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

Journal of High Energy Physics, Apr 2017

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 126 H 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 45 F 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 M U 0 = 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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Standard coupling unification in SO(10), hybrid seesaw neutrino mass and leptogenesis, dark matter, and proton lifetime predictions

Received: August Standard coupling uni cation in SO(10), hybrid seesaw neutrino mass and leptogenesis, dark matter, M.K. Parida 0 1 2 4 5 Bidyut Prava Nayak 0 1 2 4 5 Rajesh Satpathy 0 1 2 4 5 Ram Lal Awasthi 0 1 2 3 5 0 Knowledge City , Sector 81, SAS Nagar, Manauli 140306 , India 1 Khandagiri Square , Bhubaneswar 751030 , India 2 Siksha `O' Anusandhan University 3 Indian Institute of Science Education and Research 4 Centre of Excellence in Theoretical and Mathematical Sciences 5 Open Access , c The Authors We discuss gauge coupling uni cation of SU(3)C the stability and identi cation 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 tted with the neutrino oscillation data. Predictions of the baryon asymmetry via leptogenesis are investigated through the decays of both the patterns of seesaw; neutrino; mass and leptogenesis; dark - U(1)Y descending directly from non-supersymmetric SO(10) while providing solutions to the three outstanding 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 masses. A complete avor 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 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 ArXiv ePrint: 1608.03956 1 Introduction 3.1 3.2 3.3 2 Hybrid seesaw t to neutrino oscillation data 3 Baryon asymmetry of the Universe Boltzmann equations Baryon asymmetry in the compact scenario Baryon asymmetry in the hierarchical scenario 4 Fermionic triplet as dark matter candidate General considerations with matter parity Light non-standard fermion masses from SO(10) Triplet fermion dark matter phenomenology Prospects from indirect searches 5 Gauge coupling uni cation 5.1 Uni cation with lighter fermions and scalars 6 Threshold corrections and proton lifetime prediction 6.1 6.2 Threshold e ects on the GUT scale Proton lifetime prediction 7 Summary and conclusion A Renormalization group coe cients for uni cation of gauge couplings and threshold uncertainties A.1 Decomposition of representations and beta function coe cients Particle content and beta function coe cients A.2 Super-heavy particles and coe cients for threshold e ects A.3 A discussion on charged fermion mass parametrization The standard model (SM) of particle interactions based upon the gauge symmetry SU(3)C U(1)Y has been tested by numerous experiments. Also the last piece of evidence in favour of the SM has been vindicated with the discovery of the Higgs boson at the CERN Large Hadron Collider [1, 2]. Yet the model fails to explain the three glaring physical phenomena: neutrino oscillation [3{6], baryon asymmetry of the universe (BAU) [7{12], and dark matter (DM) [13{18]. Although the electroweak part of the SM provides excellent description of weak interaction phenomenology manifesting in V A structure of neutral and charged currents, it fails to answer why parity violation is exhibited by weak interaction alone. On the fundamental side, the SM itself can not explain the disparate values of its gauge couplings. The minimal gauge theory which has the potential to unify the three gauge couplings [19, 20] and explain the origin of parity violation is SO(10) grand uni ed theory (GUT) [21, 22] that contains the Pati-Salam [23, 24] and left-right gauge theories [25{27] as its subgroups. However, it is well known that direct breaking of all non-supersymmetric (non-SUSY) GUTs [20{22] to the SM gauge theory under the assumption of minimal ne tuning hypothesis [28, 29] fails to unify the gauge couplings of the SM whereas supersymmetric GUTs like SU(5) [20] and SO(10) [21, 22] achieve this objective in a profound manner. In fact the prediction of coupling uni cation in the minimal supersymmetric standard model (MSSM) [30, 31] evidenced through the CERN-LEP data [32{35] led to the belief that a SUSY GUT [36{45] with its underlying mechanism for solutions to the gauge hierarchy problem [46{50] could be the realistic model for high energy physics. SUSY GUTs also predict wino or neutralino as popular candidates of cold dark matter (CDM). Compared to SUSY SU(5) [30], SUSY SO(10) has a number of advantages. Whereas parity violation in SO(10) has its spontaneous breaking origin, for SU(5) it is explicit and intrinsic. The right-handed neutrino (RH ) as a member of spinorial representation 16 of SO(10) mediates the well known canonical seesaw mechanism [51{56] that accounts for small neutrino masses evidenced by the neutrino oscillation data. Further the Dirac neutrino mass matrix that occurs as an important ingredient of type-I seesaw [51{56] is predicted in this model due to its underlying quark-lepton symmetry [23, 24]. In addition, the presence of the left-handed (LH) triplet scalar, rally leads to the possibility of Type-II seesaw formula for neutrino masses [56{59]. Both the heavy RH neutrinos and the LH triplet scalar have the high potential to account for BAU via leptogenesis [60{62, 87]. With R-Parity as its gauged discrete symmetry [63{66], the model also guarantees stability of dark matter. Another attractive aspect of SUSY SO(10) [67] has been its capability to make a reasonably good representation of all fermion masses and mixings at the GUT scale [68, 69]. Such a data set exhibiting b Yukawa uni cation and very approximately satisfying Georgi-Jarlskog [70] type relation is obtained using RG extrapolated values of the masses and mixings at the electroweak scale following the bottom-up approach [71{73]. In particular 2 estimation has been carried out to examine goodness of t to all fermion masses in SUSY SO(10) [69]. Other interesting aspects of the SUSY GUT such as Yukawa uni cation and a heavier gluino [74], viability of GUT-scale tribimaximal mixing [75], and uni ed description of fermion masses with quasi-degenerate (QD) neutrinos [76] have been explored. A comparison of quality of di erent models has been also discussed [77]. Recently existence of avour symmetries [78] and emergence of ordered anarchy from 5:dim: theory [79, 80], and Sparticle spectroscopy [81] have been also investigated with numerical analyses on fermion masses. However, there exists a large class of SUSY SO(10) models where a qualitative or at most a semi-quantitative representation of fermion masses have been considered adequate without 2 estimation. Examples from a very small part 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 Z0 [93], prediction of Axions [98], low-mass Z0 induced by avor symmetry [100], realization of SUSY SO(10) from M theory [99, 101], predictions of in aton mass [102], and Starobinsky type in ation [103], or quartic in ation [104] from SUSY SO(10). Generalised hidden avour symmetries have been explored without con ning to any particular type of fermion mass ts [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 uni cation of the standard gauge theory in non-supersymmetric (non-SUSY) GUTs while sacri cing the elegant solution to the gauge hierarchy problem in favour of ne-tuning to every loop order [106, 107]. 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 ne-tuning hypothesis [28, 29] fail to unify gauge couplings. Introducing gravity induced corrections through higher dimensional operators [108, 109] or additional ne-tuning of parameters with lighter scalars or fermions, gauge coupling uni cation in non-SUSY SU(5) GUT has been implemented [110{119] including RH neutrino as DM [120]. Such uni cation has been also achieved including triplet fermionic DM [121]. A color octet fermion with mass > 108 GeV which is also needed for uni cation has been suggested as a source of non-thermal DM via non-renormalizable interactions [121]. As the model does not use matter parity [122{128], the stabilising discrete symmetry for DM has to be imposed externally and appended to the GUT framework. Further, issues like neutrino masses and mixings and the baryon asymmetry of the universe have not been addressed in this model. Naturally the non-SUSY SU(5) models [108{ 117, 120, 121] have no explanation for the monopoly of parity violation in weak interaction However, with or without broken D-Parity at the GUT scale [129{132], non-SUSY SO(10) has been shown to unify gauge couplings having one or more intermediate symmetries [129{135]. Extensive investigations in such models have been reported with high intermediate scales [69, 129{141] and also with TeV scale WR; ZR bosons and veri able seesaw mechanisms [142{153]. Out of a large number of possible models that are predicted from non-SUSY SO(10) [132] fermion mass t has been investigated only in one class of models with Pati-Salam intermediate symmetry [69, 137, 139, 140] and also including additional vector-like fermions [138]. The issue of DM has been also addressed with di erent types of high scale intermediate symmetries and by introducing additional fermions or scalars beyond those needed by extended survival hypothesis [154{158] but without addressing fermion mass ts. The problem of TeV scale WR boson prediction along with DM have been also addressed in non-SUSY SO(10) by invoking external Z2 symmetry [148] without tting charged fermion masses as also in a number of other models [132, 133, 135, 136, 141{147, 159{162]. As there has been no experimental evidence of supersymmetry so far, likewise there has been also no de nite evidence of any new gauge boson beyond those of the SM. This in turn has prompted authors to implement gauge coupling uni cation with the SM gauge symmetry below the GUT scale [110{123, 125{128] by the introduction of additional particle degrees of freedom with lighter masses. A natural question in this context is how much of the advantages of the SUSY GUT paradigm is maintained in the case of non-SUSY gauge coupling uni cation models. While SUSY SO(10) is well known for its intrinsic R-Parity [63, 65] as gauged discrete symmetry [64] for the stability of dark matter, as an encouraging factor in favour of the non-SUSY GUT it has been be the corresponding discrete symmetry intrinsic to non-SUSY SO(10) where B(L) stands for baryon (lepton) number. Whereas neutralino or wino are predicted as dark matter candidates in SUSY GUTs, in non-SUSY SO(10) the DM candidates could be non-standard fermions (scalars) carrying even (odd) matter parity. In fact all SO(10) representations have been identi ed to carry de nite values of matter parity which makes the identi cation of a dark matter candidate transparent from among the non-standard scalar(fermion) representations. Thus there is enough scope within non-SUSY SO(10) to implement the DM paradigm along with an intrinsic stabilising symmetry. 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+ 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 su er from the well known gravitino problem. [164{169]. Coupling uni cation 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 45F as dark matter candidate. The presence of a color octet fermion of mass 1010 GeV has been also noted for uni cation. The proton lifetime has been predicted in this model at two-loop level of gauge coupling uni cation. However details of tting the neutrino oscillation data including derivation of Dirac neutrino mass matrix and the RH 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 modi cation due to threshold e ects that could arise from the superheavy components of various representations [170{178] need further investigation. The contents of the present paper are substantially di erent 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{181]. Both of the seesaw mechanisms are naturally predicted in matter parity based SO(10) model having their origins rooted in the Higgs representation 126H and the latter's coupling to the fermions in the spinorial representation 16 through f 16:16:126yH . Unlike a number of neutrino mass models adopted earlier, in this work we have not assumed 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{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 RH masses: (i) Compact scenario where all masses are heavier than the Davidson-Ibarra (DI) bound, and (ii) The hierarchical scenario where only the lightest N1 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 RH decays. We have carried out a complete avour 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 RH masses. With the compact pattern of RH 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 RH masses, the dominant CP asymmetry that survives the washout due to N1-decay and contributes to the desired baryon asymmetry is generated by the decay of the second generation RH 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 RH decay is found to lead to a CP-asymmetry component comparable to other dominant contributions. Thus the same heavy triplet scalar L and the RH s 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 45F rying even matter parity. Having exploited the triplet fermionic DM 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 uni cation 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 107 GeV under SU(3)C to complete the precision gauge coupling uni cation. The degrees of freedom used in this model having their origins from SO(10) representations 126H ; 10H ; 45H , and 45F are expected to contribute substantially to GUT threshold e ects on the uni cation 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 126H in its spontaneous symmetry breaking is inevitable. This in turn dictates a dominant contribution to threshold e ects on proton lifetime which has been ignored earlier but estimated in this direct breaking chain for the rst time. In addition the superheavy fermions in 45F have been noted to contribute substantially. A possibility of partial cancellation of scalar and fermionic threshold e ects 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+ 0 is found to be within the accessible range of the ongoing search limits [183{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 t. But we discuss in appendix A.3 how all fermion masses may be tted at least approximately in future without substantially a ecting this model predictions. This paper is planned in the following manner. In section 2 we discuss successful t to the neutrino oscillation data where we estimate the LH Higgs triplet and the RH In section 3.1 we present the estimations of CP-asymmetry for di erent avor states. In section 3.2 we discuss Boltzmann equations for avour based analysis. In section 3.3 and section 3.4 we present the results of nal 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 uni cation of gauge couplings and determine the uni cation 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 coe cients for gauge coupling evolution and estimation of threshold e ects. In appendix A.3 we discuss the possibility of parameterization of fermion masses. Hybrid seesaw t to neutrino oscillation data In this section we address the issue of tting 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 rst term on the right-hand side (r.h.s.) of eq. (2.1) is from the SO(10) symmetric Yukawa term Y (10):16:16:10H whereas the second and the third terms are from f:16:16:126y [67]. Also we have de ned vR vR. Although the associated RH scalar eld 126H has the respective quantum number under the LR gauge group SU(2)L G2213), it is the singlet comSimilarly the LH triplet scalar eld 126H has the transformation property L(1; 3; 1). Here is the quartic coupling of the SO(10) invariant Lagrangian resulting from the combination of 10H and 126H : 102H :126yH :126H has its origin from M 2 126yH 126H . Other notations are self explanatory. The hybrid formula for the light neutrino mass matrix is the sum of type-I and type-II seesaw contributions [29] HT i 2 LH. The Higgs triplet mass-squared term = f vL where vL = vRve2w=M 2 is the induced VEV of triplet scalar There is the well known standard ansatz to t 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 = MZ using experimental data on charged fermion masses we choose up-quark or down-quark mass diagonal bases in two di erent scenarios. We then evolve them upto the GUT scale = MU using bottom-up approach [71{73]. At this scale we assume equality of the up-quark and the Dirac neutrino mass matrices, MD ' Mu, 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, 2 t to all fermion masses and mixings in SUSY SO(10) or in non-SUSY SO(10) with G224 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 MD close to Mu [141] as in our Although in the present case of non-SUSY SO(10) breaking directly to the SM gauge theory, fermion mass t is not our goal in this paper, we have discussed the issue in We further assumed that MD(MMGUT ) MD( ) for all lower mass scales 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 3C [71{73] contribution, this approximation does not in uence our nal result substantially. Another reason is that for leptogenesis we need Dirac neutrino Yukawa couplings at intermediate scales, (106{1012) GeV where the renormalisation group (RG) running e ects are expected to be smaller in the top-down approach. Thus in the down quark diagonal basis under the assumption of negligible RG e ects = MZ 00:01832+0:00441i 0:65882+0:27319i 1 3:32785+0:00019i 81:8543 1:64 We repeat the above procedure in the up-quark diagonal basis at = MZ instead of the down quark diagonal basis leading to (1:5027+0:0038i)10 9 (7:51+3:19i)10 61 M D(u)(GeV) = B@(1:5027+0:0038i)10 9 (7:51+3:19i)10 6 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 Mu(MZ ) 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 Md(MZ ) are non-vanishing. In the alternative case, called the d-quak diagonal basis, we have chosen Md(MZ ) diagonal for which all nine elements of Mu(MZ ) are non-vanishing. In the case of up-quark diagonal basis, however, the non-diagonal elements of Mu(MMGUT ) acquire non-vanishingly small corrections due to RG e ects in the bottom-up approach and this is approximated as the Dirac-neutino mass matrix M D(u)(MGUT). This explains the appearance of non-diagonal elements appearing in eq. (2.4). It may be noted further that the RG-corrections in the Dirac neutrino mass matrix M D(u) for evolutions from = MGUT down to relevant lower scales have been ignored as they are expected to be much smaller. 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 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 MD 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 should be only two distinct positive de nite 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 xing the lightest neutrino mass and the VEV vL in a chosen hierarchy of light neutrino masses, the precise forms of the two solutions with positive de nite f are evaluated upto the desired precision. These solutions are presented in gure 1 for two sets of values of = 0:1 and In gure 1 we have presented these solutions for the normally ordered values of active light neutrino masses. Solutions in the top row of the gure have strongly hierarchical heavy RH neutrino masses, lightest of them being MN1 collider experiments, and the heaviest MN3 O(1012) GeV. We call such solutions of RH O(103 5) GeV, testable in future neutrino masses to represent a hierarchical spectrum scenario. Solutions in the bottom row of the gure 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(109 11) GeV which is far away from direct detection limit of any collider experiment. masses increase with for the compact spectrum scenario while it almost stays una ected in the hierarchical spectrum scenario. Also the theory should continue to remain perturbative on the quartic coupling in the case when the three neutrino masses are normally ordered. The top row represents a hierarchical spectrum solution of RH neutrinos and the bottom row represents a not so hierarchical scenario which we call as compact spectrum solution. The values of M L = 1012 GeV and vR = 1015:5 GeV have been kept xed. The value of the quartic coupling used here has been = 0:1(0:001) for the left panel (right panel). acquiring N1-dominated leptogenesis because increasing ( 1) for the above value of M will make MN1 < 109 GeV and N1- dominated leptogenesis will not be possible. In the compact spectrum scenario we estimate the f matrix in the d-diagonal basis MD = M D(d) f = B@0:4617 0 0:385 + 0:1291i 0:4617 0:4922i 3:509 + 1:080i 1 0:4922i 4:626 + 0:1567i 22:80 + 0:3317iC For the same parameters in the compact spectrum scenario but with M D(u) in u-diagonal basis given in eq. (2.4), we derive MD = M D(u) 0 0:3175 + 0:0904i 0:1232 f = B@ 0:1232 1:587 + 0:2599i C 0:6918i 1:587 + 0:2599i In the hierarchical spectrum scenario, similarly, we have the two matrices for f MD = M D(d) MD = M D(u) f = B@ 4:0194 + 1:5783i 4:0194 + 1:5783i 1 0 0:000025 + 0:000008i 0:00019 f = B 0:56091 + 0:0092i 0:95702 0:00177i 0:95702 Despite widely varying magnitudes of di erent elements in the matrix, the mass eigenvalues quark and d quark diagonal bases are not very di erent in both the compact spectrum and the hierarchical spectrum scenarios. Therefore, we have presented only one set of solutions for the RH masses in gure 1. It is quite encouraging to note that despite the GUT scale value of vR, the type-II term does not upset the type-I seesaw term in the hybrid formula, rather both of them contribute signi cantly to the light neutrino mass matrix. We will explore the plausibility of su cient leptogenesis using the hybrid seesaw mechanism of this model to explain BAU. Baryon asymmetry of the Universe In this section at rst we estimate the leptonic CP- asymmetry generated in decays of both 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{208]. The avour independent calculation of asymmetry is applicable at high temperatures when all the charged lepton mediated interactions are out of equilibrium i.e. T & 1012 GeV. Flavour dependent analysis [202] becomes necessary for leptogenesis at lower temperatures. In hierarchical spectrum scenarios we have MN1 which violates the Davidson-Ibarra bound [209] badly, therefore it can not produce required amount of avour independent lepton asymmetry. Instead it washes out the asymmetry produced at the early stage in N2;3 decays. In the recent studies [202, 210{213] it has been shown that under such circumstances the next heavy neutrino N2 can produce the required asymmetry, if MN2 & 1010 GeV and there exists a heavier N3. If the asymmetry produced by N2 is not completely washed out by lightest neutrino N1, 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 RH decay. Since for a large region of the parameter space we have shown that MN1 1012 GeV, the asymmetry will depend on avour dynamics. i = "iN + "i : iN = k6=i k6=i diagrams represent vertex corrections and the second diagram represents self-energy correction. The avoured CP-asymmetry in the decay of Ni to a lepton l is generated in the lepton avor generation , and is de ned as [216, 217, 229] i = (Ni ! l + H ) + (Ni ! l + H) gure 2. The total asymmetry is sum of the two contributions L [62] as shown in The rst line of this expression contains lepton number violating terms while the second line is the lepton number conserving but violates lepton avour. Here, Y^ = Y Uf is the Dirac Yukawa coupling in the right-handed neutrino diagonal mass basis and Uf is the unitary matrix diagonalizing f . The loop functions in the asymmetry expression are [217] g(x) = h(x) = p Here by retaining the Wigner-Eckart term in the loop function we can handle degenerate mass scenario without hitting singularity, which is possible in compact spectrum scenario in our model (see gure 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 Ni decay from the L mediated diagram is [62] i = | Δiα10-6 ε top left(right)-panel correspond to d(u)-quark diagonal basis for = 0:1. The bottom left(right) panel correspond to d(u)-quark diagonal basis but for which gets contribution proportional to the trilinear coupling mass term . Its loop function is larger for smaller M L can not be made arbitrarily small without decreasing or increasing vL which is constrained to be below GeV from electroweak (EW) precision constraints. Decreasing would decrease CP asymmetry linearly. estimated the avored CP-asymmetry for di erent values of the lightest neutrino mass in the normally ordered hierachical case of light neutrino masses. Change in the mass of m 1 alters f and thus changes the masses and mixings of RH s. Flavour asymmetries for Ni avour are shown in gure 3 for compact spectrum case and in gure 4 for the hierarchical spectrum case of RH s. We note that variation in quartic coupling changes CP-asymmetry signi cantly, particularly in the hierarchical spectrum scenario. The tree level decay widths are una ected 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 CP ) 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{221] or combination of triplet scalar and right| Δiα10-6 ε masses. The top left (right)-panel correspond to d(u)-quark diagonal basis for = 0:1. The bottom left(right) panel correspond to d(u)-quark diagonal basis but for handed neutrinos [62] as shown in gure 5 for our model. The CP-asymmetry generated L decay and mediated by RH is written as [62] = 2 log(1 + M 2 =M N2k ) : We note that, since vR ' 1015:5 GeV and M L ' 1012 GeV, either of the two terms in the denominator of " is large enough to keep the CP -asymmetry fairly small for the parameters under consideration. For example, if three right-handed neutrino masses are MNk = (6:6990; 13:869; 1431) 109 GeV, the three CP-asymmetries due to Nk decays from CP-asymmetries from the third diagram are: j Nk j = (5:2 Compared to these numbers, the CP-asymmetry due to j j = 2:1 M1;2, the asymmetry generated at the early stage will be washed out at the production phase of lighter RH s. 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. Boltzmann equations The evolution of number density is obtained by solving the set of Boltzmann equations. The co-moving number density is YX nX =s. The Boltzmann equations for heavy neutrinos number density are [190] Ki(Di(z) + Si(z)) YNi (z) X Wi(z) A stands for the total asymmetry stored in the fermionic such that Ki = P Ki . In eq. (3.7) the equilibrium number density [190, 215] is de ned as Ki = YNeqi = 135 (3) R2z2K2(Riz) T Mi 135 (3) i ! requires the lightest right-handed neutrino to acquire mass MN1 & 4 decay rates are Di(z) = R2zK1(Riz)=K2(Riz) where K1 and K2 are the rst and the seci ond order modi ed Bessel functions [215, 222], respectively. The scattering terms Si(z) account for Higgs-mediated 108 GeV [179{181] inverse decay contribution is WiID(z) = 1 R4z3K1(Riz): The unit lepton number changing L = 1 scattering contributing to washout is WiS(z) = matrices are [213] scattering involving top quark are included in the evolution of asymmetry [222]. We have ignored the o -shell part of L = 2 process in the washout term which is a good approximation as long as MNi =1013 Ki [223]. We have also omitted the not contribute to the washout but can a ect the abundance of heavy neutrinos. When avor e ects are taken into account, they also tend to redistribute the lepton asymmetry among avors. These e ects are of higher order in the neutrino Yukawa couplings and are expected to have little impact on the nal 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 Ni eR ! qR qR0 and NiNi ! f f tend to keep the heavy neutrinos in thermal equilibrium, thus reducing the generated lepton asymmetry. This e ect is practically negligible because RH s are much lighter than the RH gauge bosons. We also ignore such avour e ects [224] which are relevant for resonant leptogenesis. Baryon asymmetry in the compact scenario In this scenario the tau lepton avour state decouples while the electron and muon states are still coupled. Thus, a avour dependent analysis is necessary. In the two avour case = "ie + "i , Ki;e+ = Kie + Ki , and the avour coupling A = 417=589 120=589 ! ; C = 224=589 : In this case the baryon asymmetry is expressed as asymmetry in to baryon asymmetry by non-perturbative sphaleron process [225, 226]. The results of BBN [227] and PLANCK [11, 12] experiments are Y B = Y BBBN = (8:10 Y PBlanck = (8:58 Compared to these somewhat higher value of BAU obtained from WMAP 7 years' data has been reported in ref. [228]. The washout coe cients Ki in the compact spectrum scenario of RH neutrino masses for the lightest neutrino mass m 1 = 0:00127 eV and 2 [0:0001; 0:5] are plotted in gure 6. We see that there are two to four orders of variation in the washout for the above allowed in both the d-diagonal (left panel) and the u-diagonal (right panel) cases. We 0.0001 0.001 0.0001 0.001 Other parameters are kept xed as described in the text. list the washout parameters for = 0:1 in the case of the d-quark diagonal basis In the u-quark diagonal basis the washout parameters are K = B@2:77 K = B@1:46 Our observations in the two cases are summarized below. (a) The d-quark diagonal basis. We note that Ki = P 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 nal asymmetry freeze occurs at the decoupling of washout with zf lepton asymmetry is approximated as [229] (7{10). In the single avour analysis the YNeq1 (z = 0) [229] gives Using the values of Ki from gure 6 and "1 = P "1 from gure 7 we can easily achieve the required lepton asymmetry. In fact it may lead to a constraint on quartic coupling . (b) The u-diagonal basis. We note that, since K1 = P 1, this is a very weak washout regime. Ignoring thermal e ect on CP-asymmetry and assuming zero initial 0.0001 0.001 0.0001 0.001 Other parameters are kept xed as described in the text. avours (double-dot-dashed blue curve) and dashed curve) for the u-quark diagonal basis and compact spectrum RH mass scenario. Left (right) panel correspond to non-zero (zero) initial thermal abundance. The quartic coupling If there is already an initial amount of asymmetry left over, say through N2 decay, it will not be washed out because the system is in weak washout regime. But with zero initial abundance, YN1 (z = 0) = 0 [229] We note that even if we assume initial thermal abundance Y eq(0) 0:0039, the CPasymmetry "1 10 6 ( gure 7) and K 10 3 ( gure 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 avour independent analysis in the u-quark diagonal scenario with zero initial abundance of YN1 fails to give the required asymmetry. On the other hand a avor dependent analysis can enhance the asymmetry. The avour dependent lepton asymmetry is analyzed using Boltzmann equations (3.7) and is shown in gure 8 for u-quark diagonal basis. Thus in avoured analysis we nd that nal lepton vR=1015.5 GeV spectrum scenario with Dirac neutrino mass matrix determined in the d-quark diagonal basis as described in the text. asymmetry is independent of initial abundance and is close to the experimental value for < 0:05. This explicitly shows that N2 decay contributes to lepton asymmetry which is not completely washed out in the N1 decay. The reason for doing avoured analysis is that there are enhancements in the nal asymmetry compared to the un avoured case. Using d-quark diagonal basis shows the variation of total asymmetry with respect to quartic coupling for a xed value of the scalar triplet mass M = 1012 GeV, vR = 1015:5 GeV, and the lightest neutrino Baryon asymmetry in the hierarchical scenario The Davidson-Ibarra bound is not respected in the hierarchical spectrum scenario of RH (see gure 1). In such a case there is the possibility of leptogenesis if asymmetry is produced by the decay of N2. Lower bound on the lightest RH is passed to MN2 & 1010 GeV. The N2-dominated leptogenesis can be successful if there is a heavy neutrino, or triplet scalar with MN3 ; M L > MN2 , and the washout from the lightest RH (N1) is circumvented. 109 GeV the lepton avour states become incoherent and the washout acts separately on each avour asymmetry. We need to solve Boltzmann equations at the i is very small compared to CP-asymmetry due to N2;3 decays. The decay and washout are also suppressed by a factor M12=M32( the scenario M3 & 1012 GeV 10 15) and M12=M22( M2 > 109 GeV 10 10). Also we note that in M1, the role of N3 becomes indistinct by 10 + 5y+ 1 10 + 10y + 1 + 24 24 + 15 + 15y 5 +5y+10y+45y +10 + 45 5y+ 45+15y +50y + 10 + 1 1+ 24 + 10y+ 10 + 40 + 40y + 75 + 5 +5y Energy Scale Particle content SM+(1; 3; 0)F SM + (1; 3; 0)F + (8; 1; 0)F SM + (1; 3; 0)F + (8; 1; 0)F + (1; 3; 1)H MZ MT MT MO 0 41=10 1 0 41=10 1 0 41=10 1 043=101 B 7=6 C 0199=50 27=10 44=51 B@ 9=10 0199=50 27=10 44=51 B@ 9=10 163=6 12 CA 0199=50 27=10 44=51 B@ 9=10 163=6 12 CA 083=10 171=10 44=51 B@57=10 275=6 Particle content and beta function coe cients In this subsection we present the particle content used in various ranges of mass scales as shown in table 2 and the corresponding beta-function coe cients which have contributed for the gauge coupling uni cation, leptogenesis, and dark matter as shown in table 3. SU(5) (3C ; 2L; 1Y ) SU(5) (3C ; 2L; 1Y ) (1, 3/2, 5/2) (1/2, 0, 1/5) (1, 3/2, 5/2) (1/2, 0, 1/5) (1, 3/2, 1/10) (1, 3/2, 1/10) the sake of convenience, the would-be goldstone modes of all super-heavy gauge bosons have been provided from the scalar representation 45H . Super-heavy particles and coe cients for threshold e ects In this subsection we identify the super-heavy particle contents of various SO(10) representations with their quantum numbers and beta function coe cients under the SM gauge group. These coe cients shown in table 4, table 5, and table 6 have been used for the estimation of threshold e ects on proton lifetime predictions. A discussion on charged fermion mass parametrization While all single step descents of SUSY GUTs leading to MSSM exhibit almost profound gauge coupling uni cation, there has been several attempts in SUSY SO(10) to explain fermion masses of three generations of quarks and leptons along with the attractive phenomena like b Yukawa uni cation. In certain other cases approximate SU(5) (3C ; 2L; 1Y ) (15, 24, 12/5) (12, 8, 24/5) (12, 8, 24/5) 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 2- t [69] as pointed out in section 1, a much larger number of other research papers have con ned 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 ts in the direct breaking model of SUSY SO(10) [69], non-SUSY models need at least one intermediate gauge symmetry to ensure gauge coupling uni cation within the constraint of extended survival hypothesis [28, 29]. Also unlike the MSSM or SUSY SO(10), the RG extrapolated values of charged fermion masses through either SM or twoHiggs doublet model in the bottom-up approach [71{73] do not exhibit a precise b Yukawa uni cation at the scale 1016 GeV. Unlike the attempts to present all fermion masses in SUSY SO(10) through 2 t and non-SUSY case with SU(4)C 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 uni cation itself under the minimal ne-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 tting the charged fermion masses while con ning 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 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 uni cation, and GUT scale parity restoration. The Higgs representations 10H ; 126H , and 120H are known to contribute to fermion masses through the corresponding renormalizable Yukawa interactions. We include two copies of 10H elds in the corresponding renormalizable part of the Yukawa Lagrangian L(N1R) = L(N2R) = The Yukawa term f 16:16:126H has been found to be speci cally suitable in approximately satisfying the GJ type relations in the down quark and charged lepton sectors. Convenwhich 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 t in the present model may be the smallness of the value of the matrix elements for successful predictions of baryon asymmetry in this model. We provide below how this di culty can be circumvented in two di erent 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 avor avon elds [282, 283]. Without introducing any such additional elds or symmetries, our attempt here is con ned 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 tions with mi0j and lepton mass matrices can be parameterized as: bution is suppressed by a factor MGUT . Noting that 10H (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 2 t is desired in future works. The second Yukawa interaction in eq. (A.3) containing 45H has an e ective (2; 2; 15)H component that is contained in 126 and its contribution is symmetric. It is important to note that at the GUT scale L(N2R) gives a suppressed factor that adequately quali es it to parameterize the needed additional correc10 5)vew. Thus, at the GUT scale the quark Mu = Gu + Fu; MD = Gu Md = Gd + Fd; Ml = Gd where Gu = Y (u) < 10Hu >, Gd = Y (d) < 10Hd >, Fp F(2)10 4: < 10Hp >, p = u; d. Details of fermion mass parametrization goes in a manner similar to those discussed in [94{96, 142, 144{147]. (ii). Renormalizable correction. Through renormalizable interaction, the improvement of fermion mass parametrization is also suggested by the introduction of a second 126H representation [144, 146, 147]. We denote this and its corresponding components under G224 as 1260H 0R(1; 3; 10); 0(2; 2; 15); : : :. In contrast to the 126H whose mass has been ne tuned to be at M L 1012 GeV for the implementation of the type-II seesaw component of the hybrid seesaw formula, leptogenesis, and coupling uni cation, all the components of 1260H are naturally assigned masses near the GUT scale consistent with extended survival hypothesis [28, 29]. Also no VEV is needed to be assigned to 0R >= 0, since the corresponding role of gauge symmetry breaking has been taken over by < R(1; 3; 10) >= vR 126H . Thus the presence of the second Higgs representation 1260H does not a ect the type-II seesaw and the RH neutrino masse parameters of type-I in the hybrid seesaw formula of eq. (2.2). Even upto the two-loop level it does not a ect the gauge coupling uni cation of the present model. Denoting the corresponding SO(10) invariant Yukawa term as f 016:16:(126)0, we have renorIt is well known that such corrections provide reasonable parameterization of the fermion masses of the rst and second generations. With degeneracy of all superheavy components of 1260H , its threshold corrections to uni cation scale and proton lifetime are vanishingly small [281]. Similarly, if the renormalizable antisymmetric contributions to fermion mass matrices due to Yukawa interaction of a 120H SO(10) are included, its threshold e ects on uni cation 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 rst nonrenormalizable term in L(N1R) may be can be very well replaced by renormalizable Yukawa contribution h(120)16:16:120H . also included for still further improvement. Further, the antisymmetric NRY due to LNR The next question is whether this parametrization signi cantly a ects the predicted In SO(10) there are two maximal subgroups of rank 5: the Pati-Salam group G224 and the Gfl). When SU(4)C G224 is unbroken, the assumed boundary condition is exact. Similarly it is well known that in the presence of Gfl symmetry gauge symmetries are also broken and the boundary condition is approximate to the extent MD = 4Fu. This suggests that u 4Fu=mtop should be a small number in case fermion mass t is also included as a required ingredient in this model. For a very is almost exactly satis ed near the GUT scale bottom-up approach within the SM paradigm [71{73]: 1015:56 GeV by values obtained in the With the dominance of the element (Fd)22 in the (22) elements of down-quark and charged lepton mass matrices, j(Fd)22j j(Gd)22j, gives (Fd)22 30 MeV and a fractional change We have checked that even afte applying these corrections satisfying the rst of GJ relation in eq. (A.1), our solutions and predictions on baryon asymmetry made in this work are not signi cantly a ected. Also they remain largely una ected as long as the corrections to the elements of the Dirac neutrino mass matrix MD 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 di erent up type and down type doublets to remain massless to form the standard Higgs doublet. 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M. K. Parida, Bidyut Prava Nayak, Rajesh Satpathy, Ram Lal Awasthi. Standard coupling unification in SO(10), hybrid seesaw neutrino mass and leptogenesis, dark matter, and proton lifetime predictions, Journal of High Energy Physics, 2017, 75, DOI: 10.1007/JHEP04(2017)075