Table of Contents Author Guidelines Submit a Manuscript
Advances in High Energy Physics
Volume 2015, Article ID 152394, 26 pages
Research Article

Radiative Corrections to from Three Generations of Majorana Neutrinos and Sneutrinos

1Instituto de Física de Cantabria (CSIC-UC), 39005 Santander, Spain
2Departamento 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 SCOAP3.


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) [57], with the addition of heavy right-handed Majorana neutrinos, and where the well-known seesaw mechanism of type I [813] 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 [1821] 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 [2331]), nonnegligible contributions to electric dipole moments of charged leptons [3234], 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 [4648] have been supplemented by the leading and subleading two-loop corrections; see [49] and references therein. Together with leading three-loop corrections [5053] 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, [5658]), 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 [813] 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, 6265]. 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 [7075] 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.

Figure 1: Generic one-loop Feynman-diagrams contributing to the computation of the one-loop new corrections to the Higgs boson mass form neutrinos and sneutrinos. Here .

Following a similar analysis here as the one performed in [16] for the one generation case, it is illustrative to expand the renormalized self-energy in powers (notice that only even powers of are present in this expansion [16]) of :where means terms in the expansion of in powers of . For the present case with three generations represents shortly products of two Dirac matrices, such as or ; equivalently, refers to combinations of four matrices.

The first term in this expansion is independent of both and and represents, therefore, the pure gauge contribution (i.e., the result for ), which is already present in the MSSM. On the other hand, the term proportional to is actually of order (see [16] 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 terms. Thus we have In the one generation case, the Dirac mass is related to the light, , and heavy, , neutrino physical masses by [16] . In the three generations case, a similar functional dependence of with the physical masses in and in is found, as it is explicitely manifested in the parametrization of (20). This means that the Yukawa contribution in (70c), being proportional to , grows with the Majorana masses, therefore leading to potential nondecoupling 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 [16], and recently in [17], showing that the presence and relevance of the term in (70c) depend on the chosen renormalization scheme for .

In order to better understand where these differences come from, it is interesting to look first for the 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 system. By computing the one-loop contributions from the diagrams in Figure 1 we have obtained the following analytical result for the contributions from three generations of neutrinos and sneutrinos to the bare self-energy:In this expression, for shortness, we have set , and we have considered the most simple case with just one single soft mass scale in the slepton sector, , with . is defined in (59) and is again the renormalization scale. The corresponding result for the is obtained from the above formula by replacing , .

First of all, it should be noted that the result in (71) is a pure radiative correction with an overall factor given bySecondly, a good check of our computation in (71) is that by setting to zero all the entries in the matrix except for one in the diagonal (for instance, ) we recover the result of the one generation case, in full agreement with the expressions in Appendix  A of [17] (with and ).

The result in (71) shows, most importantly, that the bare self-energy has a nonnegligible 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 (71), one arrives at the following expressions for the counterterms in the various schemes:Then, one can easily find the relation among the corresponding renormalized values, at this same level of approximation. Using, for instance, the renormalized value in the OS scheme, , which is independent, as the reference value to be compared with in this illustrative exercise, we getFinally, using the computed expressions at of the bare self-energy and the counterterms one obtains the renormalized self-energy at this same order. In the case of the scheme we getwhich can be rewritten in terms of simply asNotice that there are no terms proportional to in (76), since they are cancelled by the , , , and counterterms. We have numerically studied the accuracy of these approximate 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 with respect to the full expressions for all the explored parameter space (including for nonzero values of , , and ).

It is also straight forward to check that by setting properly the matrix entries in (75) and (76) we recover again the proper results for the one generation case, in agreement with [16, 17].

Similarly, one can derive the corresponding expressions in the other considered schemes. In the we getAnd 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 [16, 17]

In summary, in this section we have analyzed the relevant differences among the various schemes for and the wave function renormalizations, and these differences have been understood in terms of 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 Figure 2. (a) displays the renormalized self-energies in three schemes that are independent: OS, DEC, and . In all the cases we plot the full one-loop result from neutrinos and sneutrinos evaluated at the tree Higgs mass, , as a function of . In this example the instabilities that are found in the OS scheme are clearly visible, in comparison with the stability of the and DEC schemes. These “dips” are due to thresholds encountered in the loop diagrams and, as can be seen in Figure 2, appear at values approximately twice each one of the soft SUSY-breaking parameters . We have checked that these “dips” are indeed very narrow and profound. For aritrary close values to threshold they go to due to the fact that the imaginary part of the standard one-loop function [77] is not differentiable at threshold. These instabilities occur as long as width effects are not taken into account. We also see that, for the input values in this plot, the numerical values for the renormalized self-energies of the OS, DEC, and 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 independent schemes. This difference of the HM has been studied in [17] in the one generation case and it has been understood in terms of the substantially different contributions in the pure gauge part, that is, of , which are numerically relevant. For instance, comparing the HM with the DEC approximate results for the mass correction in [17], the first one is a factor of larger than the last one (for , e.g., this yields a factor of 2.8). We have found agreement with this numerical factor in our numerical results for , in the region out of the instabilities.

Figure 2: Comparison among the various schemes. (a) The renormalized self-energies evaluated at in the OS, DEC, and as functions of , for  GeV,  eV, and  GeV. (b) The predictions of the mass differences , for (dashed lines) and OS and DEC (solid lines), as functions of and for several choices of the Majorana masses, (GeV): (in light blue); (in purple), and (in green). The rest of input parameters are fixed as in (79). is defined in (44).

Figure 2(b) compares the predictions for the Higgs mass correction among the different renormalization schemes in various examples with different choices for the Majorana masses and their hierarchies. Again the full one-loop renormalized self-energies are considered and the simple formula for the Higgs mass correction in (44) is used. In this plot we have chosen the as the reference scheme to be compared with, such that represents the difference in the prediction of the mass correction in the scheme with respect to the prediction in the 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 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 1015 GeV. The comparison with the scheme, whose result is dependent, shows that, in order to get a prediction close to the other schemes, within say a 1 GeV interval, a value of at the near proximity of the highest Majorana mass should be chosen.

5. Numerical Analysis of

In this final section we show some numerical results for the one-loop corrections to the light Higgs boson mass, (via (44)). Using the DEC scheme, the OS scheme or another scheme that decouples the heavy mass scales completely would yield small effects (except where the numerical instabilities occur as demonstrated in Section 3.3). Since every scheme, however, has its advantages and disadvantages as discussed in Section 3.3 we choose here to use the scheme. The numerical results in other schemes can be inferred from these by using the results in the previous section. While by definition not showing full decoupling, the 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 nonlogarithmic finite terms are not removed in this scheme translates into a finite contribution of which will leave a nonvanishing radiative contribution from the neutrinos and sneutrinos into the Higgs mass correction. Furthermore, we are interested in different scenarios where the Majorana masses can range from the extreme large values of order 1014-1015 GeV down to low values of order 103 GeV and, correspondingly, we will explore these scenarios keeping explicitly the contributions from particles. Consequently, the numerical analysis is performed as a function of all relevant parameters that will be varied in a wide range: the masses of the light neutrinos, the masses of the heavy Majorana neutrinos, and the mixing provided by the 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:The masses of the other two light neutrinos are obtained from and the mass differences given in (22), implying that these light neutrinos of our reference case are quasidegenerate.

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  GeV. Here it should be noted that in the non(s)neutrino part of the calculation a renormalization of and the wave function of the two Higgs doublets has been used (with ). The choice of a different renormalization scale in the estimate of within the MSSM has been discussed at length in the literature (see, for instance, [67, 68]), but it is not relevant for the present work given the fact that we are using this as a given value (fixed here to 125 GeV) and we are estimating just the shift with respect to this value due to the new sectors (given by (44)).

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.(i)Normal hierarchy (NH) is as follows: 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:(ii)Inverted hierarchy (IH) is as follows: is the lightest neutrino, and its mass will be our input value. The mass of the other two neutrinos again is fixed by the experimental mass differences:with and 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 (i.e., ) matrix in terms of the input parameters of the light sector, and , and of the heavy sector, and , 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 nonperturbative. In order to ensure that   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 tootherwise, the point in the parameter space is rejected.

5.1. 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, , with the Majorana mass as in the one generation case [16]. 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 has been set to 0.1 eV and 1014 GeV, respectively. The result for the one-generation case delivered in such a way is shown in Figure 3(a). In Figure 3(b) it is shown the behavior of the three generations case with three equaly heavy neutrino masses; that is, . 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 (70a)where corresponds to setting all the Yukawa couplings to zero and is the remaining contribution. Within our approximation of (44), they are related to the renormalized self-energy as follows: It should also be noted that, similarly to the one generation case, the full mass correction changes from positive values in the low region to negative values in the region of large  GeV. In particular, for the reference values in (79), it is  GeV.