With Low-Mass Z , RH Neutrinos, and Verifiable LFV, LNV And

Eur. Phys. J. C (2015) 75:183 DOI 10.1140/epjc/s10052-015-3385-x

Regular Article - Theoretical Physics

New mechanism for Type-II seesaw dominance in SO(10) with low-mass Z, RH neutrinos, and veriﬁable LFV, LNV and proton decay

Bidyut Prava Nayaka, Mina Ketan Paridab Centre of Excellence in Theoretical and Mathematical Sciences, Siksha ‘O’ Anusandhan University, Khandagiri Square, Bhubaneswar 751030, Odisha, India

Received: 30 October 2014 / Accepted: 31 March 2015 / Published online: 1 May 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract The dominance of Type-II seesaw mechanism models are based upon the underlying assumption that neutri- for the neutrino masses has attracted considerable atten- nos are Majorana fermions that may manifest in the detection tion because of a number of advantages. We show a novel of events in neutrino-less double beta (0νββ) decay experi- approach to achieve Type-II seesaw dominance in non- ments on which a number of investigations are in progress supersymmetric SO(10) grand unification where a low-mass [1–9]. Theories of neutrino masses and mixings are placed on Z boson and specific patterns of right-handed neutrino a much stronger footing if they originate from left–right sym- masses are predicted within the accessible energy range of metric (LRS) [10–13] grand unified theories such as SO(10) the Large Hadron Collider. In spite of the high value of the where, besides grand unification of three forces of nature, 8 9 seesaw scale, ML 10 –10 GeV, the model predicts new P (for parity) and CP-violations have spontaneous-breaking dominant contributions to neutrino-less double beta decay in origins, the fermion masses of all the three generations are the WL –WL channel close to the current experimental limits adequately fitted [14], all the 15 fermions plus the right- via exchanges of heavier singlet fermions used as essential handed neutrino (N) are unified into a single spinorial rep- ingredients of this model even when the light active neutrino resentation 16 and the canonical (≡Type-I) seesaw formula masses are normally hierarchical or invertedly hierarchical. for neutrino masses is predicted by the theory. More recently We obtain upper bounds on the lightest sterile neutrino mass a non-SUSY SO(10) origin of cold dark matter has also been ms 3.0GeV,2.0 GeV and 0.7GeV for normally hierar- suggested [15–19]. Although a Type-I seesaw formula was chical, invertedly hierarchical and quasi-degenerate patterns also proposed by using extensions of the SM [20–27], it is of light-neutrino masses, respectively. The underlying non- well known that this was advanced even much before the unitarity effects lead to lepton flavour violating decay branch- atmospheric neutrino oscillation data [28] and it is interesting ratios within the reach of ongoing or planned experiments ing to note that Gell-Mann, Ramond and Slansky had used the and the leptonic CP-violation parameter nearly two order left–right symmetric SO(10) theory and its Higgs represen- larger than the quark sector. Some of the predicted values tations 10H , 126H to derive it. A special feature of left–right on the proton lifetime for p → e+π 0 are found to be within (LR) gauge theories and SO(10) grand unification is that the the currently accessible search limits. Other aspects of model canonical seesaw formula for the neutrino masses is always applications including leptogenesis etc. are briefly indicated. accompanied by a Type-II seesaw formula [29–31]forthe Majorana neutrino mass matrix, M = II + I , 1 Introduction ν mν mν (1) I =− 1 T , mν MD MD (2) Experimental evidence on tiny neutrino masses and their MN II large mixings has attracted considerable attention as regards mν = f vL (3) physics beyond the standard model (SM) leading to differ- ( ) ent mechanisms for neutrino mass generation. Most of these where MD MN is the Dirac (RH-Majorana) neutrino mass, vL is the induced vacuum expectation value (VEV) of the a e-mail: [email protected] left-handed (LH) triplet L , and f is the Yukawa coupling of b e-mail: [email protected] the triplet. Normally, because of the underlying quark–lepton 123 183 Page 2 of 24 Eur. Phys. J. C (2015) 75 :183 symmetry in SO(10), MD is of the same order as Mu, the up- in the quark sector has been suggested to be Type-II seesaw quark mass matrix. Then the neutrino oscillation data forces dominance [40–43] via renormalisation group evolution of 11 the canonical seesaw scale to be large, MN ≥ 10 GeV. quasi-degenerate neutrino masses that holds in supersymmet- Similarly the Type-II seesaw scale is also large. With such ric quark–lepton unified theories [10,11] or SO(10) and for high seesaw scales, these two mechanisms in SO(10) cannot large values of tan β, which represents the ratio of the VEVs be directly verified at low energies or by the Large Hadron of up-type and down type Higgs doublets. In an interesting Collider (LHC) except for the indirect signature through the approach to understand neutrino mixing in SUSY theories, light active neutrino mediated 0νββ decay and, possibly, lep- it has been shown [44] that the maximality of atmospheric togenesis. neutrino mixing is an automatic consequence of Type-II see- It is well known that the theoretical predictions of branch- saw dominance and b–τ unification that does not require ing ratios for LFV decays such as μ → eγ , τ → μγ , quasi-degeneracy of the associated neutrino masses. A num- τ → eγ and μ → eee¯ closer to their experimental limits are ber of consequences of this approach have been explored to generic features of SUSY GUTs even with high seesaw scales explain all the fermion masses and mixings by utilising Type- but, in non-SUSY models with such seesaw scales, they are II seesaw, or a combination of both Type-I and Type-II see- far below the experimental limits. Recently they have been saw [45–52] through SUSY SO(10). As a further interesting also predicted to be experimentally accessible along with property of Type-II seesaw dominance, it has been recently low-mass WR, Z R bosons through TeV scale gauged inverse shown [53–55] without using any flavour symmetry that the seesaw mechanism [32] in SUSY SO(10). In the absence of well-known tri-bimaximal mixing pattern for neutrino mix- any evidence of supersymmetry so far, alternative non-SUSY ings is simply a consequence of rotation in the flavour space. SO(10) models have been found with predictions of substan- Although several models of Type-II seesaw dominance in tial LFV decays and TeV scale Z bosons. Although two- SUSY SO(10) have been investigated, precision gauge cou- step breakings of LR gauge theory was embedded earlier in pling unification is distorted in most cases.1 All the charged non-SUSY GUTs with low-mass Z (For earlier work on, Z fermion mass fittings in the conventional one-step breaking boson in GUTs embedding two-step breaking of left-right of SUSY GUTs including fits to the neutrino oscillation data gauge symmetry, see [33–35]), its successful compliance require the left-handed triplet to be lighter than the Type-I with neutrino oscillation data has been possible in the context seesaw scale. The gauge coupling evolutions, being sensitive of inverse seesaw mechanism and predictions of LFV decays to the quantum numbers of the LH triplet L (3, −2, 1) under [36], or with the predictions of low-mass WR, Z R bosons, an SM gauge group, tend to misalign the precision unification LFV decays, observable neutron oscillations and dominant in the minimal scenario achieved without the lighter triplet. LNV decay via an extended seesaw mechanism [37]. The Two kinds of SO(10) models have been suggested for possibility of an LHC accessible low-mass Z has been also ensuring precision gauge coupling unification in the presence investigated recently in the context of heterotic string models of Type-IIseesaw dominance. In the first type of SUSY model 17 [38]. Another attractive aspect of non-SUSY SO(10) is rare [57], SO(10) breaks at a very high scale, MU ≥ 10 GeV, kaon decay and neutron–antineutron oscillation, which has to SUSY SU(5), which further breaks to the minimal super- been discussed in a recent work with an inverse seesaw mech- symmetric standard model (MSSM) at the usual SUSY GUT- 16 anism for light-neutrino masses and TeV scale Z bosons but scale MU ∼ 2 × 10 GeV. Type-II seesaw dominance is having much larger WR mass not accessible to LHC [39]. achieved by fine tuning the mass of the full SU(5) multiplet The viability of the model of Ref. [32] depends on the dis- 15H containing the L (3, −2, 1) to remain at the desired , = 11 13 covery of TeV scale SUSY, TeV scale WR Z R bosons and Type-II scale ML 10 –10 GeV. Since the full mul- TeV scale pseudo-Dirac neutrinos. The viability of the non- tiplet 15H is at the intermediate scale, although the evolu- SUSY model of Ref. [36] depends on the discovery of TeV tions of the three gauge couplings of the MSSM gauge group μ> scale low-mass Z R boson and heavy pseudo Dirac neutrinos deflect from their original paths for ML , they con- in the range 100–1200 GeV; both types of models predict a verge exactly at the same scale MU as the MSSM unification proton lifetime within the Super-K search limit. The falsifi- scale but with a slightly larger value of the GUT coupling ability of the non-SUSY model of Ref. [39] depends upon leading to a marginal reduction of the proton-lifetime pre- any one of the following predicted observables: the TeV scale diction compared to SUSY SU(5). In the second class of Z R boson, dominant neutrino-less double beta decay, heavy models, applicable to a non-SUSY or split-SUSY case [56], Majorana type sterile and right-handed neutrinos, neutron the grand unification group SO(10) breaks directly to the SM oscillation and rare kaon decays. Whereas the neutrino mass generation mechanism in all these models is through a gauged 1 inverse seesaw mechanism, our main thrust in the present A brief review of different SUSY SO(10) models requiring Type-II seesaw, or an admixture of Type-I and Type-II for fitting fermion masses work is the Type-II seesaw. A key ansatz to resolve the issue is given in Ref. [53–55]. and a brief review of distortion occurring to of a large mixing in the neutrino sector and small mixing precision gauge coupling unification is given in Ref. [56]. 123 Eur. Phys. J. C (2015) 75 :183 Page 3 of 24 183

16 gauge symmetry at the GUT-scale MU ∼ 2 × 10 GeV, and note that the model predicts a new observable contribution by tuning the full SU(5) scalar multiplet 15H to have degen- to 0νββ decay in the WL –WL channel. The key ingredients to = 11 13 erate masses at ML 10 –10 GeV, the Type-II seesaw achieve Type-II seesaw dominance by complete suppression dominance is achieved. The question of precision unifica- of Type-I seesaw contribution are addition of one SO(10) tion is answered in this model by pulling out all the super- singlet fermion per generation (Si , i = 1, 2, 3) and utilisa- partner scalar components of the MSSM but by keeping all tion of the additional Higgs representation 16H to generate the fermionic superpartners and the two Higgs doublets near the N–S mixing term in the Lagrangian through the Higgs- the TeV scale. In the non-SUSY case the TeV scale fermions Yukawa interaction. The underlying leptonic non-unitarity can also be equivalently replaced by complex scalars carrying effects lead to substantial LFV decay branching ratios and the same quantum numbers. The proton-lifetime prediction leptonic CP-violation accessible to ongoing search experi- + 0 35 is τP (p → e π ) 10 years in this model. ments. We derive a new formula for the half-life of 0νββ In the context of LR gauge theory, the Type-II seesaw decay as a function of the fermion singlet masses and extract mechanism was originally proposed with manifest left–right lower bound on the lightest sterile neutrino mass from the symmetric gauge group SU(2)L × SU(2)R × U(1)B−L × existing experimental lower bounds on the half-life of differ- SU(3)C × D (g2L = g2R)(≡ G2213D) where both the ent experimental groups. For certain regions of the parame- left- and the right-handed triplets are allowed to have the ter space of the model, we also find the proton lifetime for same mass scale as the LR symmetry breaking (or parity p → e+π 0 to be accessible to ongoing or planned experi- breaking) scale [58]. With the emergence of D-parity and its ments. breaking leading to decoupling of parity and SU(2)R break- Compared to earlier existing SO(10)-based Type-II see- ings [59,60], a new class of asymmetric LR gauge group saw dominant models whose RH neutrino masses are in the also emerged: SU(2)L × SU(2)R × U(1)B−L × SU(3)C inaccessible range and new gauge bosons are in the mass 15 17 (g2L = g2R)(≡G2213) where the left-handed triplet acquired range 10 –10 GeV, the present model predictions on LHC a larger mass than the RH triplet leading to the Type-I see- scale Z , light and heavy Majorana type sterile neutrinos, RH saw dominance and suppression of Type-II seesaw in SO(10) Majorana neutrino masses in the range 100–10, 000 GeV [61]. It is possible to accommodate both types of intermedi- accessible to LHC in the WL − WL channel through dilepton ate symmetries in non-SUSY SO(10), but these models make production, the LFV branching ratios closer to experimental negligible predictions for the branching ratios of charged limits, and dominant 0νββ decay amplitudes caused by ster- LFV processes and they leave no other experimental signa- ile neutrino exchanges provide a rich testing ground for new tures verifiable at low or LHC energies except 0νββ decay. physics signatures. The purpose of this work is to show that in a class of mod- This paper is organised as follows. In Sect. 2.wegivean els descending from non-SUSY SO(10) or from Pati–Salam outline of the model and discuss gauge coupling unification gauge symmetrty, Type-II seesaw dominance at intermediate along with proton lifetime predictions. In Sect. 3 we derive a 8 9 scales (M 10 –10 GeV) but with MN ∼ O(1)–O(10) Type-II seesaw dominance formula and show how the model TeV can be realised by cancellation of the Type-I seesaw predicts RH neutrino masses from fits to the neutrino oscil- contribution along with the prediction of a Z boson at the lation data. In Sect. 4 we discuss the derivation of the Dirac ∼ O(1)–O(10) TeV scale, accessible to the LHC, where neutrino mass matrix from the GUT-scale fit to the fermion U(1)R × U(1)B−L breaks spontaneously to U(1)Y through masses. In Sect. 5 we discuss predictions on lepton flavour the VEV of the RH triplet component of Higgs scalar con- violation and leptonic CP violation due to the underlying tained in 126H that carries B − L =−2. non-unitarity effects. In Sect. 6 we discuss briefly the ana- Although two-step breakings of LR gauge theory was lytic derivation of amplitudes on lepton number violation. In embedded earlier in non-SUSY GUTs with low-mass Z Sect. 7 we discuss predictions on the effective mass parame- (for earlier work on, Z boson in GUTs embedding two-step ters and half-life for 0νββ; we also obtain the singlet fermion breaking of left-right gauge symmetry, see [33–35]), its suc- mass bounds. We also indicate very briefly some plausible cessful compliance with neutrino oscillation data has been model applications including effects on electroweak preci- possible in the context of an inverse seesaw mechanism [36]. sion observables, Z–Z mixings, dilepton production and lep- We also discuss how the Type-II seesaw contribution dom- togenesis in Sect. 8. We summarise and conclude our results inates over the linear seesaw formula. Whereas in all previ- in Sect. 9. ous Type-II seesaw dominance models in SO(10), the RH Majorana neutrino masses have been very large and inaccessible for accelerator energies, the present model pre- 2 Unification with TeV scale Z dicts these masses in the LHC accessible range. In spite of large values of the WR boson and the doubly charged In this section we devise two symmetry breaking chains of ++,++ Higgs boson L R masses, it is quite interesting to non-SUSY SO(10) theory, one with LR symmetric gauge 123 183 Page 4 of 24 Eur. Phys. J. C (2015) 75 :183 theory with unbroken D-parity and another without D-parity χR(1, 2, −1, 1) ⊂ 16H . The symmetry breaking in the last at the intermediate scale. In the subsequent sections we will step is implemented through the SM Higgs doublet con- compare the ability of the two models to accommodate Type- tained in the bidoublet φ(2, 2, 0, 1) ⊂ 10H of SO(10). This II seesaw dominance to distinguish one model from the other. is the minimal Higgs structure of the model, although we φ ⊂ As necessary requirements, we introduce one SO(10)-singlet will utilise two different Higgs doublets u 10H1 and , = , , φ ⊂ per generation (Si i 1 2 3) and Higgs representations d 10H2 for fermion mass fits. In Model I, the GUT- 126H and 16H in both models. symmetry breaks to LR gauge symmetry G2213(g2L = g2R) in such a way that the D-parity breaks at the GUT-scale and is 2.1 Models from SO(10) symmetry breaking decoupled from SU(2)R breaking, which occurs at the intermediate scale. This is achieved by giving a GUT scale VEV to Different steps of symmetry breaking are given below for the the D-parity odd singlet-scalar component in (1, 1, 0, 1)H ⊂ following two models: (1, 1, 15)H ⊂ 45H where the first, the second and the third Model I submultiplets are under G2213, the Pati–Salam symmetry G224 and SO(10), respectively. In this case by adopting the D- (MU =MP ) SO(10) −→ SU(2)L × SU(2)R × U(1)B−L × SU(3)C [G2213] parity breaking mechanism [59,60] in SO(10), normally the ( +) ( , , − , ) ⊂ MR LH triplet component L 3 1 2 1 126H and the LH −→ SU(2) × U(1) × U(1) − × SU(3) [G ] L R B L C 2113 doublet component χ (2, 1, −1, 1) ⊂ 16 acquire masses ( 0 ) L H MR −→ SU(2)L × U(1)Y × SU(3)C [SM] at the GUT-scale, while the RH triplet and RH doublet com- ( ) MZ ponents, R(1, 3, −2, 1) ⊂ 126H χR(1, 2, −1, 1) ⊂ 16H , −→ SU(3)C × U(1)Q , can be made much lighter. We have noted that in the pres- Model II ence of a colour octet at lower scales, found to be necessary in Model I as well as in Model II, precision gauge coupling (MU ) SO(10) −→ SU(2)L × SU(2)R × U(1)B−L × SU(3)C [G2213D] is achieved even if the parameters of the Higgs potential are ( += ) MR MP tuned so as to have the LH triplet mass at intermediate scale, −→ SU(2)L × U(1)R × U(1)B−L × SU(3)C [G2113] M 108–109 GeV. The presence of (3, 1, −2, 1) ( 0 ) L L MR −→ SU(2)L × U(1)Y × SU(3)C [SM] at the intermediate scale plays a crucial role in achieving ( ) MZ Type-II seesaw dominance as will be explained in the fol- −→ SU(3)C × U(1)Q . lowing section. The necessary presence of lighter LH triplets In Model II, SU(2)L ×SU(2)R ×U(1)B−L ×SU(3)C ×D [≡ in GUTs with or without vanishing B − L value for physi- G2213D](g2L = g2R) is obtained by breaking the GUT- cally appealing predictions was pointed out earlier in achiev- symmetry and by giving the VEV to the D-parity even sin- ing observable matter–anti-matter oscillations [63,64], in the glet (1, 1, 0, 1) ⊂ (1, 1, 15) ⊂ 210H [59,60,62] where the context of low-scale leptogenesis [65], and Type-II seesaw first, the second and the third set of quantum numbers of the dominance in SUSY, non-SUSY and split-SUSY models scalar components are under G2213P , the Pati–Salam sym- [56,57], and also for TeV scale LR gauge theory originat- metry G224 and SO(10), respectively. As a result, the Higgs ing from SUSY SO(10) grand unification [32]. sector is symmetric below μ = MU , leading to equality ( +) ( +) between the gauge couplings g2L MR and g2R MR .In 2.2 Renormalisation group solutions to mass scales this case the LR discrete symmetry (≡ parity) survives down to the intermediate scale, MR+ = MP . The second step of In this section while safeguarding precise unification of symmetry breaking is implemented by assigning a VEV to gauge couplings at the GUT-scale, we discuss allowed solu- the neutral component of the right-handed (RH) Higgs triplet tions of renormalisation group equations (RGEs) for the σ ( , , , ) ⊂ − = , + R 1 3 0 1 45H that carries B L 0. The third mass scales MU MR and MR0 as a function of the mass step of breaking to SM is carried out by assigning a VEV MC of the lighter colour octet C8(1, 1, 0, 8) ⊂ 45H .The O( ) 0 ( , , − , ) of 5–10 TeV to the G2113 component R 1 1 2 1 Higgs scalars contributing to RG evolutions are presented contained in the RH triplet R(1, 3, −2, 1) ⊂ 126H carry- in Table 1 for Model I. In Model II, in addition to the Higgs ing B − L =−2. This is responsible for the RH Majorana scalars shown in Table 1, the masses of the left-handed scalars = = 0 χ ( , , − , ) σ ( , , , ) neutrino mass generation MN fVR where VR R L 2 1 1 1 and L 3 1 0 1 are naturally constrained † μ = + = = and f is the Yukawa coupling of 126 to SO(10) spino- to be at MR MP , the parity violation scale. . . † rial fermionic representation: f 16 16 126H. We introduce The renormalisation group (RG) coefficients for the min- SO(10) invariant N–S mixing mass via the Yukawa inter- imal cases are given in Appendix A to which those due to . . † = action yχ 16 1 16H and obtain the mixing mass M yχ Vχ the colour octet scalar in both models and the LH triplet L = χ 0 where Vχ R by noting that under G2113 the submul- in Model I in their suitable ranges of the running scale have χ 0 ( , / , − , ) tiplet R 1 1 2 1 1 is contained in the G2213 doublet been added. 123 Eur. Phys. J. C (2015) 75 :183 Page 5 of 24 183

Table 1 Higgs scalars and their quantum numbers used in Model I in the respective ranges of mass scales. The mass of colour octet C8(1, 0, 0, 8) has been used in the range 104–1011 GeV contributing to the variation of predicted proton lifetime discussed below in Sect. 2.3. In Model II, in addition to these scalars, the masses of the left-handed scalars χ ( , , − , ) σ ( , , , ) μ = + = L 2 1 1 1 and L 3 1 0 1 are constrained to be at MR = MP , the parity violation scale ) μ ( -1 Mass scale (μ) Symmetry Higgs scalars (Model I) i α − 0 Φ( , , ) MZ MR G213 2 1 1 Φ1(2, 1, 0, 1), Φ2(2, 1, 0, 1), 0 − + χ ( , / , − , ), ( , , − , ), MR MR G2113 R 1 1 2 1 1 R 1 1 2 1 C8(1, 0, 0, 8)

Φ1(2, 2, 0, 1), Φ2(2, 2, 0, 1)), χ ( , , − , ), ( , , − , ), + − R 1 2 1 1 R 1 3 2 1 MR MU G2213 L (3, 1, −2, 1), C8(1, 0, 0, 8), σ ( , , , ) R 1 3 0 1 μ log10 ( /GeV)

Fig. 1 Two-loop gauge coupling uniﬁcation in the SO(10) symmetry Table 2 Allowed values of mass scales as solutions of RGEs for gauge = 15.62 + = 9.7 = breaking chain with MU 10 GeV and MR 10 , ML couplings for Model I with ﬁxed value of the LH triplet mass M = 108 GeV with a low-mass Z boson at M0 = 10 TeV for Model I 108 GeV R 0 + α−1 τ MR (TeV) MC (GeV) MR (GeV) MG (GeV) G p (years) 70

10 104.5 109 1016.9 41.1 5.4 × 1039 60 10 105 108.9 1016.74 41.4 1.1 × 1039 7 9 16.4 37 10 10 10 10 41.7 8.4 × 10 50 10 1010.9 109.7 1015.63 41.9 3.2 × 1034

7.8 8.8 16.4 × 37 ) 40 51010 10 41.5 9 10 μ ( -1 i

α 30

Model I: 20 = 8 As shown in Table 2 for Model I, with ML 10 GeV the 8 10 G2213 symmetry is found to survive down to MR+ = (10 – 1010) GeV with larger or smaller unification scale depending 0 upon the colour octet mass. In particular we note one set of 2 4 6 8 10 12 14 16 18 μ solutions, log10( /GeV) 9.7 15.62 M 0 = 10 TeV, MR+ = 10 GeV, MU = 10 GeV, Fig. 2 Two-loop gauge coupling unification in the SO(10) symmetry R . . = 15 62 + = 8 7 = 8 , = 10.9 . breaking chain with MU 10 GeV and MR 10 GeV with a ML 10 GeV MC 10 GeV (4) 0 = low-mass Z boson at MR 10 TeV for Model II As explained in the following sections, this set of solutions are found to be attractive both from the prospects of achiev- the unification of the four gauge couplings of the G inter- ing Type-II seesaw dominance and detecting proton decay 2213 15 mediate gauge symmetry. at Hyper-Kamiokande. with MU = 6.5 × 10 GeV when 11 the colour octet mass is at MC ∼ 10 GeV. As discussed Model II: In addition to the Higgs scalars of Table 1,this below, the proton lifetime in this case is closer to the current model has the masses of left-handed scalars χL (2, 1, −1, 1) = ( σ ( , , , ) μ = + = = experimental limit. With the allowed values of MR0 5– and L 3 1 0 1 naturally at MR MP , the par- 10) TeV, this model also predicts MZ (1.2–3.5) TeV in ity violation scale. As shown in Table 3 for Model II, the the accessible range of the LHC. As discussed in Sect. 3, G2213D symmetry is found to survive down to MR+ = MP = 8.2 15 because of the low mass of the Z boson associated with TeV 10 GeV with MU = 6.5×10 GeV when the colour octet 8 scale VEV of VR, the Type-II seesaw mechanism predicts mass is at MC = 10 GeV. As discussed below, the proton RH neutrino masses which can be testified at the LHC or lifetime in this case is closer to the current experimental limit. future high energy accelerators. One example of RG evolution of gauge couplings is . = + = 8 7 The RG evolution of gauge couplings for the set of mass showninFig.2 for MR0 10 GeV, MR 10 GeV, 8 15 scales given in Eq. (4) is presented in Fig. 1, showing clearly MC = 10 GeV and MU = 6.5 × 10 GeV. Clearly the fig- 123 183 Page 6 of 24 Eur. Phys. J. C (2015) 75 :183

Table 3 Allowed mass scales as 0 + α−1 τ solutions of renormalisation MR (TeV) MC (GeV) MR (GeV) MG (GeV) G p (years) group equations for Model II as 4.5 7.886 16.15 . × 36 described in the text 10 10 10 10 40.25 4 3 10 10 105.5 107.89 1016.04 40.64 1.6 × 1036 10 108 108.789 1015.62 41.49 3.9 × 1034 10 108.5 108.8 1015.5 41.69 1.12 × 1034 5105.8 107.2 1015.83 41.15 2.3 × 1035

ure shows a precise uniﬁcation of the three gauge couplings of the intermediate gauge symmetry G2213P at the GUT-scale. For all other solutions given in Table 3, the RG evolutions and uniﬁcation of the gauge couplings are similar. In both + = models, with the allowed values of MR MR0 5–10 , TeV, the numerical values of the gauge couplings g2L g1R /years) p τ and gB−L predict [66–68] ( 10 log MZ = (1.2—3.5)TeV. (5)

2.3 Proton-lifetime prediction

In this section we discuss predictions on the proton lifetimes in the two models and compare them with the current Super- log10(m8/GeV) Kamiokande limit and reachable limits by future experiments such as the Hyper-Kamiokande [69–72]. Currently, Fig. 3 Variation of proton lifetime as a function of color octet mass MC for Model I (upper curve) and Model II (lower curve). The horizontal the Super-Kamiokande detector has reached the search limit line is the present experimental limit (τ ) ( → +π 0) ≥ . × 34 . p expt. p e 1 4 10 years (6) . The proposed 5 6 Megaton years Cherenkov water detector 2 4 64π fπ M at Hyper-Kamiokande is expected to probe a lifetime [69–72] τ = Γ −1(p → e+π 0) = U p 4 m p gG (τ ) ( → +π 0) ≥ . × 35 . p Hyper−K. p e 1 3 10 years (7) 1 × , (9) | |2|¯α |2( + + )2 × The width of the proton decay for p → e+π 0 is expressed AL H 1 D F R as [73–75] 4 2 2 m g where the factor Fq = 2(1 +|Vud| ) 7.6 for SO(10). Γ(p → e+π 0) = p × G π 2 4 64 fπ MU Now using the given values of the model parameters the predictions on the proton lifetimes for both models are given ×|A |2|¯α |2(1 + D + F)2 × R, (8) L H in Tables 2 and 3. We ﬁnd that for proton-lifetime predic- =[( 2 + 2 )( +| |2)2] = where R ASR ASL 1 Vud for SO(10), Vud tions accessible to Hyper-Kamiokande detector, it is neces- 0.974 = is the (1, 1) element of VCKM for quark mixings, sary to have an intermediate value of the color octet mass 8.6 10.8 ASL(ASR) is the short-distance renormalisation factor in the MC ≥ 10 GeV in Model II and MC ≥ 10 GeV in left (right) sectors and AL = 1.25 = the long distance renor- Model I. The predicted proton lifetime as a function of the malisation factor. MU is the degenerate mass of the 24 super- colour octet mass is shown in Fig. 3 both for Model I and heavy gauge bosons in SO(10), α¯ H the hadronic matrix ele- for Model II. These analyses suggest that a low colour octet ment, m p the proton mass = 938.3MeV, fπ the pion decay mass in the TeV scale and observable proton lifetime within constant, 139 MeV,and the chiral Lagrangian parameters are the Hyper-Kamiokande limit are mutually exclusive. If LHC D = 0.81 and F = 0.47. With αH =¯αH (1 + D + F) = discovers a colour octet within its achievable energy range, 0.012 GeV3 obtained from lattice gauge theory computa- proton decay searches would need far bigger detector than tions, we get AR AL ASL AL ASR 2.726 for both the Hyper-K detector. On the other hand the absence of a models. The expression for the inverse decay rates for the colour octet at the LHC would still retain the possibility of models is expressed as observing proton decay within the Hyper-K limit. 123 Eur. Phys. J. C (2015) 75 :183 Page 7 of 24 183

3 Type-II seesaw dominance

In this section we discuss prospects of having a Type-II seesaw dominated neutrino mass formula in the two SO(10)- based models discussed in Sect. 2.

3.1 Derivation of Type-II seesaw formula

( = Wehave added to the usual spinorial representations 16Fi i 1, 2, 3) for fermion representations in SO(10) and one fermion singlet per generation S (i = 1, 2, 3).TheG i 2213 (a) (b) symmetric Yukawa Lagrangian descending from SO(10) symmetry can be written as Fig. 4 Feynman diagrams for induced contributions to VEVs of the LH triplet (diagram a) and the LH doublet (diagram b) in Model I and L = ψ ψ Φ + (ψc ψ + ψc ψ ) Yuk Yi L R i f R R R L L L Model II = , i 1 2 +yχ ψ S χR + ψ S χL + (h.c.) , (10) R L order to generate an N–S mixing mass term MNS between Φ ⊂ ( , ) ⊂ where 1,2 10H1,H2 are two bidoublets, L R the RH neutrino and the sterile fermion where we have 126F and (χL ,χR) ⊂ 16H . As discussed in Sect. 2,the the 3 × 3matrixM= yχ Vχ . We define the other 3 × 3 −→ (1) spontaneous breaking of G2213 G2113 takes place by mass matrices MD = Y vu and MN = fVR.Wealso the VEV of the RH triplet σR(1, 3, 0, 1) ⊂ 45H carrying include induced small contributions to the VEV of the LH − = B L 0, which does not generate any fermion mass term. Higgs triplet vL = L (3, 0, −2, 1) and the LH Higgs Φ v = χ ( , , − , ) As we discuss below, when the Higgs scalar i , R and doublet χL L 2 0 2 1 leading to the possibili- χ ν = v R acquire VEVs spontaneous symmetry breakings lead- ties –S mixing with ML yχ χL and the induced Type- −→ −→ ( ) × ( ) II ing to G2113 SM U 1 em SU 3 C occur and II seesaw contribution to LH neutrino masses mν = f vL = χ 0 generate an N–S mixing mass term M yχ R by the given in Eq. (20). The induced VEVs are shown in the left v = χ 0 v = 0 induced VEVs. In addition, χL L and L L and right panels of Fig. 4. We have also derived them by are automatically generated even though the LH doublet χL actual potential minimisation, which agrees with the dia- and the RH triplet L are assigned vanishing VEVs directly. grammatic contribution. Including the induced VEV contri- In models with the inverse seesaw [76,77] or extended see- butions, the mass term due to the Yukawa Lagrangian can be saw [36,37,78–82] mechanisms, a bare mass term of the written as T singlet fermions μS S S occurs in the Lagrangian. Being unrestricted as a gauge singlet mass term in the Lagrangian, 1 T Lmass = (MDνN + MN N N + M NS determination of its value has been left to phenomenological 2 II T analyses in neutrino physics. Larger values of the parameter + ML νS + h.c) + mν ν ν. (11) near the GUT-Planck scale [83] or at the intermediate scale [84,85] have also been exploited. On the other hand, fits to the In the (ν, S, N C ) basis the generalised form of the 9×9 neu- neutrino oscillation data through an inverse seesaw formula tral fermion mass matrix after electroweak symmetry break- by a number of authors have been shown to require much ing can be written as smaller values of μS [32,36,37,78] (see, for instance [86]), ⎛ ⎞ II [87–91]. Even phenomenological implications of its vanish- mν ML MD ⎜ ⎟ ing value have been investigated recently in the presence of ⎜ ⎟ M = ⎝M T 0 M T ⎠ , (12) other non-standard and non-vanishing fermion masses [92– L T 95]inthe9× 9 mass matrix. Very small values of μS are MD MMN justified on the basis of ’t Hooft’s naturalness criterion, repre- = Φ = v = χ 0 = senting a mild breaking of global lepton number symmetry of where MD Y , MN f R,M yχ R , ML χ 0 μ = the SM [96]. While we consider the implication of this term yχ L and we have used s 0. In this model the symmetry later in this section, at first we discuss the emerging neutrino breaking mechanism and the VEVs are such that MN > mass matrix by neglecting it. In addition to the VEVs dis- M MD. The RH neutrino mass being the heaviest fermion cussed in Sect. 2 for gauge symmetry breaking at different mass scale in the Lagrangian, this fermion is at first integrated stages, we assign the VEV to the neutral component of RH out leading to the effective Lagrangian at lower scales [65, Higgs doublet of 16H with χR(1, 1/2, −1/2, 1) =Vχ in 97–101], 123 183 Page 8 of 24 Eur. Phys. J. C (2015) 75 :183 − L = II + 1 T νT ν 300–1000 GeV. Similar expressions as in Eq. (17) are also eff mν MD MD α β MN αβ obtained by minimisation of the scalar potential. 1 T + ML + MD M να Sm + Smνα MN α m 3.2 Suppression of linear seesaw and dominance of Type-II + 1 T T . M M Sm Sn (13) seesaw MN mn

Whereas the heaviest RH neutrino mass matrix MN separates Now we discuss how a linear seesaw term is suppressed without trivially, the other two 3 × 3 mass matrices Mν and mS out fine tuning of certain parameters in Model I but with fine are extracted through various steps of block diagonalisation tunning of the same parameters in Model II. The expression [37].The details of the various steps are given in Appendix for the neutrino mass is given in Eq. (15) where the first, the B and the results are second and the third terms are Type-II seesaw, double seesaw and linear seesaw formulas for the light-neutrino masses. Out M = II + −1 T − ( −1 T ) ν mν MD MN MD MD MN MD of these, for all parameters allowed in both models (Model I + ( T −1 )−1 T and Model II), the double seesaw term will be found to be far ML M MN M ML − − more suppressed compared to the other two terms. Therefore − M (M T M 1 M)−1(M T M 1 M T ) L N N D (14) we now discuss how the linear seesaw term is suppressed − ( −1 )( T −1 )−1 T , compared to the Type-II seesaw term allowing the domi- MD MN M M MN M ML =− −1 T +··· . nance of the latter. In Model I, gauge coupling unification has ms MMN M = ∼ ≥ 15.6 been achieved such that MP MχL MU 10 GeV, m N = MN . = 8 ∼ ∼ + ∼ 9 ML 10 GeV where M1 M2 MR 10 GeV. From the first of the above three equations, it is clear that the Using these masses in Eq. (15), we find that even with ∼ . . Type-I seesaw term cancels out [65,97–101] with another of Cχ 0 1–1 0 opposite sign resulting from block diagonalisation. Then the −18 −17 vχ ∼ 10 eV–10 eV, generalised form of the light-neutrino mass matrix turns out L (19) v . − . . to be L 0 1eV 0 5eV

−1 T −1 T Such induced VEVs in Model I suppress the second and the Mν = f v + M M M (M ) M L L N L third terms in Eq. (15) making the model quite suitable for −[ T −1 + T ( T )−1]. ML MD M ML MD M (15) Type-II seesaw dominance although the Model II needs fine − tuning in the induced contributions to the level of Cχ ≤ 10 5 With ML = yχ vχ , which induces ν–S mixing, the second L as discussed below. term in this equation is the double seesaw formula and the In Model II, M ∼ Mχ ∼ M ∼ 109 GeV,and without third term is the linear seesaw formula, which are similar to L L P any fine tuning of the parameters in Eq. (16), we obtain v ∼ those derived earlier [84,85]. L −10 v ∼ × −6 ∼ 10 GeV. From Eq. (17) we get χL Cχ 10 GeV From the Feynman diagrams, the analytic expressions for −7 10 GeV for Cχ ∼ 0.1. With (MD)(3,3) ≤ 100 GeV and the induced VEVs are M D 0.1–1, the most dominant third term in Eq. (15)gives V M − v ∼ R λ v2 + λ v2 , Mν ≥ 10 8 GeV. This shows that fine tuning is needed in the L 1 1 2 2 (16) M2 ≤ −5 L parameters occurring to reduce Cχ 10 to suppress linear Vχ seesaw and permit Type-II seesaw dominance in Model II vχ ∼ λ M v + λ M v , L M2 1 1 1 2 2 2 whereas the Type-II seesaw dominance is achieved in Model χL I with Cχ 0.1–1.0 without requiring any such fine tuning. Vχ MR+ vwk = Cχ , 2 (17) In what follows we will utilise the Type-II seesaw dominated Mχ L neutrino mass formula to study neutrino physics,2 neutrino- where vwk ∼ 100 GeV, and less double beta decay and lepton flavour violations in the context of Model I although they are similar in Model II λ v + λ v 1 M1 1 2 M2 2 Cχ = . (18) (M + vw ) R k 2 Following a similar block diagonalisation procedure to Appendix B, μ T In Eq. (17), v (i = 1, 2) are the VEVs of two electroweak but in the presence of S S S in the Yukawa Lagrangian with mass i > >> ,μ ⊂ ( ) ordering MN M MD S results in the appearance of the doublets each originating from separate 10H SO 10 as M = v + inverse seesaw part of the full neutrino mass matrix, ν f L , T explained in the following section, and M1 M2 are Higgs ( MD )μ ( MD ) . M S M Although we plan to investigate the implications of this trilinear coupling masses which are normally expected to be formula in a future work, for the present purpose we assume μS 0 of order MR+ . In both models VR = 5–10 TeV and Vχ ∼ such that Type-II seesaw dominance prevails. 123 Eur. Phys. J. C (2015) 75 :183 Page 9 of 24 183 subject to the fine tuning constraint on Cχ . Thus the light- NH neutrino mass is dominated by the Type-II seesaw term ⎛ ⎞ 0.117 + 0.022i −0.124 − 0.003i 0.144 + 0.025i f = ⎝−0.124 − 0.003i 0.158 − 0.014i −0.141 + 0.017i ⎠ , Mν f vL . (20) 0.144 + 0.025i −0.141 + 0.017i 0.313 − 0.00029i (25)

3.3 Right-handed neutrino mass prediction ˆ |MN |=diag(160, 894, 4870) GeV. (26) Global ﬁts to the experimental data [102–105] on neutrino IH ⎛ ⎞ oscillations have determined the mass squared differences . − . . + . − . + . σ 0 390 0 017i 0 099 0 01i 0 16 0 05i and mixing angles at 3 level, f = ⎝ 0.099 + 0.01i 0.379 + 0.02i 0.176 + 0.036i⎠ , 2 2 −0.16 + 0.05i 0.176 + 0.036i 0.21 − 0.011i sin θ12 = 0.320, sin θ23 = 0.427, 2 (27) sin θ13 = 0.0246,δCP = 0.8π, − (21) m2 = 7.58 × 10 5eV2, ˆ sol |MN |=diag(4880, 4910, 131) GeV. (28) − |m |2 = 2.35 × 10 3eV2. atm QD For normally hierarchical (NH), inverted hierarchical (IH) ⎛ ⎞ . + . . + . − . + . and quasi-degenerate (QD) patterns, the experimental values 2 02 0 02i 0 0011 0 02i 0 019 0 3i = ⎝ . + . . + . . + . ⎠ . of mass squared differences can be ﬁtted by the following f 0 0011 0 02i 2 034 0 017i 0 021 0 21i − . + . . + . . − . values of the light neutrino masses: 0 019 0 3i 0 021 0 21i 1 99 0 04i (29) mˆ ν = (0.00127, 0.008838, 0.04978) eV (NH) For v = 0.1eV,wehave = (0.04901, 0.04978, 0.00127) eV (IH) L | ˆ |= ( . , . , . ) , = (0.2056, 0.2058, 0.2) eV (QD) (22) MN diag 21 46 20 34 18 87 TeV (30) but for v = 0.5 eV we obtain We use the diagonalising Pontecorvo–Maki–Nakagawa– L ˆ Sakata (PMNS) matrix. The UPMNS matrix is given by |MN |=diag(4.3, 4.08, 3.77) TeV. (31)

These RH neutrino masses predicted with vL = 0.1eVfor ⎛ ⎞ v = . −iδ the NH and IH cases and with L 0 5 eV for the QD case c13c12 c13s12 s13e ⎝ iδ iδ ⎠ are clearly veriﬁable by the LHC. −c23s12 − c12s13s23e c12c23 − s12s13s23e s23c13 , iδ iδ s12s23 − c12c23s13e −c12s23 − s12s13c23e c13c23 (23) 4 The Dirac neutrino mass matrix and we determine it using mixing angle and the leptonic Dirac phase from Eq. (21) The Dirac neutrino mass matrix which has a quark–lepton symmetric origin [10,11] plays a crucial role in the predictions of lepton ﬂavour violations (LFVs) [32,36]aswellas ⎛ ⎞ 0.814 0.55 −0.12 − 0.09i lepton number violations (LNVs) as pointed out very recently ⎝ ⎠ UPMNS = −0.35 − 0.049i 0.67 − 0.034i 0.645 . [37,78]. The determination of the Dirac neutrino mass matrix . − . − . − . . ( ) 0 448 0 057i 0 48 0 039i 0 74 MD MR0 at the TeV seesaw scale is done; it was discussed (24) in [36,106–108].

ˆ = † M ∗ Now inverting the relation mν UPMNS νUPMNS where mˆ ν is the diagonalised neutrino mass matrix, we determine 4.1 Extrapolation to the GUT-scale Mν for three different cases and further determine the cor- μ = responding values of the f matrix using f = Mν/vL where The RG extrapolated values at the GUT-scale are, with we use the predicted value of vL = 0.1eV.Notingthat MGUT, = = M /v MN fVR ν VR L , we have also derived the eigen- 0 = . , 0 = . , 0 = . , ˆ me 0 00048 GeV mμ 0 0875 GeV mτ 1 8739 GeV values of the RH neutrino mass matrix MN as the positive i 0 = . , 0 = . , 0 = . , square root of the ith eigenvalue of the Hermitian matrix md 0 0027 GeV ms 0 0325 GeV mb 1 3373 GeV † 0 = . , 0 = . , 0 = . , MN MN . We have the following solutions: mu 0 001 GeV mc 0 229 GeV mt 78 74 GeV (32) 123 183 Page 10 of 24 Eur. Phys. J. C (2015) 75 :183

0 ± The VCKM matrix at the GUT-scale is given by In both models, the WR bosons and the doubly charged ⎛ ⎞ Higgs scalars, both left-handed (LH) and right-handed (RH), 0.97 0.22 −0.0003 − 0.003i ∼ 8 9 0 = ⎝− . − . . . ⎠ . are quite heavy with MWR M 10 –10 GeV. These VCKM 0 22 0 0001i 0 97 0 036 0.008 − 0.003i −0.035 + 0.0008i 0.99 make negligible contributions arising out of the RH current effects and Higgs exchange effects on LFV or LNV (33) decay amplitudes. In the two models considered here, the For fitting the charged fermion masses at the GUT-scale, in flavour eigenstate of any LH neutrino ν can be represented in terms of the mass eigenstates ν , S and N . From details of addition to the two complex 10H1,2 representations with their i i i respective Yukawa couplings Y1,2, we also use the higher- the model parametrisation discussed in Sects. 3–5,wehave dimensional operator [32,36] found the corresponding mixing matrices with active neutri- νN νS κ nos, V = MD/MN ≡ X N and V = MD/M ≡ X S, ij . 2 16i16j10H45H45H (34) MG νN νS ν = N νi + V Ni + V Si , In the above equation the product of three Higgs scalars acts N [ − (ηN + ηS)]U , † 1 PMNS as an effective 126H operator [32]. With MG MPl or N † 2 −3 η = (X N .X )/2, M Mstring, this is suppressed by (MU /MG ) 10 – N −5 ηS = ( . †)/ . 10 for a GUT-scale VEV of 45H . Then the formulas for X S X S 2 (38) different charged fermion mass matrices are These mixings modify the standard weak interaction Lagran- M = G + F, M = G + F, u u d d gian in the LH sector by small amounts but they could = − , = − . Me Gd 3F MD Gu 3F (35) be in the experimentally accessible range (see, for instance Following the procedure given in [36], the Dirac neutrino [109]). In the LH sector the charged current weak interaction mass matrix at the GUT-scale is found to be Lagrangian is ⎛ ⎞ . . − . . − . g2L μ νN νS 0 014 0 04 0 01i 0 109 0 3i L =−√ Wμe¯γ P N ν + V N + V S ( )=⎝ . + . . . + . ⎠ . CC L i i i MD MR0 0 04 0 01i 0 35 2 6 0 0007i GeV 2 0.1 + 0.3i 2.6 − 0.0007i 79.20 +h.c. (39) (36) The implications of these terms for LFV and LNV effects have been discussed below. From Eq. (38) it is clear that N is non-unitary. We assume the N–S mixing matrix M 5 Lepton flavour violation to be diagonal for the sake of simplicity and economy of parameters, In the present non-SUSY SO(10) models, even though the neutrino masses are governed by high scale Type-II see- M = diag (M , M , M ). (40) saw formula, the essential presence of singlet fermions that 1 2 3 implement the Type-II seesaw dominance by cancelling out η = the Type-I seesaw contribution gives rise to experimentally Note that the non-unitarity deviation is characterised by ηS + ηN M >> M observable LFV decay branching ratios through their loop , which in the limit N turns out to be mediation. The heavier RH neutrinos in this model, being S 1 † −2 † in the range of ∼1–10 TeV mass, also contribute, but less η η = X S.X = MD M M , 2 S D significantly than the singlet fermions. The charged current ∗ M M 1 Dαk Dβk weak interaction Lagrangian in this model can be written in ηαβ = . (41) 2 M2 the generalised form. k=1,2,3 k

5.1 Estimation of non-unitarity matrix For the degenerate case, Mi = MDeg(i = 1, 2, 3),gives

Using the ﬂavour basis, the general form of the charged cur- 2 η = 1GeV rent weak interaction Lagrangian including both V ± A cur- 2 MDeg rents in Model I and Model II is ⎛ ⎞ 0.0394 0.146 − 0.403i 4.17 − 11.99i 1 μ μ ×⎝ . + . . . − . ⎠ . L =−√ γ ν + γ 0 146 0 403i 3 602 105 8 0 002i CC g2L α L μ α L WL g2R α R μ Nα R WR 2 α=e,μ,τ 4.173 + 11.9i 105.805 + 0.002i 3139.8 +h.c. (37) (42) 123 Eur. Phys. J. C (2015) 75 :183 Page 11 of 24 183

For the general non-degenerate case of M, we saturate the The estimations presented in Eq. (47) show that in a wider − upper bound |ηττ| < 2.7 × 10 3 [110–113] to derive range of the parameter space, the leptonic CP violation parameter could be nearly two orders larger than the CKM- . . . 1 0 1026 + 7 0756 + 6762 4 = . × −3. CP violation parameter for quarks. 2 2 2 2 7 10 (43) 2 M1 M2 M3 By inspection, this equation gives the lower bounds

M1 > 4.35 GeV, M2 > 36.2GeV, M3 > 1120 GeV, 6 Neutrino-less double beta decay (44) Even with the vanishing bare mass term μ = 0inthe = S and for the degenerate case MDeg 1213 GeV. For the par- Yukawa Lagrangian of Eq. (10), the singlet fermions S (i = = = i tially degenerate case of M1 M2 M3, the solutions 1, 2, 3) acquire Majorana masses over a wide range of val- can be similarly derived as in Ref. [36] and one example is ues and, in the leading order, the corresponding mass matrix ( , , . ) M 100 100 1319 67 GeV. given in Eq. (14)ism =−M 1 M T . As far as the light- S MN neutrino mass matrix is concerned, it is given by the Type- 5.2 Branching ratio and CP violation II seesaw formula of Eq. (20), which is independent of the Majorana mass matrix mS of singlet fermions. But the com- One of the most important outcomes of non-unitarity effects bined effect of substantial mixing between the light neutri- is expected to become manifest through ongoing experimen- nos and the singlet or the RH neutrinos and also between τ → γ τ → μγ tal searches for LFV decays such as e , , the singlet neutrinos and the RH neutrinos result in the new μ → γ e . In these models the RH neutrinos and the singlet Majorana neutrino mass insertion terms in the Feynman dia- fermions contribute to the branching ratios [32,36,114–120]. grams. Out of these the mass insertion m due to the singlet >> S Because of the condition MN M, neglecting the RH neu- fermions in the Feynman diagram gives rise to new domi- trino exchange contribution compared to the sterile fermion nant contributions to the amplitude and the effective mass singlet contributions, our estimations for the different cases parameter for 0νββ even in the WL –WL channel. This may of the M values are presented below. These values are many be contrasted with the conventional Type-II seesaw domi- orders larger than the standard non-SUSY contributions and nated non-SUSY SO(10) models with only three generations are accessible to ongoing or planned searches [121–126]. For of standard fermions in 16i(i = 1, 2, 3), where there are the degenerate case no such contributions to 0νββ decay. The generalised form J 12 =− . × −6, of the charged current interaction Lagrangian for leptons in eμ 2 1 10 ± J 23 =− . × −6, this model including both V A currents has been given in eμ 2 4 10 (45) Eq. (37). J 23 = . × −4, ± μτ 1 4 10 As stated above, in Model I and Model II, the WR bosons − J 31 =1.2 × 10 4, and the doubly charged Higgs scalars, both the left-handed μτ ∼ and the right-handed, are quite heavy with MWR M we have the predicted values of the branching ratios 108–109 GeV.These make negligible contributions due to the − νββ BR(μ → eγ)= 6.43 × 10 17, RH current effects and Higgs exchange effects for the 0 − decay amplitude. The most popular standard and conven- (τ → eγ)= . × 16, (46) − − BR 8 0 10 tional contribution in the W –W channel is due to light − L L BR(τ → μγ ) = 2.41 × 10 12. neutrino exchanges. But one major new point in this work − − Because of the presence of non-unitarity effects in the present is that even in the WL –WL channel, the singlet fermion model, the leptonic CP-violation turns out to be similar to exchange, allowed within the Type-II seesaw dominance Refs. [36,110–113,127–135]. The moduli and phase of non- mechanism, can yield a much more dominant contribution to the 0νββ decay rate. For the exchange of singlet fermions unitarity and CP-violating parameter for the degenerate case ˆ (S j ), the Feynman diagram is shown in Fig. 5.Forthe of the present models are ˆ exchange of heavier RH Majorana neutrinos (Nk), the dia- |η |= . × −8, eμ 2 73 10 gram is the same as this ﬁgure but with the replacement of the δ = . , VνS → VνN → eμ 1 920 mixing matrix and masses by and mSi MNi . −7 The heavier RH neutrino exchange contributions are found |ηeτ |=4.54 × 10 , to be negligible compared to the singlet fermion exchange δ τ = 1.78, e contributions. In the mass basis, the contributions to the |η |= . × −5, μτ 2 31 10 decay amplitudes by ν and S and N exchanges are estimated −7 δμτ = 2.39 × 10 . (47) as 123 183 Page 12 of 24 Eur. Phys. J. C (2015) 75 :183

n p ee = Vνν 2 , mν ei mνi (52) W i L 2 | |2 e − ee νS p L m = V , (53) S ei m i Si 2 | |2 m = −MM −1MT ee νN p s N m = V , (54) N ei m i Ni

− with eL W = ee + ee + ee. L meff mν mS mN (55) p n Here mSi is the eigenvalue of the S-fermion mass matrix mS, and the magnitude of neutrino virtuality momentum |p|= Fig. 5 Feynman diagrams for neutrino-less double beta decay contri- ˆ 120–200 MeV. As the predicted values of the RH neutrino bution with virtual Majorana neutrinos νˆi and Si in the WL –WL -channel. For the RH neutrino exchange the Feynman diagram is the same but masses carried out in Sect. 3 have been found to be large, νS νN with the replacements V → V , Si → Ni which makes their contribution to the 0νββ decay amplitude negligible, we retain only contributions due to light neutrino and singlet fermion exchanges. The estimated values of the Vνν 2 LL 1 ei mνi effective mass parameters due to the S-fermion exchanges Aν ∝ , (48) M4 p2 and light neutrino exchanges are shown separately in Fig. 6 WL i=1,2,3 where the magnitudes of the corresponding mass eigenvalues 2 VνS used have also been indicated. ALL ∝ 1 ej , S 4 (49) M mS WL j=1,2,3 j 7.1 Numerical estimations of effective mass parameters 2 VνN ALL ∝ 1 ej , Using the equations of the normalised mass parameters [37], N 4 (50) M m N WL j=1,2,3 j we estimate numerically the nearly standard contribution due to the light neutrino exchanges and the dominant non- where |p|190 MeV represents the magnitude of standard contributions due to the singlet fermion exchanges. the neutrino virtuality momentum [136–146]. Using the uncertainties in the nuclear matrix elements [144–147]we 7.1.1 A. Nearly standard contribution have found it to have values in the range |p|=120– νν 200 MeV. In this model the new mixing matrix N ≡ V = (1 − η) Uν contains an additional non-unitarity effect due to the non- vanishing η [37]. Using MDeg = 1213 GeV in the degenerate case, we estimate 7 Effective mass parameter and half-life

Nei = (0.81437, 0.54858, 0.1267 + 0.0922i). (56) Adding together the 0νββ decay amplitudes arising out of η |η | < −3 light neutrino exchanges, singlet fermion exchanges and the Since all the -parameters are constrained by αβ 10 , it is expected that |N ||U | for any other choice of M. heavy RH neutrino exchanges in the WL –WL channel from ei ei η Eq. (49), and using suitable normalisations [144–147], we In the leading approximation, by neglecting the αi contri- express the inverse half-life thus: butions, the effective mass parameter in the WL –WL channel with light neutrino exchanges is expressed as −1 M0ν 0ν 0ν ν 2 ee ee ee 2 ee 2 T / G | | |(mν + m + m )| , mν = N mˆ i 1 2 01 m S N ei e i = |( ee + ee + ee)|2, K0ν mν m m 2 iα1 2 iα2 S N (c12c13) mˆ 1e + (s12c13) mˆ 2e = | |2. K0ν meff (51) + 2 iδ ˆ , s13e m3 (57) 0ν = . × −14 −1 M0ν = α In the above equation G01 0 686 10 years , ν where we have introduced the two Majorana phases 1 and −25 −1 −2 2.58 − 6.64, K0ν = 1.57 × 10 years eV and the three α2. As discussed subsequently in this section, they play cru- effective mass parameters for light neutrino, singlet fermion cial roles in preventing the cancellation between two differ- and heavy RH neutrino exchanges are ent effective mass parameters. Using α1 = α2 = 0 and the 123 Eur. Phys. J. C (2015) 75 :183 Page 13 of 24 183 experimental values of light-neutrino masses and the Dirac For comparison, the effective mass parameters in the stan- phase δ = 0.8π from Eq. (21), the light neutrino exchanges dard case without singlet fermions have been also given. It is have their well known values, clear that for allowed masses of the model, the non-standard ⎧ contributions to the effective mass parameters can be much ⎨ 0.0039 eV NH, more dominant compared to the standard values irrespective |mee|= 0.04805 eV IH, (58) ν ⎩ of the mass patterns of the light-neutrino masses: NH, IH, or 0.23 eV QD. QD.

7.1.2 B. Dominant non-standard contributions 7.2 Cancellation between effective mass parameters ( ) ν The ei element of the –S mixing matrix is [37] When plotted as a function of singlet fermion mass eigenvalue mS , the resultant effective mass parameter shows a M 1 VνS = D , cancellation for a certain region of the parameter space, the ei (59) M ei cancellation being prominent in the QD case. Like the light- neutrino masses, the singlet fermion masses mS are also M i where the Dirac neutrino mass matrix D has been given expected to have two Majorana phases. When all Majorana in Eq. (36), and the diagonal elements are estimated using phases are absent, both in the light active neutrino as well as the non-unitarity equation as discussed in the previous sec- in the singlet fermion sectors, it is clear that in the sum of tion. We derive the relevant elements of the mixing matrix the two effective mass parameters there will be a cancella- VνS using the structures of the Dirac neutrino mass matrix tion between the light active neutrino and the singlet fermion M D given in Eq. (36) and values of the diagonal elements of contributions because of the inherent negative sign of the M = (M , M , M ) 1 2 3 satisfying the non-unitarity constraint non-standard contribution. Our estimations for the NH, IH S in Eq. (43). The eigenvalues of the -fermion mass matrix and QD patterns of light-neutrino mass hierarchies are dis- m S are estimated for different cases using the structures of cussed separately. the RH Majorana neutrino mass matrices given in Eqs. (26), (28) and (30) in the formula m =−M 1 M T .Itisclear S MN that in the effective mass parameter the non-standard contri- 7.2.1 A. Effective mass parameter for NH and IH active bution due to sterile fermion exchange has a sign opposite to neutrino masses that due to light neutrino exchange, and also its magnitude is inversely proportional to the sterile fermion mass eigenval- In Fig. 7, we show the variation of the resultant effective mass ues. In the NH case the estimated effective mass parameters parameter with mS1 for NH and IH patterns of active light- are shown in Fig. 6 where the values of the diagonal ele- neutrino masses. It is clear that for lower values of mS1 ,the singlet fermion exchange term continues to dominate. For ments of M and the eigenvalues of ms have been speciﬁed. larger values of mS1 the resultant effective mass parameter tends to be identical to the light-neutrino mass contribution due to the vanishing non-standard contribution. We note that |(eV) ee |m |(eV) eff |m

m1(eV)

Fig. 6 Variation of the effective mass parameters with lightest LH neutrino mass. The dominant non-standard contributions due to fermion |m |(GeV) singlet contributions are shown by three horizontal lines with corre- s1 sponding mass values in GeV units. The subdominant effective mass parameters due to the NH and IH cases shown are similar to the standard Fig. 7 Variation of effective mass parameter of 0νββ decay with the values mass of the lightest singlet fermion for |p|=190 MeV 123 183 Page 14 of 24 Eur. Phys. J. C (2015) 75 :183 the values |meff |=0.5–0.1 eV can easily be realised for at m0 = 0.2 eV. The middle line corresponds to α1 = π, | |= | |= α = = . mS1 3–5 GeV in the NH case but for mS1 1–2 GeV 2 0atm0 0 5 eV. The lower line corresponds to in the IH case. α1 = 0, α2 = π at m0 = 0.3 eV. We ﬁnd that because of the introduction of the appropriate Majorana phases the dips in the two curves have disappeared. 7.2.2 B. Effective mass parameter for QD neutrinos

The variation of the effective mass with mS1 for the QD case 7.3 Half-life as a function of singlet fermion masses with one experimentally determined Dirac phase δ = 0.8π and assumed values of two unknown Majorana phases is In order to arrive at a plot of the half-life against the light- given in Fig. 8. The upper panel of Fig. 8 shows the vari- est singlet fermion mass in different cases, at ﬁrst we esti- α = α = ation with 1 2 0 for different choices of the com- mate the mass eigenvalues of the three singlet fermions for = . . . mon light-neutrino mass m0 0 5eV,03 eV and 0 2eV different allowed combinations of the N–S mixing matrix for the upper, middle and the lower curves, respectively, elements satisfying the non-unitarity constraint of Eq. (43) where the cancellations are clearly displayed in the regions and by using the RH neutrino mass matrices predicted for = . . of ms1 0 4–1 5GeV. However, before such a cancellation the NH, IH and QD cases from Eqs. (26), (28), (30) and (31). occurs, the dominance of the singlet exchange contribution These solutions are shown in Table 4. is clearly shown to occur in the regions of lower values of We then derive expressions for the half-life taking into > mS1 . For larger values of mS1 5 GeV, the singlet exchange account the contributions of the two different amplitudes or contribution tends to be negligible and the light QD neu- effective mass parameters arising out of the light neutrino trino contribution to meff is recovered. In the lower panel and the singlet fermion exchanges, leading to of Fig. 8, the upper curve corresponds to α1 = π, α2 = π 2 −2 0ν ms1 T / = |1 + X + Y | , (60) 1 2 4 4 K0ν|p| (MD/M)e1 where (M /M) 2 m (M /M) 2 m X = D e2 S1 + D e3 S1 , (61) ( / ) 2 ( / ) 2 MD M e1 ms2 MD M e1 mS3 m = ee S1 . |(eV) Y mν (62) 2( / )2 eff p MD M e1

|m ee Here we have used the expression for mν given in Eq. (52). In Eq. (60), Y = 0 gives complete dominance of the singlet fermion exchange term. However, this formula for the half-life is completely different from the one obtained using inverse seesaw dominance in SO(10) [39]. In the present |m |(GeV) model in the leading order, the predicted half-life depends s1 directly on the square of the lightest singlet fermion mass and it is independent of the RH neutrino mass, which is non-diagonal. But in [39], the half-life of neutrino-less double beta decay is directly proportional to the fourth power of the lightest singlet fermion mass and the square of the lightest right-handed neutrino mass, leading to a different |(eV) result. eff |m 7.3.1 A. Half-life in the NH and IH cases

We have computed the half-life for NH and IH patterns of active neutrino masses, taking the contributions of singlet fermion as well as light active neutrino exchanges. This is |m |(GeV) s1 shown in the upper panel for the NH case and in the lower Fig. 8 Variation of the effective mass of 0νββ decay with the mass of panel for the IH case in Fig. 9. the lightest singlet fermion for QD light neutrinos with one Dirac phase Taking both the X term and the Y term in Eq. (60), we (upper) and with one Dirac phase and two Majorana phases (lower) ﬁnd that for a smaller value of mS1 , the contribution due 123 Eur. Phys. J. C (2015) 75 :183 Page 15 of 24 183

Table 4 Eigenvalues of singlet fermion masses for different allowed N–S mixing matrix elements and for NH, IH and QD patterns of light-neutrino masses

M (GeV) mˆs (NH) (GeV) M (GeV) mˆs (IH) (GeV) M (GeV) mˆs (QD) (GeV)

(40,400,1180) (1.2,502,883) (40,450,1280) (0.4,54.32,7702) (100,600,1500) (0.5,17.7,109)) (100,400,1180) (7.65,515,909)) (60,450,1280) (0.9,54.4,7705) (130,600,1500) (0.8,17.7,109) (150,400,1180) (16,533,951) (70,450,1280) (1.2,54.4,7706) (200,600,1500) (1.97,17.7,109) (200,400,1180) (25,558,1011) (100,450,1280) (2.5,55,7715) (300,600,1500) (4.4,17.7,109) (250,400,1180) (35,588,1093) (300,450,1280) (22,56,7831) (350,600,1500) (6.05,17.7,109) (300,400,1180) (43,622,1200) (400,450,1280) (36.2,59,7933) (400,600,1500) (8,17.7,109) (350,400,1180) (50,659,1331) (450,450,1280) (42,64,7996) (500,600,1500) (12.3,17.7,109) (600,600,1500) (17.7,17.7,109)

to the current experimental bound even if the light-neutrino masses have the NH pattern of masses. Similarly the cor- ≤ responding bound for the IH case is mS1 2 GeV. But in a recent paper [39] which has an inverse seesaw dominant yrs.)

25 neutrino mass, the corresponding bound for the NH and IH ≤ (10 case is mS1 14 GeV. 1/2 T

7.3.2 B. Lifetime prediction with QD neutrino masses.

For the QD masses of the light active neutrinos, we considered the X term and Y term of Eq. (60) i.e. includ- |m |(GeV) s1 ing both the sterile neutrino exchange and the light neutrino exchange contributions. For the light-neutrino effective mass parameter occurring in Y , we have considered three different cases with common light-neutrino mass values m0 = 0.2eV, 0.3eV and 0.5eV, resulting in the three different curves shown in the upper and the lower panels yrs.)

25 of Fig. 10. In the upper panel, only the experimentally deter-

(10 mined Dirac phase δ = 0.8π has been included in the PMNS 1/2

T mixing matrix for light QD neutrinos, while ignoring the two Majorana phases (α1 = α2 = 0). In the lower panel, while keeping δ = 0.8π for all the three curves, the Majorana phases have been chosen as indicated against each of them. As the sterile neutrino exchange amplitude given in Eq. (53) |m |(GeV) is inversely proportional to the eigenvalue of the correspond- s1 ing sterile neutrino mass mSi , even in the quasi-degenerate Fig. 9 Variation of half-life of 0νββ decay with the sterile neutrino case this contribution is expected to dominate for allowed mass for the NH (upper)andtheIH(lower) patterns of the light active small values of mS . This fact is reflected in both figures | |= i neutrino masses for p 190 MeV given in Fig. 10. When the Majorana phases are ignored, this dominance gives a half-life less than the current bounds < . = . < . to the sterile neutrino is dominated for both the NH and for mS1 0 5 GeV when m0 0 5eV,butformS1 0 7 = . . the IH cases. But with the increase in the value of mS1 , GeV when m0 0 2–0 3 eV. When the Majorana phases are the half-life increases, showing its decreasing strength. The included preventing cancellation between the two contribu- < . predicted half-life curve saturates the experimental data at tions, these crossing points are changed to mS1 0 7GeV = . < . = . mS1 3 GeV and mS1 2 GeV for the NH and the IH cases, when m0 0 3eV,butmS1 1 0 GeV when m0 0 2– respectively. The interesting predictions are that if the light- 0.5 eV. Repeating the same procedure as for Ref. [39] which ≤ est sterile neutrino mass satisfies the bound mS1 3GeV, is based upon inverse seesaw dominance, the corresponding νββ ≤ . then the 0 decay should be detected with half-life close bound for the QD case is mS1 12 5GeV. 123 183 Page 16 of 24 Eur. Phys. J. C (2015) 75 :183

8.1 Electroweak precision observables and other constraints

We have shown that dominant contributions to 0νββ decay yrs.)

25 are possible for the ﬁrst generation sterile neutrino masses ˆ ∼ O( ) ˆ ∼ (10 mS1 1 GeV. For larger values of this mass mS1

1/2 5–10 GeV partial cancellation between the effective mass T parameters due to light neutrino and sterile neutrino exchanges occurs depending upon the choices of different Majo- rana phases. Various lighter sterile mass eigenvalues relevant for 0νββ decay are shown in Table 4 in the NH, IH and QD cases. It is pertinent to discuss the inﬂu- |ms |(GeV) 1 ence of these lighter masses on the precision electroweak observables. For the choices of parameters permitted by observable LFV and/or dominant LNV, the sterile fermion masses of ˆ < = , the ﬁrst two generations could be msi 45 GeV, i 1 2, yrs.) whereas in the absence of dominant LNV decay, the mass 25 eigenvalues could be even larger, mˆ 500 GeV. When (10 si

1/2 they are in the range of 1–45 GeV, we have estimated T the corresponding corrections on the electroweak observ- ν VνS = ( MD ) ables. The –S mixing M νS is well determined in our model and all the relevant ν–S mixings are easily deduced using Eqs. (36) and (43). In the allowed kinematical region, we have estimated the partial decay |m |(GeV) s1 widths, Fig. 10 Variation of half-life of 0νββ decay with the mass of the light- Γ( → ) = Γ νν¯ | VνS |4 ( = , ), est singlet fermion for QD light neutrinos including one Dirac phase Z Si Si Z α,i i 1 2 (63) (upper curve) and one Dirac phase and two Majorana phases (lower α curve) Γ νν¯ = . VνS = where the standard value Z 0 17 GeV and α,i (MD/M)α,i with α = νe,νμ,ντ and i = 1, 2, 3. We then −14 In the present case, the peaks in the half-life prediction obtain Γ(Z → S1 S1) = 1.2 × 10 GeV for the NH, IH −11 appear because of cancellation between the two effective and QD cases, and Γ(Z → S2 S2) = 6.6 × 10 GeV for parameters. The inclusion of the Majorana phases annuls the QD case only. Similarly we have estimated the partial the cancellation, resulting in the constructive addition of the decay width two effective mass parameters and reduced values of the ν ν half-life accessible to ongoing searches. For larger values Γ(W → lS ) = Γ l | V S |2 (i = 1, 2), (64) >> i W α,i of mS1 20 GeV, the sterile neutrino contribution to the α 0νββ amplitude becomes negligible and the usual contribu- and we obtained Γ(W → eS1) Γ(W → eS2) = tions due to light quasi-degenerate neutrinos are recovered. −9 3.5 × 10 GeV, Γ(W → μS1) Γ(W → μS2) = −7 1.8 × 10 GeV and Γ(W → τ S1) Γ(W → τ S2) = 1.0 × 10−5 GeV. These and other related estimations cause 8 Brief discussion of other aspects and leptogenesis negligible effects on the electroweak precision observables [150] primarily because of the small ν–S mixings deter- Here we discuss briefly constraints imposed on the model mined by the model analyses. In addition to these insignifi- by electroweak precision observables and predictions on the cant tree level corrections, new physics effects may affect the order of magnitude of baryon asymmetry of the universe electroweak observables indirectly via oblique corrections through resonant leptogenesis [148]. We also point out the through loops’ leading corrections to the Peskin–Takeuchi occurrence of small Z–Z mixings, while indicating briefly S, T, U parameters [151–153]. Although the computation of a possible application for dilepton production. Since details these loop effects are beyond the scope of the present paper, it of analyses and predictions on these aspects are beyond the may be interesting to estimate how the new fermions through scope of this paper, they will be presented elsewhere [149]. their small mixings with active neutrinos may affect the lep- 123 Eur. Phys. J. C (2015) 75 :183 Page 17 of 24 183 tonic and the invisible decay widths of the Z-boson, the W- ble of giving rise to two jets. It is interesting to note that mass and other observables [149]. our model predicts a rich structure of like sign dilepton In this model the neutral generator corresponding to heavy production through the mediation of Ni or Si , or both. Z is a linear combination of the U(1)R and U(1)B−L gener- From the details of the model parametrisations discussed ators, while the other orthogonal combination is the U(1)Y in Sects. 3–5, we have found the corresponding mixing generator of the SM (for earlier work on, Z boson in GUTs matrices with charged leptons defined through Eq. (39)dis- VνN embedding two-step breaking of left-right gauge symmetry, cussed in Sect. 5. We have estimated the elements e1 − . + . VνN . − . see [33–35]), [66–68]. The Z–Z mixing in such theories 0 0000727 i0 000203 and μ2 0 000813 i0 001148, 2 is computed through the generalised formula tan θzz = which would contribute to the production cross sections of 2− 2 ± ± ± ± M0 MZ √ MW pp → e e jjX and pp → μ μ jjX by the exchange where M0 = . In our model, since the M2 −M2 ρ cos θW Z 0 0 of RH neutrinos, the cross sections being proportional to LH triplet (3, −1, 1) has a very small VEV, v = 0.1– L L the modulus squares of these mixings. Similarly, we have 0.5eV<< Vew, the model is consistent with the tree level νS Vμ 0.0003191, which can also contribute to produc- value ρ = 1. The radiative corrections due to the 125GeV 2 0 tion process pp → μ±μ± jjX by the exchange of the sec- Higgs of the SM and the top quark yield ρ 1.009 [154]. ond sterile neutrino mass eigenstate. The first sterile neu- The new neutral gauge boson Z in principle may have an trino is too light to mediate the dilepton production pro- additional influence on the electroweak precision parame- cess detectable inside the LHC detector itself. Thus the ters as well as the Z-pole parameters if M << O(1) TeV Z LHC evidence of dilepton production signals may indicate [66–68,155–157]. The most recent LHC data has given the the presence of heavy Majorana neutrinos (see, for instance lower bound M ≥ 1.6TeV[158–160]. Since our model Z [109]). Details of the predictions will be reported elsewhere is based on an extended seesaw mechanism, we require [149]. VR >> Vew = 246GeV and this implies MZ >> MZ but accessible to LHC. Under this constraint MZ ∼ O(5– 10) TeV are the most suitable predictions of both mod- 8.3 Leptogenesis els discussed in this work. As some examples, using such values of MZ and the reported values from the Particle This model may have a wider range of possibilities for 2 Data Group [161–163]ofsin θW = 0.23126 ± 0.00005, leptogenesis via decays of Higgs triplets [167], or through MW = 80.385 ± 0.015GeV, MZ = 91.1876 ± 0.0021 GeV, the decays of LHC scale Majorana fermions N or S. ρ0 = 1.01, we obtain θzz = 0.00131 ± 0.0003, 0.0005 ± Although a rigorous estimation including solutions of the 0.00012, 0.0003 ± 0.00008 and 0.0002 ± 0.00006 for Boltzmann equations is beyond the scope of this work MZ = 2.0TeV, 5.0TeV, 7.5TeV and 10TeV, respectively. (it will be addressed elsewhere [149]), we discuss here Because of the smallness of the values, these mixings are con- briefly only a plausible case with a very approximate esti- sistent with the electroweak precision observables including mation of the CP-asymmetry parameter and the order of the Z-pole data [66–68,155–157,164,165]. Some of these magnitude of the baryon asymmetry through the decays masses may also be in the accessible range of the ILC [166]. of two nearly degenerate Majorana masses of sterile neu- Details of the experimental constraints on Z–Z mixings trinos. For resonant leptogenesis through the decays of a as a function of the Z masses will be investigated else- pair of quasi-degenerate RH neutrinos, relevant formulas where [149]. for CP-asymmetry and baryon asymmetry have been sug- ˆ ∼ O( ) gested in [148]. Noting that mS1 1 GeV is important for a dominant contribution to 0νββ decay and the 8.2 Possibility of dilepton signals at LHC N–S mixing matrix elements M2 ∼ M3 O(1) TeV are capable of predicting experimentally accessible LFV In both models considered in this work, there are two types decays in our model, we choose an interesting region of of heavy Majorana neutrinos: (i) the RH neutrinos with the parameter space M diag.(146, 3500, 3500) GeV in masses MN ≥ O(1–10) TeV, (ii) some of the three ster- i the quasi-degenerate case of S2 and S3. Then, using the ile neutrinos with masses mˆ S−i << MN . In principle both i G2113 breaking VEV VR O(10) TeV, the results of of these classes of fermions are capable of contributing Sect. 3.3 in the QD case of active neutrinos and Eq. (14), we to dilepton production at LHC through the sub-processes obtain → ± → ± ± + pp WL l l jjX where, for example, the WL produced from pp collision gives rise to a charged lepton mˆ = diag.(1.0, 595.864..., 595.864...)GeV, (65) + Si l and a Ni or Si in the first step by virtue of the latter’s mixing with the charged leptons given in Eq. (39). where the ellipses on the RHS indicate higher degrees of The particle Ni or Si can then produce a second charged quasi-degeneracy of the two masses the model tolerates. In − lepton of the same sign and a WL boson that is capa- order to estimate lepton asymmetry caused by the decay of 123 183 Page 18 of 24 Eur. Phys. J. C (2015) 75 :183

expected to cause the most dominant wash-outs, are sub- stantially depleted for the heavier quasi-degenerate Majo- rana masses of the decaying fermions. The depletion fac- δ2 tor is proportional to i leading to an effective wash- eff out factor Ki , which replaces Ki for the ith decaying Majorana fermion,

|ˆmS −ˆmS | δ = i j (i = j), i Γ Si eff δ2 . Ki i Ki (68) We find a sizeable baryon asymmetry in the following two cases: (i) In the case of finite perturbation theory, the mˆ 2 Γ 2 S j S j term in the denominator of R has been noted to be absent [148] leading to a singular term in the CP-asymmetry. (ii) In (mˆ −ˆm ) Fig. 11 Tree and one-loop diagrams for the Sk decay contributing to |ˆ −ˆ | << Γ / = Si S j the limit when mSi mS j S j 2, R 2 Γ . the CP-asymmetry. All fermion–Higgs couplings in the diagrams are S j of the form Vh wherewehavetheh = N–l–Φ Yukawa coupling and V M/MN (i) In finite perturbation theory

ˆ 1 Γ j heavy sterile fermions Sk(k = 2, 3) via their mixing with R = , 2 (mˆ S −ˆmS ) the heavier RH neutrinos, the corresponding Feynman dia- j k grams at the tree and one-loop levels, including the vertex mˆ δ = − S j , and self-energy diagrams, are shown in Fig. 11. jk 1 ˆ mSk The fermion–Higgs coupling in all the diagrams is Im[(y† y)2 ] Vh instead of the standard Higgs-Yukawa coupling h = ε = kj . Sk † (69) 16π|y y|kkδ jk MD/Vwk, where V M/MN , MD is given in Eq. (36) and j Vwk 174 GeV. The widths of these sterile fermions are Γ . Γ . S2 16 3 keV and S3 14 0 MeV. In order to exploit Similar formulas have been used by a number of authors the quasi-degeneracy of the second and the third generation in the case of decays of quasi-degenerate RH neutrinos fermions in resonant leptogenesis, we use the formula for CP- [169] and, speciﬁcally, in the context of SO(10) [170]. 7 asymmetry generated due to interference between the tree For the decay of S2 for which K2 = 2.7 × 10 ,using ( ˆ −ˆ ) × −7 and the self-energy graphs [148], mS2 mS3 2 10 GeV, we obtain † 2 Im[(y y) ] ε = 0.824, ε = kj R S2 Sk |y† y| |y† y| = . × −10. j jj kk YB 1 5 10 (70) ( ˆ 2 −ˆ2 ) ˆ Γ mS mS mSi S j The ﬁne tuning in the quasi-degenerate masses can be R = i j , (66) (mˆ 2 −ˆm2 )2 +ˆm2 Γ 2 reduced by one order if we use the effective wash-out fac- Si S j Si S j ( ˆ −ˆ ) . × −6 tor. For example using mS2 mS3 1 35 10 GeV, where y = M/MN h, h = MD/Vwk and Vwk 174 GeV. we get δ2 = 0.008, leading to For a computation of the baryon asymmetry Y withagiven B ε = . , wash-out factor K , we have also utilised the formula sug- S2 0 0357 eff = . × 5, gested in Ref. [148]: K2 1 92 10 ε = . × −10. Sk YB 9 3 10 (71) YB , 200Kk = . × 10 Γ For the decay of S3 for which K3 2 4 10 ,using Sk −6 2 −8 K = , (67) (mˆ S −ˆmS ) 10 GeV, we obtain δ 5 × 10 , k H(mˆ ) 2 3 3 Sk leading to ( ˆ ) ˆ H mSk being the Hubble parameter at temperature mSk .As ε = × −5, in TeV scale leptogenesis models, here also we encounter S3 8 10 large wash-out factors, which, in some cases, tend to damp K eff = 575.4, (72) out the baryon asymmetry generation. However, it has been 3 = . × −10. shown [168] that all the L = 2 processes lΦ → l¯Φ†, YB 7 3 10 123 Eur. Phys. J. C (2015) 75 :183 Page 19 of 24 183

Γ > |( ˆ −ˆ )| (ii) In the larger width limit. k 2 mSk mS j , neutrino masses naturally negligible in Model I, although in Model II it needs additional fine tuning. In spite of the 2(mˆ −ˆm ) high values of the Type-II seesaw scale M 108–109 Sk S j , L R GeV >> M , the models predict new dominant contribu- Γk Z − tions to 0νββ decay in the W –W channel mediated by 2 1 L L 4(mˆ S −ˆmS ) × 1 + k j , sterile neutrinos which acquire Majorana masses. The pre- Γ 2 μ → γ τ → μγ k dicted LFV decay branching ratios for e , I [( † )2 ] and τ → eγ , are found to be accessible to ongoing and m y y kj εS = mˆ S R. (73) planned experiments. We discuss the impact on the resultant k 8π|y† y| Γ k j kk k effective mass parameter and the 0νββ half-life, showing the cancellation between light-neutrino exchange and ster- This case can be more efficiently implemented for S 3 ile neutrino exchange contributions. The cancellation occurs decay, which has Γ 14 MeV and K = 2.4 × 1011. S3 3 because of the opposite signatures of the two effective mass In this case the depletion in K is quite effective. Using 3 parameters due to light neutrino exchange and the sterile neu- (mˆ −ˆm ) 10−6 GeV, we obtain δ2 5 × 10−11, S2 S3 3 trino exchange when the effects of the Majorana phases are leading to ignored. We derive an analytic formula for the half-life of ε = × −7, S3 3 10 0νββ decay as a function of the singlet fermion masses, pre- eff = . , dicting a lower bound on the lightest sterile neutrino mass K3 10 3 (74) − eigenvalue from the current experimental data on the lower Y = 1.1 × 10 10. B bounds. We find that the half-life close to the current lower Thus we have shown very approximately that the model may bound or even lower can be easily accommodated even with be capable of accommodating the order of magnitude of the NH or IH patterns of light-neutrino masses. We find that baryon asymmetry of the universe that requires fine tun- the QD nature of the light-neutrino masses is not a necessary ing of the mass difference of the two sterile neutrino in criterion to satisfy as regards existing lower bounds on the − − the range 10 6–10 7 GeV. In a separate paper we plan to half-life estimated by different experimental groups. Even look into an improvement in these approximate solutions if the light active neutrino masses are NH or IH, a half-life 25 and other possible channels of leptogenesis including the prediction T1/2 (2 − 5) × 10 years is realisable if the impact of the present model on electroweak precision observ- lightest sterile neutrino mass mS1 2–3 GeV. Depending ables and detection possibilities of RH neutrinos, S-fermions upon the common mass of the light QD neutrinos, the model ≤ × 25 ≤ ( . and the Z at collider energies such as those of LHC and also predicts a lifetime T1/2 2 10 years for mS1 0 5– ILC [149]. 1.0) GeV.A large cancellation between the two contributions is found to occur in the quasi-degenerate case of the light active neutrinos in the regions of a sterile neutrino mass, 9 Summary and conclusion mS1 2–8GeV. The bounds obtained in the sterile neutrino mass in these Type-II seesaw dominant models are signifi- In this work we have investigated the prospect of having cantly smaller than the bounds obtained in the inverse seesaw a new Type-II seesaw dominated neutrino mass generation model [39]. As the sterile neutrino contribution to the 0ν2β mechanism in non-SUSY SO(10) GUT by a novel proce- decay is inversely proportional to the corresponding mass dure by introducing one additional singlet fermion per gen- eigenvalues, the smallness of the lightest mass eigenvalues eration. Following the popular view that the only meaning- causes dominant contributions compared to those by light ful fermion masses in the Lagrangian must have dynami- neutrinos in NH, IH and QD cases. For the same reason the cal origins, and taking the non-dynamical singlet fermion new contributions are damped out for large sterile neutrino mass μS to be negligible, one of the models (Model I) dis- mass eigenvalues. Because of the underlying Type-II see- cussed is found to exhibit Type-II seesaw dominance and it saw formula for neutrino masses, heavy RH neutrino masses predicts a TeV scale Z boson accessible to LHC without in the range O(100) GeV–O(10000) GeV and with specified any drastic fine tuning in the corresponding Yukawa sector. heavy-light neutrino mixings are also predicted, which can For Model II the desired Type-II seesaw dominance requires be testified at the LHC and future high energy accelerators. an additional fine tuning up to one part in a million. The The proton lifetime predictions for p → e+π 0 for some would-be dominant Type-I seesaw contribution to neutrino regions of the parameter space are also accessible to ongo- masses in both models cancels out. The induced contribution ing experimental searches, especially for intermediate mass to the ν–S mixing mass term ML is shown to be damped values of the colour octet scalar which has been found to be out because of the GUT-scale mass of the LH doublet in necessary for gauge coupling unification. Further we have 16H , which renders the linear seesaw contribution to light verified that the lighter S1 or S2 states in the models have 123 183 Page 20 of 24 Eur. Phys. J. C (2015) 75 :183 negligible effects on the values of the electroweak precision Table 5 One-loop and two-loop beta function coefficients for gauge observables at the tree level although loop effects through coupling evolutions described in the text taking the second Higgs dou- the Peskin–Takeuchi parameters S, T, U will be investigated blet mass at 1 TeV elsewhere. Approximate estimations show the occurrence Symmetryai bij of small Z–Z mixings, apparently consistent with Z-pole ⎛ ⎞ 199/50, 27/10, 44/5 ⎝ ⎠ and non-Z-pole data. The possibility of dilepton signals at G213 −19/6, 41/10, −7 9/10, 35/6, 12 11/10, 9/2, −26 LHC in the WL –WL channel is briefly noted in both models, ⎛ ⎞ while an approximate estimation indicates the possibility of 8, 1, 3/2, 12 ⎜ / , / , / , ⎟ baryon asymmetry generation through leptogenesis due to − , / , / , − ⎜3 2 33 57 63 8 12⎟ G2113 3 57 12 37 8 7 ⎝ / , / , / , ⎠ the decay of quasi-degenerate sterile Majorana fermions at 9 2 63 8 209 16 4 9/2, 3/2, 1/2, 26 ⎛ ⎞ the TeV scale. The details and rigorous estimations on dilep- , , / , , , 31 6 39 2 12 ton signals, leptogenesis, estimation of S T U parameters ⎜ 6, 115/6, 3/2, 12 ⎟ G −2, −3/2, 29/4, −7 ⎜ ⎟ and the impact of Z–Z mixings on the Z-pole and non-Z- 2213 ⎝ 81/2, 6, 181/8, 4 ⎠ 9/2, 9/2, 1/2, −26 pole data including electroweak precision observables are ⎛ ⎞ currently under investigation and will be reported separately 319/6, 6, 57/4, 12 ⎜ 6, 319/6, 57/4, 12 ⎟ [149]. G −3/2, −3/2, 15/2, −7 ⎜ ⎟ 2213D ⎝171/4, 171/4, 239/4, 4⎠ 9/2, 9/2, 1/2, −26 Acknowledgments M. K. P. thanks Thomas Hambye for discussion and the Department of Science and Technology, Govt. of India, for the research project, SB/S2/HEP-011/2013. B. N. thanks Siksha ’O’ ⎛ ⎞ Anusandhan University for a research fellowship. II mν ML MD ⎜ ⎟ Open Access This article is distributed under the terms of the Creative M = ⎜M T 0 M ⎟ , (76) Commons Attribution 4.0 International License (http://creativecomm ⎝ L ⎠ ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, T T M M MN and reproduction in any medium, provided you give appropriate credit D to the original author(s) and the source, provide a link to the Creative where ML = yχ vχ , M = yχ vχ , MN = f vR and MD is Commons license, and indicate if changes were made. L R Funded by SCOAP3. the Dirac neutrino mass matrix as discussed in Sect. 4. Assuming a generalised unitary transformation from mass basis to flavour basis gives 10 Appendix A |ψ flavour = V |ψ mass (77)

10.1 Beta function coefficients for RG evolution of gauge or couplings ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ν Vνν VνS VνN νˆ α ⎜ αi αj αk ⎟ i ⎝ ⎠ = V Sν V SS V SN ⎝ ˆ ⎠ The RGEs for gauge couplings are Sβ ⎝ βi β j βk ⎠ S j (78) C V Nν V NS V NN ˆ Nγ γ i γ j γ k Nk ∂ gi ai 3 1 3 2 μ = g + bijg g , (75) ∂μ 16π 2 i (16π 2)2 i j with j V†MV∗ = Mˆ = Mˆ ; Mˆ ; Mˆ . diag νi S j Nk (79) where ai (bij) are one-loop (two-loop) beta function coefficients. Their values for the Model I and Model II are given Here Mν is the 9×9 neutral fermion mass matrix in a flavour in Table 5. basis with α, β, γ running over three generations of light- neutrinos, sterile-neutrinos and right handed heavy-neutrinos ˆ in their respective flavour states and Mν is the diagonal mass 11 Appendix B matrix with (i, j, k = 1, 2, 3) running over corresponding mass states. 11.1 Block diagonalisation and determination of Mν In the first step of block diagonalisation, the full neutrino ˆ mass matrix is reduced to a block-diagonal form MBD and In this section we discuss the various steps of block diago- in the second step we further block diagonalise to obtain the nalisation in order to calculate the light-neutrino mass, ster- three matrices as three different block-diagonal elements, ile neutrino mass and right-handed neutrino mass and their MBD = diag(Mν, mS, m N ) whose each diagonal element mixings. The complete 9×9 mass matrix in the flavour basis is a 3 × 3 matrix. In our estimation, we have used the mass {ν , , C } > , , v L SL NR is hierarchy MN M MD ML f L . Finally in the third 123 Eur. Phys. J. C (2015) 75 :183 Page 21 of 24 183 step we discuss a complete diagonalisation to arrive at the We denote physical masses and their mixings. ZB M = , (90) eff CD − 11.2 Determination of MBD = v − 1 T , Z f L MD MN MD (91) −1 B = ML − MD M M, (92) With the two unitary matrix transformations Q1 and Q2, N = T − T −1 T , C ML M MN MD (93) † ∗ ˆ − Q MνQ = MBD, (80) = T 1 . D M MN M (94) where 11.4 Determination of Q2

Q = Q1 Q2, (81) The remaining mass matrix Meff can be further block diagonalised using another transformation matrix, i.e. the product matrix Q = Q1 Q2 directly give MBD ˆ M from Mν Here MBD and MBD are the intermediate block- † ∗ ν 0 S Meff S = (95) diagonal and full block-diagonal mass matrices, respectively, 0 mS such that in Eq. (11.2) M 0 Mˆ = eff (82) BD 0 m S 0 N Q = , (96) 2 01 and − 1 ∗ T ∗ 1 2 P P P ⎛ ⎞ S = ∗ . (97) −P T 1 − 1 P T P Mν 00 2 ⎝ ⎠ MBD = 0 mS 0 . (83) Using Eq. (97)inEq.(95), we get through Eqs. (90)–(94) 00m N † = ( T −1 )−1 T −1 T − T P M MN M M MN MD ML = −1 T − −1 −1 11.3 Determination of Q1 M MD M MN M ML (98)

In the leading order parametrisation the standard form of Q where we have used that yχ is symmetric, leading to 1 is II −1 T Mν = m + M M M ν D N D 1 ∗ ∗ 1 − R RT R − ( −1 T ) + ( T −1 )−1 T Q = 2 , (84) MD MN MD ML M MN M ML 1 −RT 1 − 1 RT R∗ 2 − ( T −1 )−1( T −1 T ) ML M MN M M MN MD × − ( −1 )( T −1 )−1 T , where R is a 6 3 dimensional matrix. We have MD MN M M MN M ML − m =−MM 1 M T +··· . (99) † = −1 T , T = ( T , T ) S N R MN MD M K J (85) − The 3×3 block-diagonal mixing matrix Q2 has the following J = MM −1 K = M M 1 I = KJ−1 = M M−1. N D N D (86) form: ⎛ ⎞ Q 1 − 1 II† I 0 Therefore, the transformation matrix 1 can be written S 0 2 Q = = ⎝ −I † 1 − 1 I † I 0⎠ (100) purely in terms of the dimensionless parameters J and K , 2 01 2 001 ⎛ ⎞ 1 † 1 † 1 − KK − KJ K − ⎜ 2 2 ⎟ wherewehaveusedEq.(86) to define I = KJ 1 = Q = ⎝ − 1 JK† 1 − 1 JJ† J ⎠ ,(87) −1 1 2 2 MD M . − † − † − 1 ( † + † ) K J 1 2 K K J J Complete diagonalisation and physical neutrino masses The 3×3 block-diagonal matrices Mν , mS and m N can further be while the light and heavy mass matrices are diagonalised to give physical masses for all neutral leptons by a 9 × 9 unitary matrix U as − − f v M M M 1 M T M M 1 M ⎛ ⎞ M = L L − D N D D N , eff − − (88) Uν 00 M T 0 M T M 1 M T M T M 1 M ⎝ ⎠ L N D N U = 0 US 0 , (101) = +··· . m N MN (89) 00UN 123 183 Page 22 of 24 Eur. Phys. J. C (2015) 75 :183 where the 3 × 3 unitary matrices Uν, US and UN satisfy 24. R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980) 25. J.J. Schechter, J.W.F. Valle, Phys. 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