# High Scale Type-II Seesaw, Dominant Double Beta Decay within Cosmological Bound and LFV Decays in SU(‎5)

Authors: M. K. Parida; Rajesh Satpathy
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

Very recently novel implementation of type-II seesaw mechanism for neutrino mass has been proposed inSU(5) grand unified theory with a number of desirable new physical phenomena beyond the standard model. Introducing heavy right-handed neutrinos and extra fermion singlets, in this work we show how the type-I seesaw cancellation mechanism works in this SU(5) framework. Besides predicting verifiable LFV decays, we further show that the model predicts dominant double beta decay with normal hierarchy or inverted hierarchy of active light neutrino masses in concordance with cosmological bound. In addition a novel right-handed neutrino mass generation mechanism, independent of type-II seesaw predicted mass hierarchy, is suggested in this work.


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## Body

## 1. Introduction

Renormalizable standard model (SM) predicts neutrinos to be massless whereas oscillation experiments prove them to be massive [1–5]. All the generational mixings have been found to be much larger than the corresponding quark mixings. Theoretically [6–13] neutrino masses are predicted through various seesaw mechanisms [14–43]. In a minimal left-right symmetric [44–47] grand unified theory (GUT) like SO(10) [48, 49] where parity (P) violation in weak interaction is explained along with fermion masses [50–55], a number of these seesaw mechanisms can be naturally embedded while unifying the three forces of the SM [56–72]. More recently precision gauge coupling unification has been successfully implemented in direct symmetry breaking of SO(18)→SM which may have high potential for new physics [73].TheSO(10) model that predicts the most popular canonical seesaw as well as the type-II seesaw has also the ability to explain baryon asymmetry of the universe via leptogenesis through heavy RH neutrino [74] or Higgs triplet decays [75–77]. But because of underlying quark lepton symmetry [44], the type-I seesaw scale as well as RHν masses are so large that the model predicts negligible lepton flavor violating (LFV) decays like μ→eγ, τ→μγ, τ→eγ, and μ→ee¯e. Similarly direct mediation of large mass of scalar triplet required for type-II seesaw gives negligible contribution to lepton number violating (LNV) and lepton flavor violating (LFV) decays. Ever since the proposal of left-right symmetry, extensive investigations continue in search of experimentally observable double beta decay [78–82] in the WR-WR channel [83, 84]. Adding new dimension to such lepton number violating (LNV) process, the like-sign dilepton production has been suggested as a possible means of detection of WR-boson at accelerator energies [85], particularly the LHC [43]. However, no such signals of TeV scale WR- have been detected so far. Even if WR mass and seesaw scales are large and inaccessible for direct verification, neutrinoless double beta decay (0ν2β) in the WL-WL channel [20, 86–91] is predicted close to observable limit with τββ≥1025 yrs provided light neutrino masses predicted by such high scale seesaw mechanisms are quasidegenerate (QD) with each mass mν≥O(0.2) eV [78] and their sum >0.6 eV. But as noted by the recent Planck data such QD type masses violate the cosmological bound [92].(1)Σν≡Σim^i≤0.23eV.The fact that such QD type ν masses violate the cosmological bound may be unravelling another basic fundamental reason why detection of double beta decay continues to elude experimental observation for several decades. On the other hand, if neutrinos have smaller NH or IH type masses, there is no hope for detection of these LNV events in near future with RHν extended SM. In other words, predicting observable double beta decay in the WL-WL channel with left-handed helicities of both the beta particles has been a formidable problem confronting theoretical and experimental physicists. However, it has been shown that in case of dynamical seesaw mechanism generating Dirac neutrinos the seesaw scale is accessible for direct experimental verification [93].The path breaking discovery of inverse seesaw [25–31] with one extra singlet fermion per generation not only opened up the neutrino mass generation mechanism for direct experimental tests, but also lifted up lepton flavor violating (LFV) decays [94] from the abysmal depth of experimental inaccessibility of negligible branching ratios (Br.(lα→lβγ)~10-50) to the illuminating salvation of profound observability (Br.≃10-8-10-16) [95–100] which has been discussed extensively [99, 101–118]. Despite inverse seesaw, observable double beta decay in the WL-WL channel and the non-QD type neutrino masses remained mutually exclusive until both the RH neutrinos and singlet fermions (Si) were brought into the arena of LFV and LNV conundrum through the much needed extension of the Higgs sector. The King-Kang [119] mechanism cancelled out the ruling supremacy of canonical seesaw which was profoundly exploited in SO(10) models with the introduction of both the SO(10) Higgs representations 16H and 126H† [13, 84, 120–125] with successful prediction of observable double beta decay in the WL-WL channel [20, 86]. Very interestingly, even though high scale type-II seesaw can govern light neutrino masses of any hierarchy, possibility of observable LFV and double beta decay prediction in the WL-WL channel irrespective of light neutrino mass hierarchies has been realized at least theoretically [13, 125].The purpose of this work is to point out that there are new interesting physics realizations with suitable extension of a non-SUSYSU(5) GUT model proposed recently [126] where type-II seesaw, precision coupling unification, verifiable proton decay, scalar dark matter, and vacuum stability have been already predicted. However with naturally large type-II seesaw scale >109.2 GeV, observable double beta decay accessible to ongoing experiments [78–82] is possible in this model too with QD type neutrinos only of common mass with m0≥0.2 eV like many other high scale seesaw models as noted above. In this work we make additional prediction that dominant double beta decay in the WL-WL channel can be realized with NH or IH type hierarchy consistent with much lighter neutrino masses mi<<0.2 eV. Thus, this realization is consistent with cosmological bound of (1). Although such possibilities were realized earlier in SO(10) with TeV scale WR or Z′ bosons as noted above, in SU(5) without the presence of left-right symmetry and associated gauge bosons, we have shown here for the first time that the dominant double beta decay is mediated by a sterile neutrino (Majorana fermion singlet) of O(1) GeV mass of first generation. The model further predicts LFV decay branching ratios only 4–5 orders smaller than the current experimental limits. An additional interesting part of the present work is the first suggestion of a new mechanism for heavy RHν mass generation that permits these masses to have hierarchies independent of conventional type-II seesaw prediction. Thus highlights of the present model are as follows:(i)
first implementation of type-I seesaw cancellation mechanism leading to the dominance of type-II seesaw inSU(5);(ii)
prediction of verifiable LFV decays only4-5 orders smaller than the current experimental limits;(iii)
prediction of dominant double beta decay in theWL-WL channel close to the current experimental limits for light neutrino masses of NH or IH type in concordance with cosmological bound;(iv)
suggestion of a new right-handed neutrino mass generation mechanism independent of type-II predicted mass hierarchy;(v)
precision gauge coupling unification with verifiable proton decay which is the same as discussed in [126].This paper is organised in the following manner. In Section2 we briefly review the SU(5) model along with gauge coupling unification and predictions of the intermediate scales. In Section 3 we discuss how type-I seesaw formula for active neutrino masses cancels out giving rise to dominance of type-II seesaw and prediction of another type-I seesaw formula for sterile neutrino masses. Fit to neutrino oscillation data is discussed in Section 4. In Section 5 we suggest a new mechanism of RHν mass generation. Prediction on LFV decay branching ratios is discussed in Section 6. Lifetime prediction for double beta decay is presented in Section 7. In Section 8 we discuss the results of this work and state our conclusion. Block diagonalization procedure is explained in more detail in Appendix A.


## 2. A Non-Supersymmetric SU(‎5) Model

### 2.1. Extension of SU(‎5)

As noted in [127], inclusion of the scalar κ(3,0,8)⊂75H with mass Mκ=109.23GeV in the extended non-SUSY SU(5) achieves precision gauge coupling unification. Then it has been shown in [126] that type-II seesaw ansatz for neutrino mass is realized by inserting the entire Higgs multiplet 15H⊂SU(5) containing the LH Higgs triplet ΔL(3,-1,1) at the same mass scale M15H=1012GeV. (2)κ3,0,8⊂75H,Mκ=109.23GeV,ΔL3,-1,1⊂15H,M15H=1012GeV,ξ1,0,1,Mξ~O1TeV.The scalar singletξ(1,0,1) has played two crucial interesting roles of stabilising the SM scalar potential as well as serving as WIMP DM candidate. The introduction of 15H at any scale >109.23 GeV in this model maintains precision coupling unification. In the present model we extend the model further by the inclusion of the following fermions and an additional scalar χS(1,0,1):(i)
three right-handed neutrino singletsNi(i=1,2,3), one for each generation, with masses to be fixed by this model phenomenology;(ii)
three left-handed Majorana fermion singletsSi(i=1,2,3), one for each generation, similar to those introduced in case of inverse seesaw mechanism [25–31];(iii)
a Higgs scalar singletχS(1,0,1) to generate S-N mixings through its VEV.Being singlets under the SM gauge group, they do not affect precision gauge coupling unification of [126].


### 2.2. Coupling Unification, GUT Scale, and Proton Lifetime

As already discussed [126, 127] using renormalization group equations for gauge couplings and the set of Higgs scalars of (2), precision unification has been achieved with the PDG values of input parameters [128–130] on sin2⁡θW(MZ), αS(MZ) resulting in the following mass scales and the GUT fine-structure constant αG: (3)MU=1015.23GeV,Mκ=109.23GeV,MΔL=M15H=1012GeV,1αG=37.765.Using threshold effects due to superheavy Higgs scalars [131–138], proton lifetime prediction for p→e+π0 turns out to be in the experimentally accessible range [139]:(4)τpp→e+π0=(1.01×1034±0.44-5.5×1035±0.44yrs.Extensive investigations with number ofSU(5) GUT extensions have been carried out with proton lifetime predictions consistent with experimental limits [140–149]. But implementation of type-II seesaw dominance due to type-I seesaw cancellation resulting in dominant LFV and LNV decays as discussed below is new especially in the context of non-SUSY SU(5).


## 2.1. Extension of SU(‎5)

As noted in [127], inclusion of the scalar κ(3,0,8)⊂75H with mass Mκ=109.23GeV in the extended non-SUSY SU(5) achieves precision gauge coupling unification. Then it has been shown in [126] that type-II seesaw ansatz for neutrino mass is realized by inserting the entire Higgs multiplet 15H⊂SU(5) containing the LH Higgs triplet ΔL(3,-1,1) at the same mass scale M15H=1012GeV. (2)κ3,0,8⊂75H,Mκ=109.23GeV,ΔL3,-1,1⊂15H,M15H=1012GeV,ξ1,0,1,Mξ~O1TeV.The scalar singletξ(1,0,1) has played two crucial interesting roles of stabilising the SM scalar potential as well as serving as WIMP DM candidate. The introduction of 15H at any scale >109.23 GeV in this model maintains precision coupling unification. In the present model we extend the model further by the inclusion of the following fermions and an additional scalar χS(1,0,1):(i)
three right-handed neutrino singletsNi(i=1,2,3), one for each generation, with masses to be fixed by this model phenomenology;(ii)
three left-handed Majorana fermion singletsSi(i=1,2,3), one for each generation, similar to those introduced in case of inverse seesaw mechanism [25–31];(iii)
a Higgs scalar singletχS(1,0,1) to generate S-N mixings through its VEV.Being singlets under the SM gauge group, they do not affect precision gauge coupling unification of [126].


## 2.2. Coupling Unification, GUT Scale, and Proton Lifetime

As already discussed [126, 127] using renormalization group equations for gauge couplings and the set of Higgs scalars of (2), precision unification has been achieved with the PDG values of input parameters [128–130] on sin2⁡θW(MZ), αS(MZ) resulting in the following mass scales and the GUT fine-structure constant αG: (3)MU=1015.23GeV,Mκ=109.23GeV,MΔL=M15H=1012GeV,1αG=37.765.Using threshold effects due to superheavy Higgs scalars [131–138], proton lifetime prediction for p→e+π0 turns out to be in the experimentally accessible range [139]:(4)τpp→e+π0=(1.01×1034±0.44-5.5×1035±0.44yrs.Extensive investigations with number ofSU(5) GUT extensions have been carried out with proton lifetime predictions consistent with experimental limits [140–149]. But implementation of type-II seesaw dominance due to type-I seesaw cancellation resulting in dominant LFV and LNV decays as discussed below is new especially in the context of non-SUSY SU(5).


## 3. Cancellation of Type-I and Dominance of Type-II Seesaw

Due to introduction of heavy RHνs in the present model which were absent in [126], it may be natural to presume a priori that besides type-II seesaw, type-I seesaw may also contribute substantially to light neutrino masses and mixings. But it has been noted that there is a natural mechanism to cancel out type-I seesaw contribution while maintaining dominance of inverse seesaw [84, 119, 120, 122–124] or type-II seesaw or even linear seesaw [13, 125] as the case may be. Briefly we discuss below how this cancellation mechanism operates in the present extended model resulting in type-II seesaw dominance even in the presence of heavy RHνs.The SM invariant Yukawa Lagrangian of the model is(5)LYuk=Ylψ¯LψRϕ+fψLcψLΔL+yχN¯CSχs+12MNN¯CN+h.c.Using the VEVs of the Higgs fields and denoting M=yχχS=yχVχ, MD=Yϕ, a 9×9 neutral fermion mass matrix has been obtained which, upon block diagonalization, yields 3×3 mass matrices for each of the light neutrino (να), the right-handed neutrino (Nα), and the sterile neutrino (Sα) [120, 124, 125]. The block diagonalization of 9×9 neutral fermion mass matrix was presented in useful format in [32] but without cancellation of type-I seesaw. Later on, this diagonalization procedure has been effectively utilized to study the type-I seesaw cancellation mechanism in SO(10) models [13, 120, 124, 125].In this model the left-handed tripletΔL and RH neutrinos MN being much heavier than the other mass scales with MΔL≫MN≫M≫MD are at first integrated out from the Lagrangian leading to(6)-Leff=mνII+MD1MNMDTαβναTνβ+MD1MNMTαmνα¯Sm+Sm¯να+M1MNMTmnSmTSn,which, in the ν,S basis, gives the 6×6 mass matrix(7)Meff=MDMN-1MDT+mνIIMDMN-1MTMMN-1MDTMMN-1MT,while the 3×3 heavy RH neutrino mass matrix MN is the other part of the full 9×9 neutrino mass matrix. This 9×9 mass matrix M~BD which results from the first step of block diagonalization procedure as discussed above and in the Appendix is(8)W1†MνW1∗=M~BD=Meff00MN.Defining(9)X=MDM-1,Y=MMN-1,Z=MDMN-1,the transformation matrixW1 has been derived as shown in (10) [120, 125] (10)W1=1-12ZZ†-12ZY†Z-12YZ†1-12YY†Y-Z†-Y†1-12Z†Z+Y†Y.After the second step of block diagonalization, the type-I seesaw contribution cancels out and gives in the ν,S,N basis(11)W2†M~BDW2∗=MBD=mν000mS000mN,where W2 has been derived in (12) [120, 125]. We have used the bare mass of Si and VEV of χL(2,-1/2,1) to be vanishing, i.e., μS=0, <χL>=0, to get the form suitable for this model building. (12)W2=1-12XX†X0-X†1-12X†X0001In (11) the three 3×3 matrices are (13)mν=mνII=fvL(14)mS=-MMN-1MT(15)mN=MN,the first of which is the well-known type-II seesaw formula and the second is the emergence of the corresponding type-I seesaw formula for the singlet fermion mass. The third of the above equations represents the heavy RHν mass matrix.In the third step,mν, mS, and mN are further diagonalized by the respective unitary matrices to give their corresponding eigenvalues:(16)Uν†mνUν∗=m^ν=diagm1,m2,m3,US†mSUS∗=m^S=diagmS1,mS2,mS3,UN†mNUN∗=m^N=diagMN1,MN2,MN3.The complete mixing matrix [32, 120] diagonalizing the above 9×9 neutrino mass matrix occurring in (8) and in (A.1) of the Appendix turns out to be(17)V≡Vαiνν^VαjνS^VαkνN^VβiSν^VβjSS^VβkSN^VγiNν^VγjNS^VγkNN^(18)=1-12XX†UνX-12ZY†USZUN-X†Uν1-12X†X+YY†USY-12X†ZUN0-Y†US1-12Y†YUN,as shown in the Appendix. In (18), X=MDM-1, Y=MMN-1, and Z=MDMN-1.The mass of the singlet fermion is acquired through a type-I seesaw mechanism:(19)mS=-M1MNMTwhere M is the N-S mixing mass term in the Yukawa Lagrangian (6).


## 4. Type-II Seesaw Fit to Oscillation Data

### 4.1. Neutrino Mass Matrix from Oscillation Data

Using diagonalization of neutrino mass matrix(mν) by the PMNS matrix UPMNS(20)mν=UPMNSdiag⁡m1,m2,m3UPMNST,where mi(i=1,2,3) denote the mass eigen values. For neutrino mixings we use the abbreviated cyclic notations ti=sin⁡θjk, ci=cos⁡θjk, where i,j,k are cyclic permutations of generational numbers 1,2,3. Following the standard parametrisation we denote the PMNS matrix [128–130](21)UPMNS=c3c2t3c2t2e-iδD-t3c1-c3t1t2eiδDc3c1-t3t1t2eiδDt1c2t3t1-c3c1s2eiδD-c3t1-t3c1t2eiδDc1c2diag⁡eiαM/2,eiβM/2,1,where δD is the Dirac CP phase and (αM,βM) are Majorana phases.Here we present numerical analyses within3σ limit of the neutrino oscillation data in the type-II seesaw framework [126]. As we do not have any experimental information about Majorana phases, they are determined by means of random sampling; i.e., from the set of randomly generated values, each confined within the maximum allowed limit of 2π, only one set of values for (αM,βM) is chosen. Very recent analysis of the oscillation data has determined the 3σ and 1σ limits of Dirac CP phase δD [1]. The best fit values of δD in the normally ordered (NO) and invertedly ordered (IO) cases are near 1.2π and 1.5π, respectively, which we utilize for the sake of simplicity. A phenomenological model analysis has yielded δD=±1.32π [150].Global fit to the oscillation data [1] is summarized below including respective parameter uncertainties at 3σ level:(22)θ12∘=34.5±3.25,θ23∘NO=41.0±7.25,θ23∘IO=50.5±7.25,θ13∘NO=8.44±0.5,θ13∘IO=8.44±0.5,δDπNO=1.40±1.0,δDπIO=1.44±1.0,Δm212=7.56±0.545×10-5eV2,Δm312NO=2.55±0.12×10-3eV2,Δm312IO=2.49±0.12×10-3eV2.We denote the cosmologically constrained parameter, the sum of the three active neutrino masses, as (23)Σν=Σim^i.For normally hierarchical (NH), inverted hierarchical (IH), and quasidegenerate (QD) patterns, the experimental values of mass squared differences have been fitted by the following values of light neutrino masses and the respective values of the cosmological parameter Σν.(24)m^ν=0.00127,0.008838,0.04978eVNHΣν=0.059888eV,m^ν=0.04901,0.04978,0.00127eVIHΣν=0.059888eV,m^ν=0.2056,0.2058,0.2eVQD,Σν=0.6114eV.Using oscillation data and best fit values of the mixings, we have also determined the PMNS mixing matrix numerically:(25)UPMNS=0.8160.56-0.0199-0.0142i-0.354-0.0495i0.675-0.0346i0.6500.450-0.0568i-0.485-0.0395i0.75.


### 4.2. Determination of Majorana Yukawa Coupling Matrix

Now inverting the relationm^ν=UPMNS†MνUPMNS∗ where m^ν is the diagonalized neutrino mass matrix, we determine Mν for three different cases and further determine the corresponding values of the f matrix using f=Mν/vL where we use the predicted value of vL=0.1 eV.NH (26) f = 0.117 + 0.022 i - 0.124 - 0.003 i 0.144 + 0.025 i - 0.124 - 0.003 i 0.158 - 0.014 i - 0.141 + 0.017 i 0.144 + 0.025 i - 0.141 + 0.017 i 0.313 - 0.00029 iIH (27) f = 0.390 - 0.017 i 0.099 + 0.01 i - 0.16 + 0.05 i 0.099 + 0.01 i 0.379 + 0.02 i 0.176 + 0.036 i - 0.16 + 0.05 i 0.176 + 0.036 i 0.21 - 0.011 iQD (28) f = 2.02 + 0.02 i 0.0011 + 0.02 i - 0.019 + 0.3 i 0.0011 + 0.02 i 2.034 + 0.017 i 0.021 + 0.21 i - 0.019 + 0.3 i 0.021 + 0.21 i 1.99 - 0.04 iRandomly chosen Majorana phases [126] αM=74.84∘, βM=112.85∘, and the central value of the Dirac phase δD=218∘ have been used in this analysis. Using the well-known definition of the Jarlskog-Greenberg [151, 152] invariant, (29)JCP=-t3c3t2c22t1c1sin⁡δD,and keeping δD at its best fit values, we have estimated the predicted allowed ranges of the CP-violating parameter in both cases: (30)JCP=0.0175-0.0212NHJCP=0.0302-0.0365IHwhere the variables have been permitted to acquire values within their respective 3σ ranges of the oscillation data. Besides these there are nonunitarity contributions which have been discussed extensively in the literature.


### 4.3. Scaling Transformation of Solutions

In general there could be type-II seesaw models characterizing different seesaw scales and induced VEVs matching the given set of neutrino oscillation data represented by the same neutrino mass matrix. For two such models,(31)mν=f1vL1=f2vL2,Then the f- matrix in one case is determined up to good approximation in terms of the other from the knowledge of the two seesaw scales.(32)f1≃f2MΔ1MΔ2At MΔ(1)=1012 GeV our solutions are the same as in [126]. In view of this scaling relation, we can determine the values of the Majorana Yukawa matrix in the present case from the estimations of [126]. For example, if we choose MΔ(1)=1010 GeV in the present case compared to MΔ(2)=1012 GeV in [126], we rescale the solutions of [126] by a factor 10-2 to derive solutions in the present case. Thus graphical representations of solutions are similar to those of [126] for MΔ(2)=1012 GeV which we do not repeat here. The values of magnitudes of fij at any new scale are obtained by rescaling them by the appropriate scaling factor while the phase angles remain the same as in [126].


### 4.4. Dirac Neutrino Mass Matrix

The Dirac neutrino mass matrixMD plays crucial role in predicting LFV and LNV decays. In certain SO(10) models [50, 51, 125] this is usually determined by fitting the charged fermion masses at the GUT scale and equating it with the up-quark mass matrix. The fact that MD0≃Mu0 at the GUT scale follows from the underlying quark lepton symmetry [44] of SO(10). In SU(5) itself, however, there is no such symmetry to predict the structure of MD in terms of quark matrices. Also in this SU(5) model we do not attempt any charged fermion mass fit at the GUT scale or above it. Since the Dirac neutrino mass matrix is not predicted by the SU(5) symmetry itself, for the sake of simplicity and to derive maximal effects on LFV and LNV decays, we assume MD0 to be equal to the up-quark mass matrix Mu0 at the GUT scale. Noting that N is SU(5) singlet fermion, in the context of relevant Yukawa interaction Lagrangian(33)-LYuk=YN5-F·1F·5H+Yu10F·10F·5H+⋯+h.c.,this assumption is equivalent to alignment of the two Yukawa couplings:(34)YN≃Yu.This alignment is naturally predicted in SO(10) or SO(18) [73], but in the present SU(5) case it is assumed.We realize this matrixMD using renormalization group equations for fermion masses and gauge couplings and their numerical solutions [153–155] starting from the PDG values [128–130] of fermion masses at the electroweak scale. Following the bottom-up approach and using the down-quark diagonal basis, the quark masses and the CKM mixings are extrapolated from low energies using renormalization group (RG) equations [153–157]. After assuming the approximate equality MD0≃Mu0 at the GUT scale where Mu0 is the up-quark mass matrix, the top-down approach is exploited to run down this mass matrix MD0 using RG equations [153]. Then the value of MD near 1-10 TeV scale turns out to be (35)MD≃0.0140.04-0.01i0.109-0.3i0.04+0.01i0.352.6+0.0007i0.1+0.3i2.6-0.0007i79.20GeV.As already noted above, although on the basis ofSU(5) symmetry alone there may not be any reason for the rigorous validity of (35), in what follows we study the implications of this assumed value of MD to examine maximum possible impact on LFV and LNV decays discussed in Sections 6 and 7. Another reason is that the present assumption on MD may be justified in direct SO(10) breaking to the SM which we plan to pursue in a future work.


## 4.1. Neutrino Mass Matrix from Oscillation Data

Using diagonalization of neutrino mass matrix(mν) by the PMNS matrix UPMNS(20)mν=UPMNSdiag⁡m1,m2,m3UPMNST,where mi(i=1,2,3) denote the mass eigen values. For neutrino mixings we use the abbreviated cyclic notations ti=sin⁡θjk, ci=cos⁡θjk, where i,j,k are cyclic permutations of generational numbers 1,2,3. Following the standard parametrisation we denote the PMNS matrix [128–130](21)UPMNS=c3c2t3c2t2e-iδD-t3c1-c3t1t2eiδDc3c1-t3t1t2eiδDt1c2t3t1-c3c1s2eiδD-c3t1-t3c1t2eiδDc1c2diag⁡eiαM/2,eiβM/2,1,where δD is the Dirac CP phase and (αM,βM) are Majorana phases.Here we present numerical analyses within3σ limit of the neutrino oscillation data in the type-II seesaw framework [126]. As we do not have any experimental information about Majorana phases, they are determined by means of random sampling; i.e., from the set of randomly generated values, each confined within the maximum allowed limit of 2π, only one set of values for (αM,βM) is chosen. Very recent analysis of the oscillation data has determined the 3σ and 1σ limits of Dirac CP phase δD [1]. The best fit values of δD in the normally ordered (NO) and invertedly ordered (IO) cases are near 1.2π and 1.5π, respectively, which we utilize for the sake of simplicity. A phenomenological model analysis has yielded δD=±1.32π [150].Global fit to the oscillation data [1] is summarized below including respective parameter uncertainties at 3σ level:(22)θ12∘=34.5±3.25,θ23∘NO=41.0±7.25,θ23∘IO=50.5±7.25,θ13∘NO=8.44±0.5,θ13∘IO=8.44±0.5,δDπNO=1.40±1.0,δDπIO=1.44±1.0,Δm212=7.56±0.545×10-5eV2,Δm312NO=2.55±0.12×10-3eV2,Δm312IO=2.49±0.12×10-3eV2.We denote the cosmologically constrained parameter, the sum of the three active neutrino masses, as (23)Σν=Σim^i.For normally hierarchical (NH), inverted hierarchical (IH), and quasidegenerate (QD) patterns, the experimental values of mass squared differences have been fitted by the following values of light neutrino masses and the respective values of the cosmological parameter Σν.(24)m^ν=0.00127,0.008838,0.04978eVNHΣν=0.059888eV,m^ν=0.04901,0.04978,0.00127eVIHΣν=0.059888eV,m^ν=0.2056,0.2058,0.2eVQD,Σν=0.6114eV.Using oscillation data and best fit values of the mixings, we have also determined the PMNS mixing matrix numerically:(25)UPMNS=0.8160.56-0.0199-0.0142i-0.354-0.0495i0.675-0.0346i0.6500.450-0.0568i-0.485-0.0395i0.75.


## 4.2. Determination of Majorana Yukawa Coupling Matrix

Now inverting the relationm^ν=UPMNS†MνUPMNS∗ where m^ν is the diagonalized neutrino mass matrix, we determine Mν for three different cases and further determine the corresponding values of the f matrix using f=Mν/vL where we use the predicted value of vL=0.1 eV.NH (26) f = 0.117 + 0.022 i - 0.124 - 0.003 i 0.144 + 0.025 i - 0.124 - 0.003 i 0.158 - 0.014 i - 0.141 + 0.017 i 0.144 + 0.025 i - 0.141 + 0.017 i 0.313 - 0.00029 iIH (27) f = 0.390 - 0.017 i 0.099 + 0.01 i - 0.16 + 0.05 i 0.099 + 0.01 i 0.379 + 0.02 i 0.176 + 0.036 i - 0.16 + 0.05 i 0.176 + 0.036 i 0.21 - 0.011 iQD (28) f = 2.02 + 0.02 i 0.0011 + 0.02 i - 0.019 + 0.3 i 0.0011 + 0.02 i 2.034 + 0.017 i 0.021 + 0.21 i - 0.019 + 0.3 i 0.021 + 0.21 i 1.99 - 0.04 iRandomly chosen Majorana phases [126] αM=74.84∘, βM=112.85∘, and the central value of the Dirac phase δD=218∘ have been used in this analysis. Using the well-known definition of the Jarlskog-Greenberg [151, 152] invariant, (29)JCP=-t3c3t2c22t1c1sin⁡δD,and keeping δD at its best fit values, we have estimated the predicted allowed ranges of the CP-violating parameter in both cases: (30)JCP=0.0175-0.0212NHJCP=0.0302-0.0365IHwhere the variables have been permitted to acquire values within their respective 3σ ranges of the oscillation data. Besides these there are nonunitarity contributions which have been discussed extensively in the literature.


## 4.3. Scaling Transformation of Solutions

In general there could be type-II seesaw models characterizing different seesaw scales and induced VEVs matching the given set of neutrino oscillation data represented by the same neutrino mass matrix. For two such models,(31)mν=f1vL1=f2vL2,Then the f- matrix in one case is determined up to good approximation in terms of the other from the knowledge of the two seesaw scales.(32)f1≃f2MΔ1MΔ2At MΔ(1)=1012 GeV our solutions are the same as in [126]. In view of this scaling relation, we can determine the values of the Majorana Yukawa matrix in the present case from the estimations of [126]. For example, if we choose MΔ(1)=1010 GeV in the present case compared to MΔ(2)=1012 GeV in [126], we rescale the solutions of [126] by a factor 10-2 to derive solutions in the present case. Thus graphical representations of solutions are similar to those of [126] for MΔ(2)=1012 GeV which we do not repeat here. The values of magnitudes of fij at any new scale are obtained by rescaling them by the appropriate scaling factor while the phase angles remain the same as in [126].


## 4.4. Dirac Neutrino Mass Matrix

The Dirac neutrino mass matrixMD plays crucial role in predicting LFV and LNV decays. In certain SO(10) models [50, 51, 125] this is usually determined by fitting the charged fermion masses at the GUT scale and equating it with the up-quark mass matrix. The fact that MD0≃Mu0 at the GUT scale follows from the underlying quark lepton symmetry [44] of SO(10). In SU(5) itself, however, there is no such symmetry to predict the structure of MD in terms of quark matrices. Also in this SU(5) model we do not attempt any charged fermion mass fit at the GUT scale or above it. Since the Dirac neutrino mass matrix is not predicted by the SU(5) symmetry itself, for the sake of simplicity and to derive maximal effects on LFV and LNV decays, we assume MD0 to be equal to the up-quark mass matrix Mu0 at the GUT scale. Noting that N is SU(5) singlet fermion, in the context of relevant Yukawa interaction Lagrangian(33)-LYuk=YN5-F·1F·5H+Yu10F·10F·5H+⋯+h.c.,this assumption is equivalent to alignment of the two Yukawa couplings:(34)YN≃Yu.This alignment is naturally predicted in SO(10) or SO(18) [73], but in the present SU(5) case it is assumed.We realize this matrixMD using renormalization group equations for fermion masses and gauge couplings and their numerical solutions [153–155] starting from the PDG values [128–130] of fermion masses at the electroweak scale. Following the bottom-up approach and using the down-quark diagonal basis, the quark masses and the CKM mixings are extrapolated from low energies using renormalization group (RG) equations [153–157]. After assuming the approximate equality MD0≃Mu0 at the GUT scale where Mu0 is the up-quark mass matrix, the top-down approach is exploited to run down this mass matrix MD0 using RG equations [153]. Then the value of MD near 1-10 TeV scale turns out to be (35)MD≃0.0140.04-0.01i0.109-0.3i0.04+0.01i0.352.6+0.0007i0.1+0.3i2.6-0.0007i79.20GeV.As already noted above, although on the basis ofSU(5) symmetry alone there may not be any reason for the rigorous validity of (35), in what follows we study the implications of this assumed value of MD to examine maximum possible impact on LFV and LNV decays discussed in Sections 6 and 7. Another reason is that the present assumption on MD may be justified in direct SO(10) breaking to the SM which we plan to pursue in a future work.


## 5. Right-Handed Neutrino Mass in SU(‎5) vs SO(‎10)

### 5.1. RHν Mass in SO(‎10)

The fermions responsible for type-I and type-II seesaw are the LH leptonic doublets and the RH fermionic singlets of three generations. InSO(10) the left-handed lepton doublet (ν,l)T and the right-handed neutrino N are in the same spinorial representation 16F. (36)ν,lT⊂16F,N⊂16F.The Higgs representation 126H†⊂SO(10) contains both the left-handed and the right-handed triplets carrying B-L=-2,(37)126H†=ΔL3,1,-2,1+ΔR1,3,-2,1+⋯where the quantum numbers are under the left-right symmetry group SU(2)L×SU(2)R×U(1)B-L×SU(3)C. The common Yukawa coupling f10 in the Yukawa term (38)-LYuk10=f1016F16F126H†generates the dilepton-Higgs triplet interactions both in the left-handed and right-handed sectors giving rise to type-I and type-II seesaw mechanisms. The RH neutrino mass is generated through the VEV of the neutral component of the of ΔR(39)MN=f10ΔR0.The type-II seesaw contribution to light neutrino mass is(40)Mν=f10vLwhere vL is the corresponding induced VEV of ΔL(41)vL=λ10ΔR0vew2MΔL2.Here λ10 is the quartic coupling in the part of the scalar potential (42)V10=λ10ΔL†ΔRϕ†ϕ⊂λ10126H†126H10H10H. Thus with type-II seesaw dominance, the predicted heavy RH neutrino masses in SO(10) follow the same hierarchical pattern as the active light neutrino masses (43)MN1:MN2:MN3∷m1:m2:m3.


### 5.2. RHν Mass in SU(‎5)

Feynman diagram for type-II seesaw mechanism in the presentSU(5) model is shown in Figure 1.Figure 1
Feynman diagram representing type-II seesaw mechanism for neutrino mass generation inSU(5). Scalar fields ϕ,σS, and ΔL represent SM Higgs doublet, singlet, and LH triplet as defined in the text. This diagram defines the trilinear coupling mass μΔ=λσS.In contrast toSO(10) where the LH leptonic doublet and the RHν are in one and the same representation 16F, in SU(5) they are in different representations(44)ν,lT⊂5¯F,N⊂1F.In SU(5), while the dilepton-Higgs interaction is given by(45)-LYukll=f5¯F5¯F15H,the RH neutrino mass is generated through(46)-LYukNN=12fNNNσS+h.c.The fact that N is a singlet under SU(5) forces σS to be a singlet too. Further this singlet σS must carry B-L=-2 as its VEV generates the heavy Majorana mass (47)MN=fNσS.In sharp contrast to SO(10) where the LH triplet ΔL and the RH triplet ΔR scalars contained in the same representation 126H† generate the type-II seesaw and MN, the situation in SU(5) is different. Since LH triplet ΔL(3,-1,1) mediating type-II seesaw belongs to Higgs representation 15H⊂SU(5) and σS belongs to a completely different representation (which is a singlet ⊂SU(5)), the two relevant Majorana type couplings in general may not be equal (48)fN≠f.Also this assertion is further strengthened if we do not assume SU(5) to be a remnant of SO(10). Then the RH neutrino mass hierarchy can be decoupled from the type-II seesaw prediction. It is interesting to note that in SU(5)(49)vL=μΔvew2MΔ2where μΔ is the trilinear coupling in the potential term (50)Vtri=μΔΔLϕϕ+h.c.The VEV of this singlet σS can explain the dynamical origin of such trilinear coupling through its VEV vσ=σS(51)μΔ=λvσ,where λ is the quartic coupling in the potential term(52)Vql=λσSΔLϕϕ+h.c.(53)⊂λσS15H5H5H+h.cwhere the second line represents the SU(5) origin. For GUT scale U(1)B-L symmetry breaking driving VEV vσ≃MGUT in the natural absence of any intermediate symmetry, it is possible to ensure μΔ≃MΔL for (54)λ≃MΔLMGUT.Thus the SU(5) model gives similar explanation for quartic coupling as in direct breaking case of SO(10). But the predicted hierarchy of RHν masses in SU(5) may not, in general, follow the same hierarchical pattern as given by SO(10) shown in (43). This is precisely so because (43) follows from the fact that the same matrix UPMNS diagonalizes both the LH and the RH neutrino mass matrices which is further rooted in the fact that the same Majorana coupling f10 that generates the type-II seesaw mass term also generates MN. But because of the general possibility in SU(5) that fN≠f, the RHνs may acquire a completely different pattern depending upon the value of fN. Unlike SO(10), these masses emerging from SU(5) are also allowed to be quite different from the type-II seesaw scale.Even if the value ofvσ may be needed to be near MΔL, the value of MN is allowed to be considerably lower by fine-tuning the value of fN. Our LFV and LNV decay phenomenology as discussed below may need MN=1-10 TeV which is realizable using this new technique in SU(5). In contrast SO(10) needs U(1)R×U(1)B-L [84, 125, 157] or SU(2)R×U(1)B-L gauge symmetry and hence new gauge bosons near the TeV scale to generate such RHν masses which should be detected at LHC [84, 125]. Thus a new mechanism for RHν mass emerges here by noting the coupling fN≠f which has the potential to generate RHν masses over a wide range of values MN~100-1015 GeV. Then the RHν mass predictions in the two GUTs in the presence of type-II seesaw dominance are as follows:Type-II Seesaw Dominated SO(10) (55) M N i ≃ m i M Δ L 2 v e w 2 .Type-II Seesaw Dominated SU(5) (56) M N i = O 10 G e V - O M Δ L .Here mi,i=1,2,3, are the three mass eigen values of light neutrinos. It is to be noted that mi is absent in the RHS of (56) in the SU(5) case.


### 5.3. Realization of Mass Hierarchies

Here we discuss how the stated hierarchy in Section3(57)MΔL≫MN≫M≫MD,for type-II seesaw dominance, is realized through fine tuning. At first noting that the mass squared term for 15H⊂SU(5) in the scalar potential, M15215H†15H, is SU(5) invariant, M15 can have any mass below the GUT scale subject to proton stability and gauge coupling unification. Since 15H, unlike 5H, does not have Yukawa interactions with SM fermions, the Higgs mediated proton decay is suppressed. We now explain why we have used M15=MΔL=1012 GeV. In our model it is possible to assign any mass M15=MΔL>Mη where the mass of η(3,0,8)⊂75H is Mη=109.23 GeV. Because of the presence of η(3,0,8) at such intermediate mass value, precision gauge coupling unification is achieved which has been discussed separately [126, 127]. Following the standard symmetry breaking SU(5)→ SM through the GUT scale VEV of the SM singlet scalar in the adjoint representation 24H, VGUT=24H0, a SU(5) invariant scalar potential Vη gives the mass of η(3,0,8)(58)Vη=M75275H2+m24,7524H75H2+λ24,7524H275H2⊃M752+m24,75VGUT+λ24,75VGUT2η2.This leads to Mη2=M752+m(24,75)VGUT+λ(24,75)VGUT2. Here M75~m(24,75)~VGUT~MGUT. Thus fine-tuning any one of these four parameters can give Mη=109.23 GeV.  The presence of 15H below Mη destabilizes unification but protects it for M15>Mη. This has led to the chosen value of MΔL=M15=1012 GeV. We have noted in the following section that the value of M3<1 TeV violates the observed bound on the nonunitarity parameter ηττ<2.7×10-3 leading to the lower bound on M in the degenerate case (59)M1=M2=M3≥1250GeV.Here M=diag⁡(M1,M2,M3). Noting the definition (60)M=yχχS1,0,1=yχVχ,we now argue that even for GUT scale mass of χS and its VEV Vχ=VGUT, it is possible to satisfy (59). For Vχ~VGUT~1015 GeV we need a small fine-tuned value of Yukawa coupling (61)yχ>10-12,which satisfies M≥1 TeV but does not affect any known fermion mass. This shows the interesting possibility that even without having a low mass nonstandard Higgs χS, it is possible to realize the extended seesaw with type-II seesaw dominance. However, if we insist on yχ≤1, we need Vχ≥1 TeV and MχS≥1 TeV which is realizable as χS is a SU(5) scalar singlet. As we have assumed MD=Mu, this gives at GUT scale on extrapolation [153] (62)MD33~mtop≃85GeV.Here mtop is the top-quark mass. Thus achieving precision unification and type-II seesaw dominance has already given us MDelta>Mη whereas fine-tuning the Majorana coupling, fN>10-11, has yielded MN>104 GeV. Combining these with (59) and (62) gives the hierarchical relation of (57).


## 5.1. RHν Mass in SO(‎10)

The fermions responsible for type-I and type-II seesaw are the LH leptonic doublets and the RH fermionic singlets of three generations. InSO(10) the left-handed lepton doublet (ν,l)T and the right-handed neutrino N are in the same spinorial representation 16F. (36)ν,lT⊂16F,N⊂16F.The Higgs representation 126H†⊂SO(10) contains both the left-handed and the right-handed triplets carrying B-L=-2,(37)126H†=ΔL3,1,-2,1+ΔR1,3,-2,1+⋯where the quantum numbers are under the left-right symmetry group SU(2)L×SU(2)R×U(1)B-L×SU(3)C. The common Yukawa coupling f10 in the Yukawa term (38)-LYuk10=f1016F16F126H†generates the dilepton-Higgs triplet interactions both in the left-handed and right-handed sectors giving rise to type-I and type-II seesaw mechanisms. The RH neutrino mass is generated through the VEV of the neutral component of the of ΔR(39)MN=f10ΔR0.The type-II seesaw contribution to light neutrino mass is(40)Mν=f10vLwhere vL is the corresponding induced VEV of ΔL(41)vL=λ10ΔR0vew2MΔL2.Here λ10 is the quartic coupling in the part of the scalar potential (42)V10=λ10ΔL†ΔRϕ†ϕ⊂λ10126H†126H10H10H. Thus with type-II seesaw dominance, the predicted heavy RH neutrino masses in SO(10) follow the same hierarchical pattern as the active light neutrino masses (43)MN1:MN2:MN3∷m1:m2:m3.


## 5.2. RHν Mass in SU(‎5)

Feynman diagram for type-II seesaw mechanism in the presentSU(5) model is shown in Figure 1.Figure 1
Feynman diagram representing type-II seesaw mechanism for neutrino mass generation inSU(5). Scalar fields ϕ,σS, and ΔL represent SM Higgs doublet, singlet, and LH triplet as defined in the text. This diagram defines the trilinear coupling mass μΔ=λσS.In contrast toSO(10) where the LH leptonic doublet and the RHν are in one and the same representation 16F, in SU(5) they are in different representations(44)ν,lT⊂5¯F,N⊂1F.In SU(5), while the dilepton-Higgs interaction is given by(45)-LYukll=f5¯F5¯F15H,the RH neutrino mass is generated through(46)-LYukNN=12fNNNσS+h.c.The fact that N is a singlet under SU(5) forces σS to be a singlet too. Further this singlet σS must carry B-L=-2 as its VEV generates the heavy Majorana mass (47)MN=fNσS.In sharp contrast to SO(10) where the LH triplet ΔL and the RH triplet ΔR scalars contained in the same representation 126H† generate the type-II seesaw and MN, the situation in SU(5) is different. Since LH triplet ΔL(3,-1,1) mediating type-II seesaw belongs to Higgs representation 15H⊂SU(5) and σS belongs to a completely different representation (which is a singlet ⊂SU(5)), the two relevant Majorana type couplings in general may not be equal (48)fN≠f.Also this assertion is further strengthened if we do not assume SU(5) to be a remnant of SO(10). Then the RH neutrino mass hierarchy can be decoupled from the type-II seesaw prediction. It is interesting to note that in SU(5)(49)vL=μΔvew2MΔ2where μΔ is the trilinear coupling in the potential term (50)Vtri=μΔΔLϕϕ+h.c.The VEV of this singlet σS can explain the dynamical origin of such trilinear coupling through its VEV vσ=σS(51)μΔ=λvσ,where λ is the quartic coupling in the potential term(52)Vql=λσSΔLϕϕ+h.c.(53)⊂λσS15H5H5H+h.cwhere the second line represents the SU(5) origin. For GUT scale U(1)B-L symmetry breaking driving VEV vσ≃MGUT in the natural absence of any intermediate symmetry, it is possible to ensure μΔ≃MΔL for (54)λ≃MΔLMGUT.Thus the SU(5) model gives similar explanation for quartic coupling as in direct breaking case of SO(10). But the predicted hierarchy of RHν masses in SU(5) may not, in general, follow the same hierarchical pattern as given by SO(10) shown in (43). This is precisely so because (43) follows from the fact that the same matrix UPMNS diagonalizes both the LH and the RH neutrino mass matrices which is further rooted in the fact that the same Majorana coupling f10 that generates the type-II seesaw mass term also generates MN. But because of the general possibility in SU(5) that fN≠f, the RHνs may acquire a completely different pattern depending upon the value of fN. Unlike SO(10), these masses emerging from SU(5) are also allowed to be quite different from the type-II seesaw scale.Even if the value ofvσ may be needed to be near MΔL, the value of MN is allowed to be considerably lower by fine-tuning the value of fN. Our LFV and LNV decay phenomenology as discussed below may need MN=1-10 TeV which is realizable using this new technique in SU(5). In contrast SO(10) needs U(1)R×U(1)B-L [84, 125, 157] or SU(2)R×U(1)B-L gauge symmetry and hence new gauge bosons near the TeV scale to generate such RHν masses which should be detected at LHC [84, 125]. Thus a new mechanism for RHν mass emerges here by noting the coupling fN≠f which has the potential to generate RHν masses over a wide range of values MN~100-1015 GeV. Then the RHν mass predictions in the two GUTs in the presence of type-II seesaw dominance are as follows:Type-II Seesaw Dominated SO(10) (55) M N i ≃ m i M Δ L 2 v e w 2 .Type-II Seesaw Dominated SU(5) (56) M N i = O 10 G e V - O M Δ L .Here mi,i=1,2,3, are the three mass eigen values of light neutrinos. It is to be noted that mi is absent in the RHS of (56) in the SU(5) case.


## 5.3. Realization of Mass Hierarchies

Here we discuss how the stated hierarchy in Section3(57)MΔL≫MN≫M≫MD,for type-II seesaw dominance, is realized through fine tuning. At first noting that the mass squared term for 15H⊂SU(5) in the scalar potential, M15215H†15H, is SU(5) invariant, M15 can have any mass below the GUT scale subject to proton stability and gauge coupling unification. Since 15H, unlike 5H, does not have Yukawa interactions with SM fermions, the Higgs mediated proton decay is suppressed. We now explain why we have used M15=MΔL=1012 GeV. In our model it is possible to assign any mass M15=MΔL>Mη where the mass of η(3,0,8)⊂75H is Mη=109.23 GeV. Because of the presence of η(3,0,8) at such intermediate mass value, precision gauge coupling unification is achieved which has been discussed separately [126, 127]. Following the standard symmetry breaking SU(5)→ SM through the GUT scale VEV of the SM singlet scalar in the adjoint representation 24H, VGUT=24H0, a SU(5) invariant scalar potential Vη gives the mass of η(3,0,8)(58)Vη=M75275H2+m24,7524H75H2+λ24,7524H275H2⊃M752+m24,75VGUT+λ24,75VGUT2η2.This leads to Mη2=M752+m(24,75)VGUT+λ(24,75)VGUT2. Here M75~m(24,75)~VGUT~MGUT. Thus fine-tuning any one of these four parameters can give Mη=109.23 GeV.  The presence of 15H below Mη destabilizes unification but protects it for M15>Mη. This has led to the chosen value of MΔL=M15=1012 GeV. We have noted in the following section that the value of M3<1 TeV violates the observed bound on the nonunitarity parameter ηττ<2.7×10-3 leading to the lower bound on M in the degenerate case (59)M1=M2=M3≥1250GeV.Here M=diag⁡(M1,M2,M3). Noting the definition (60)M=yχχS1,0,1=yχVχ,we now argue that even for GUT scale mass of χS and its VEV Vχ=VGUT, it is possible to satisfy (59). For Vχ~VGUT~1015 GeV we need a small fine-tuned value of Yukawa coupling (61)yχ>10-12,which satisfies M≥1 TeV but does not affect any known fermion mass. This shows the interesting possibility that even without having a low mass nonstandard Higgs χS, it is possible to realize the extended seesaw with type-II seesaw dominance. However, if we insist on yχ≤1, we need Vχ≥1 TeV and MχS≥1 TeV which is realizable as χS is a SU(5) scalar singlet. As we have assumed MD=Mu, this gives at GUT scale on extrapolation [153] (62)MD33~mtop≃85GeV.Here mtop is the top-quark mass. Thus achieving precision unification and type-II seesaw dominance has already given us MDelta>Mη whereas fine-tuning the Majorana coupling, fN>10-11, has yielded MN>104 GeV. Combining these with (59) and (62) gives the hierarchical relation of (57).


## 6. Lepton Flavor Violations

Using SM extensions there has been extensive investigation of lepton flavor violating phenomenalα→lβ+γ,(α≠β) and other processes like μ→ee¯e including unitarity violations [101–108, 110–116]. In the flavor basis we use the standard charged current Lagrangian(63)LCC=-12∑α=e,μ,τg2Ll¯αLγμναLWLμ+h.c.In predicting the LFV branching ratios we have used the relevant formulas of [101] and assumed a simplifying diagonal structure for M,(64)M=diag.M1,M2,M3.Then (64), in combination with (35), gives the elements of the ν-S mixing matrix (65)VlS=MDe1M1MDe2M2MDe3M3MDμ1M1MDμ2M2MDμ3M3MDτ1M1MDτ2M2MDτ3M3.The S-N mixing matrix, (66)VSN=MMN,is relatively damped out since MN≫M. In fact the type-I cancellation condition MN>>M>>MD ensures this damping. Noting that the physical neutrino flavor state να is a mixture of ν^, S^, and N^, (67)να=Uαiν^i+VαilSS^+VαiSNN^i.Here U~UPMNS and the other two mixings violate unitarity. For large MN≫M the third term in the RHS of (67) can be dropped leading to the unitarity violation parameter η(68)U′≃1-ηUPMNS.Here (69)ηαβ=12X·X†αβ,X=MDM.There has been extensive discussion on the constraint imposed on this parameter [108, 111]. The largest out of these is ηττ≤0.0027. Theoretically (70)12∑iMDτi·MDτi∗Mi2≤0.0027.In the completely degenerate case of S-N mixing, M1=M2=M3=M, we get the following. (71)M≥1250GeVThe RH neutrinos in the present model being degenerate with massesMNi≫mSi have much less significant contributions than the singlet fermions. The predicted branching ratios being only few to four orders less than the current experimental limits [95] are verifiable by ongoing searches,(72)BRμ→eγ=1.19×10-16,BRτ→eγ=1.69×10-14,BRτ→μγ=1.8×10-12.For the sake of completeness we present the variation of LFV decay branching ratios as a function of the lightest neutrino mass in Figure 2.Figure 2
Variation of LFV decay branching ratios as a function of the lightest neutrino mass. Colored horizontal lines representI:BR(τ→μγ), II:BR(τ→eγ), and III:BR(μ→eγ).In this approach the LFV decay rate mediated by theWL boson in the loop depends predominantly upon N-S mixing matrix M and the Dirac neutrino mass matrix MD, although subdominantly upon the RHν mass matrix MN. However in the high scale type-II seesaw ansatz followed here LFV decay rate is independent of light neutrino masses. This behavior of LFV decay rates is clearly exhibited in Figure 2 where the three branching ratios have maintained constancy with the variation of mν.


## 7. DominantWL-WL Channel Double Beta Decay within Cosmological Bound

### 7.1. Double Beta Decay Mediation by Sterile Neutrinos

In the absence ofWR bosons and right-handed ΔR±± in SU(5), there is no contribution to right-handed double beta production. The gauge coupling unification constraint sets the lower bound on the masses of left-handed doubly charged Higgs bosons ΔL±± to be MΔL≃M15H>109.23 GeV. As such they have negligible contributions for direct mediations of 0νββ process with left-handed electrons. Thus the only significant contributions in the WL-WL channel could be through the mediation of ν, S, and N. Feynman diagrams for 0νββ decay amplitude in the WL-WL channel due to the exchanges of Majorana fermions ν, S, and N are shown in Figure 3. In Figure 4 we also present Feynman diagram for 0νββ decay amplitude due to the sterile neutrino exchange where its mass insertion has been explicitly indicated. Mass eigen values of different sterile neutrinos for different sets of (M1,M2,M3) consistent with constraints on unitarity violating parameters ηαβ are presented in Table 1. We have used the singlet fermion mass seesaw formula of (15) and MN1=MN2=MN3=4000 GeV. These solutions are displayed in Figure 5.Table 1
Prediction of singlet fermion masses for different values of(M1,M2,M3) where we have used MN1=MN2=MN3=4000 GeV.
M (GeV) m ^ s (GeV) (60,1200,1200) ( 0.9,360,360 ) (70,1200,1200) ( 1.22,360,360 ) (80,1200,1200) ( 1.60,360,360 ) (90,1200,1200) ( 2.00,360,360 ) (100,1200,1200) ( 2.50,360,360 ) (110,1200,1200) ( 3.00,360,360 ) (120,1200,1200) ( 3.60,360,360 ) (130,1200,1200) ( 4.22,360,360 ) (140,1200,1200) ( 4.90,360,360 ) (150,1200,1200) ( 5.62,360,360 )Figure 3
Feynman diagrams representing neutrinoless double beta decay due to exchanges of all three types of Majorana fermionsν, S, and N.
(a) (b) (c)Figure 4
Feynman diagram representing neutrinoless double beta decay amplitude due to exchanges of singlet fermionsSi(i=1-3) with explicit mass insertion ms=-M1/MNMT.Figure 5
Prediction of singlet fermion mass eigen values as a function ofN-S mixing mass parameters Mi(i=1,2,3) for MNi=4TeV(i=1,2,3). The horizontal red colored line represents solutions for two other eigen values for M2=M3=1200 GeV.We use normalizations necessary for different contributions [158–164] due to exchanges of light neutrinos, sterile neutrinos, and the heavy RH neutrinos in the WL-WL channel. They lead to the inverse half-life [120, 124, 125],(73)T1/20ν-1≃G01Mν0νme2Mνee+MSee+MNee2,=K0νMνee+MSee+MNee2,=K0νMeff2.Here G01=0.686×10-14yrs-1, Mν0ν=2.58-6.64, and K0ν=1.57×10-25yrs-1eV-2. In (73) the three effective mass parameters have been defined as(74)Mνee=∑iVeiνν2mνi(75)MSee=∑iVeiνS2p2m^Si(76)MNee=∑iVeiνN2p2MNi,with(77)Meff=Mνee+MSee+MNee.The quantitym^Si is the i-th eigen value of the S- fermion mass matrix mS. The magnitude of neutrino virtuality momentum p has been estimated to be in the allowed range p=120 MeV-200 MeV [158–164]. The contributions of the RHνs, being much heavier than the singlet fermions, have been neglected.


### 7.2. Singlet Fermion Assisted Enhanced Double Beta Decay Rate

We use neutrino oscillation data to estimateMνee for NH and IH cases with the values of Dirac phase and Majorana phases as discussed above. We further use the values of Mi from Table 1 and Figure 5 and the Dirac neutrino mass matrix from (35) to estimate MSee while treating the RHν mass at its assumed degenerate value of MNi=4TeV(i=1,2,3). The variation of effective parameter mee as a function of lightest neutrino mass is shown in Figure 6 when ms1=2 GeV.Figure 6
Variation of effective mass parameter as a function of lightest active neutrino massm1 for ms1=2 GeV. For comparison, predictions in the standard model supplemented by light neutrino masses of NH type are shown by green dot-dashed curve. For IH pattern of mass hierarchy the standard prediction is shown by red dot-dashed curve.As noted from the analytic formulas, the effective mass parameter in the singlet fermion dominated case, being inversely proportional toms1, will proportionately decrease with increasing value of the mediating particle mass. This feature has been shown in Figure 7. We present predictions of double beta decay half-life as a function of the singlet fermion mass in Figure 8. It is clear that while for ms1=2 GeV the half-life saturates the current experimental limit, for larger values of ms1 the half-life is found to increase. Neglecting heavy RHν contributions but including those due to the lightest sterile neutrino and the IH type light neutrinos, our predictions of half-life as a function of the lightest sterile neutrino mass are shown in Figure 9.Figure 7
Same as Figure6 but for ms1=4.0 GeV.Figure 8
Prediction of double beta decay half-life as a function of sterile neutrino massms1 GeV (blue shaded curve) where the NH type light neutrino and the sterile neutrino exchange contributions have been included. Effects of much larger masses (mS2,mS3)≫mS1 have been neglected. The spread in the curve reflects uncertainty in the virtuality momentum p=120-190 MeV. For comparison, the standard prediction with NH and IH pattern of light neutrino mass hierarchies is shown by the two respective horizontal lines. The bottom most thick red horizontal line closest to the X-axis represents overlapping experimental bounds from different groups [78–82].Figure 9
Same as Figure8 but with contributions of IH type light neutrinos combined with lightest sterile neutrinos.Predicted lifetimes are seen to decrease with increasing sterile neutrino mass. The sterile neutrino exchange contribution completely dominates over light neutrino exchange contributions formS1=1.3-7 GeV in case of IH but for mS1=1.5-20 GeV in case of NH. At mS1≃1.5GeV both types of solutions saturate the current laboratory limits reached by different experimental groups.


## 7.1. Double Beta Decay Mediation by Sterile Neutrinos

In the absence ofWR bosons and right-handed ΔR±± in SU(5), there is no contribution to right-handed double beta production. The gauge coupling unification constraint sets the lower bound on the masses of left-handed doubly charged Higgs bosons ΔL±± to be MΔL≃M15H>109.23 GeV. As such they have negligible contributions for direct mediations of 0νββ process with left-handed electrons. Thus the only significant contributions in the WL-WL channel could be through the mediation of ν, S, and N. Feynman diagrams for 0νββ decay amplitude in the WL-WL channel due to the exchanges of Majorana fermions ν, S, and N are shown in Figure 3. In Figure 4 we also present Feynman diagram for 0νββ decay amplitude due to the sterile neutrino exchange where its mass insertion has been explicitly indicated. Mass eigen values of different sterile neutrinos for different sets of (M1,M2,M3) consistent with constraints on unitarity violating parameters ηαβ are presented in Table 1. We have used the singlet fermion mass seesaw formula of (15) and MN1=MN2=MN3=4000 GeV. These solutions are displayed in Figure 5.Table 1
Prediction of singlet fermion masses for different values of(M1,M2,M3) where we have used MN1=MN2=MN3=4000 GeV.
M (GeV) m ^ s (GeV) (60,1200,1200) ( 0.9,360,360 ) (70,1200,1200) ( 1.22,360,360 ) (80,1200,1200) ( 1.60,360,360 ) (90,1200,1200) ( 2.00,360,360 ) (100,1200,1200) ( 2.50,360,360 ) (110,1200,1200) ( 3.00,360,360 ) (120,1200,1200) ( 3.60,360,360 ) (130,1200,1200) ( 4.22,360,360 ) (140,1200,1200) ( 4.90,360,360 ) (150,1200,1200) ( 5.62,360,360 )Figure 3
Feynman diagrams representing neutrinoless double beta decay due to exchanges of all three types of Majorana fermionsν, S, and N.
(a) (b) (c)Figure 4
Feynman diagram representing neutrinoless double beta decay amplitude due to exchanges of singlet fermionsSi(i=1-3) with explicit mass insertion ms=-M1/MNMT.Figure 5
Prediction of singlet fermion mass eigen values as a function ofN-S mixing mass parameters Mi(i=1,2,3) for MNi=4TeV(i=1,2,3). The horizontal red colored line represents solutions for two other eigen values for M2=M3=1200 GeV.We use normalizations necessary for different contributions [158–164] due to exchanges of light neutrinos, sterile neutrinos, and the heavy RH neutrinos in the WL-WL channel. They lead to the inverse half-life [120, 124, 125],(73)T1/20ν-1≃G01Mν0νme2Mνee+MSee+MNee2,=K0νMνee+MSee+MNee2,=K0νMeff2.Here G01=0.686×10-14yrs-1, Mν0ν=2.58-6.64, and K0ν=1.57×10-25yrs-1eV-2. In (73) the three effective mass parameters have been defined as(74)Mνee=∑iVeiνν2mνi(75)MSee=∑iVeiνS2p2m^Si(76)MNee=∑iVeiνN2p2MNi,with(77)Meff=Mνee+MSee+MNee.The quantitym^Si is the i-th eigen value of the S- fermion mass matrix mS. The magnitude of neutrino virtuality momentum p has been estimated to be in the allowed range p=120 MeV-200 MeV [158–164]. The contributions of the RHνs, being much heavier than the singlet fermions, have been neglected.


## 7.2. Singlet Fermion Assisted Enhanced Double Beta Decay Rate

We use neutrino oscillation data to estimateMνee for NH and IH cases with the values of Dirac phase and Majorana phases as discussed above. We further use the values of Mi from Table 1 and Figure 5 and the Dirac neutrino mass matrix from (35) to estimate MSee while treating the RHν mass at its assumed degenerate value of MNi=4TeV(i=1,2,3). The variation of effective parameter mee as a function of lightest neutrino mass is shown in Figure 6 when ms1=2 GeV.Figure 6
Variation of effective mass parameter as a function of lightest active neutrino massm1 for ms1=2 GeV. For comparison, predictions in the standard model supplemented by light neutrino masses of NH type are shown by green dot-dashed curve. For IH pattern of mass hierarchy the standard prediction is shown by red dot-dashed curve.As noted from the analytic formulas, the effective mass parameter in the singlet fermion dominated case, being inversely proportional toms1, will proportionately decrease with increasing value of the mediating particle mass. This feature has been shown in Figure 7. We present predictions of double beta decay half-life as a function of the singlet fermion mass in Figure 8. It is clear that while for ms1=2 GeV the half-life saturates the current experimental limit, for larger values of ms1 the half-life is found to increase. Neglecting heavy RHν contributions but including those due to the lightest sterile neutrino and the IH type light neutrinos, our predictions of half-life as a function of the lightest sterile neutrino mass are shown in Figure 9.Figure 7
Same as Figure6 but for ms1=4.0 GeV.Figure 8
Prediction of double beta decay half-life as a function of sterile neutrino massms1 GeV (blue shaded curve) where the NH type light neutrino and the sterile neutrino exchange contributions have been included. Effects of much larger masses (mS2,mS3)≫mS1 have been neglected. The spread in the curve reflects uncertainty in the virtuality momentum p=120-190 MeV. For comparison, the standard prediction with NH and IH pattern of light neutrino mass hierarchies is shown by the two respective horizontal lines. The bottom most thick red horizontal line closest to the X-axis represents overlapping experimental bounds from different groups [78–82].Figure 9
Same as Figure8 but with contributions of IH type light neutrinos combined with lightest sterile neutrinos.Predicted lifetimes are seen to decrease with increasing sterile neutrino mass. The sterile neutrino exchange contribution completely dominates over light neutrino exchange contributions formS1=1.3-7 GeV in case of IH but for mS1=1.5-20 GeV in case of NH. At mS1≃1.5GeV both types of solutions saturate the current laboratory limits reached by different experimental groups.


## 8. Summary, Discussion, and Conclusion

A recently proposed scalar extension of minimal non-SUSYSU(5) GUT has been found to realize precision gauge coupling unification, high scale type-II seesaw ansatz for neutrino masses, and prediction of a WIMP scalar DM candidate that also completes vacuum stability of the scalar potential. But the LFV decays are predicted to have negligible rates inaccessible to ongoing searches in foreseeable future. Likewise experimentally verifiable double beta decay rates measurable by different search experiments are possible only for quasidegenerate neutrino mass spectrum with large common mass scale m0>0.2 eV or ∑i‍mi>0.6 eV. This violates the recently measured cosmological bound ∑i‍mi≤0.23 eV. In order to remove these theoretical shortcomings in the context of SU(5), we have extended this model by the addition of three RHνs, three extra Majorana fermion singlets Si(i=1,2,3), and a scalar singlet χS(1,0,1) that generates N-S mixing mass term through its vacuum expectation value. In the original theory of type-I seesaw cancellation mechanism, although the choice of particles is the same as Ni, Si, and χS(1,0,1), the neutrino mass is given by double seesaw [119]. Further, there is no grand unification of gauge couplings or prediction of proton decay in this model [119], and the scalar potential of the model has vacuum instability. In addition the Ni are not gauged. The model does not predict dominant contributions to double beta decay for NH or IH type neutrino masses. In non-SUSY SO(10) models of unification of three forces implementing the cancellation of type-I seesaw [84, 120, 124], the TeV scale RH neutrinos are gauged but the neutrino masses are controlled by inverse seesaw. But in [13, 125] the RHνs are gauged and the neutrino mass formula is linear seesaw or type-II seesaw [13]. In all type-II seesaw dominated SO(10) models, the RHν masses have the same hierarchy as the left-handed neutrino masses: MN1:MN2:MN3∷m1:m2:m3. This happens precisely because the left-handed and the right-handed dilepton Yukawa interactions originate from the same SO(10) invariant term: f16F16F126†. In SU(5), however, as the LH triplet ΔL(3,-1,1) generating type-II seesaw and the singlet σS(1,0,1) generating RHνs belong to different scalar representations, 15H⊂SU(5) and 1H⊂SU(5), respectively, they can possess different Majorana couplings in their respective Yukawa interactions:fllΔLC and fNσSNN. Because of this reason the generated RHν masses through MN=fNσs no longer follows the predicted type-II seesaw predicted hierarchical pattern. Then the allowed fine-tuning fN<<f permits RH neutrino mass scale MN~O(1-10) TeV even though, unlike SO(10) models, there are no possibilities of low mass WR or Z′ bosons at this scale. The apprehension of unacceptably large active neutrino mass generation through type-I seesaw mechanism is rendered inoperative through the well-established procedure of cancellation mechanism that is also shown to operate profoundly in this SU(5) model. Such RHνs generating N-S mixing mass M≃O(100-1000) GeV now reproduce the well-known results on LFV decay branching ratios only 4-5 orders lower than the current experimental limit as well as the extensively investigated nonunitarity effects. Through the sterile neutrino canonical seesaw formula emerging from this cancellation mechanism (in the presence of Ni), mS=-M(1/MN)MT, this mechanism predicts their masses over a wide range of values, ms1=O(1-100) GeV and ms2,ms3~O(10-1000) GeV. The lightest sterile neutrino mass ms1 now predicts dominant double beta decay in the WL-WL channel through the ν-S mixing close to the current experimental limits even though the light neutrino masses are of NH or IH type (mi<<0.2 eV) which satisfy the cosmological bound. For larger values of mS1 the predicted decay rate decreases and the sterile neutrino contribution becomes negligible for msi>>50 GeV. In the limiting case when all the singlet fermion masses have such large values, the double beta decay rates asymptotically approach the respective standard NH or IH type contributions. The new mechanism of RHν mass generation also allows the second and the third generation sterile neutrino masses to be quasidegenerate (QD) near 1-10 TeV scale while keeping mS1~1-10 GeV suitable for dominant double beta decay mediation. There is a possibility that such TeV scale QD masses while maintaining observable predictions on LFV decays can effectively generate baryon asymmetry of the universe via resonant leptogenesis [125]. A scalar singlet DM can be easily accommodated as discussed in [126, 165, 166] while resolving the issue of vacuum stability. Irrespective of scalar DM, the model can also accommodate a Majorana fermion singlet dark matter [167] which can emerge from the additional fermionic representation 24F⊂SU(5).The prediction of new fermions has an additional advantage over scalars as these masses are protected by leptonic global symmetries [168]. Also the prediction of such Majorana type sterile neutrinos can be tested by high energy and high luminosity accelerators through their like-sign dilepton production processes [169]. For example, at LHC they can mediate the process pp→WLX→l±l±jjX where the jets could be manifested as mesons. It would be quite interesting to examine emergence of such SU(5) theory as a remnant of SO(10) or E6 GUTs.We conclude that even in the presence of SM as effective gauge theory descending from a suitableSU(5) extension, it is possible to predict experimentally accessible double beta decay rates in the WL-WL channel satisfying the cosmological bound on active neutrino masses as well as verifiable LFV decays. The RHν masses can be considerably different from those constrained by conventional type-I or type-II seesaw frameworks which are instrumental in predicting interesting physical phenomena even if there are no nonstandard heavy gauge bosons anywhere below the GUT scale.



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