. The derived Feynman Rules for both neutrinos and sneutrinos are collected in App. A.

Radiative corrections to the Higgs mass
Contrary to the SM, in the MSSM two Higgs doublets are required, H 1 and H 2 , which can be decomposed as: The Higgs spectrum contains two CP-even neutral bosons (h, H), one CP-odd neutral boson (A), two charged bosons, (H ± ), and three unphysical Goldstone bosons, (G, G ± ), and are related to the components of H 1 and H 2 via the orthogonal transformations: In the Feynman diagrammatic (FD) approach and assuming CP conservation, the higherorder corrected CP-even Higgs boson masses in the MSSM are derived by finding the poles of the (h, H)-propagator matrix. The inverse of this matrix is given by: where the tree-level masses of the CP-even Higgs bosons are given by andΣ denotes the renormalized self-energy. The poles of the propagator ∆ Higgs are obtained by solving the equation It has been shown [9] that the mixing between these two Higgs bosons can be neglected in a good approximation for the neutrino/sneutrino contributions. Morevoer, if the oneloop contributions due to neutrinos and sneutrinos are small in comparison with the pure MSSM contributions, the correction to the light CP-even Higgs boson mass from the neutrino/sneutrino sector can be can be approximated by: HereΣ ν/ν hh denotes the one-loop corrections to the renormalized Higgs-boson self-energy from the neutrinos/sneutrinos sector, and M h denotes the higher-order corrected light CP-even Higgs boson mass, calculated with the help of FeynHiggs [21,23,[30][31][32]. In this way ∆M h approximates the new corrections arising from the new neutrino/sneutrino sectors with respect to the MSSM corrected Higgs mass, as shown in [9]. Therefore, in this paper we will use (49) in order to compute the one-loop radiative corrections to the lightest Higgs boson mass.

Renormalized Higgs boson self-energy
At one-loop level, the renormalized self-energies can be expressed through the unrenormalized self-energies, Σ(p 2 ), the field renormalization constants, δZ, and the mass counterterms, δm 2 : The mass counterterms arise from the Higgs potential. We introduce the following counterterms: M A denotes the mass of the CP-odd Higgs boson, T h,H are the tadpoles in the Higgs potential, i.e. the terms linear in the fields h, H, respectively. Choosing δM 2 Z , δM 2 W , δT h , δT H , δM 2 A and δ tan β as independent counterterms, we can express the Higgs mass counterterms as follows: where we have used the tree level relation . On the other hand, the field renormalization constants read, If we choose to give one renormalization constant to each Higgs doublet, we obtain the relations, (55c) Using the renormalization of the vacuum expectation values v i of the Higgs doublets, the tan β counterterm can be expressed in terms of the field renormalization constants: This last relation is based on the fact that the divergent parts of δv 1 /v 1 and δv 2 /v 2 are equal, so one can set: The validity of this equation has been discussed in [33]. 11

Renormalization conditions
Since there are six independent counterterms, six renormalization conditions are needed. For the masses, we choose an on-shell renormalization condition: which sets the mass counterterms to, where the gauge bosons self energies are to be understood as the transverse parts of the full self-energies.
The tadpole condition requires that the tadpole coefficients must vanish in all orders, implying at the one-loop level, so we choose the tadpole counterterms as, where T h,H(1) denotes the one loop contributions to the respective Higgs tadpole graph.
On the other hand, tan β is just a Lagrangian parameter, it is not a directly measurable quantity. Therefore, there is no obvious relation of this parameter to a specific physical observable which would favor a particular renormalization scheme. Furthermore, the choice of one particular renormalization scheme sets the actual definition of tan β, its physical meaning and its relation to observables, as it happens within the SM for the weak mixing angle θ W .

Renormalization schemes for tan β
There are different possible renormalization schemes for tan β, as has been extensively discussed in the literature, see for instance, the discussion in [34,35]. Notice that, due to the relation in (57), the renormalization scheme for tan β is closely related to the scheme for the field renormalization constants δZ H 1 and δZ H 2 . Next, we will review some different choices for the renormalization of tan β that have been considered previously in the literature.

DR scheme
One possibility is to use the field counterterms to remove just the terms proportional to the divergence in dimensional reduction. This defines the most frequently used scheme, the so-called DR scheme: where we have used the notation Σ ≡ ∂Σ/∂p 2 . Following (57), the tan β counterterm is then given by: The notation [ ] div used here means that one takes just the terms that are proportional to the divergence ∆, which is defined, as it is usual in dimensional regularization/reduction, by: where is related to the dimension d by d = 4 − and γ E is the Euler constant. Notice that we have not specified the particular momentum p 2 at which Σ is evaluated in eqs.(63) because these [ ] div terms are not p 2 dependent. In this scheme, there is still a remaining dependence of the renormalized Green functions on the renormalization scale µ DR , which has to be fixed to a "proper" value. This choice will be discussed in more detail below.
The DR scheme is often used in the literature, because it is process independent and numerically stable, although it induces a gauge dependence on the tan β parameter already at one-loop level [35]. It was also shown in [35] that for the particular case of R ξ gauges the ξ dependence cancels at one-loop resulting in a gauge invariant result. Nevertheless, this numerical stability could be lost in presence of large scales, such as the Majorana mass, since large logarithmic corrections, proportional to log(m 2 M /µ 2 DR ), could appear.

Modified DR scheme (mDR)
In models where there is one mass scale much larger than the rest of the mass scales, the remaining dependence on the µ DR scale in the DR scheme is associated to the large scale. In our case of study, the large scale is the Majorana mass (or Majorana masses in the case they are different for each of the three generations), and this will give rise to new terms in the radiative corrections involving the neutrino Yukawa coupling that are proportional to log(m 2 M /µ 2 DR ). These logarithmic terms can give large contributions for large Majorana masses, worsening the convergence of the perturbative expansion.
However, these terms can be absorbed in the tan β and field counterterms including not only the terms proportional to the divergence ∆ but also those large logarithms. This choice defines the modified DR scheme (mDR), which sets the tan β and field counterterms as follows [9]: δZ mDR where the notation [ ] mdiv means that one now takes only the terms proportional to ∆ m ≡ ∆ − log(m 2 M /µ 2 DR ). One can see that if there is only one large scale, this scheme corresponds effectively to the choice µ DR = m M in the DR scheme, namely: 13 In a general type I seesaw with three generations, however, there will be different Majorana masses, m M 1 , m M 2 and m M 3 , so the choice of the "proper" renormalization scale µ DR becomes more involved. Besides, there are also new additional (soft) mass scales from the sneutrino sector, which can be different for the three generations, and these could also a priori enter in a non-negligible way into the renormalization procedure. This will be discussed in more detail below. This scheme conserves the good properties that the DR scheme has, but is safe from large logarithmic contributions. Consequently, this option is often used in the literature when a large scale is present in the problem. One well known example in the MSSM is the case of radiative corrections from the top-stops loops, where the usual scale chosen in this mDR scheme is the top mass. Another well known example are the loop corrections to the beta function in QCD with massive fermions. In fact such a modified DR scheme was precisely first proposed in that QCD context in order to implement properly the matching conditions when crossing through the various thresholds, which relate the value of the strong coupling constant for the case of n f + 1 active flavors with the one with n f active flavors. In this QCD case the matching scale is chosen to be precisely the mass of this fermion "+1" that is crossed by (see, for instance, [36]).

On-shell scheme
An on-shell (OS) renormalization requires the derivative of the renormalized self-energy to cancel at the physical mass: At one loop level, the physical masses in (67) can be consistently replaced by the corresponding tree masses, so the field renormalization constants are set to: Using (55), we can write the following relations: which yields for the tan β counterterm, using (57), Although the OS scheme is interesting due to its intuitive physical interpretation, it can lead to large corrections to the Higgs boson self-energy, which could spoil the convergence of the perturbative expansion [34,35]. Moreover, it also induces gauge dependence at one-loop level and, contrary to the DR scheme, this dependence remains even if one chooses the class of R ξ gauges [35].

Decoupling scheme (DEC)
As we will see explicitly in the next section, the mDR scheme removes the large logarithmic terms, but there are still non-logarithmic finite terms present, which could give nondecoupling effects. Comparison among It has been recently proposed [10] that those finite terms can be removed by hand, forcing the decoupling to happen. This decoupling (DEC) scheme is defined as: δZ DEC

Higgs mass scheme (HM)
Another possibility is to demand that some physical quantity, e.g., the mass m H , is given at one loop level by its tree level expression: This condition defines the Higgs mass (HM) scheme and fixes, from (52c), the tan β counterterm to: The HM scheme, as any other scheme that is defined in terms of physical masses, provides manifestly a gauge-independent definition of tan β [35]. However, it is not numerically stable either, as has been shown in [35], so the convergence of the perturbative expansion is again not ensured.

Analytic results and analysis of the relevant terms
We have used the Feynman diagrammatic (FD) approach to calculate the one-loop corrections from the neutrino/sneutrino sector to the MSSM Higgs boson masses. The full one-loop neutrino and sneutrino corrections to the self-energies, Σ ν/ν hh and Σ ν/ν HH , entering the computation have been evaluated with the help of FeynArts [37] and FormCalc [38]. The relevant Feynman rules for the present computation with three generations of Majorana neutrinos and sneutrinos have been derived from the Lagrangians of section 2 and expressed in terms of the physical basis. The results are collected in App. A. These Feynman rules have also been inserted into a new model file which is available upon request.
The generic one-loop Feynman diagrams that enter in the computation of the renormalized self-energies are collected in Fig. 1. They include the two-point (one-point) diagrams in the Higgs self-energies (tadpoles), and the two-point diagrams in the Z boson self-energy. Here the notation is: φ refers to all physical neutral Higgs bosons, h, H, and A; n i (where i = 1, . . . , 6) refers to all physical neutrinos;ñ i (where i = 1, . . . , 12) refers to all physical sneutrinos; and Z refers to the Z gauge boson. Following a symmilar analysis here as the one performed in [9] for the one generation case, it is illustrative to expand the renormalized self-energy in powers 2 of m D : where Σ ν/ν (p 2 )  [9] for details), so it is suppressed by the Majorana mass; higher order terms in this expansion are also suppressed by inverse powers of the Majorana mass. Thus, the new relevant contributions, coming from the neutrino and sneutrino sectors are those governed by the Yukawa couplings, and can arise only from the order O(m 2 D ) terms. Thus we have: In the one generation case, the Dirac mass is related to the light, m ν , and heavy, m N , neutrino physical masses by [9] (20). This means that the Yukawa contribution in (75c), being proportional to m 2 D , grows with the Majorana masses, therefore leading to potential non-decoupling effects with respect to these masses. The question now is whether such a term is present in the renormalized self-energy and, in that case, if it is numerically relevant. This issue was first analyzed for the one generation case in [9], and recently in [10], showing that the presence and relevance of the O(m 2 D ) term in (75c) depends on the chosen renormalization scheme for tan β.
In order to better understand where these differences come from, it is interesting to look first for the O(m 2 D ) terms in the bare self-energy, where the choice of the renormalization scheme does not enter. We will focus here on the lightest CP-even Higgs boson self-energy, but the conclusions will be the same for the full (h, H) system. By computing the one-loop contributions from the hh diagrams in Fig. 1 we have obtained the following analytical result for the O(m 2 D ) contributions from three generations of neutrinos and sneutrinos to the bare self-energy: In this expression, for shortness, we have set a ν = b ν = µ = 0, and we have considered the most simple case with just one single soft mass scale in the slepton sector, mL i = mR j =: m SUSY , with i = 1, 2, 3. ∆ is defined in (64) and µ DR is again the renormalization scale. The corresponding result for the Σ HH is obtained from the above formula by replacing cos α → sin α, sin α → − cos α.
First of all, it should be noted that the result in (76) is a pure O(Y 2 ν ) radiative correction with an overall factor given by: Secondly, a good check of our computation in (76) is that by setting to zero all the entries in the m D ij matrix except for one in the diagonal (for instance, m D 11 ) we recover the result of the one generation case, in full agreement with the expressions in the Appendix A of [10] (with m D 11 = m D and m M 1 = m M ). The result in (76) shows, most importantly, that the bare self-energy has a non-negligible O(m 2 D ) term, which grows logarithmically with the Majorana masses. Nevertheless, as we have already said, we will analyze whether such a term is present or not in the renormalized self-energy. If one assumes that the Yukawa contribution from neutrinos/sneutrinos to the bare self-energy is approximated by the previous result in (76), one arrives at the following O(m 2 D ) expressions for the tan β counterterms in the various schemes: Then, one can easily find the relation among the corresponding renormalized tan β values, at this same level of approximation. Using, for instance, the renormalized value in the OS scheme, tan β OS , which is µ DR independent, as the reference value to be compared with in this illustrative exercise, we get: Finally, using the computed expressions at O(m 2 D ) of the bare self-energy and the counterterms one obtains the renormalized self-energy at this same order. In the case of the DR scheme we get: A cos 2 (α − β) cos 2 β + 2p 2 cos 2 α −M 2 Z sin β sin(α + β) 2 1 + cos 2 β cos α − sin 2β sin α , (80) which can be rewritten in terms of m h,tree simply as: Notice that there are no terms proportional to m 2 SUSY in (81), since they are cancelled by the δT h , δT H , δM 2 A and δM 2 Z counterterms. We have numerically studied the accuracy of these approximate O(m 2 D ) results, both for the renormalized self-energy and the finite contribution in the bare self-energy, and compared with their corresponding full results. We have found that they constitute extremely good approximations, leading to relative differences below 10 −4 w.r.t. the full expressions for all the explored parameter space (including for non-zero values of a ν , b ν and µ).
It is also straight forward to check that by setting properly the m D matrix entries in (80) and (81) we recover again the proper results for the one generation case, in agreement with [9] and [10].
Similarly, one can derive the corresponding O(m 2 D ) expressions in the other considered schemes. In the mDR we get: And in the OS, DEC and HM we get the expected decoupling behavior at this order, in agreement with the results for the one generation case in [9] and [10]: In summary, in this section we have analyzed the relevant differences among the various schemes for tan β and the wave function renormalizations, and these differences have been understood in terms of O(m 2 D ) contributions to the self-energies. Once we have set clearly these differences, it is a simple exercise to find the prediction in one scheme and then extract from it the prediction in another scheme.
We illustrate numerically the most relevant differences among the various schemes in Fig. 2. The plot on the left displays the renormalized self-energies in three schemes that are µ DR independent: OS, DEC and mDR. In all the cases we plot the full one-loop result from neutrinos and sneutrinos evaluated at the tree Higgs mass, p 2 = m 2 h,tree , as a function of M A . In this example the instabilities that are found in the OS scheme are clearly visible, in comparison with the stability of the mDR and DEC schemes. These "dips" are due to thresholds encountered in the loop diagrams when M A is twice the mass of one of the light sneutrinos. We also see that, for the input values in this plot, the numerical values for the renormalized self-energies of the OS, DEC and mDR are quite close to each other. In particular, in the region out of the dips, the OS and DEC values are practically identical. We have also checked that the numerical results in the HM scheme (not shown) also manifest instabilities and, furthermore, they turn out to be substantially different than in the other µ DR independent schemes. This difference of the HM has been studied in [10] in the one generation case and it has been understood in terms of the substantially different contributions in the pure gauge part, i.e of O(m 0 D ), which are numerically relevant. For instance, comparing the HM with the DEC approximate results for the mass correction in [10], the first one is a factor of (cos 2 2β) −1 larger than the last one (for tan β = 2, e.g., this yields a factor of 2.8). We have found agreement with this numerical factor in our numerical results forΣ represents the difference in the prediction of the mass correction in the scheme x respect to the prediction in the mDR scheme. Firstly, we have found again that the results of the OS and the DEC schemes are practically indistinguishable. We also see that for the input values explored in this plot, the predictions in these OS and DEC schemes differ from the predictions in the mDR scheme in 1 GeV at most, and this largest difference is for the case when the heaviest Majorana mass is at the largest considered value of 10 15 GeV. The comparison with the DR scheme, whose result is µ DR dependent, shows that, in order to get a prediction close to the other schemes, within say a 1 GeV interval, a value of µ DR at the near proximity of the highest Majorana mass should be chosen.
For the following numerical analysis in the next section we will use the mDR scheme as our default. The numerical results in other schemes can therefore be inferred from these by using the results in this section. Our choice in favor of the mDR scheme is based on the fact that it combines several of the desired properties: stability, perturbativity and gauge invariance at the one-loop level. Besides, this scheme is safe of large logarithms introduced by the large Majorana scales. The fact that the non-logarithmic finite terms are not removed in this scheme, translates into a finite contribution of O(m 2 D ) which will leave a non-vanishing radiative contribution from the neutrinos and sneutrinos into the Higgs mass correction. This will be analyzed numerically in full detail in the next section. By choosing instead another scheme like OS, DEC or HM, this finite correction does not appear in the Higgs mass correction. However, in the light of the relations given in (78) and in (79) one can suspect that these O(m 2 D ) corrections could (re-)appear in another observable 3 (for instance, in leptonic Higgs decays) via the induced contributions from the corresponding definition of tan β. These different choices for renormalization of tan β, including totally, partially, or not including the threshold corrections from heavy particles is an open issue, and it might be related to the question of whether tan β is a low energy or a high energy parameter in this theory. Unfortunately, this issue is not yet set and therefore there should be a priori no fundamental preference for a particular choice in the renormalization scheme of tan β, in absence of an experimental measurement for this parameter.

Numerical analysis of ∆M h
In this section we show the full one-loop numerical results for the Higgs mass correction ∆M h , that is evaluated here using (49) in the mDR scheme, as a function of the relevant parameters: the masses of the light neutrinos, the masses of the heavy Majorana neutrinos and the mixing provided by the R matrix in the case of three generations, as well as the MSSM parameters. Unless stated otherwise, we set the parameters to the following reference values: (84) 3 We wish to thank W.Hollik for some illuminating discussions on this point 21 We assume that the other MSSM parameters, in particular from the top/scalar top sector, which do not affect our results, give a corrected Higgs mass of M h ∼ 125 GeV. The masses of the other two light neutrinos are obtained from m ν 1 and the mass differences given in (22), implying that these light neutrinos of our reference case are quasi-degenerate. Two different scenarios for the mass hierarchy of the light neutrinos can be set, the normal hierarchy (NH) case and the inverted hierarchy (IH) case: • Normal hierarchy (NH): ν 1 is the lightest neutrino, and its mass will be our input value. The mass of the other two neutrinos are fixed by the experimental mass differences: • Inverted hierarchy (IH): ν 3 is the lightest neutrino, and its mass will be our input value. The mass of the other two neutrinos again, are fixed by the experimental mass differences: with ∆m 2 21 and ∆m 2 32 are given in section 2. The default choice used below is the NH case, and the IH case will be especially indicated.
Notice that we are using the Casas-Ibarra parametrization (20) that provides a prediction of the full 3 × 3 v 2 Y ν (i.e. m D ) matrix in terms of the input parameters of the light sector, m ν i and θ ij , and of the heavy sector m M i and θ i , and the last two can take in principle any value. Therefore the size of the Yukawa couplings that we are generating is related directly to these parameters, and in consequence they can be large and even non-perturbative. In order to ensure that Y ν is inside the perturbative region, for every set of input parameters we first check that all of the entries of the Yukawa matrix fulfill a perturbative condition that we set here to otherwise, the point in the parameter space is rejected.

Relation with the one-generation case
As a first check of our three generations code, we have reproduced with this code the same behavior of the Higgs mass correction, ∆M h , with the Majorana mass as in the one generation case [9]. The connection with the one generation case is done by setting the corresponding absent entries in the Dirac mass matrix to zero. For this analysis, the mass of the light and heavy Majorana neutrinos have been set to 0.1 eV and 10 14 GeV respectively. The result for the one-generation case delivered in such a way is shown in the left plot of Fig. 3. In the right plot it is shown the behavior of the three generations case with three equaly heavy neutrino masses, i.e. m M i = m M . As expected, we obtain that the Higgs mass corrections in the three generations case are three times the ones of the one generation case. Notice that we have separated the contributions to the full mass correction coming from the gauge and the Yukawa parts, according to (75a): where (∆M h ) gauge corresponds to setting all the Yukawa couplings to zero and (∆M h ) Yukawa is the remaining contribution. Within our approximation of (49), they are related with the renormalized self energy as follows: It should also be noted that, similarly to the one generation case, the full mass correction As mentioned before, the gauge part of the Higgs mass correction represents the common part with the MSSM, therefore in the following, we will focus the discussion mainly in the Yukawa part which is the new contribution, denoted here and from now on shortly as∆M h .

Sensitivity of the Higgs mass correction to the relevant SUSY parameters
We next study the effects on ∆M h of the other parameters entering the calculation: tan β, M A , mL i , mR i , a ν , b ν and µ. In order to explore these behaviours of ∆M h with the relevant MSSM parameters in presence of three Majorana neutrinos and their SUSY partners, we run with one of the parameters while the others are set to the reference values given in (84).
The behaviour of the one-loop corrections to the lightest Higgs boson mass in the mDR scheme with these relevant parameters are shown in figs 4 and 5.
We start with the analysis of the behaviour with tan β, which is shown in Fig. 4. In the left plot the behavior of the full mass correction as well as the gauge and Yukawa parts are shown. In the right plot we focus on the Yukawa contribution to the mass correction. The biggest negative correction∆M h is obtained for the lowest considered value of tan β = 2; so in the following, motivating tan β = 2 as our reference value. The numerical results for other choices of tan β in the remaining plots of this work can be easily inferred from this plot on the right. The dependence on the pseudoscalar Higgs boson mass is analyzed in Fig. 5. For M A larger than 200 GeV the behavior with M A is nearly flat. The dependence on the soft SUSYbreaking mass of the "left handed" SU (2) doublet, mL is also flat as shown in Fig. 5. The behavior with the soft mass of the "right handed" sector mR in a range similar to the other soft SUSY-breaking parameters is shown in Fig. 5. In addition, also values of mR closer to m M i are explored in this figure. The correction to the Higgs boson h mass stays flat with mR up to about mR ∼ 10 13 GeV. Above this mass scale the correction grows rapidly, reaching ∆M h ∼ −1 GeV at mR ∼ 10 14 GeV, in agreement with the results found for the one generation case in [9].
We have also checked that the behavior of ∆M h with the remaining parameters, a ν , b ν and µ, in the intervals −1000 GeV < a ν < 1000 GeV, 100 GeV < b ν < 10 4 GeV and −1000 GeV < µ < 1000 GeV are also flat as in the case of the low mass values of mR. The behaviors of∆M h with all these parameters agree as well with the results obtained in the one-generation case [9].

Sensitivity of the Higgs mass corrections to the light neutrinos
In this section we analyze the sensitivity of the mass correction to the mass hierarchy of the light neutrinos. Here we investigate the two cases of NH and IH, where the values of the rest of the parameters are fixed to the ones of our reference scenario given in (84). Fig. 6 shows the behavior of the Yukawa part of the mass correction with the mass of the lightest neutrino, ν 1 and ν 3 for the NH (solid lines) and IH (dashed lines), respectively. We show the Yukawa contribution to the mass correction (vertical left axis) as well as the sum of the three neutrino masses (vertical right axis) for each value as a function of the lightest neutrino mass. We conclude that, even though the numerical result of∆M h for both hierarchies are quite similar, the Higgs mass corrections found in the IH case are slightly bigger than the ones of the NH case.

Sensitivity of the Higgs mass corrections to the heavy neutrino masses
In this section the behaviors of the mass correction with the masses of the heavy Majorana neutrinos as well as with the R matrix are analyzed. As mentioned before, the R matrix of any other real R matrix different from the reference value R = 1. This independence on the particular real R value can be understood from the fact that, as we have mentioned before, and with the definition of m D given in (20), we find: As the three light neutrinos in (84) are quasi degenerate, m diag ν ≈ m ν 1 1, and since here R is a real and orthogonal matrix, then (90) becomes independent on R, i.e.: In contrast, when a complex R matrix is implemented, the result in (91) is no longer true and∆M h grows with the size of both Re(θ i ) and Im(θ i ), as can be seen in the right panel of Fig. 7. There we plot∆M h for three different values θ 1 = 0, π/2e iπ/8 , π/2e iπ/4 while the other two angles, i.e. θ 2 and θ 3 , are set to zero. We have checked that similar growing behaviors with the other complex θ 2,3 angles are found.
Next we study the case where there is a hierarchy between the three heavy Majorana neutrino masses. In order to perform a complete analysis with hierarchical heavy neutrinos, we have scanned the Majorana masses m M 1 and m M 2 in the range 10 12 m M 1,2 10 14 GeV for two different values of m M 3 . As a result, we have obtained the two contour plots that are shown in Fig. 9. Due to the fact that we are assuming in practice that the light neutrinos are quasi degenerate and that there is no mixing among the heavy Majorana neutrinos (R = 1), the behavior of the Higgs mass correction is symmetric in all the three Majorana masses and consequently, the biggest correction is obtained when the three masses are equal and set to the highest value, i.e. 10 14 GeV in these plots. We have checked that once the value of m M 3 lies below 10 12 GeV there is no appreciable sensitivity to that mass, so the result will be the same as in the left panel of Fig. 9. Similarly to the previous degenerate case, there is not sensitivity to the choice of the real R matrix in the hierarchical case either, as can be understood from the result in (91) that also holds here. Therefore, the results in Fig. 9 are valid for all values of real R. Finally we analyze the imprints of the mixing of the hierarchical heavy neutrinos in∆M h when a complex R matrix is implemented. Fig. 10 shows the∆M h contours in the general case of three Majorana masses, m M 1,2,3 , and when one of the three θ i angles is fixed to 3π/4 e iπ/4 while the other two are set to zero. As before, the biggest correction is obtained when all the three Majorana masses are degenerate and have their biggest considered value of 10 14 GeV. The symmetry shown in Fig. 10 with respect to the three masses m M 1,2,3 is a consequence of the quasi degeneracy assumed of the three light neutrinos. When the three θ i angles are non zero and complex,∆M h becomes considerably larger than in the real case, as can be seen in Fig. 11 where we have chosen as an illustrative example, θ 1 = 3π/8 e iπ/4 , θ 2 = π/2 e iπ/5 and θ 3 = 3π/4 e iπ/7 . The larger the arguments of the angles θ i are, the larger ∆M h becomes. However, the size of these θ i , as well as the size of the m M i , are constrained by perturbativity of the Yukawa coupling. In this context it should be remembered that large corrections for∆M h would not be reliable within the approximation used here of (49).

Conclusions
In this paper we have presented the full one-loop radiative corrections to the renormalized CP-even Higgs boson self-energies and to the lightest Higgs boson mass, M h , from the three-generations in the neutrino-sneutrino sector within the context of the MSSM-seesaw. This work extends and completes the previous calculation in the simplified one-generation case [9]. The most interesting features in this MSSM-seesaw are that the neutrinos, contrary to other fermions, are assumed to be Majorana particles, and that the origin for the light neutrino masses, again in contrast to the other fermions, are generated by means of the seesaw mechanism with the addition of heavy right handed neutrinos with large Majorana masses. As a by-product, we have included here the complete set of Feynman rules in this MSSMseesaw for the three-generation (s)neutrino case relevant for this work (again extending and completing Ref. [9]). This includes the vertices for the interactions of the neutrinos and sneutrinos with the Higgs sector and with the Z boson. These Feynman rules have been presented in terms of all the physical masses and mixing angles of the particles involved, in particular in the mass eigen basis of the light and heavy Majorana neutrinos, as well as their light and heavy SUSY partners.
Our computation is a complete one-loop calculation in the Feynman diagrammatic approch without any simplifying assumptions. The corresponding analytical results are also presented in terms of the physical neutrinos, sneutrinos, Z, and Higgs bosons masses. Particular attention has been paid to the renormalization of tan β and of the wave func-tion of the two Higgs doublets. As was discussed previously in the literature, the dependence of the prediction of M h on the Majorana mass scales depends strongly on the choice of the tan β renormalization. Various schemes have been analyzed, where each scheme exhibits advantages and disadvantages. Especially, the "modified DR" scheme (mDR) was contrasted to other schemes, like the "more physical" OS and HM schemes and the "decoupling" scheme (DEC). The latter one leads, hence its name, to a full decoupling of the heavy Majorana mass scales in∆M h , which we confirm here for the three generations case. The dependence on the heavy Majorana scales is hidden however inside the renormalized tan β DEC and it is not totally irrelevant. Since this tan β DEC should be treated as a low energy parameter of the theory, the radiatively induced threshold effects from the heavy Majorana scales could reappear in another low energy observable, via this tan β DEC value. Regarding the comparison with the "more physical" schemes, like OS and HM, we have seen that they can lead to potentially unstable numerical behavior in certain regions of the MSSM-seesaw parameter space. Therefore the convergence of the perturbative expansion is not ensured and these are not convenient schemes for these kind of scenarios with heavy scales. We have also found that the use of the "more traditional" DR scheme is not convenient either, since there is an extremely high sensitivity to the choice of the µ DR scale. When this µ DR scale is set to the high Majorana scale, then the large logarithmic contributions disappear and one reaches a more stable result. The absence of large logarithmic contributions, log(m M i /µ DR ), is automatically implemented in the mDR scheme. This mDR scheme, on the other hand, is closely related to the schemes used in RGE calculations, connecting the GUT scale with the electroweak scale, and it leads to numerically stable predictions for M h , gauge invariant to one-loop, while exhibiting a residual dependence of M h on the heavy Majorana mass scales. The analytic structure of those terms have been derived and fully analyzed here for the three-generations case.
In the numerical investigation we have analyzed the neutrino/sneutrino corrections to the renormalized CP-even Higgs self-energies and M h with respect to all the involved masses and parameters: m M i , tan β, M A , mL i , mR i , a ν , m ν i , θ i and b ν . These analyses have been performed in the mDR scheme. A clear prescription has also been presented to pass from this scheme to the other introduced schemes. We have ensured that our numerical scenarios are in agreement with experimental data by using the Casas-Ibarra parametrization of the neutrino sector and choosing the relevant values, e.g. of neutrino mass differences, according to the most recent experimental results. We have investigated both, the normal and the inverted hierarchy. The pure gauge contributions, which are already present in the MSSM and grow with tan β, can amount about ∆M h ∼ 150 MeV, in the low tan β ∼ 2 region of interest here, i.e. about half of the current experimental uncertainty. These corrections arise from the sneutrino sector only, and thus are independent of the assumed hierarchy in the neutrino sector. The remaining contributions,∆M h , which are sensitive to the heavy neutrinos/sneutrinos via the Yukawa couplings, are larger than the pure gauge contributions in presence of very heavy scales, and are in contrast larger at the lower values of tan β.
In particular, we have studied the size of the corrections to M h w.r.t. the Majorana mass scales (where no dependence would have been found in the DEC scheme). The largest corrections are found in the degenerate case and for the largest allowed Majorana mass values. In the present work these maximum values have been set to 10 15 GeV in order to respect the perturbativity condition on the Yukawa couplings. In the large re-gion of 10 14 GeV < ∼ m M < ∼ 10 15 GeV we find sizable negative corrections of up to ∆M h ∼ O(−5) GeV. We have also checked that the correction in the three degenerate generations case is three times the correction of the one generation case. The corrections are found to be proportional to the square of the neutrino Dirac mass scale, specifically to the combination m † D m D , and therefore can be substantially enhanced when complex θ i parameters are taken into account. However, the above commented perturbativity requirements on the Yukawa of couplings will always restrict the size of the mass correction.
In this Appendix we collect the Feynman rules derived from the interaction Lagrangian terms of section 2 within the MSSM-seesaw that are relevant for the present work. They represent the interactions between the neutrinos and sneutrinos with the MSSM neutral Higgs bosons and between the neutrinos and sneutrinos with the Z gauge bosons. All the Feynman rules are written here in the physical basis. Here c w ≡ cos θ W and we have shortened the notation as in (11), i.e. U ij ≡ U i,j ,Ũ ij ≡Ũ i,j .

Neutrinos
Three-point couplings of two Majorana neutrinos to one MSSM Higgs boson and of two Majorana neutrinos to the Z gauge boson.
iV L hn i n j P L + iV R hn i n j P R = − ig cos α 2M W sin β U * m+3,i (m † D ) mn U * nj P L + + U mi (m D ) mn U n+3,j P R + (i ↔ j) iV L Hn i n j P L + iV R Hn i n j P R = − ig sin α 2M W sin β U * m+3,i (m † D ) mn U * nj P L + + U mi (m D ) mn U n+3,j P R + (i ↔ j) iV L An i n j P L + iV R An i n j P R = iV L Zn i n j γ µ P L = −ig 2c W U mi U * mj γ µ P L + (i ↔ j)

Sneutrinos
Three-point couplings of two sneutrinos to one MSSM Higgs boson and of two sneutrinos to the Z gauge boson. All the couplings not shown here vanish.