Standard coupling unification in SO(10), hybrid seesaw neutrino mass and leptogenesis, dark matter, and proton lifetime predictions

We discuss gauge coupling unification of SU(3)C × SU(2)L × U(1)Y descending directly from non-supersymmetric SO(10) while providing solutions to the three out-standing problems of the standard model: neutrino masses, dark matter, and the baryon asymmetry of the universe. Conservation of matter parity as gauged discrete symmetry for the stability and identification of dark matter in the model calls for high-scale spontaneous symmetry breaking through 126H Higgs representation. This naturally leads to the hybrid seesaw formula for neutrino masses mediated by heavy scalar triplet and right-handed neutrinos. Being quadratic in the Majorana coupling, the seesaw formula predicts two distinct patterns of right-handed neutrino masses, one hierarchical and another not so hierarchical (or compact), when fitted with the neutrino oscillation data. Predictions of the baryon asymmetry via leptogenesis are investigated through the decays of both the patterns of RHν masses. A complete flavor analysis has been carried out to compute CP-asymmetries including washouts and solutions to Boltzmann equations have been utilised to predict the baryon asymmetry. The additional contribution to vertex correction mediated by the heavy left-handed triplet scalar is noted to contribute as dominantly as other Feynman diagrams. We have found successful predictions of the baryon asymmetry for both the patterns of right-handed neutrino masses. The SU(2)L triplet fermionic dark matter at the TeV scale carrying even matter parity is naturally embedded into the non-standard fermionic representation 45F of SO(10). In addition to the triplet scalar and the triplet fermion, the model needs a nonstandard color octet fermion of mass ∼ 5 × 107 GeV to achieve precision gauge coupling unification at the GUT mass scale MU0 = 1015.56 GeV. Threshold corrections due to superheavy components of 126H and other representations are estimated and found to be substantial. It is noted that the proton life time predicted by the model is accessible to the ongoing and planned experiments over a wide range of parameter space.


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of a huge list are [36-39, 39, 40, 40, 41, 41, 42, 42-45, 73, 82-88, 88-96, 98-104]. Even while confronting other challenging problems through SUSY SO (10), explanation of neutrino data only has been considered adequate; some examples out of many such works in this direction include derivation of new seesaw mechanism with TeV scale Z [93], prediction of Axions [98], low-mass Z induced by flavor symmetry [100], realization of SUSY SO (10) from M − theory [99,101], predictions of inflaton mass [102], and Starobinsky type inflation [103], or quartic inflation [104] from SUSY SO (10). Generalised hidden flavour symmetries have been explored without confining to any particular type of fermion mass fits [105]. Despite many attractive qualities of SUSY GUTs including the resolution of the gauge hierarchy problem, no experimental evidence of supersymmetry has been found so far. This has led to search for gauge coupling unification of the standard gauge theory in non-supersymmetric (non-SUSY) GUTs while sacrificing the elegant solution to the gauge hierarchy problem in favour of fine-tuning to every loop order [106,107]. As stated above, single step breakings of all popular non-SUSY GUTs including SU(5) [20] and SO (10) [21,22] under the constraint of the minimal fine-tuning hypothesis [28,29] fail to unify gauge couplings.
Compared to SUSY GUTs, the non-SUSY GUTs do not have the problems associated with the Higgsino mediated proton decay [88,163] while the canonical proton decay mode p → e + π 0 has been accepted as the hall mark of predictions of non-SUSY GUTs since more than four decades. Further, the non-SUSY GUT also does not suffer from the well known gravitino problem. [164][165][166][167][168][169].
Coupling unification in the single step breaking of non-SUSY SO (10) has been addressed in an interesting paper by Frigerio and Hambye (FH) [125] by exploiting the intrinsic matter parity of SO (10) leading to triplet fermion in 45 F as dark matter candidate. The presence of a color octet fermion of mass ≥ 10 10 GeV has been also noted for unification. The proton lifetime has been predicted in this model at two-loop level of gauge coupling unification. However details of fitting the neutrino oscillation data including derivation of Dirac neutrino mass matrix and the RHν mass spectrum have not been addressed. Likewise related details of derivation of the baryon asymmetry of the universe via leptogenesis has been left out from the purview of discussion. An added attractive aspect of the model is the discussion of various methods, both renormalizable and non-renormalizable, by which the triplet fermionic DM can have TeV scale mass. Although proton lifetime has been predicted from the two-loop determination of the GUT scale, important modification due to threshold effects that could arise from the superheavy components of various representations [170][171][172][173][174][175][176][177][178] need further investigation.
The contents of the present paper are substantially different from earlier works in many respects. We have discussed the matching with the neutrino oscillation data in detail where, instead of type-I seesaw, we have used hybrid seesaw which is a combination of both type-I and type-II [179][180][181]. Both of the seesaw mechanisms are naturally predicted in matter parity based SO(10) model having their origins rooted in the Higgs representation 126 H and the latter's coupling to the fermions in the spinorial representation 16 through f 16.16.126 † H . Unlike a number of neutrino mass models adopted earlier, in this work we have not assumed JHEP04(2017)075 dominance of any one of the two seesaw mechanisms over the other. For the purpose of the present work we have determined the Dirac neutrino mass matrix at the GUT scale from the extrapolated values of charged fermion masses [71][72][73] and exploiting the exact quark lepton symmetry [23,24] at that scale. With a view to investigating basis dependence of leptogenesis, the Dirac neutrino mass estimation has been carried out in two ways: by using the u-quark diagonal basis as well as the d-quark diagonal basis. Using these in the hybrid seesaw formula which is quadratic in the Majorana coupling f gives two distinct patterns of mass eigen values for the heavy RHν masses: (i) Compact scenario where all masses are heavier than the Davidson-Ibarra (DI) bound, and (ii) The hierarchical scenario where only the lightest N 1 mass is below the DI bound. Thus each of these sets of RH neutrino masses corresponds to two types of Dirac neutrino mass matrices or Yukawa couplings which play crucial roles in the determination of CP-asymmetry resulting from RHν decays. We have carried out a complete flavour analysis in determining the CP asymmetries. We have also exploited solutions of Boltzmann equations in every case to arrive at the predicted results on baryon asymmetry. Successful ansatz for baryogenesis via leptogenesis is shown to emerge for each pattern of RHν masses. With the compact pattern of RHν mass spectrum, this occurs when the Dirac neutrino masses are determined in the u-quark or the d-quark diagonal basis. However, in the hierarchical scenario of RHν masses, the dominant CP asymmetry that survives the washout due to N 1 -decay and contributes to the desired baryon asymmetry is generated by the decay of the second generation RHν where the Dirac neutrino mass corresponds to the u-quark diagonal basis. Because of the heavier mass of the LH triplet scalar, although its direct decay to two leptons [62] gives negligible contribution to the generated CP-asymmetry, the additional vertex correction generated by its mediation to the RHν decay is found to lead to a CP-asymmetry component comparable to other dominant contributions. Thus the same heavy triplet scalar ∆ L and the RHνs which drive the hybrid seesaw formula for neutrino masses and mixings are shown to generate the leptonic CP asymmetry leading to the experimentally observed value for the baryon asymmetry of the universe over a wide range of the parameter space in the model.
For the embedding of the suggested triplet fermionic DM [182] in SO(10) [125], we assume it to originate from the non-standard fermionic representation 45 F ⊂ SO(10) carrying even matter parity. Having exploited the triplet fermionic DM Σ F (1, 3, 0) and the LH triplet Higgs scalar ∆ L (1, 3, −1) mediating the hybrid seesaw for neutrino masses and leptogenesis, we justify the presence of these light degrees of freedom as ingredients for coupling unification through their non-trivial contribution to the SU(2) L × U(1) Y gauge coupling evolutions. In addition, we need lighter scalar or fermionic octets with mass ∼ 5 × 10 7 GeV under SU(3) C to complete the precision gauge coupling unification.
The degrees of freedom used in this model having their origins from SO(10) representations 126 H , 10 H , 45 H , and 45 F are expected to contribute substantially to GUT threshold effects on the unification scale through their superheavy components even without resorting to make the superheavy gauge boson masses non-degenerate as has been adopted in a number of earlier works for proton stability. It is important to note that if we accept the stabilising symmetry for DM to be matter parity, then the participation of 126 H ⊂ SO (10) in its spontaneous symmetry breaking is inevitable. This in turn dictates a dominant JHEP04(2017)075 contribution to threshold effects on proton lifetime which has been ignored earlier but estimated in this direct breaking chain for the first time. In addition the superheavy fermions in 45 F have been noted to contribute substantially. A possibility of partial cancellation of scalar and fermionic threshold effects is also pointed out. Although it is challenging to rule out the present model by proton decay experiments, the predicted proton lifetime in this model for the p → e + π 0 is found to be within the accessible range of the ongoing search limits [183][184][185][186][187][188][189] for a wider range of the parameter space.
Unlike the case of direct breaking of SUSY SO (10) to MSSM [69] or non-SUSY SO(10) through Pati-Salam intermediate symmetry [69], but like very large number of cases of model building in non-SUSY GUTs, it is not our present goal to address charged fermion mass fit. But we discuss in appendix A.3 how all fermion masses may be fitted at least approximately in future without substantially affecting this model predictions.
This paper is planned in the following manner. In section 2 we discuss successful fit to the neutrino oscillation data where we estimate the LH Higgs triplet and the RHν masses. In section 3.1 we present the estimations of CP-asymmetry for different flavor states. In section 3.2 we discuss Boltzmann equations for flavour based analysis. In section 3.3 and section 3.4 we present the results of final baryon asymmetry. In section 4 we discuss why the neutral component of fermionic triplet is a suitable dark matter candidate. In section 5 we discuss unification of gauge couplings and determine the unification scale. In section 6 we discuss proton lifetime prediction including GUT-threshold uncertainties. In section 7 we summarize and state conclusions. In appendix A.1 and appendix A.2 we provide renormalization group coefficients for gauge coupling evolution and estimation of threshold effects. In appendix A.3 we discuss the possibility of parameterization of fermion masses.

Hybrid seesaw fit to neutrino oscillation data
In this section we address the issue of fitting the neutrino masses and mixings as determined from the neutrino oscillation data by the hybrid seesaw formula. We then infer on the masses of heavy left-handed triplet and RH neutrinos necessary for leptogenesis.
After SO(10) breaking, the relevant part of the Lagrangian under SM symmetry is The first term on the right-hand side (r.h.s.) of eq. (2.1) is from the SO(10) symmetric Yukawa term Y (10) . 16 where v L = λv R v 2 ew /M 2 ∆ is the induced VEV of triplet scalar ∆ L , and M D ≡ Y ν v ew . There is the well known standard ansatz to fit fermion masses in SO(10) along the line of [67]. To estimate the Dirac mass matrix in this work we have carried out one-loop renormalization group evolution of Yukawa couplings in the bottom-up approach using PDG values of all charged fermion masses. At the electroweak scale µ = M Z using experimental data on charged fermion masses we choose up-quark or down-quark mass diagonal bases in two different scenarios. We then evolve them upto the GUT scale µ = M U using bottom-up approach [71][72][73]. At this scale we assume equality of the up-quark and the Dirac neutrino mass matrices, M D M u , which holds upto a very good approximation in SO(10) due to its underlying quark-lepton symmetry [23,24].
As pointed out in section 1, χ 2 fit to all fermion masses and mixings in SUSY SO(10) or in non-SUSY SO(10) with G 224 intermediate symmetry requires a small departure from this assumption [69,137,139,140]. On the other hand a very recent derivation of neutrino mass and mixing sum-rules has been found to require M D close to M u [141] as in our case. Although in the present case of non-SUSY SO(10) breaking directly to the SM gauge theory, fermion mass fit is not our goal in this paper, we have discussed the issue in appendix A.3.
We further assumed that M D (M M GUT ) ∼ M D (µ) for all lower mass scales µ < M GUT . We could have done better to estimate the Dirac mass matrix at the electroweak scale by following the top-down approach but since it does not get appreciable correction due to the absence of the strong gauge coupling α 3C [71][72][73] contribution, this approximation does not influence our final result substantially. Another reason is that for leptogenesis we need Dirac neutrino Yukawa couplings at intermediate scales, µ ∼ (10 6 -10 12 ) GeV where the renormalisation group (RG) running effects are expected to be smaller in the top-down approach.
Thus in the down quark diagonal basis under the assumption of negligible RG effects we have at µ = M Z M (d)

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For the sake of clarity it might be necessary to explain how the mass matrix structure given in eq. (2.4) emerges with very small non-diagonal elements. In the bottom-up approach for the RG evolution of Yukawa matrices, we have assumed the up-quark mass matrix M u (M Z ) to be diagonal in one case at the electroweak scale which we designate as up-quark diagonal basis. In this case naturally all elements of the down quark mass matrix M d (M Z ) are non-vanishing. In the alternative case, called the d-quak diagonal basis, we have chosen M d (M Z ) diagonal for which all nine elements of M u (M Z ) are non-vanishing. In the case of up-quark diagonal basis, however, the non-diagonal elements of M u (M M GUT ) acquire non-vanishingly small corrections due to RG effects in the bottom-up approach and this is approximated as the Dirac-neutino mass matrix M The Dirac neutrino mass matrices given in eq. eq. (2.3) and eq. (2.4) are used in the second term of the right-hand side (r.h.s.) of eq. (2.2) where in the left-hand side (l.h.s.) we use the value of light neutrino mass matrix for the normally ordered case with m ν 1 = 0.00127 eV and the best fit values for other parameters [191]. We have also assumed that Majorana phases are zero at all mass scales.
We then search for solutions for the Majorana coupling f or, equivalently, the values of RH neutrino masses. Due to strongly hierarchical structure of M D matrix, it is impractical to assume the dominance of the type-I or the type-II term in the hybrid seesaw formula of eq. (2.2). Since eq. (2.2) is quadratic in f , it has two solutions for every eigenvalue and thus giving a total of 2 3 = 8 plausible solutions [190]. But for a given M D and m ν there should be only two distinct positive definite solutions. We estimated these solutions for f using the neutrino oscillation data of ref. [191] as input and numerical iteration. A robust iterative numerical estimation of f matrix is performed to match the oscillation data. Thus by fixing the lightest neutrino mass and the VEV v L in a chosen hierarchy of light neutrino masses, the precise forms of the two solutions with positive definite f are evaluated upto the desired precision. These solutions are presented in figure 1 for two sets of values of quartic coupling, λ = 0.1 and λ = 0.001.
In figure 1 we have presented these solutions for the normally ordered values of active light neutrino masses. Solutions in the top row of the figure have strongly hierarchical heavy RH neutrino masses, lightest of them being M N 1 ∼ O(10 3−5 ) GeV, testable in future collider experiments, and the heaviest M N 3 ∼ O(10 12 ) GeV. We call such solutions of RH neutrino masses to represent a hierarchical spectrum scenario. Solutions in the bottom row of the figure are not so hierarchical and the RH neutrinos only span three orders of magnitude of mass range. We call the solutions of this type given in the bottom row to represent a compact spectrum scenario. Lightest of RH neutrino in this scenario is ∼ O(10 9−11 ) GeV which is far away from direct detection limit of any collider experiment. In arriving at these solutions we assumed the LH triplet scalar mass M ∆ L = 10 12 GeV, GUT symmetry breaking VEV v R = 10 15.5 GeV, and the value of the quartic coupling λ = 0.1 (left panel) and 0.001 (right panel). We note that the RHν masses increase with decrease in λ for the compact spectrum scenario while it almost stays unaffected in the hierarchical spectrum scenario. Also the theory should continue to remain perturbative on JHEP04(2017)075 acquiring N 1 -dominated leptogenesis because increasing λ(∼ 1) for the above value of M ∆ will make M N 1 < 10 9 GeV and N 1 -dominated leptogenesis will not be possible.
In the compact spectrum scenario we estimate the f matrix in the d-diagonal basis using eq. (2.6)

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In the hierarchical spectrum scenario, similarly, we have the two matrices for f Despite widely varying magnitudes of different elements in the matrix, the mass eigenvalues in the u− quark and d− quark diagonal bases are not very different in both the compact spectrum and the hierarchical spectrum scenarios. Therefore, we have presented only one set of solutions for the RHν masses in figure 1. It is quite encouraging to note that despite the GUT scale value of v R , the type-II term does not upset the type-I seesaw term in the hybrid formula, rather both of them contribute significantly to the light neutrino mass matrix. We will explore the plausibility of sufficient leptogenesis using the hybrid seesaw mechanism of this model to explain BAU.

Baryon asymmetry of the Universe
In this section at first we estimate the leptonic CP-asymmetry generated in decays of both RHν and ∆ L . The dynamically generated lepton asymmetry gets converted into baryon asymmetry due to sphaleron interaction [192]. Leptogenesis is discussed in various papers [58,160,[193][194][195][196][197][198][199][200][201][202][203][204][205][206][207][208]. The flavour independent calculation of asymmetry is applicable at high temperatures when all the charged lepton mediated interactions are out of equilibrium i.e. T 10 12 GeV. Flavour dependent analysis [202] becomes necessary for leptogenesis at lower temperatures. In hierarchical spectrum scenarios we have M N 1 ∼ 10 3−5 GeV which violates the Davidson-Ibarra bound [209] badly, therefore it can not produce required amount of flavour independent lepton asymmetry. Instead it washes out the asymmetry produced at the early stage in N 2,3 decays. In the recent studies [202,[210][211][212][213] it has been shown that under such circumstances the next heavy neutrino N 2 can produce the required asymmetry, if M N 2 10 10 GeV and there exists a heavier N 3 . If the asymmetry produced by N 2 is not completely washed out by lightest neutrino N 1 , it survives and gets converted to baryon asymmetry. On the other hand, in the compact spectrum scenario, the lightest RH neutrino is well within the Davidson-Ibarra bound, therefore the asymmetry can be produced in the lightest RHν decay. Since for a large region of the parameter space we have shown that M N 1 10 12 GeV, the asymmetry will depend on flavour dynamics. Figure 2. One-loop Feynman diagrams for the decay of RH neutrino N i . The first and the third diagrams represent vertex corrections and the second diagram represents self-energy correction.

CP-asymmetry
The flavoured CP-asymmetry in the decay of N i to a lepton l α is generated in the lepton flavor generation α, and is defined as [216,217,229] . (3.1) One loop decay contributions of N i are mediated by either N k =i or ∆ L [62] as shown in figure 2. The total asymmetry is sum of the two contributions The asymmetry produced in the N i decay due to N k =i appearing in the loop is [216,217] 3) The first line of this expression contains lepton number violating terms while the second line is the lepton number conserving but violates lepton flavour. Here,Ŷ ν = Y ν U * f is the Dirac Yukawa coupling in the right-handed neutrino diagonal mass basis and U f is the unitary matrix diagonalizing f . The loop functions in the asymmetry expression are [217] g Here by retaining the Wigner-Eckart term in the loop function we can handle degenerate RHν mass scenario without hitting singularity, which is possible in compact spectrum scenario in our model (see figure 1). Note that in the degenerate regime CP asymmetry gets largest contribution from self-energy term and may reach to a value of O(1). The CP -asymmetry produced in N i decay from the ∆ L mediated diagram is [62] ε λ=0.1  which gets contribution proportional to the trilinear coupling mass term µ. Its loop function is larger for smaller M ∆ L . But M ∆ L can not be made arbitrarily small without decreasing µ or increasing v L which is constrained to be below GeV from electroweak (EW) precision constraints. Decreasing µ would decrease CP asymmetry linearly.
Keeping the GUT scale value of v R = 10 15.5 GeV and M ∆ L = 10 12 GeV we have estimated the flavored CP-asymmetry for different values of the lightest neutrino mass in the normally ordered hierachical case of light neutrino masses. Change in the mass of m ν 1 alters f and thus changes the masses and mixings of RHνs. Flavour asymmetries for N i decay into α flavour are shown in figure 3 for compact spectrum case and in figure 4 for the hierarchical spectrum case of RHνs. We note that variation in quartic coupling changes CP-asymmetry significantly, particularly in the hierarchical spectrum scenario. The tree level decay widths are unaffected by the presence of the scalar triplet ∆ L in the scheme.
The presence of the heavy scalar triplet ∆ L in our theory adds another source of CPasymmetry ( ∆ ) which is produced by the decay of the triplet scalar itself into two like-sign or neutral leptons [62]. Though one triplet scalar is enough to generate the active neutrino masses and mixings through type-II seesaw, the asymmetry production in ∆ L decay needs either more than one triplet scalars [218][219][220][221]    handed neutrinos [62] as shown in figure 5 for our model. The CP-asymmetry generated due to ∆ L decay and mediated by RHν is written as [62] We note that, since v R 10 15.5 GeV and M ∆ L 10 12 GeV, either of the two terms in the denominator of ε ∆ is large enough to keep the CP -asymmetry fairly small for the parameters under consideration. For example, if three right-handed neutrino masses are M N k = (6.6990, 13.869, 1431) × 10 9 GeV, the three CP-asymmetries due to N k decays from the first two diagrams of figure 2 . Compared to these numbers, the CP-asymmetry due to ∆ L decay of figure 5 is | ∆ | = 2.1 × 10 −12 . Also, since M ∆ L M 1,2 , the asymmetry generated at the early stage will be washed out at the production phase of lighter RHνs. Henceforth, we will ignore the ∆ L asymmetry in our numerical estimations [219]. In the next subsection we will estimate the lepton asymmetry using Boltzmann equations for the system. Figure 5. Left handed triplet scalar one-loop decay.

Boltzmann equations
The evolution of number density is obtained by solving the set of Boltzmann equations. The co-moving number density is Y X ≡ n X /s. The Boltzmann equations for heavy neutrinos number density are [190] where ∆ α ≡ B/3 − L α , and Y ∆α stands for the total ∆ α asymmetry stored in the fermionic flavours, and z = M 1 /T . The washout parameter for various flavors is such that K i = α K iα . In eq. (3.7) the equilibrium number density [190,215] is defined as requires the lightest right-handed neutrino to acquire mass M N 1 4 × 10 8 GeV [179][180][181] where H 1.66g * M 2 N k /(M Pl z 2 k ) is the Hubble expansion rate. The thermally averaged decay rates are D i (z) = R 2 i zK 1 (R i z)/K 2 (R i z) where K 1 and K 2 are the first and the second order modified Bessel functions [215,222], respectively. The scattering terms S i (z) account for Higgs-mediated ∆L = 1 scatterings involving top quark and anti-quark as The unit lepton number changing ∆L = 1 scattering contributing to washout is

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The ∆L = 1 scattering and related washout from Higgs and lepton mediated inelastic scattering involving top quark are included in the evolution of asymmetry [222].
We have ignored the off-shell part of ∆L = 2 process in the washout term which is a good approximation as long as M N i /10 13 K i [223]. We have also omitted the ∆L = 0 scattering such as N i N j → ll, N i N j → HH * , N i l → N j l, N il → N jl which do not contribute to the washout but can affect the abundance of heavy neutrinos. When flavor effects are taken into account, they also tend to redistribute the lepton asymmetry among flavors. These effects are of higher order in the neutrino Yukawa couplings and are expected to have little impact on the final baryon asymmetry. We further neglected the scalar triplet related washout processes, gauge scatterings, spectator processes, and the higher order processes like 1 → 3 and 2 → 3. The heavy gauge bosons processes such as N i e R →q R q R and N i N i → ff tend to keep the heavy neutrinos in thermal equilibrium, thus reducing the generated lepton asymmetry. This effect is practically negligible because RHνs are much lighter than the RH gauge bosons. We also ignore such flavour effects [224] which are relevant for resonant leptogenesis.

Baryon asymmetry in the compact scenario
In this scenario the tau lepton flavour state decouples while the electron and muon states are still coupled. Thus, a flavour dependent analysis is necessary. In the two flavour case In this case the baryon asymmetry is expressed as where the factor 12/37 is due to partial conversion of ∆ α asymmetry in to baryon asymmetry by non-perturbative sphaleron process [225,226]. The results of BBN [227] and PLANCK [11,12] experiments are (3.14) Compared to these somewhat higher value of BAU obtained from WMAP 7 years' data has been reported in ref. [228]. The washout coefficients K iα in the compact spectrum scenario of RH neutrino masses for the lightest neutrino mass m ν 1 = 0.00127 eV and λ ∈ [0.0001, 0.5] are plotted in figure 6. We see that there are two to four orders of variation in the washout for the above allowed range of λ in both the d-diagonal (left panel) and the u-diagonal (right panel) cases. We  list the washout parameters for λ = 0.1 in the case of the d-quark diagonal basis In the u-quark diagonal basis the washout parameters are (3.16) Our observations in the two cases are summarized below.
(a) The d-quark diagonal basis. We note that K i = α K iα ∼ (300-4000). Therefore the system is in strong washout regime for most of the parameter space. The asymmetry is determined by a balance between production and destruction. The final asymmetry freeze occurs at the decoupling of washout with z f ∼ (7-10). In the single flavour analysis the lepton asymmetry is approximated as [229] Y ∆L (∞) Using the values of K i from figure 6 and ε 1 = α ε 1α from figure 7 we can easily achieve the required lepton asymmetry. In fact it may lead to a constraint on quartic coupling λ.
(b) The u-diagonal basis. We note that, since K 1 = α K 1α 1, this is a very weak washout regime. Ignoring thermal effect on CP-asymmetry and assuming zero initial abundance in the weak washout regime with initial thermal abundance Y N 1 (z = 0) = Y eq N 1 (z = 0) [229] gives   If there is already an initial amount of asymmetry left over, say through N 2 decay, it will not be washed out because the system is in weak washout regime. But with zero initial abundance, We note that even if we assume initial thermal abundance Y eq N 1 (0) ∼ 0.0039, the CPasymmetry ε 1 ∼ 10 −4 − 3 × 10 −6 (figure 7) and K ∼ 10 −7 − 10 −3 ( figure 6). Therefore the generated asymmetry would be determined by initial abundance and, in the zero initial abundance scenario, the required lepton asymmetry can not be produced for any parameter value. Therefore the flavour independent analysis in the u-quark diagonal scenario with zero initial abundance of Y N 1 fails to give the required asymmetry.
On the other hand a flavor dependent analysis can enhance the asymmetry. The flavour dependent lepton asymmetry is analyzed using Boltzmann equations (3.7) and is shown in figure 8 for u-quark diagonal basis. Thus in flavoured analysis we find that final lepton asymmetry is independent of initial abundance and is close to the experimental value for λ < 0.05. This explicitly shows that N 2 decay contributes to lepton asymmetry which is not completely washed out in the N 1 decay. The reason for doing flavoured analysis is that there are enhancements in the final asymmetry compared to the unflavoured case. Using d-quark diagonal basis figure 9 shows the variation of total asymmetry with respect to quartic coupling for a fixed value of the scalar triplet mass M ∆ = 10 12 GeV, v R = 10 15.5 GeV, and the lightest neutrino mass m ν 1 = 0.00127 eV in normalyy ordered case. Similar is the effect in the u-quark diagonal basis.

Baryon asymmetry in the hierarchical scenario
The Davidson-Ibarra bound is not respected in the hierarchical spectrum scenario of RHν (see figure 1). In such a case there is the possibility of leptogenesis if asymmetry is produced by the decay of N 2 . Lower bound on the lightest RHν is passed to M N 2 10 10 GeV. The N 2 -dominated leptogenesis can be successful if there is a heavy neutrino, or triplet scalar with M N 3 , M ∆ L > M N 2 , and the washout from the lightest RHν (N 1 ) is circumvented. Since M N 1 10 9 GeV the lepton flavour states become incoherent and the washout acts separately on each flavour asymmetry. We need to solve Boltzmann equations at the production phase with z 2 = M 2 /T , and at the washout phase with z 1 = M 1 /T [213]. We note from the figure 4 that the CP-asymmetry due to N 1 decay ε i = α ε iα is very small compared to CP-asymmetry due to N 2,3 decays. The decay and washout are also suppressed by a factor M 2 1 /M 2 3 (∼ 10 −14 − 10 −15 ) and M 2 1 /M 2 2 (∼ 10 −9 − 10 −10 ). Also we note that in the scenario M 3 10 12 GeV M 2 > 10 9 GeV M 1 , the role of N 3 becomes indistinct by JHEP04(2017)075 the time asymmetry is produced due to N 2 decay and when washout is active. Thus N 1,3 do not contribute to asymmetry generation at the N 2 decay phase and we can write The flavour coupling matrices in the production phase are the same as given in eq. (3.12). For T 10 9 GeV, the muon Yukawa interaction also gets equilibrated. Then the flavour coupling matrices are [213,230]  The washout factors and the CP-asymmetries for different flavours as a function of quartic coupling are shown in figure 10 and in figure 11, respectively, for the d-quark diagonal (left panel) and the u-quark diagonal (right panel) bases in each case. Notice that in the d-quark diagonal basis K 1α 1 for α = µ, τ . Therefore any such type of flavoured asymmetry produced during N 2 decay will be washed out during the N 1 decay. But since K 1e 2 the corresponding flavoured asymmetry will be washed out only partially. However, in the u-quark diagonal basis, K 1α (α = e) 1. Therefore the corresponding flavour asymmetries produced during N 2 decay would survive. Also noting that in this basis K 1e ∼ 2.8, the e-asymmetry generated by the N 2 decay will be only partially washed out by the N 1 decay. Also, noting from figure 11 that ε 2τ is significantly large, it may produce the required amount of asymmetry. The complete flavoured analysis scenario is discussed below.
With the washout caused due to the N 1 decay, the solutions to Boltzmann equations can be achieved by the substitution 2 → 1 everywhere. Since N 2,3 abundance has vanished below 10 9 GeV, the corresponding equations are redundant. We also note from figure 4 that the CP-asymmetries ε iα are negligibly small, therefore the first term in the r.h.s. of corresponding equation in eq. (3.21) in the N 1 decay can be ignored when K 1 is not very     large. This results in the redundancy of the equation for N 1 in eq. (3.20) and we need to solve only The washout from the lightest RHν is more efficient which acts on the whole of the generated asymmetry. We found that in the d-quark diagonal basis, the asymmetry Y ∆α produced by the N 2 decay as shown in figure 12 is itself much smaller than the experimentally observed asymmetry. There is no way to enhance it at the stage of N 1 decay in the case of d-quark diagonal basis leading to insufficient asymmetry. We also note from figure 10 and figure 11 that variation in quartic coupling is not going to help in enhancing the depleted asymmetry.
On the other hand in the u-quark diagonal basis K 1e ∼ 2 and K 1µ(τ ) 1, the asymmetries may survive the washout during the N 1 decay. In figure 13 we have shown solutions to JHEP04(2017)075  . Noting that |A ee | ∼ 1 but |A eµ | ∼ |A eτ | 1, and C e ∼ C µ ∼ C τ 1 gives the rising behaviour of |Y ∆e | from eq. (3.20) as K 1e ∼ 2. But because of the negligible values of K 1µ and K 1τ , eq. (3.20) gives constant behaviours for |Y ∆µ | and |Y ∆τ | as shown in the right-panel of figure 13.
Using type-I seesaw and N 2 dominated flavoured leptogenesis it has been shown that parts of e and µ asymmetries, designated as phantom terms [213], can completely escape washouts due to the lightest RHν N 1 decay. Such phantom terms can give large contribu-

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tion to the asymmetry resulting in a large B − L asymmetry generation by the N 1 wash outs. The N 2 dominated leptogenesis generated due to such terms has been termed as "phantom" leptogenesis. In this work [213] each of the phantom terms being proportional to the N 2 abundance, the phantom terms vanish in the case of zero initial number density of the heavier RHν i,e N 2 .
However in a subsequent investigation [214] phantom terms have been shown to emerge as a generic feature of flavoured leptogenesis. They have to be taken into account even for initially vanishing RHν abundances. In the strong washout regime the phantom terms have been also shown to give a contribution independent of initial conditions.
In the present case with hybrid seesaw as the origin of neutrino masses and leptogenesis, we find that even though we have ignored any such phantom term in the three flavour analysis, the N 1 decay does not wash out the produced asymmetry at all. Also since K 1e ∼ 1 it helps increasing Y ∆e during the second phase of decay. Thus the conclusion of this analysis is that, in the hierarchical spectrum of RHνs, the production of the observed baryon asymmetry of the universe in heavy neutrino decays is favoured when Dirac mass matrix is such that it is derived from a GUT in the flavour basis satisfying To summarize this section, we have attempted to generate the right value of BAU through lepton asymmetry produced by the hybrid seesaw mechanism where the three heavy RHνs and a LH triplet scalar decay directly or act as mediators in the one-loop Feynman diagrams. Two classes of heavy RHν spectra are found to be predicted by the neutrino oscillation data: compact and hierarchical. We have carried out complete flavor dependent analysis in both these cases. We have also examined the possibility of basis dependence that determines the Dirac neutrino mass matrix at the GUT scale by choosing either the u-quark diagonal basis, or the d-quark diagonal basis. Rigorous solutions to the Boltzmann equations are exploited in every case. In the compact spectrum case, the decay of the lightest RHν which is heavier than the Davidson-Ibarra bound, produces the desired BAU in both the choices of the Dirac neutrino mass matrix. This is shown in figure 8 and figure 9. In the hierarchical spectrum scenario the lightest RHν is much lighter than the Davidson-Ibarra bound. The right value of CP-asymmetry is generated predominantly by the decay of heavier RHν N 2 that also survives the wash out caused by the lightest N 1 . Successful generation of BAU shown in figure 13 is possible with the Dirac neutrino mass matrix determined in the u-quark diagonal basis. Although direct decay of the LH scalar triplet itself does not produce the lepton asymmetry to produce the required BAU, its one loop mediation to the RHν decay vertex correction generates the desired asymmetry which is comparable to other contributions. Thus the role of the LH triplet predicted by the matter parity based SO(10) model is emphasized in the generation of BAU.

General considerations with matter parity
Usually the prospective DM candidates are accommodated in model extensions by imposing additional discrete symmetries for their stability. But as noted in section 1 an encouraging aspect of non-SUSY SO(10) is that [122][123][124] matter parity is available as JHEP04 (2017)075 an intrinsic gauged discrete symmetry if the neutral component of the RH higgs triplet ∆ R (1, 3, −2, 1) ⊂ 126 H ⊂ SO(10) is assigned GUT scale VEV to break the gauge symmetry leading to the SM Lagrangian. As the Higgs particle possesses even value of |B − L|, the vacuum with SM gauge symmetry conserves matter parity P M = (−1) 3(B−L) . This enables to identify the SO(10) representations to be identified with odd value of P M for 16, 144, 560, . . . but with even P M for 10, 45, 54, 120, 126, 210, 210 , 660 . . .. Then it turns out that the would-be DM fermions must be in the non-standard fermionic representations 10 F , 45 F , 54 F , 120 F , 126 F , 210 F . . .. Thus the smallest representation to provide a doublet fermion with hypercharge Y = ±1 is 10 F and the hyperchargeless triplet needed for this model building is in the next larger representation 45 F ⊂ SO (10).
Similarly if it is desired to construct models with scalars as DM candidates, they must belong to the odd P M scalar representations 16 H , 144 H . . .. Whereas the phenomenology of scalar DM has been emphasized in [122,123], the triplet fermionic DM has been found suitable in model construction in [125,231]. In addition, the color octet fermions have been found to be essential at high scale M C 8 ≥ 10 10 GeV [125]. The importance of various other types of DM along with the triplet fermions of both types of chiralities has been also discussed in high intermediate scale models [154][155][156][157][158].
An important advantage of using triplet or doublet fermions over scalars as DM is that in the limit of zero chiral fermion masses, a U(1) global lepton symmetry of the SM is restored. Thus a value of the fermion mass substantially lighter than the GUT scale is naturally protected by this global symmetry in the 't Hooft sense. [232]. On the other hand if a scalar component is used as DM, its mass lighter than the GUT scale has to be obtained by additional fine-tuning in the Lagrangian. Also matter parity conservation forbids it from acquiring any VEV.

Light non-standard fermion masses from SO(10)
In this model with the SM gauge symmetry below the GUT scale, a triplet fermionic DM candidate with zero hypercharge appears to be more appropriate with its mass of the order of TeV scale for gauge coupling unification as would be shown below in section 5. The neutral component of fermionic triplet Σ F (1, 3, 0) ⊂ 45 F ⊂ SO(10) would act as a cold dark matter candidate. For accurate coupling unification we also need a Majorana-Weyl type color octet fermion C 8 (8, 1, 0) at lower scale. Using Yukawa interaction via higher dimensional non-renormalizable operators, the light triplet fermion mass ⊂ 45 F has been obtained in ref. [125]. But both the lighter values of masses of the triplet fermion and the octet fermion can be obtained easily from the renormalizable SO(10) Yukawa Lagrangian at the GUT scale. In the notation 45 F = A F , 54 H = E, and 210 H = Φ, the relevant GUT scale Lagrangian is

Triplet fermion dark matter phenomenology
The phenomenology of a hyperchargeless triplet fermionic DM in the non-SUSY model is similar to that of the wino DM in MSSM and SUSY GUTs. This has been extensively investigated recently [233] and also continues to be a subject of current importance [234]. It is worthwhile to mention here different constraints on their masses derived from direct and indirect searches because of their relevance to the present model building. The even matter parity of fermion triplet DM Σ F (1, 3, 0), compared to odd (even) matter parity of standard fermion (Higgs scalar), guarantees stability of the DM by ruling out Yukawa interactions with SM particles. This may make it difficult for the detection of the triplet fermionic DM at the LHC and other hadron colliders.
(i). Triplet fermion mass from relic density. The only interaction of the DM fermion with standard model particles is through gauge interaction that leads to the well known mass difference m Σ + − m Σ 0 = 166 MeV [233]. where we have denoted the mass of the charged (neutral) component of Σ F (1, 3, 0) as m Σ + (m Σ 0 ). Each of its two charged components has been estimated to be heavier by ∼ 166 MeV [233]. Within the 3σ uncertainty, the observed DM relic abundance is 0.095 < Ω DM h 2 < 0.125 where h = Hubble parameter.

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For the triplet mass m Σ much larger than the W -boson mass, the Sommerfeld resonance enhancement plays a crucial role in the annihilations of components of the Σ F leading to the observed DM relic abundance. Neglecting mass difference between the charged and neutral components, the relevant cross section taking into account the annihilation and co-annihilation of all triplet components has been derived [125], where v = relative velocity of DM particles. The Sommerfeld enhancement enters into the annihilation process because of the fact that the triplet components are non-relativistic at the freezeout temperature. Matching the theoretical prediction within the 3σ uncertainty of the observed value of the relic density Ω DM [235] results in the triplet mass m Σ = 2.75 ± 0.15 TeV [182,233,236] whereas a value of m Σ = 3.0-3.2 TeV has been also estimated [234]. A non-thermal production of Σ 0 relic density due to the decay of color octet fermion, C 8 (8, 1, 0) F , has been recently discussed in [121]. Quite recently only the neutral components of DM candidates at the TeV scale originating from RH fermionic triplets, rather than the LH triplets, have been suggested to be produced at high temperature through non-equilibrium thermal production process in non-SUSY SO (10) where the charged components acquire larger intermediate scale masses [154][155][156][157][158]. The direct detection, indirect detection, and collider search for triplet fermion DM at p − p collider have been analysed in [237]. Phenomenology of wino DM in the mass range 500-2000 GeV which has much similarity with this non-SUSY triplet fermionic DM, ρ 3 , has been also discussed recently [238].
(ii). Direct detection and collider signatures. In general, for elastic scattering of a DM particle (which is electrically neutral) off nucleons either a standard Higgs or a Z-boson exchange is needed in the t-channel of the dominant tree diagrams. In the absence of such couplings of Σ 0 , a sub-dominant process occurs by the exchange of two virtual W ± bosons in a box diagram [233]. This process leads to suppression of spin independent cross section by 2-3 orders below the experimentally detectable value. However, such predicted cross sections are measurable with improvement of detector sensitivities [239]. The inelastic scattering with a charged component (Σ + or Σ − ) is prevented because of kinematic constraints since the mass difference, m Σ + − m Σ 0 = 166 MeV, is about three orders of magnitude above the kinetic energy of Σ and also much above the proton-neutron mass difference, m n −m p ∼ 2 MeV. If the triplet fermion has mass ∼ 400 GeV, its contribution to the spin independent cross section is found to suffer more deviation from the LUX direct bound [240]. Prospects of observing signatures of the triplet fermion DM at colliders have been investigated in [237,[241][242][243][244][245]. For m Σ ∼ 2.7 TeV and integrated luminosity of 100fb −1 , the DM pair production cross section at LHC in the channel pp → ΣΣX has been shown to result in only one event [241][242][243]. For better detection capabilities upgradation of LHC with twice energy and more luminosity has been suggested [244,245].
For detection at e + e − collider that requires a collision energy of at least twice the DM mass, observation of Σ + Σ − pair production is predicted via Z boson exchange [233,241].

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The neutral pair Σ 0 Σ 0 * can be also produced, although at a suppressed rate, through one-loop box diagram mediated by two virtual W bosons. After production such charged components would provide a clean signal as they would manifest in long lived charged tracks due to their decays via standard gauge boson interactions, Σ ± → W ± → Σ 0 π ± , or Σ ± → W ± → Σ 0 l ± ν l (l = e, µ). The production of e ± and µ ± charged leptons but the absence of τ ± due to kinematical constraint may be another distinguishing experimental signature of the triplet fermionic DM. The decay length of such displaced vertices is clearly predicted [233,241] to be L Σ ± 5.5 cm.
A contrasting feature regarding the fate of the produced neutral component of the triplet fermion DM, Σ 0 ⊂ SO(10), different from the prediction of [241][242][243], has been observed in ref. [125]. In the case of ref. [241][242][243] it has been suggested that the corresponding Σ 0 can decay into leptons. But it has been noted in the context of the matter parity conserved SO(10) model [125] that the decay product Σ 0 is stable because of its matter parity. As such the production of this neutral component of the triplet fermion DM originating from SO(10) will be signalled through missing energy [125]. This stability feature of Σ 0 with its TeV scale mass has negligible impact on electroweak precision variables. These interesting features are applicable also in the present model under investigation.

Prospects from indirect searches
PAMELA [246][247][248] and FERMI/LAT [249] experiments concluded the positron excess in case of the WIMP as DM candidate which is again confirmed by recent AMS-02 [250] data [251]. The electron and positron flux is still significant in the measurement of FERMI/LAT. There are various constraints on the wino dark matter from different search channels such as antiprotons, leptons, dark matter halo from diffuse galactic gamma rays, high latitude gamma-ray spectra, galaxy clusters, dwarf spheroids, gamma-ray line feature, neutrinos from the galactic halo, CMB constraints, and antideuterons [234]. In the case of the antiproton search channel the wino dark matter having mass close to the resonance i.e, 2.4 TeV, and thin zone of diffusion is consistent with the antiproton measurement. The wino dark matter having mass near the resonance produces very small amount of leptons and large amount of positrons at very low energy scale. This DM can not solve cosmic ray (CR) lepton puzzle because the lepton data can rule out the very proximity of resonance. The galactic γ rays impose a stringent limit on the wino DM model. With the inclusion of the γ ray constraint, the limit on the wino DM changes. If the mass of DM is 2.5 TeV and it is in a thin diffusion zone, then it is excluded by the γ ray data for a wide variation of galactic CR propagation. There is also a very significant limit on the wino dark matter from high latitude γ ray spectra. For a 2.5 TeV wino DM the expected 10 year cross section is 1.5×10 −25 cm 3 s −1 including DM substructures [234]. Possible signatures of DM annihilations are given from γ ray observations [252,253] towards nearby galaxy clusters but observations in ref. [254][255][256][257][258][259] have not seen any significant limits from γ ray excess. The wino dark matter having mass 2.4 TeV can be ruled out in this search channel whereas all the other masses are allowed in the dwarf spheroids channel [234]. The winos with masses heavier than 2 TeV are excluded by the HESS [254] data at 95% CL. A new method to search for the indirect signals of DM annihilation is obtained due to the motion of high energy neutrons towards JHEP04(2017)075 the galactic center. Wino models having the mass 2.4 TeV can be observed in this search channel [234]. There is also a constraint on the wino dark matter due to the CMB temperature and polarization power spectra. Taking WMAP-5 [260] data and with 98% CL, the DM masses in the region 2.3 TeV to 2.4 TeV have been excluded. With WMAP-9 [263] the excluded limit is 2.25-2.46 TeV. But the combined search of WMAP-9 with ACT [261,262] excludes the mass range of 2.18-2.5 TeV. To search for the dark matter, the most effective channel is through antideuterons. Due to the smaller signal to back ground ratio at mass 2.5 TeV, the resultant signal is very low with high uncertainty. With the theoretical and experimental progress, there may be stringent limit on the wino dark matter [234].
In our model the triplet fermionic thermal DM resulting from any one of the nonstandard fermionic representations 45 F , 54 F , or 210 F would be adequate although we have preferred to choose the minimal of these three representations in order to minimise the impact on GUT threshold uncertainties as discussed in section 6.

Gauge coupling unification
In this section we discuss gauge coupling unification at the two-loop level using lighter scalar and fermionic degrees of freedom motivated by solutions to the neutrino masses by hybrid seesaw, dark matter and leptogenesis. At first exact unification of the three gauge couplings is realized using a triplet scalar ∆ L (1, 3, 0) at M ∆ = 10 12 GeV, a triplet fermion Σ F (1, 3, −1) at M T ∼ 500-1000 GeV, and, in addition, a color octet fermion of Majorana-Weyl type at M C 8 ∼ 5 × 10 7 GeV. We then estimate threshold effects on the GUT scale due to various superheavy components in the theory. We discuss proton life prediction in the model including these threshold uncertainties.

Unification with lighter fermions and scalars
We use the standard renormalization group equations (RGEs) for the evolution of the three gauge couplings [19] and their integral forms are where M Σ = triplet fermionic DM mass scale, M ∆ = LH triplet mass mediating type-II seesaw, and M C 8 = additional fermion octet mass scale found to be necessary to achieve exact unification of the three gauge couplings at two-loop level. The one-loop coefficients a . . i in their respective ranges of mass scales are shown in table 3 in the appendix. The terms Θ i , Θ i , and Θ i are the two-loop contributions in the three different ranges of the mass scales with the respective coefficients B , , ij given in table 3.
In terms of the experimentally determined parameters at the electroweak scale [266]: sin 2 θ W (M Z ) = 0.23126 ± 0.00005, α(M Z ) = 1./127.9, and α S (M Z ) = 0.1187 ± 0.0017, we define From the RGEs of eq. (5.1), the corresponding RGEs for P S and P Θ are obtained. These two are then solved to yield formulas for the two mass scales M U and M ∆ ln In eq. (5.4) Apart from depending upon the RG coefficients, the quantities C S ans C Θ in eq. (5.4) depend upon the lighter mass scales M Σ and M C 8 In deriving the analytic formulas in eq. (5.4) we have ignored the two-loop terms for the sake of simplicity although they have been included in numerical estimations of mass scales involved. It is clear that in eq. (5.4) the first two terms in the r.h.s. for the two mass scales M U and M ∆ represent the one-loop contributions but the third term in each case represents the corresponding threshold correction.
At first retaining only one-loop and the two-loop contributions we find excellent unification of the three gauge couplings for M Σ = 500-1000 GeV, M C 8 ∼ 5 × 10 7 GeV and M ∆ = 10 12 GeV. This is shown in figure 14.
In this model we have found the necessity of either two color octet scalars S 8 (8, 1, 0) or a single octet fermion C 8 (8, 1, 0) at mass M C 8 ∼ 5 × 10 7 GeV, in addition to the triplet fermionic DM candidate Σ F (1, 3, 0) and the LH triplet scalar ∆ L (1, 3, −1). This color octet fermion is thus safely above the cosmologically allowed limit [264,265]. The two-loop prediction of the GUT scale and the gauge coupling are and hence on the proton lifetime predictions. In this estimation at first we assume all the superheavy DM components in 45 F to be exactly degenerate with the GUT scale leading to their vanishing threshold effects. In the next step we estimate the fermionic contribution by following the same procedure [170][171][172][173][174][175][176][177][178]. We next evaluate the functions λ i (M U ) involving small logs caused due to super-heavy scalar components in the loop. These are contained in the SO(10) Higgs representations 10 H , 45 H , and 126 H . We further introduce the "partially degenerate" assumption on the super-heavy component masses of Higgs scalars which has been found to be useful in handling large representations especially in SO(10) [133]. Under this assumption all superheavy scalar masses belonging to a given representation have a common degenerate mass. Then using decompositions of representations shown in the appendix we find λ 1 = 17/5 + 4η (10) + (0)η (45) + 136η (126) , λ 2 = 6 + 4η (10) + 2η (45) + 140η (126) , λ 3 = 8 + 4η (10) + 3η (45)  Using eq. (6.2) in eq. (6.1) and maximizing the uncertainty [133] gives where M SH is the super-heavy Higgs mass scale and M 0 U represents the two-loop solution of eq. (5.7) without threshold effects. Similarly excluding the light triplet DM component Σ F (1, 3, 0), the rest of the fermionic component of the representation 45 F contribute to the threshold effects

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We also note that the degenerate super-heavy gauge bosons contribute a very small correction with a positive sign M U M 0 U V = 10 0.0227 . (6.5) In general following Coleman-Weinberg [267] idea, M SH could vary quite naturally within the range M U /10 to 10M U . As the the super-heavy fermionic components are unaffected by such corrections it may be natural to treat their masses to be degenerate at the GUT scale or at a degenerate mass M F around M U . In the first case they do not contribute to threshold corrections to the corrected unification scale. We have considered the general case with degenerate mass M F = (1/10 → 10)M F . Adding all corrections together we get Treating this as the mass of super-heavy gauge bosons mediating proton decay, we next estimate proton lifetime prediction in the model.

Proton lifetime prediction
As the unification scale predicted by this model has an uncertainty naturally dictated by the matter parity motivated SO(10) model, it would be interesting to examine its impact on proton life time predictions for p → e + π 0 for which there are ongoing dedicated experimental searches [88,183,[277][278][279] with measured value of the lower limit on the life time [186][187][188][189]280] τ expt. Including strong and electroweak renormalization effects on the d = 6 operator and taking into account quark mixing, chiral symmetry breaking effects, and lattice gauge theory estimations, the decay rates for the two models are [88,[271][272][273][274][275][276], (10), V ud = 0.974 = the (1, 1) element of V CKM for quark mixings, and A SL (A SR ) is the short-distance renormalization factor in the left (right) sectors. In eq. (6.8) A L = 1.25 = long distance renormalization factor but A SL A SR = 2.542. These are numerically estimated by evolving the dim.6 operator for proton decay by using the anomalous dimensions of ref. [271][272][273][274][275] [268][269][270], we obtain A R A L A SL A L A SR 3.18 and the expression for the inverse decay rate is, where the GUT-fine structure constant α G = 0.0263 and the factor F q = 2(1+|V ud | 2 ) 2 7.6 for SO (10). This formula reduces to the form given in [88,231] and sets the lower limit for the non-SUSY GUT scale to be M U ≥ 10 15.5 GeV from the lower limit of eq. (6.7). Now using the estimated values of the model parameters eq. (6.9) gives, τ SO(10) p 1.8 × 10 34±3.712η S ±1.012η F yrs. (6.10) As an example, a super-heavy scalar mass splitting by a factor 2(1/2) from the GUT scale gives η S = 0.3(−0.3) leading to τ p ∼ 1.8×10 34±1.11 yrs even if all fermion masses are at M 0 U . Similarly if all super-heavy scalar masses are degenerate at the unification scale M 0 U , the super-heavy fermions with their mass splitting factor 2(1/2) lead to τ p ∼ 1.8 × 10 34±0.3 yrs. These lifetimes are clearly above the current experimental limit but accessible to ongoing searches. The proton lifetime predictions as a function of η = η S or η = η F are shown in figure 15 for the p → e + π 0 decay mode.
It is clear that most of the uncertainties arise out of the GUT threshold corrections due to the larger Higgs representation 126 H which plays the central role in determining the contents of dark matter and their stability in the non-SUSY SO(10) by preserving matter parity as gauged discrete symmetry. We note that such uncertainties which are crucial for proton decay searches have been estimated here for the first time. The DM motivated JHEP04(2017)075 SO(10) also predicts additional threshold corrections to proton lifetime predictions especially due to fermions. Although this may enhance the uncertainty further, in one class of solutions the model also offers an interesting new possibility compared to GUTs without fermionic dark matter. The fermionic threshold corrections may contribute to cancel out a substantial part of the scalar threshold effects in another class of solutions which are shown by the blue curve marked S − F in figure 15. With this cancellation, the proton decay has somewhat more probability for detection by the ongoing searches.

Summary and conclusion
In this work we have attempted unification of gauge couplings of the non-SUSY standard gauge theory by addressing solutions to three of its outstanding problems: neutrino masses, dark matter, and baryon asymmetry of the Universe (BAU). To achieve these objectives we have exploited an interesting breaking pattern of non-SUSY SO(10) by assigning GUT scale VEV to the representation 126 H where matter parity is conserved as a natural gauged discrete symmetry of the SM that guarantees dark matter stability. As the origin of dark matter candidates, the model classifies non-standard fermionic or scalar representations of non-SUSY SO(10) carrying even or odd matter parity containing suitable components of dark matter. It predicts the type-I ⊕ type-II as the hybrid seesaw formula for neutrino masses driven by LH scalar triplet ∆ L (1, 3, −1) and heavy RH neutrinos. This formula has been used here to fit the neutrino oscillation data that predicts the heavy masses of the scalar triplet and the RHν masses. We have carried out this fitting procedure using values of the Dirac neutrino mass matrix derived in two ways by assuming u-quark diagonal or the d-quark diagonal bases. For a given intermediate mass value of the scalar triplet, induced VEV, and Dirac neutrino mass matrix, this seesaw formula being quadratic in Majorana neutrino Yukawa coupling f , predicts two distinct sets of RHν masses: (i) Compact spectrum where all three masses are heavier than the Davidson-Ibarra (DI) bound, and (ii) Hierarchical spectrum where only N 1 is lighter than the DI bound. These solutions provide a variety of results on the surviving lepton asymmetries after washout factors are adequately taken into account. We have carried out a complete flavor analysis of the RHν decays and exploited solutions to Boltzmann equations in every case to arrive at the model predictions on the baryon asymmetry. Although the decay of the LH scalar triplet in this model is found to yield negligible CP-asymmetry, it contributes quite significantly through the new Feynman diagram it generates for the vertex correction of RHν decays. In fact this contribution to the CP-asymmetry is found to be as dominant as other contributions without triplet mediation. The decay of the lightest RHν in the compact spectrum scenario predicts the values of BAU in agreement with the existing data when the Dirac neutrino mass determination is associated with either the u-quark diagonal basis or the d-quark diagonal basis. In the case of hierarchical spectrum of RH neutrinos, the right value of BAU is predicted by the N 2 decay where the Dirac neutrino mass is associated with the u-quark diagonal basis. This has been found possible even if the initial condition satisfies vanishing N 2 abundance.

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With the matter parity available as the stabilising discrete symmetry for dark matter, the neutral component of hyperchargeless triplet fermion Σ F (1, 3, 0) ⊂ 45 F ⊂ SO(10) having even matter parity is well accommodated as a candidate for thermal dark matter at TeV scale whose phenomenology has been discussed extensively in the literature and summarized here. Having thus addressed solutions to the three outstanding problems of the SM as stated above, we implemented unification of the three gauge couplings which needed a fermionic color octet of mass M C 8 ∼ 5 × 10 7 GeV, in addition to the heavy Higgs scalar triplet, and the fermionic triplet dark matter. The two-loop solutions yielded excellent unification with the predicted GUT scale value M U = 10 15.56+0.0288 GeV where the small positive fraction in the exponent is due to degenerate masses of all superheavy gauge bosons at M 0 U that causes nearly 30% increase in the proton lifetime prediction over its two-loop prediction. Noting the compelling requirement of the scalar representation 126 H to drive the symmetry breaking in this SO(10) model, its superheavy components predict substantial GUT threshold effects on the unification scale and proton lifetime. We have also estimated threshold corrections on the predicted proton lifetime due to superheavy fermions in 45 F . An interesting possibility of cancelling out a substantial part of threshold corrections due to scalars by fermions has been pointed out. We find that a large region of the parameter space can be explored by the ongoing searches on proton decay p → e + π 0 .
In conclusion we find that in the non-supersymmetric standard gauge theory, the predictions for neutrino masses, dark matter, baryon asymmetry of the universe, unification of gauge couplings, and proton lifetime accessible to ongoing searches can be successfully implemented through direct breaking of non-SUSY SO(10) with particle content inherent to matter parity conservation. The only additional particle needed beyond these requirements for coupling unification is a color octet Weyl fermion (or a pair of complex color octet scalars) which also belong to the SO(10) GUT representation. The introduction of the scalar triplet ∆ L at the intermediate scale brought in naturally by matter parity conservation in SO(10) causes remarkable changes in the model predictions over its conventional values. The very fact of successful implementation of the current programme in SO(10) resolves the issue of parity violation as a monopoly of weak interaction.

A Renormalization group coefficients for unification of gauge couplings and threshold uncertainties
In the appendix A.1 below we provide various decompositions of SO(10) representations under different subgroups relevant for the present work. In appendix A.2 we give different beta function coefficients along with particle content for different mass scales.

A.1 Decomposition of representations and beta function coefficients
In this appendix we present decompositions of non-SUSY SO(10) representations under SU (5)  Energy Scale Particle content  Table 3. One-loop and two-loop beta function coefficients in the respective ranges of mass scales.

A.1.1 Particle content and beta function coefficients
In this subsection we present the particle content used in various ranges of mass scales as shown in (0, 1, 3/5) Table 4. Decomposition of the complex 10 representation under SU(5) and one-loop coefficients. (1, 3/2, 1/10)

A.2 Super-heavy particles and coefficients for threshold effects
In this subsection we identify the super-heavy particle contents of various SO(10) representations with their quantum numbers and beta function coefficients under the SM gauge group. These coefficients shown in table 4, table 5, and table 6 have been used for the estimation of threshold effects on proton lifetime predictions.

A.3 A discussion on charged fermion mass parametrization
While all single step descents of SUSY GUTs leading to MSSM exhibit almost profound gauge coupling unification, there has been several attempts in SUSY SO (10)   validity of some of the Georgi-Jarlskog [70] type mass relations have been found to hold at the GUT scale. While some recent works have presented very attractive details of data analysis with χ 2 -fit [69] as pointed out in section 1, a much larger number of other research papers have confined to partially quantitative or qualitative representations of the charged fermion masses as these latter types of investigations focus on other challenging issues of particle physics. Compared to such interesing results on fermion mass fits in the direct breaking model of SUSY SO(10) [69], non-SUSY models need at least one intermediate gauge symmetry to ensure gauge coupling unification within the constraint of extended survival hypothesis [28,29]. Also unlike the MSSM or SUSY JHEP04(2017)075 SO(10), the RG extrapolated values of charged fermion masses through either SM or two-Higgs doublet model in the bottom-up approach [71][72][73] do not exhibit a precise b − τ Yukawa unification at the scale µ ∼ 10 16 GeV. Unlike the attempts to present all fermion masses in SUSY SO(10) through χ 2 fit and non-SUSY case with SU(4) C × SU(2) L × U(1) R intermediate symmetry [69], to our knowledge no such analysis appears to have been done so far in the direct breaking of non-SUSY SO (10) where gauge coupling unification itself under the minimal fine-tuning constraint [28,29] is highly challenging. In attempts to confront more challenging problems in SUSY or non-SUSY SO(10), a number of recent works have ignored the question of fitting the charged fermion masses while confining mainly to only neutrino masses and mixings, or at most a qualitative presentation of charged fermion masses [39-42, 73, 79-88, 88-96, 98-104]. However, even though a χ 2 fit [69] is not our present goal, we point out how the charged fermion masses may be parameterized within this direct breaking model of non-SUSY SO(10) while successfully encompassing standard model paradigm at lower scales, neutrino masses, baryon asymmetry, dark matter, gauge coupling unification, and GUT scale parity restoration. The Higgs representations 10 H , 126 H , and 120 H are known to contribute to fermion masses through the corresponding renormalizable Yukawa interactions. We include two copies of 10 H fields in the corresponding renormalizable part of the Yukawa Lagrangian The Yukawa term f 16.16.126 H has been found to be specifically suitable in approximately satisfying the GJ type relations in the down quark and charged lepton sectors. Conventionally, the same matrix f also contributes to the RH neutrino mass matrix M N = f v R which plays a crucial role in the type-I and type-II seesaw components of the hybrid seesaw formula used in this work. Therefore, the prime concern for charged fermion mass fit in the present model may be the smallness of the value of the matrix elements f ij ∼ O(10 −6 (i, j = 1, 2) as shown in eq. (2.5), eq. (2.6), eq. (2.7), and eq. (2.8) needed for successful predictions of baryon asymmetry in this model. We provide below how this difficulty can be circumvented in two different ways: (i) Non-renormalizable, and (ii) Renormalizable; any one of these can be added to L (10) for charged fermion mass parametrization.
(i). Non-renormalizable Yukawa correction. There have been attempts to represent fermion masses in SUSY SO(10) via non-renermalizable interactions with additional flavor symmetries and flavon fields [282,283]. Without introducing any such additional fields or symmetries, our attempt here is confined to the non-SUSY SO(10) gauge symmetry and the Higgs representations of the model. We note that the following non-renormalizable Yukawa (NRY) interactions are allowed where M G = Planck scale M Planck , or the String scale M String . The first Yukawa contribution is suppressed by a factor M GUT M G ∼ 10 −2 − 10 −3 . Noting that 10 H × 45 H ⊃ 120 H ⊃ ξ (2,2,15), it contributes to non-diagonal elements of all Dirac type mass matrices antisymmetrically which we ignore in this qualitative explanation, but can be included if a χ 2 fit is desired in future works. The second Yukawa interaction in eq. (A.3) containing 10 H × 45 H × 45 H has an effective (2, 2, 15) H component that is contained in1 26 and its contribution is symmetric. It is important to note that at the GUT scale L (2) NR gives a suppressed factor that adequately qualifies it to parameterize the needed additional corrections with m 0 ij ∼ F ij Thus, at the GUT scale the quark and lepton mass matrices can be parameterized as: where G u = Y (u) < 10 Hu >, G d = Y (d) < 10 H d >, F p ∼ F (2) 10 −4 . < 10 Hp >, p = u, d.
(ii). Renormalizable correction. Through renormalizable interaction, the improvement of fermion mass parametrization is also suggested by the introduction of a second 126 H representation [144,146,147]. We denote this and its corresponding components under G 224 as 126 H ⊃ ∆ L (3, 1, 10), ∆ R (1, 3,10), ξ (2, 2, 15), . . .. In contrast to the ∆ L ⊂ 126 H whose mass has been fine tuned to be at M ∆ L ∼ 10 12 GeV for the implementation of the type-II seesaw component of the hybrid seesaw formula, leptogenesis, and coupling unification, all the components of 126 H are naturally assigned masses near the GUT scale consistent with extended survival hypothesis [28,29]. Also no VEV is needed to be assigned to ∆ R either i,e we fix < ∆ R >= 0, since the corresponding role of gauge symmetry breaking has been taken over by < It is well known that such corrections provide reasonable parameterization of the fermion masses of the first and second generations. With degeneracy of all superheavy components of 126 H , its threshold corrections to unification scale and proton lifetime are vanishingly small [281]. Similarly, if the renormalizable antisymmetric contributions to fermion mass matrices due to Yukawa interaction of a 120 H ⊂ SO(10) are included, its threshold effects on unification scale and proton lifetime would be also vanishingly small due to degeneracy of the components. Alternatively the fermion mass parametrization may be improved further by including both the renormalizable and non-renormalizable contributions in eq. (A.4). In addition, the antisymmetric contribution through the first nonrenormalizable term in L We have checked that even afte applying these corrections satisfying the first of GJ relation in eq. (A.1), our solutions and predictions on baryon asymmetry made in this work are not significantly affected. Also they remain largely unaffected as long as the corrections to the elements of the Dirac neutrino mass matrix M D are either less or at most of the same order as those given in section 2. After the GUT symmetry breaking to the SM gauge theory we have assumed only one linear combination of different up type and down type doublets to remain massless to form the standard Higgs doublet.