Advances in High Energy Physics

Volume 2017 (2017), Article ID 4023493, 24 pages

https://doi.org/10.1155/2017/4023493

## Singlet Fermion Assisted Dominant Seesaw with Lepton Flavor and Number Violations and Leptogenesis

Centre of Excellence in Theoretical and Mathematical Sciences, SOA University, Khandagiri Square, Bhubaneswar 751030, India

Correspondence should be addressed to M. K. Parida

Received 23 July 2016; Revised 7 December 2016; Accepted 21 December 2016; Published 21 March 2017

Academic Editor: Alexey A. Petrov

Copyright © 2017 M. K. Parida and Bidyut Prava Nayak. 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

Embedding type I seesaw in GUTs, left-right gauge theories, or even in extensions of the SM requires large right-handed neutrino masses making the neutrino mass generation mechanism inaccessible for direct experimental tests. This has been circumvented by introducing additional textures or high degree of fine-tuning in the Dirac neutrino or right-handed neutrino mass matrices. In this work we review another new mechanism that renders type I seesaw vanishing but other seesaw mechanisms dominant. Such mechanisms include extended seesaw, type II, linear, or double seesaw. The linear seesaw, double seesaw, and extended seesaw are directly verifiable at TeV scale. New observable predictions for lepton flavor and lepton number violations by ongoing searches are noted. Type II embedding in (10) also predicts these phenomena in addition to new mechanism for leptogenesis and displaced vertices mediated by gauge singlet fermions.

#### 1. Introduction

Although the standard model (SM) of strong, weak, and electromagnetic interactions has enjoyed tremendous success through numerous experimental tests, it has outstanding failures in three notable areas: neutrino masses, baryon asymmetry of the universe, and dark matter. Two of the other conceptual and theoretical difficulties are that the SM cannot explain disparate values of its gauge couplings nor can it account for the monopoly of parity violation in weak interaction leaving out strong, electromagnetic, and gravitational interactions. In hitherto unverified extensions of the SM in the scalar, fermion, or gauge sectors, however, there are different theories for neutrino masses [1–11], which mainly exploit various seesaw mechanisms [9, 11–48].

The most popular method of neutrino mass generation has been through type I or canonical seesaw mechanism [12–17] which was noted to apply in the simplest extension of the SM through right-handed (RH) neutrinos encompassing family mixings [17]. Most of the problems of the SM have potentially satisfactory solutions in the minimal left-right symmetric (LRS) [21, 49–52] grand unified theory based on [53–70]. Although the neutrino masses measured by the oscillation data [71–74] are most simply accommodated in if both the left-handed (LH) and the right-handed (RH) neutrinos are Majorana fermions, alternative interpretations in favour of Dirac neutrino masses within the standard model paradigm have been also advanced [11, 75–78]. Currently a number of experiments [79–84] on neutrinoless double beta () decay are in progress to resolve the issue on the Dirac or Majorana nature of neutrinos [21–23].

Another set of physical processes under active experimental investigation are charged lepton flavor violating (LFV) decays, , , , and , where the minimally extended SM embracing small neutrino masses and GIM mechanism predict branching ratios many orders smaller than their current experimental limits [85–96]. However supersymmetric theories possess high potential to explain LFV decays closer to the current limits [97–104].

A special feature of left-right gauge theories [49–52] and grand unified theory (GUT) [53–60] is that the canonical seesaw formula [12–17] for Majorana neutrino masses is invariably accompanied by the type II seesaw formula [18–22] The parameters entering into this hybrid seesaw formula have fundamentally appealing interpretations in Pati-Salam model [49, 50] or GUT. In (1) is Dirac (RH Majorana) neutrino mass, is the induced vacuum expectation value (VEV) of the LH triplet , breaking VEV of the RH triplet , and is the Yukawa coupling of the triplets of . The same Yukawa coupling also defines the RH neutrino mass . Normally, because of the underlying quark-lepton symmetry in or Pati-Salam model, is of the same order as , the up quark mass matrix, that drives the canonical seesaw scale to be large, GeV. In the LR theory based upon , charged lepton mass matrix. The neutrino oscillation data then pushes this seesaw scale to GeV. Similarly the type II seesaw scale is also around this mass. With such high seesaw scales in nonsupersymmetric (non-SUSY) model or LRS theory, there is no possibility of direct experimental verification of the seesaw mechanism or the associated boson mass in near future. Likewise, the predicted LFV decay rates are far below the experimental limits.

On the other hand, if experimental investigations at the Large Hadron Collider (LHC) [105, 106] are to confirm [107] TeV scale [9, 108–111] or boson production [112, 113] accompanied by RH Majorana neutrinos in the like-sign dilepton channel with jets, , this should be also consistent with the neutrino oscillation data [71–74] through seesaw mechanisms. This is possible if the relevant seesaw scales are brought down to ~(1–10) TeV (in the RH neutrino extension of the SM, explanation of neutrino oscillation data with baryon asymmetry of the universe has been addressed for (1–10) GeV type I seesaw scale [114–116]; interesting connections with double beta decay and dilepton production with displaced vertices at LHC have been also discussed with such low canonical seesaw scales [117] and the models need fine-tuning of the Yukawa coupling which is about 3 orders smaller than the electron Yukawa coupling; neutrino mass generation with standard model paradigm and their interesting applications have been discussed in a recent review [11]). Observable displaced vertices within SM extension have been also discussed in [118, 119] and references therein.

The scope and applications of type I seesaw to TeV scale boson models have been discussed in the recent interesting review [9]. In such models D-Parity is at first broken at high scale that makes the left-handed triplet much heavier than the -mass but keeps unbroken down to much lower scale [55–57]. This causes the type II seesaw contribution of the hybrid seesaw formula of (1) to be severely damped out in the LHC scale models where type I seesaw dominates. But because is also at the TeV scale, the predicted type I seesaw contribution to light neutrino mass turns out to be times larger than the experimental values unless it is adequately suppressed while maintaining its dominance over type II seesaw. Such suppressions have been made possible in two ways: (i) using fine-tuned values of the Dirac neutrino mass matrices [9, 117, 120–122] and (ii) introducing specific textures to the fermion mass matrices and/or [98, 108, 123–136].

Possible presence of specific textures of constituent matrices in the context of inverse, linear, or type I seesaw models has been also explored [137–145]. More recently the hybrid seesaw ansatz for matter parity conserving has been applied to explain neutrino masses, dark matter, and baryon asymmetry of the universe without invoking any texture or intermediate scale in the nonsupersymmetric framework [146].

Even without going beyond the SM paradigm and treating the added RH neutrinos in type I seesaw as gauge singlet fermions at ~GeV scale, rich structure of new physics has been predicted including neutrino masses, dark matter, and baryon asymmetry of the universe. The fine-tuned value of the associated Dirac neutrino Yukawa coupling in these models is [114–116].

There are physical situations where type II seesaw dominance, rather than type I seesaw or inverse seesaw, is desirable [17–23, 112, 147–162].

In the minimal case, being a mechanism driven by intermediate scale mass of LH triplet, type II seesaw may not be directly verifiable; nevertheless it can be clearly applicable to TeV scale models in non-SUSY to account for neutrino masses [112] provided type I contribution is adequately suppressed. However, as in the fine-tuning of Dirac neutrino mass in the type I seesaw case in LR models, the induced VEV needed for type II seesaw can also be fine-tuned using more than one electroweak bidoublets reducing the triplet mass to lower scales accessible to accelerator tests. Looking to (1) and the structure of the induced VEV , the most convenient method of suppressing type I seesaw with respect to type II seesaw is to make the type I seesaw scale larger and the triplet mass much smaller, . This requires the breaking scale or . SUSY and non-SUSY models have been constructed with this possibility and also in the case of split-SUSY [157, 160] where GeV. Obviously such models have no relevance in the context of TeV scale or bosons accessible to LHC searches.

Whereas the pristine type I or type II seesaw are essentially high scale formulas inaccessible for direct verification and need fine-tuning or textures to bring them down to the TeV scale, the well known classic inverse seesaw mechanism [24] which has been also discussed by a number of authors [25–34] is essentially TeV scale seesaw. It has the high potential to be directly verifiable at accelerator energies and also by ongoing experiments on charged lepton flavor violations [85–96].

Even without taking recourse to string theories, in addition to the three RH neutrinos (), one more gauge singlet fermion per generation () is added to the SM where the Lagrangian contains the mixing mass term . The heavy Majorana mass term is absent for RH neutrinos which turn out to be heavy pseudo Dirac fermions. The introduction of the global lepton symmetry breaking term in the Lagrangian, , gives rise to the well known classic inverse seesaw formula [24] Naturally small value of in the ’t Hooft sense [163] brings down the inverse seesaw mechanism to the TeV scale without having the need to fine tune the associated Dirac mass matrix or Yukawa couplings. The presence of texture zeros in the constituent matrices of different types of seesaw formulas has been investigated consistently with neutrino oscillation data [138–140, 142, 143].

Recently models have been discussed using TeV scale heavy pseudo Dirac neutrinos [109, 164–168] where dominant RH Majorana mass term is either absent in the Lagrangian or negligible. In a contrasting situation [37, 38, 43], when , the seesaw scale , and the model avoids the presence of any additional intermediate symmetry. While operating with the SM paradigm, it also dispenses with the larger Higgs representation in favour of much smaller one leading to the double seesaw. The heavy RH neutrinos with GeV turn out to be Majorana fermions instead of being pseudo Dirac. While this is an attractive scenario in SUSY [37, 38], the coupling unification is challenging in the non-SUSY .

As discussed above if type I seesaw is the neutrino mass mechanism at the TeV scale, it must be appropriately suppressed either by fine-tuning or by introducing textures to the relevant mass matrices [9]. On the other hand if type II seesaw dominance in LR models or is to account for neutrino masses, , boson masses must be at the GUT-Planck scale in the prevailing dominance mechanisms [157, 160].

In view of this, it would be quite interesting to explore, especially in the context of nonsupersymmetric , possible new physics implications when the would-be dominant type I seesaw cancels out exactly and analytically from the light neutrino mass matrix even without needing any fine-tuning or fermion mass textures in and/or . The complete cancellation of type I seesaw in the presence of heavy RH Majorana mass term was explicitly proved in [45, 46] in the context of SM extension when both and are present manifesting in heavy RH neutrinos and lighter singlet fermions. We call this as gauge singlet fermion assisted extended seesaw dominance mechanism. Since then the mechanism has been utilised in explaining baryon asymmetry of the universe via low scale leptogenesis [45, 46] and the phenomenon of dark matter (DM) [169] along with cosmic ray anomalies [170]. More recently this extended seesaw mechanism for neutrino masses in the SM extension has been exploited to explain the keV singlet fermion DM along with low scale leptogenesis [171].

In the context of LR intermediate scales in SUSY , this mechanism has been applied to study coupling unification and leptogenesis [172–174] under gravitino constraint. Application to non-SUSY LR theory originating from Pati-Salam model [175] and non-SUSY with TeV scale , bosons have been made [176, 177] with the predictions of a number of experimentally testable physical phenomena by low-energy experiments and including the observed dilepton excess at LHC [110]. In these models the singlet fermion assisted type I seesaw cancellation mechanism operates and the extended seesaw (or inverse seesaw) formula dominates.

Following the standard lore in type II seesaw dominant models, the dominant double beta decay rate in the channel is expected to be dominated by the exchange of the LH Higgs triplet carrying . As such the predicted decay rate tends to be negligible in the limit of larger Higgs triplet mass. But it has been shown quite recently [112] that the type II seesaw dominance can occur assisted by the gauge singlet fermion but with a phenomenal difference. Even for large LH triplet mass in such models that controls the type II seesaw formula for light neutrino masses and mixings, the double beta decay rate in the channel remains dominant as it is controlled by the light singlet fermion exchanges. Other attractive predictions are observable LFV decays, nonunitarity effects, and resonant leptogenesis mediated by TeV scale quasi-degenerate singlet fermions of the second and third generations. The model has been noted to have its origin in non-SUSY [112]. All the three types of gauge singlet fermions in these models mentioned above are Majorana fermions on which we focus in this review.

This article is organised in the following manner. In Section 2 we explain how the Kang-Kim mechanism [45, 46] operates within the SM paradigm extended by singlet fermions. In Section 3 we show how a generalised neutral fermion mass matrix exists in the appropriate extensions of the SM, LR theory, or . In Section 4 we show emergence of the other dominant seesaw mechanism including the extended or inverse seesaw and type II seesaw and cancellation of type I seesaw. Predictions for LFV, CP violation, and nonunitarity effect are discussed in Section 5. Predictions on double beta decay mediated by light singlet fermions in the channel are discussed in Section 6 where we have given its mass limits from the existing experimental data. Applications to resonant leptogenesis mediated by TeV scale singlet fermions in MSSM and SUSY are briefly discussed in Section 7. Singlet fermion assisted leptogenesis in non-SUSY is discussed in Section 8. This work is summarized in Section 9 with conclusions.

#### 2. Mechanism of Extended Seesaw Dominance in SM Extension

Using the explicit derivation of Kang and Kim [45, 46], here we discuss how the type I contribution completely cancels out paving the way for the dominance of extended seesaw mechanism. The SM is extended by introducing RH neutrinos and an additional set of fermion singlets , one for each generation. After electroweak symmetry breaking, the Yukawa Lagrangian in the charged lepton mass basis gives for the neutral fermions where is the Dirac neutrino mass matrix which is equal to , being the Yukawa matrix. This gives the neutral fermion mass matrix on the basis as follows: The type I seesaw cancellation leading to dominance of extended seesaw (or inverse seesaw) [24] proceeds in two steps: as , it is legitimate to integrate out the RH fields at first leading to the corresponding effective Lagrangian Then diagonalisation of the neutral fermion mass matrix including the result of gives conventional type I seesaw term and another of opposite sign leading to the cancellation. The light neutrino mass predicted is the same as in the inverse seesaw case given in (2).

It must be emphasized that the earlier realisations of the classic inverse seesaw formula [24] were possible [25–32, 34] with vanishing RH Majorana mass in (12).

Under the similar condition in which the type I seesaw cancels out the Majorana mass of the sterile neutrino and its mixing angle with light neutrinos are governed by As is naturally small, it is clear that type I seesaw now controls the gauge singlet fermion mass, although it has no role to play in determining the LH neutrino mass. These results have been shown to emerge [47, 109, 110, 112, 176, 177] from with gauge fermion singlet extensions by following the explicit block diagonalisation procedure in two steps while safeguarding the hierarchy with the supplementary condition .

#### 3. Generalised Neutral Fermion Mass Matrix

A left-right symmetric (LRS) gauge theory at higher scale () is known to lead to TeV scale asymmetric LR gauge theory via D-Parity breaking [55–57]. This symmetry further breaks to the SM gauge symmetry by the VEV of the RH triplet leading to massive , bosons and RH neutrinos at the intermediate scale . Instead of it is possible to start directly from which has been discussed at length in a number of investigations that normally leads to the type I type II hybrid seesaw formula. In the absence of additional sterile neutrinos, the neutral fermion matrix is standard form. Here we discuss how a generalised neutral fermion mass matrix that emerges in the presence of additional singlet fermions contains the rudiments of various seesaw formulas. As noted in Section 1, the derivation of the minimal classic inverse seesaw mechanism [24] has been possible in theories with gauge singlet fermion extensions of the SM [25–34, 178]. Extensive applications of this mechanism have been discussed and reported in a number of recent reviews [11, 99–102, 179, 180]. Exploring possible effects on invisible Higgs decays [181–183], prediction of observable lepton flavor violation as a hall mark of the minimal classic inverse seesaw mechanism has attracted considerable attention earlier and during recent investigations [103, 184–189]. The effects of massive gauge singlet fermions have been found to be consistent with electroweak precision observables [190–192]. Earlier its impact on a class of left-right symmetric models has been examined [193–195]. Prospects of lepton flavor violation in the context of linear seesaw and dynamical left-right symmetric model have been also investigated earlier [178].

It is well known that 15 fermions of one generation plus a right-handed neutrino form the spinorial representation of grand unified theory [53, 54]. In addition to three generation of fermions , we also include one -singlet fermion per generation . We note that such singlets under the LR gauge group or the SM can originate from the nonstandard fermion representations in such as or .

Under symmetry the fermion and Higgs representations are as follows:

*Fermions*

*Higgs*where is a D-Parity odd singlet with transformation property under . When this singlet acquires VEV , D-Parity breaks along with the underlying left-right discrete symmetry but the asymmetric LR gauge theory is left unbroken down to the lower scales. The gauge theory can further break down to the SM directly by the VEV of RH Higgs triplet or the RH Higgs doublet . The D-Parity odd (even) singlets were found to occur naturally in GUT theory [55–57]. Designating the quantum numbers of submultiplets under Pati-Salam symmetry (), the submultiplet is whereas the submultiplet is . Likewise the neutral component of the submultiplet behaves as , but that in behaves as . Thus the GUT scale symmetry breaking can occur by the VEV of in the direction , but can occur by the VEV of in the direction . Likewise can occur by the VEV of the neutral component , but can occur by the VEV of the neutral component of . As an example, one minimal chain with TeV scale LR gauge theory proposed recently in the context of like-sign dilepton signals observed at LHC is

In this symmetry breaking pattern all LH triplets and doublets is near the GUT scale, but RH triplets or doublets are near the breaking intermediate scale which could be ~(few–100) TeV. Out of two minimal models with GUT scale D-Parity breaking satisfying the desired decoupling criteria [110, 112], dominance of extended seesaw in the presence of gauge singlet fermions has been possible in [110] with single intermediate scale corresponding to TeV scale and bosons. The extended seesaw dominance in the presence of fermion singlets in has been also realised including additional intermediate symmetries and where observable proton decay, TeV scale boson and RH Majorana neutrinos, observable proton decay, oscillation, and rare kaon decay have been predicted. Interestingly the masses of boson and leptoquark gauge bosons of have been predicted at ~100 TeV which could be accessible to planned LHC at those energies where boson scale ~(100–1000) TeV matching with observable oscillation and rare kaon decay have been predicted. But the heavy RH neutrino and boson scales being near TeV scale have been predicted to be accessible to LHC and planned accelerators [176, 177]. That non-SUSY GUTs with two-intermediate scales permit a low mass boson was noted much earlier [196].

In (9), instead of breaking directly to SM, the breaking may occur in two steps where represents the gauge symmetry . This promises the interesting possibility of TeV scale boson with the constraint . Thus the model can be discriminated from the direct LR models if boson is detected at lower mass scales than the -boson. There are currently ongoing accelerator searches for this extra heavy neutral gauge boson. This has been implemented recently with type II seesaw dominance in the presence of added fermion singlets [112]. As we will discuss below both these types of models predict light neutrinos capable of mediating double beta decay rates in the channel saturating the current experimental limits. In addition resonant leptogenesis mediated by heavy sterile neutrinos has been realised in the model of [112].

The symmetric Yukawa Lagrangian descending from symmetry can be written as where are two bidoublets, and .

Including the induced VEV contribution to , the Yukawa mass term can be written as Here the last term denotes the gauge invariant singlet mass term where naturalness criteria demand to be a very small parameter. In the basis the generalised form of the neutral fermion mass matrix after electroweak symmetry breaking can be written aswhere , , , and . In this model the symmetry braking mechanism and the VEVs are such that . The LH triplet scalar mass and RH neutrino masses being at the heaviest mass scales in the Lagrangian, this triplet scalar field and the RH neutrinos are at first integrated out leading to the effective Lagrangian at lower scales [45, 46, 112] as follows:

#### 4. Cancellation of Type I Seesaw and Dominance of Others

##### 4.1. Cancellation of Type I Seesaw

Whereas the heaviest RH neutrino mass matrix separates out trivially, the other two mass matrices , and are extracted through various steps of block diagonalisation. The details of various steps are given in [112, 176, 177]. From the first of the above three equations, it is clear that the type I seesaw term cancels out with another of opposite sign resulting from block diagonalisation. Then the generalised form of the light neutrino mass matrix turns out to be In different limiting cases this generalised light neutrino mass matrix reduces to the corresponding well known neutrino mass formulas.

##### 4.2. Linear Seesaw and Double Seesaw

With that induces mixing, the second term in (15) is the double seesaw formula The third term in (15) represents the linear seesaw formula Similar formulas have been shown to emerge from single-step breaking of SUSY GUT models [39, 40, 44] which require the presence of three gauge singlet fermions.

Using the D-Parity breaking mechanism of [55, 56], an interesting model of linear seesaw mechanism in the context of supersymmetric with successful gauge coupling unification [197] has been suggested in the presence of three gauge singlet fermions. A special feature of this linear seesaw, compared to others [39, 40, 44], is that the neutrino mass formula is suppressed by the SUSY GUT scale but it is decoupled from the low breaking scale. In addition to prediction of TeV scale superpartners, the model provides another important testing ground through manifestation of extra boson at LHC or via low-energy neutrino scattering experiment [198].

##### 4.3. Type II Seesaw

When VEV or becomes negligible and in (2), type II seesaw dominates leading toAs noted briefly in Section 1, in the conventional models [157, 160] of type II seesaw dominance in , the , boson masses have to be at the GUT-Planck scale. As a phenomenal development, this singlet fermion assisted type II seesaw dominance permits breaking scale associated with or breaking (i.e., the and boson masses) accessible to accelerator energies including LHC. At the same time the heavy mixing mass terms at the TeV scale are capable of mediating observable LFV decay rates closer to their current experimental values [85–96] as discussed in Section 5. Consequences of this new type II seesaw dominance with TeV scale boson mass have been investigated in detail [112] in which charged triplet mediated LFV decay rates are negligible but singlet fermion decay rates are observable. Also predictions of observable double beta decay rates close to their experimental limits are discussed below in Section 6. While the principle of such a dominance is clearly elucidated in this derivation, the details of the model with TeV scale symmetry will be reported elsewhere.

##### 4.4. Extended Seesaw

It is quite clear that the classic inverse seesaw formula [24] of (2) for light neutrino mass emerges when the LH triplet mass is large and the VEV which is possible in a large class of non-SUSY models with left-right, Pati-Salam, and gauge groups with* D-Parity* broken at high scales [55–57] with leading to RH neutrinos as heavy pseudo Dirac fermions. Particularly in some non-SUSY examples are [109, 166] and SUSY examples are [164, 165, 199]. The mechanism operates without supersymmetry provided we reconcile with gauge hierarchy problem and some non-SUSY models are [109, 110, 175, 176].

As noted in Section 1, the derivation of classic inverse seesaw mechanism [24–34] has in (12). More recent applications in LRS and GUTs have been discussed with relevant reference to earlier works in [126, 166, 167, 200–204].

In this section we have discussed that, in spite of the presence of the heavy Majorana mass term of RH neutrino, each of the three seesaw mechanisms, (i) extended seesaw, (ii) Type II seesaw, and (iii) linear seesaw or double seesaw, can dominate as light neutrino mass ansatz when the respective limiting conditions are satisfied. Also the seesaw can operate in the presence of TeV scale or gauge symmetry originating from non-SUSY [112, 176, 177]. As the TeV scale theory spontaneously breaks to low-energy theory through the electroweak symmetry breaking of the standard model, these seesaw mechanisms are valid in the SM extensions with suitable Higgs scalars and three generations of and . For example without taking recourse to LR gauge theory type II seesaw can be embedded into the SM extension by inclusion of LH Higgs triplet with . The induced VEV can be generated by the trilinear term [205, 206]. The origin of such induced VEV in the direct breaking of is well known.

##### 4.5. Hybrid Seesaw

In the minimal , without extra fermion singlets, one example of hybrid seesaw with type I type II is given in (1). There are a number of investigations where this hybrid seesaw has been successful in parametrising small neutrino masses with large mixing angles along with in SUSY [148–153] and LR models. But the present mechanism of type I seesaw cancellation suggests a possible new hybrid seesaw formula as a combination of type II Linear Extended seesaw as revealed from (15). Neutrino physics phenomenology may yield interesting new results with this new combination with additional degrees of freedom to deal with neutrino oscillation data and leptogenesis covering coupling unification in which has a very rich structure for dark matter.

Using the D-Parity breaking mechanism of [55, 56], an interesting model of linear seesaw mechanism in the context of supersymmetric with successful gauge coupling unification [197] has been suggested in the presence of three gauge singlet fermions. A special feature of this linear seesaw, compared to others [39, 40], is that the neutrino mass formula is suppressed by the SUSY GUT scale but decoupled from the low breaking scale which can be even at ~few TeV. This serves as a testing ground through manifestation of extra boson at LHC or via low-energy neutrino scattering experiments [198]. Being a SUSY model it also predicts TeV scale superpartners expected to be visible at LHC.

##### 4.6. Common Mass Formula for Sterile Neutrinos

In spite of different types of seesaw formulas in the corresponding limiting cases the formula for sterile neutrino mass remains the same as in (6) which does not emerge from the classic inverse seesaw approach with .

We conclude this section by noting that the classic inverse seesaw mechanism was gauged at the TeV scale through its embedding in non-SUSY with the prediction of experimentally accessible boson, LFV decays, and nonunitarity effects [166]. The possibility of gauged and extended inverse seesaw mechanism with dominant contributions to both lepton flavor and lepton number nonconservation was at first noted in the context Pati-Salam model in [175] and in the context of non-SUSY in [176, 177] with type I seesaw cancellation. The generalised form of hybrid seesaw of (15) in non-SUSY with type I cancellation was realised in [112]. As a special case of this model, the experimentally verifiable phenomena like extra boson, resonant leptogenesis, LFV decays, and double beta decay rates closer to the current search limits were decoupled from the intermediate scale type II seesaw dominated neutrino mass generation mechanism. Proton lifetime prediction for mode also turns out to be within the accessible range.

#### 5. Predictions for LFV Decays, CP Violation, and Nonunitarity

The presence of nonvanishing neutrino masses with generational mixing evidenced from the oscillation data [71–74], in principle, induces charged lepton flavor violating (LFV) decays. For a recent review see [99] and references therein. The observed neutrino mixings through weak charged currents lead to nonconservation of lepton flavor numbers , , and resulting in the predictions of , , , , and a host of others [85–96]. If the non-SUSY SM is minimally extended to embrace tiny neutrino masses and mixings through GIM mechanism as the only underlying source of charged lepton flavor violation, the loop mediated branching ratio is These branching ratios turn out to be ≤10^{−53} ruling out any possibility of experimental observation of the decay rates. In high intermediate scale non-SUSY models with Dirac neutrino mass matrix similar to the up quark mass matrix , the type I seesaw ansatz for neutrino masses constrain the heavy RH neutrino masses GeV resulting in branching ratio values ≤10^{−40} which are far below the current experimental limits. Another drawback of the model is that the underlying neutrino mass generation mechanism and the predicted boson mass cannot be verified directly.

On the other hand SUSY GUTs are well known to provide profound predictions of CP-violations and LFV decay branching ratios closer to the current experimental limits in spite of their high scale seesaw mechanisms for neutrino masses. Some of the extensively available reviews on this subject are [97, 99, 100, 203, 207]. The superpartner masses near 100–1000 GeV are necessary for such predictions.

As profound applications of the classic inverse seesaw mechanism it has been noted that the presence of heavy pseudo Dirac fermions would manifest through LFV decays [33, 184–188] and also in lepton number violation [208, 209]. They are also likely to contribute to the modifications of the electroweak observables [190–192] keeping them within their allowed limits. It has been also emphasized that the SUSY inverse seesaw mechanism for neutrino masses further enhances the LFV decay rates [184]. As a direct test of the seesaw mechanism, these heavy particles with masses near the TeV scale can be produced at high energy colliders including LHC [109, 110, 112, 164, 165, 167]. Other interesting signatures have been reviewed in [11].

More interestingly a linear seesaw formula has been predicted from supersymmetric with an extra boson mass accessible to LHC [42]. The TeV scale classic inverse seesaw mechanism and gauge boson masses have been embedded in SUSY with rich structure for leptonic CP violation, nonunitarity effects, and LFV decay branching ratios accessible to ongoing experiments [164, 165]. The impact of such a model with TeV scale pseudo Dirac type RH neutrinos has been investigated on proton lifetime predictions and leptogenesis [199].

As supersymmetry has not been experimentally observed so far, an interesting conceptual and practical issue is to confront LFV decay rates accessible to ongoing experimental searches along with the observed tiny values of light neutrino masses. In this section we summarise how, in the absence of SUSY, the classic inverse seesaw and the extended seesaw could still serve as powerful mechanisms to confront neutrino mass, observable lepton flavor violation [109, 166], and, in addition, dominant lepton number violation [110, 175–177] in non-SUSY . The possibilities of detecting TeV scale bosons have been also explored recently in non-SUSY [110, 111, 175–177]. Besides earlier works in left-right-symmetric model [210] and those cited in Sections 1–4, more recent works include [61–66, 78, 211–217] and references cited in these papers.

In contrast to negligible LFV decay rates and branching ratios predicted in the non-SUSY SM modified by GIM mechanism, we discuss in Section 5.2 how the non-SUSY predicts the branching ratios in the range consistent with small neutrino masses dictated by classic inverse seesaw, extended inverse seesaw, or type II seesaw in the presence of added fermion singlets.

##### 5.1. Neutrino Mixing and Nonunitarity Matrix

The light neutrino flavor state is now a mixture of three mass eigenstates , , and as follows:where in the diagonal bases of and and . In cases where the matrices and are nondiagonal, the corresponding flavor mixing matrices are taken as additional factors to define or mixing matrices. For the sake of simplicity, treating the mixing mass matrix as diagonal, and, under the assumed hierarchy , the formula for the nonunitarity deviation matrix element has been defined in the respective cases [110, 112, 175–177, 218–231] as follows: The Dirac neutrino mass matrix needed for the fit to neutrino oscillation data through extended seesaw formula and prediction of nonunitarity effects has been derived from the GUT scale fit of charged fermion masses in the case of non-SUSY [110, 112, 166, 176, 177]. For this purpose, the available data at the electroweak scale on charged fermion masses and mixings are extrapolated to the GUT scale [232]. The fitting is done following the method of [233] by suitably adding additional contributions due to VEVs of additional bidoublets or higher dimensional operators, wherever necessary. In the inverse seesaw case with almost degenerate heavy pseudo Dirac neutrinos, has been derived in the case of SUSY with TeV scale symmetry [164, 165] and in non-SUSY with TeV scale symmetry [166]. In the case of extended seesaw dominance in non-SUSY it has been derived in [110, 175–177] whereas for type II seesaw dominance it has been derived in [112]. The value of thus derived at the GUT scale is extrapolated to the TeV scale following the top-down approach. It turns out that such values are approximately equal to the one shown in the following section in (32).

For the general nondegenerate case of the matrix, ignoring the heavier RH neutrino contributions and saturating the upper bound gives By inspection this equation gives the lower bounds , , and . And for the degenerate case GeV. GeV. For the partial degenerate case of the solutions can be similarly derived as in [112, 176] and one example is GeV.

Experimentally constrained lower bounds of the nonunitarity matrix elements are Out of several estimations of the elements of -matrix [110, 112, 166, 175–177] carried out in non-SUSY , here we give one example of [112]. Using the Dirac neutrino mass matrix from [112] and allowed solutions of , the values of the parameters and their phases as functions of are determined using (22). These results are presented in Table 1.