Advances in High Energy Physics

Volume 2015 (2015), Article ID 152394, 26 pages

http://dx.doi.org/10.1155/2015/152394

## Radiative Corrections to from Three Generations of Majorana Neutrinos and Sneutrinos

^{1}Instituto de Física de Cantabria (CSIC-UC), 39005 Santander, Spain^{2}Departamento de Física Teórica and Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

Received 16 January 2015; Revised 23 April 2015; Accepted 29 April 2015

Academic Editor: Enrico Lunghi

Copyright © 2015 S. Heinemeyer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

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, in particular concerning decoupling, is included.

#### 1. 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) [5–7], with the addition of heavy right-handed Majorana neutrinos, and where the well-known seesaw mechanism of type I [8–13] 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 [14].

In this MSSM-seesaw context, the smallness of the light neutrino masses, , appears naturally due to the induced large suppression by the ratio of the two very distant mass scales, namely, the Majorana neutrino mass that represents the new physics scale and the Dirac neutrino mass , which is related to the electroweak scale via the neutrino Yukawa couplings , by . The Higgs sector content in the MSSM-seesaw is as in the MSSM, that is, composed of two Higgs doublets. is the ratio of the two vacuum expectation values, , and . Small neutrino masses of the order of eV can be easily accommodated with large Yukawa couplings, , if the new physics scale is very large, within the range GeV. This is to be compared with the Dirac neutrino case where, in order to get similar small neutrino masses, extremely tiny and hence irrelevant, Yukawa couplings of the order of 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 [15]. 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, [16, 17] (see also [18–21] for previous evaluations). Further striking phenomenological implications [22] 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 [23–31]), nonnegligible contributions to electric dipole moments of charged leptons [32–34], and also the occurrence of sneutrino-antisneutrino oscillations [35] as well as sneutrino flavor-oscillations [36].

It is worth recalling that the seesaw mechanism is not the only way to generate neutrino masses in the context of supersymmetry (see, for instance, [37, 38]). In fact there are many well-known extensions of the MSSM that can generate small neutrino masses besides the various types of high and low scale Seesaw models (see, e.g., [39] for a review and references therein). One possible alternative to the addition of right-handed neutrinos is the incorporation of R-parity violating interactions to the MSSM, which can introduce the lepton number violating terms that are needed for the small neutrino mass generation. Indeed, R-parity violation can be produced in many ways: spontaneously, explicitly, by bilinear terms, by trilinear terms, and so forth; see, for example, [40, 41]. Another popular extension of the MSSM is the Next-to-Minimal-Supersymmetric-Standard-Model (NMSSM) (see, for instance, the review in [42]), which includes an extra chiral singlet superfield with zero lepton number, offering a solution to the so-called -problem of the MSSM and providing an extra tree level mass term to the SM-like Higgs boson which raises its mass above that of the lightest Higgs boson of the MSSM. In this NMSSM, as in the MSSM, the small neutrino masses can be generated either by allowing for R-parity violating terms or by adding extra chiral singlet superfields carrying nonvanishing lepton number (like, for instance, right-handed neutrinos). The SSM [43] can also solve the problem and generate masses for the neutrinos by adding to the MSSM right-handed neutrino superfields and R-parity breaking terms.

It should be noted that each of the above mentioned extensions of the MSSM leads to different phenomenological implications, including those in the neutrino and in the Higgs boson sectors. Our preference for the particular choice of extended MSSM with three generations of right handed neutrinos and sneutrinos, and with a seesaw mechanism of type I, is mainly because, as we have said above, it is the simplest extension of the MSSM compatible with neutrino data that naturally allows for large neutrino Yukawa couplings. It is precisely this interesting possibility of large neutrino Yukawa couplings what can induce large radiative corrections to the lightest Higgs boson mass, and thus the (s)neutrino sector phenomenology is directly linked to the Higgs sector. Other extensions of the MSSM could also induce relevant corrections to the Higgs boson mass from the additional superfields and the new input parameters associated to the neutrino mass generation. For instance, within the NMSSM, in addition to the tree level enhanced Higgs boson mass, one may generate relevant mass corrections from the TeV-scale right-handed neutrinos via their interactions with the zero-lepton-number chiral singlet superfield while having small neutrino Yukawa couplings [44]. Alternatively, one may also generate relevant corrections to the Higgs boson mass from TeV-scale right-handed neutrinos, within the context of the Inverse Seesaw Models, that allow for large Yukawa couplings but introduce in addition a small lepton number violating parameter [45].

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. While the initial evaluations and analyses of corrections to concentrated on the one-generation 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. We will focus here on the corrections to the lightest and will present the full one-loop contributions from the complete three generations of neutrinos and sneutrinos and without using any approximation. It should be noted that the extrapolation from the one generation to the three generations case cannot be trivially done due to the relevant generation mixing in the latter and, therefore, the corresponding radiative corrections must be explicitly and separately computed. A crucial 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 [16, 17] that this strongly 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 will not be possible to obtain information from a precise measurement on the Majorana mass scale. On the other hand, however, the precise prediction of in the presence of Majorana (s)neutrinos and the understanding of these corrections in the different schemes (and their respective decoupling behavior) used in the calculations, is desirable.

For the estimates of the total corrections to in the MSSM-seesaw, obviously, the one-loop 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 in the non- sector, that is, in the MSSM* without* massive neutrinos, can be summarized as follows. Full one-loop calculations [46–48] have been supplemented by the leading and subleading two-loop corrections; see [49] and references therein. Together with leading three-loop corrections [50–53] and the recently added resummation of logarithmic contributions [54], the current precision in is estimated to be ~2-3 GeV [49, 54, 55].

A summary and discussion of the previous estimates of neutrino/sneutrino radiative corrections to the Higgs mass parameters can be found in [16], 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 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 Section 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 Appendix A (to our knowledge, they are not available in the previous literature). In Section 3 we discuss the renormalization procedure and emphasize the differences between the selected renormalization schemes. The corresponding analytic analysis can be found in Section 4. A numerical evaluation and in particular the dependence on the (hierarchical) Majorana mass scales are given in Section 5. Finally, our conclusions can be found in Section 6.

#### 2. The MSSM-Seesaw Model

In order to include the proper neutrino masses and oscillations in agreement with present neutrino data (see, for instance, [56–58]), 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 [8–13] is implemented which requires in addition to the Dirac neutrino mass matrix, , the introduction of a new so-called Majorana mass matrix, . This matrix 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 [16, 35, 36] is a complex Yukawa matrix, while is a complex symmetric mass matrix. The indices represent generations (with ), the indices refer to doublets components, and . Omitting the generation indexes, for brevity, the involved superfields are as follows: is the new superfield that contains the right-handed neutrinos and their partners , while the other superfields are as in the MSSM; that is, contains the lepton doublet and its superpartner , contains the sfermion and fermion singlets , and the and are the Higgs superfields that give masses to the down and up-type (s)fermions, respectively. Here and in the following, refers to the particle-antiparticle conjugate of a fermion defined as follows:where and are the particle-antiparticle conjugation and charge conjugation, respectively.

The superfields , , and can be chosen such that and are real and nonnegative diagonal matrices, whereas , in contrast, is a general complex matrix.

The additional sneutrinos induce new relevant terms in the soft SUSY-breaking potential. Following [16, 35, 36] it can be written aswhere , are Hermitian matrices in the flavor space, is a generic complex matrix, and 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 aswhere are the vacuum expectation values (vev) of the fields, , , and . and denote the masses of the and boson, respectively.

Finally, starting with the superpotential of (1), the Yukawa couplings of the neutrinos and their corresponding mass terms can be derived:where are the two component fermion field superpartners of the corresponding scalar component of the super fields.

##### 2.1. 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 and is given bywhere we have used again the notations for generation indexes and and are the Dirac and Majorana mass matrices, respectively, which have been introduced in the previous section (4).

Notice that the particle-antiparticle conjugation operator 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),If a neutrino is a Majorana fermion it is invariant under . As a result, .

of (6) can be rewritten in a more compact form:whereis a complex symmetric matrix which can be diagonalized by an unitary matrix :Here, the diagonal elements of , , are the nonnegative square roots of the eigenvalues of .

The interaction eigenstates are the left and right handed components of the neutrino fields, and (with ), and are related to the mass eigenstates (with ) in the following way:where here and from now on we shorten the notation to . Similarly for the -conjugate relations,In the seesaw limit, that is, if (the Euclidean matrix norm is defined by for a matrix whose elements are given by ), an analytic perturbative diagonalization in blocks can be performed by expanding in powers of the dimensionless parameter matrix . This allows us to separate the light sector from the heavy sector by the introduction of a matrix:

Two independent blocks of neutrino mass matrices are obtained once this matrix is inserted in (10):

The matrix of (15) is already diagonal and its diagonal elements , , and are approximately the three respective Majorana masses, , , and . The diagonalization of the matrix of (14) is performed as usual by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary matrix [59, 60], given bywhereand the notations and have been used. Here, are the mixing angles of the light neutrinos, is the Dirac phase, and are the two Majorana phases.

As a result, the mass eigenvalues corresponding to light Majorana neutrinos and heavy Majorana neutrinos are given, respectively, by

In this work, in order to make contact with the experimental data, we have used the Casas-Ibarra parametrization [61], which provides a simple way to reconstruct the Dirac mass matrix by using as inputs the physical light and heavy neutrino masses, the matrix, and a general complex and orthogonal matrix :where and where we have considered the following parametrization:where , , and , and are arbitrary complex angles.

Thus, our set of input values consist of , , , and and for , , , and we use their suggested values from the experimental data 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 matrix:Notice that for light neutrinos with a normal hierarchy and for an inverted light neutrino hierarchy. These values are compatible with the present experimental data. Specifically, the recent global fit NuFIT 1.3 (2014) [57] setswhere 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 and three neutrinos is given, in compact form, byHere 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 :where and indexes run from 1 to 6 and and indexes run from 1 to 3.

The gauge interactions of (the have no interactions since they are singlets) with the neutral gauge boson are given, in compact form, byWhen expressed in terms of the physical neutrino basis it giveswhere the indexes and run from 1 to 6 and runs from 1 to 3.

##### 2.2. Sneutrino Mass and Interaction Lagrangians

Following [36], we will express the sneutrino mass terms in a compact matrix form by defining two six-dimensional vectors and . In this new basis, the mass Lagrangian of the sneutrinos has the formwhere and are Hermitian matrices while is a complex matrix and the three of them can be expressed in blocks of matrices as follows:where the subscripts stand for and/or . The matrices and for are general complex matrices with no restrictions, but and , for , are Hermitian matrices and complex symmetric matrices, respectively.

The expressions of the different blocks of matrices that enter in the complete sneutrino mass matrix are the following:where we have assumedwith 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 unitary matrix as follows:

The relations between the interaction eigenstates and the mass eigenstates are then given bywhere runs from 1 to 3 and from 1 to 12. Again we shorten the notation to .

Finally, the contributions from the -terms, the -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 (36), the previous Lagrangians of (37) can be expressed in terms of the physical sneutrino basis . We have omitted to write them here for brevity. The derived Feynman Rules for both neutrinos and sneutrinos are collected in Appendix A.

#### 3. Radiative Corrections to the Higgs Mass

Contrary to the SM, in the MSSM two Higgs doublets are required, and , which can be decomposed as

The Higgs spectrum contains two CP-even neutral bosons , one CP-odd neutral boson , two charged bosons , and three unphysical Goldstone bosons and is related to the components of and via the orthogonal transformations:where

In the Feynman diagrammatic (FD) approach and assuming CP conservation, the higher-order corrected CP-even Higgs boson masses in the MSSM are derived by finding the poles of the -propagator matrix. The inverse of this matrix is given bywhere the tree level masses of the CP-even Higgs bosons are given byand denotes the renormalized self-energy. The poles of the propagator are obtained by solving the equation

It has been shown [16] that the mixing between these two Higgs bosons can be neglected in a good approximation for the neutrino/sneutrino contributions. Moreover, if the one-loop 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 byHere denotes the one-loop corrections to the renormalized Higgs-boson self-energy from the neutrinos/sneutrinos sector and denotes the higher-order corrected light CP-even Higgs boson mass, calculated with the help of FeynHiggs [49, 54, 62–65]. In this way approximates the new corrections arising from the new neutrino/sneutrino sectors with respect to the MSSM corrected Higgs mass, as shown in [16]. It should be noted that the two class of mass corrections, the ones from the MSSM sectors and the ones from the new neutrino/sneutrino sectors, are separately renormalizable. Therefore, in this paper we will use (44) in order to compute the one-loop radiative corrections to the lightest Higgs boson mass.

##### 3.1. Renormalized Higgs Boson Self-Energy

At one-loop level, the renormalized self-energies can be expressed through the unrenormalized self-energies, , the field renormalization constants, , and the mass counterterms, :

The mass counterterms arise from the Higgs potential. We introduce the following counterterms: denotes the mass of the CP-odd Higgs boson and are the tadpoles in the Higgs potential, that is, the terms linear in the fields , respectively.

Choosing , , , , , and 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 readIf we choose to give one renormalization constant to each Higgs doublet,we obtain the relationsUsing the renormalization of the vacuum expectation values of the Higgs doublets,the counterterm can be expressed in terms of the field renormalization constants:

This last relation is based on the fact that the divergent parts of and are equal, so one can setThe validity of this equation has been discussed in [66].

##### 3.2. 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 towhere 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 aswhere denotes the one loop contributions to the respective Higgs tadpole graph.

On the other hand, is just a Lagrangian parameter, and 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 , its physical meaning, and its relation to observables, as it happens within the SM for the weak mixing angle .

##### 3.3. Renormalization Schemes for

There are different possible renormalization schemes for , as has been extensively discussed in the literature; see, for instance, the discussion in [67, 68]. Notice that, due to the relation in (52), the renormalization scheme for is closely related to the scheme for the field renormalization constants and . Next, we will review some different choices for the renormalization of that have been considered previously in the literature and discuss their respective advantages and disadvantages.

###### 3.3.1. 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 scheme: where we have used the notation . Following (52), the counterterm is then given byThe notation 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, bywhere is related to the dimension by and is the Euler constant. Notice that we have not specified the particular momentum at which is evaluated in (58a), (58b), and (58c) because these terms are not -dependent.

In this scheme, there is still a remaining dependence of the renormalized Green functions on the renormalization scale , which has to be fixed to a “proper” value. This choice will be discussed in more detail in the following.

The scheme is often used in the literature, because it is process independent and numerically stable by avoiding threshold effects, although it induces a gauge dependence on the parameter already at one-loop level [68]. It was also shown in [68] that for the particular case of gauges the 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 , could appear, and in these cases decoupling should be added “by hand.”

###### 3.3.2. Modified Scheme

In models where there is one mass scale much larger than the rest of the mass scales, the remaining dependence on the scale in the scheme is associated with 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 as well as numerically smaller nonlogarithmic terms. 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 and field counterterms including not only the terms proportional to the divergence but also those large logarithms. This choice defines the modified scheme (), which sets the and field counterterms as follows [16]: where the notation means that one now takes only the terms proportional to . One can see that if there is only one large scale, this scheme corresponds effectively to the choice in the scheme; namely,In a general type I seesaw with three generations, however, there will be different Majorana masses, , , and , so the choice of the “proper” renormalization scale 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 nonnegligible way into the renormalization procedure. This will be discussed in more detail below.

This scheme conserves the good properties that the scheme has, but is safe from large logarithmic contributions (while leaving the smaller nonlogarithmic contributions untouched). Consequently, this option is often used in the literature when a large scale is present in the problem. One well-known example is the loop corrections to the beta function in QCD with massive fermions. In fact such a modified 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 active flavors with the one with 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, [69]).

###### 3.3.3. 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 (62a) and (62b) can be consistently replaced by the corresponding tree masses, so the field renormalization constants are set to Using (50a), (50b), and (50c), we can write the following relations:which yields for the counterterm, using (52),

Although this OS scheme is interesting due to its intuitive physical interpretation and its decoupling properties, it can lead to large corrections to the Higgs boson self-energy, which could spoil the convergence of the perturbative expansion [67, 68]. Moreover, it also induces gauge dependence at one-loop level and, contrary to the scheme, and this dependence remains even if one chooses the class of gauges [68].

###### 3.3.4. Decoupling Scheme (DEC)

As we will see explicitly in the next section, the scheme removes the large logarithmic terms, but there are still nonlogarithmic finite terms present, which can give nondecoupling effects. It has been recently proposed [17] that those finite terms can be removed by hand, forcing the decoupling to happen. This decoupling (DEC) scheme is defined as

The convenience of this scheme in the context of effective field theories has been discussed in [17]. The advantage of this scheme is that, by construction, it implements the proper matching between the high energy theory and the intermediate energy effective theory. However, we prefer here not to use an effective field theory approach where the heavy degrees are explicitly integrated out (like the possible use of a derived one-loop effective potential), because we do not want to assume in the present computation any specific intermediate low energy effective theory, but we wish simply to ensure that the final low energy effective theory where all the non-SM particles are decoupled is indeed the SM as expected. Consequently, in our analysis we perform the one-loop computation in the full high energy theory including explicitly the heavy particles with several different mass scales involved (using an appropriate renormalization scheme) and use these masses as input parameters that will be varied in the posterior numerical analysis within a wide range from high to low energies. Correspondingly, the disadvantage of the DEC scheme is that, by assuming the MSSM as the explicit intermediate low energy effective theory, any dependence on the heavy neutrinos/sneutrinos is by construction fully removed already at the intermediate (SUSY) energy scales.

###### 3.3.5. Higgs Mass Scheme (HM)

Another possibility is to demand that some physical quantity, for example, the mass , is given at one loop level by its tree level expression:This condition defines the* Higgs mass* (HM) scheme and fixes, from (47c), the counterterm to

The HM scheme, as any other scheme that is defined in terms of physical masses, provides manifestly a gauge-independent definition of [68]. However, it is not numerically stable either, as has been shown in [68], so the convergence of the perturbative expansion is again not ensured.

#### 4. Analytic Results and Analysis of the Relevant Terms

In this section we discuss the calculation of the higher-order corrections to the light Higgs boson mass and in particular discuss analytically the decoupling behavior of the various schemes in the case of three generations of (s)neutrinos. Going from the one generation to the three generations case, due to the appearance of relevant generation mixing, the corresponding radiative corrections cannot be trivially extrapolated and they must be explicitly and separately computed.

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, and , entering the computation have been evaluated with the help of* FeynArts* [70–75] and* FormCalc* [76]. 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 Appendix A (to our knowledge, they are not available in the previous literature). 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 Figure 1. They include the two-point (one-point) diagrams in the Higgs self-energies (tadpoles) and the two-point diagrams in the boson self-energy. Here the notation is as follows: refers to all physical neutral Higgs bosons, , , and ; refers to all physical neutrinos; refers to all physical sneutrinos; and refers to the gauge boson.