Radiative Seesaw in SO(10) with Dark Matter

High energy accelerators may probe into dark matter and the seesaw neutrino mass scales if they are not much heavier than ~O(TeV). In the absence of supersymmetry, we extend a class of SO(10) models to predict well known cold dark matter candidates while achieving precision unification with experimentally testable proton lifetime. The most important prediction is a new radiative seesaw formula of Ma type accessible to accelerator tests while the essential small value of its quartic coupling also emerges naturally. This dominates over the high-scale seesaw contributions making a major impact on neutrino physics and dark matter, opening up high prospects as a theory of fermion masses.

Introduction. Over the recent years, there has been a continued surge of interests in exploring the origin of dark matter of the universe while global efforts for understanding very small masses and large mixings in the neutrino sector have been intensified. The discovery of dark matter (DM) dates back to 1933 when, from velocity measurements in the Coma cluster, Zwicky predicted the inevitable presence of large clumps of massive nonluminous matter [1] which has been reconfirmed by a number of astrophysical and cosmological observations including the WMAP [2]. Based upon the gauge group SU(2) L × U(1) Y × SU(3) C (≡ G 213 ) the standard model (SM) predicts all neutrinos to be massless and no DM candidate. More than 70 years ago, Majorana conjectured neutrinos to be their own anti-particles and a neutrino mass may signify its Majorana character uncovering the violation of well known symmetry called the lepton number [3]. The revelation of tiny neutrino masses, intimately related to the neutrino oscillation phenomena which was at first hinted through  in 1964, has been ultimately confirmed by atmospheric, solar, and reactor neutrino experiments [5]. Nearly four decades ago non-supersymmetric (non-SUSY) grand unified theories (GUTs) were proposed to unify three basic forces of nature with neat and robust prediction for proton decay, p → e + π 0 , for which there are ongoing search experiments [6,7]. Out of all GUTs, SO (10) has grown in popularity as it can predict the right order of tiny neutrino masses through a path breaking new mechanism, called the canonical (≡ type-I) seesaw mechanism, shown to be possible only if neutrinos are Majorana fermions [8]. To mention a few out of a number of other qualities, while the model can explain the origins of parity (P) and CP violations, it has the potential for fitting all fermion masses and also explain baryon asymmetry of the universe via lepton asymmetry and sphaleron effects [10]. While the heavy right-handed (RH) neutrinos in SO(10) mediate the canonical seesaw, the same theory also predicts another seesaw formula (≡ type-II) [9] but now mediated by a massive left-handed (LH) Higgs scalar triplet, ∆ L , and the two mass formulas are where Available data on light neutrino masses constrain these scales to be high, 10 13 − 10 15 GeV and, as such, large hadron collider (LHC) and future high energy accelerators can not test the underlying origin of neutrino masses. Further, the minimal non-SUSY SO(10) fails to fulfil the very purpose of unifying the SM gauge couplings for which it was designed, nor can it explain the dark matter phenomena. 1 However, supersymmetric (SUSY) SO(10) with inbuilt Fermi-Bose symmetry achieves almost precision unification [11] and predicts dark matter with potential for TeV scale seesaw mechanism [12]. But SUSY GUTs have their own shortcomings too [13]. In any case, in the absence of any evidence of SUSY so far, it is worthwhile to explore prospects of non-SUSY SO(10) while preserving precision gauge coupling unification and dark matter as the twin guiding principles.
In this Letter we show how the minimal non-SUSY SO(10) model is extended to predict DM and achieve precision unification with testable proton stability. With matter parity conservation, while type-I and type-II seesaw are automatic consequences of the model, it also generates a low-scale radiative seesaw formula of Ma type [14] accessible to accelerator tests and this formula dominates over the conventional ones causing a major impact on neutrino physics and dark matter phenomenology opening up high prospects as a theory of fermion masses in general. Comparison of prototon lifetime predictions of the present SO(10) model and the S(5) × Z 2 model [28] with the current experimental limit [7] for p → e + π 0 reveals clear distinction between the two models.
Precision unification. Prospective DM candidates are usually accommodated in model extensions by imposing additional discrete symmetries for their stability. But an encouraging aspect of non-SUSY SO(10) is that when its gauged U(1) B−L subgroup breaks spontaneously by the same mechanism as the canonical seeesaw through the high scale VEV of the right-handed Higgs triplet, ∆ R , carrying (B − L) = −2, the surviving matter parity, P M = (−1) 3(B−L) , emerging as gauged discreet symmetry Z 2 [15] can safeguard the stability of DM candiadtes once the latter are introduced into the model Lagrangian. The SO(10) representations 10,45,54,120,126, and 210 possess even matter parity, but the representations 16, 144, ... have odd matter parity, irrespective of whether they represent fermions or scalars. Consistently, the SM fermions (Higgs) carry odd (even) matter parity. Therefore, the general principle for prospective DM particles is that, subject to fulfilment of all other phenomenological constraints, they might be nonstandard fermions of even P M or scalars of odd P M . Using suitable extensions of minimal non-SUSY SO(10) while the hyperchargeless weak triplet fermion with well investigated phenomenology [16] has been predicted [18,19], independently, the inert scalar doublet has also emerged as a CDM candidate [17]. But as we find here, both the inert doublet and the fermion triplet can be made light, in addition to other non-standard fermions of SO(10) leading to a substantial impact on neutrino physics, DM phenomenology, proton decay, and fermion masses.
For precision unification, at first we take out the scalar superpartners of quarks and leptons from the well known spectrum of the minimal supersymmetric standard model (MSSM). Then the remaining non-standard degrees of freedom due to the Higgs (χ) and fermions (F i ) in the non-SUSY model at low scale are 2 In eq.(2) F φ , F χ are analogues of two Higgsino doublets, F σ , F b and F C are the analogues of wino, bino and gluino, and all the fermions except the octet have been treated as potential CDM candidates in SUSY GUTs. It is well known that without scalar superpartners, the fields in eq.(2) maitain unification of gauge coupling almost at the same scale and with the same level of precision as the MSSM but with a decreased value of the unification coupling. We note that when any one of the fields in eq.(2) with nontrivial quantum numbers is treated to be absent or made superheavy, the accuracy of precision unification at that scale is more or less reduced as in [17] while different combinations of fermions and scalars, but with exactly equivalent degrees of freedom as in eq. (2), yield unification with the same precision as in the MSSM as shown in ref. [19] 3 .

Using the SM particle masses and m
TeV, the resulting precision unification of gauge couplings in the non-SUSY theory occurs close to the MSSM GUT scale with M U = 10 15.96 GeV , α −1 G = 35.3. The closeness of the three couplings at the GUT scale is impressive, with α −1 The precision unification is guaranted in the presence of SM gauge symmetry below the GUT scale by assuming the superheavy Higgs components in each SO (10) representation to be degenerate in masses not very different from the GUT scale which reduce the GUT-threshold effects considerably [21,22].
Like the SM fields, if the light fields given in eq. (2) can also be shown to emerge from suitable SO (10) representations, then the non-SUSY GUT would be said to have realized the low-scale spectrum and this precision unification. For this purpose we exploit the nonstandard fermionic representations, 45 F (+) and 10 F (+), in addition to the Higgs representations 10 H (+), 16 H (−), 45 H (+), 126 H (+), 54 H (+), and 210 H (+) where the respective matter parity, (+) or (−), has been shown against each representation. While the three fermions F σ , F b , F C ⊂ 45 F , the fermion-doublet pair, F φ , F χ ⊂ 10 F . In order to make these nonstandard fermions light, we utilize the GUT-scale Yukawa Lagrangian where generation indices have been suppressed. Utilizing the SM singlet VEVs in 54 H , 45 H , and 210 H , we find that the model has enough parameter space to make the four non-singlet fermions of eq.(2) and two singlet fermions F i b (i = 1, 2) of the second and the third generations light by suitable tuning of the parameters in eq.(3) [24] while safeguarding precision unification in the same fashion as shown in ref. [19]. We will show that F σ and F i b (i = 1, 2) effectively replace the roles of RH neutrinos in driving the radiative seesaw. Similarly the non-standard inert doublet [25]. The presence of light fermions at low scales may be natural in non-SUSY GUTs as their masses could be protected by corresponding global symmetries. Perturbative and non-perturbative resolutions of cosmological relic density problem that might otherwise arise due to TeV scale mass of color octet fermion have been discussed earlier [19,23].
To examine the impact of conserved matter parity on neutrino mass formulas we note that while the canonical and the type-II seesaw are automatic consequeces of this model, a number of other types of formulas normally allowed in the SM extensions or different SO(10) models [12] are now disallowed since, in the following Yukawa interaction, the matter-parity violating VEV of 16 H is forbidden. Further, matter-parity conserving type-I and type-II seesaw continue to remain as the only two formulas if, in eq. (2), the second Higgs doublet χ ⊂ 10 H (+) and carries even matter parity.
Radiative seesaw. The complexion of neutrino mass changes drastically once the second Higgs doublet χ in eq.(2) originates from 16 H (−), carries odd matter parity, and aquires the status of an inert doublet [26,27]. In addition to the Yukawa interaction in eq.(4), the following part of SO (10) This leads to the low-scale Higgs potential where M Pl = 1.2 × 10 19 GeV. In the presence of precision unification with the SM gauge symmetry below the GUT scale, allowed natural value of < ∆ R >∼ 10 16 GeV. With λ g ∼ O(1), the embedding of the radiative seesaw mechanism in this SO(10) model then leads to the desired value of the quartic coupling, λ 3 ≃ 10 −5 − 10 −3 , covering the assumed value in ref. [14]. The expression for λ 3 ∝< ∆ R > in eq. (6) also serves as an anchor to type-I and type-II seesaw formulas. Thus, with the replacement of the externally imposed discrete symmetry of ref. [14] by the intrinsic matter parity (P M ) and with the replacement of RH neutrinos of ref. [14] by adjoint fermions of this model, ( , the radiatve seesaw mechanism emerges naturally. Denoting M χ R (M χ I ) as the mass of the real (imaginary) part of χ 0 , it turns out that M 2 , which is easily satisfied because of the model prediction on the smallness of λ 3 , the loop mediated radiative contribution is the same as in the derivation of Ma [14] (M rad where The formula in eq. (7) has been noted to give the resulting seesaw formulas in three limiting cases where the first, second, and the third entries hold for M In general the neutrino mass matrix has a richer structure in this model due to tree-level and radiative seesaw contributions where the three terms on the RHS are given by eq.(1) and eq.(8).
Comparison and dominance. The most natural value of 126 H -Yukawa coupling to fermions is expected to be f ≃ 1 which imparts substantial contribution also to charged fermion masses near the GUT-scale [31,32] derived from their low enegy values by renormalization group evolution [30]. In the present precision unification model, using V R ≃ M ∆ L ≃ M U ≃ 10 16 GeV in eq.(1), we have for the third light neutrino mass, m I 3 ≪ m II 3 ∼ 10 −3 λ eV, which is at least one order smaller than the experimental value. Although the type-I contribution with fine-tuned value, f ∼ 0.01, can yield the right order of neutino masses, its contribution to charged fermion masses is substantially weakened. There are SUSY SO(10) models where type-II seesaw dominance has been shown to fit the charged fermion and neutrino sectors reasonably well [32]. In the present non-SUSY model the radiative seesaw can completely dominate and fit the available neutrino oscillation data. Compared to conventional SO(10) models, the tension on f and other Yukawa couplings caused due to fitting the Dirac-neutrino masses and large neutrino mixings is absent in the present model. As a result, the Yukawa couplings of Higgs representations 126 H , 10 H , and 120 H get almost decoupled from the neutrino sector with their full potential to parametrize the charged fermion masses and mixings in a much more effective manner. This is natural as the radiative seesaw is basically designed to be more dominant as it admits much lighter seesaw scale. While details of these and a number of new SO(10) applications [29] will be reported elsewhere, we confine here to the triplet fermion DM discussed earlier using SM extensions [16,28].
Depending upon their actual masses, the Yukawa interaction in eq.(4) introduces decays, Then only the lightest of them can be a stable dark matter candidate. Thus the model offers the possibility of fermionic weak triplet, singlet, or inert scalar doublet as a CDM candidate. In eq.(1) the Dirac neutrino mass, being of the same order as the up quark mass, the experimental value of large top-quark mass pushes the canonical seesaw scale closer to the GUT scale. As there is no such constraint on the Yukawa couplings in eq.(4) and eq. (8), especially from experimental data, they can be small as has been assumed in [14]. Since the model permits additional lepton flavor violating processes compared to conventional SO(10) models, these couplings are constrained by µ → eγ and other decay rates. For example, in order to have the triplet fermionic DM, the two adjoint singlet fermions and the inert scalar doublet are needed to be heavier than the triplet fermion and we examine this possibility including the new T2K data [33]. Using the relation y αi = U αi y i , the µ → eγ decay rate constraint becomes |(|y 1 | 2 − (2/3)|y 2 | 2 − (2/7)|y 3 | 2 )| ≤ 0.672 (m χ /2.7 TeV) 2 which is different from the tribimaximal mixing constraint [28]. With λ 3 ∼ O(10 −5 ) and y i ∼ O(10 −2 ), the desired neutrino mass eigen values , m i = (0.010, 0.0135, 0.050) eV, in the hierarchial case are obtained for M i = (2.7, 3.0, 3.3) TeV and m χ = 40 TeV . But we note that while all other parameters and predicted masses remain unchanged, the mass of the inert doublet is brought down to m χ ∼ 5 TeV for y i ∼ O(10 −3 ). Inspite of nearly one order reduction in m χ , the muon decay constraint is very well satisfied because of corresponding smaller values of the allowed Yukawa couplings. It is interesting to note that these couplings, y i ∼ 10 −2 (10 −3 ), with adjoint fermions are of the same order as the charged-lepton Yukawa couplings for τ − (µ − ) in the non-SUSY SM. We also obtain solutions consistent with inverted hierarchy for y 1 ≤ y 2 ≤ y 3 in each case. The small values of y i used here do not cause any problem since more rapid rate of annihilation and coannihilation required to produce right value of relic density of the triplet fermionic DM is accomplished by gauge boson interactions [16,28]. We find that the masses of all the particles, mediating the radiative seesaw or responsible for its low scale, are in the range accessible to LHC or planned colliders.
Experimental signatures. It would be worthwhile to discuss some of the possible experimental signatures of this model which may distinguish it from the other non-SUSY GUTbased radiative seesaw model with SU(5) × Z 2 grand unification symmetry [28].
In order to make a possibly clear distinction we discuss gauge boson mediated proton decay p → e + π 0 for which there are ongoing dedicated experimental searches [7] with measured value of the lower limit on the life time, τ expt. p ≥ 1.01 × 10 34 yrs. With a choice of TeV scale particle spectrum different from eq.(2), unification of couplings has been obtained in ref. [28] with M SU (5) U = 2.65×10 15 GeV and the approximation adopted appears to predict proton lifetime substantially lower than the current experimental limit. Noting that the model prediction for the actual inverse decay rate has been underestimated, we re-evaluate it while estimating proton lifetime prediction in the present SO(10) model. Including strong and electrowek 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 [34,35], where (10), V ud = 0.974 = the (1, 1) element of V CKM for quark mixings, and A SL (A SR ) is the shortdistance renormalization factor in the left (right) sectors. In eq.(10) A L = 1.25 = long distance renormalization factor which is the same for both models, but A SL ≃ A SR = 2.414 (2.542) for SU(5) (SO (10)), M U = degenerate mass of 12 (24) superheavy gauge bosons in SU(5) (SO (10) (3.18) for SU(5) (SO (10)), and the expression for the inverse decay rates for both the models is, where the GUT-fine structure constant α G = 1/38.25 and the factor F q = 1 + (1 + |V ud | 2 ) 2 ≃ 4.8 for SU (5), but α G = 1/35.3 and F q = 2(1 + |V ud | 2 ) 2 ≃ 7.6 for SO (10). This formula reduces to the form given in [19,35] and sets the lower limit for non-SUSY SU (5) which mark clear distinction between the two models with much greater proton stability in the present model due to larger unification scale that originates from its TeV-scale spectrum. Thus, we have improved the proton lifetime estimation of [28] in the SU(5) × Z 2 model by at least one order. The significance of this estimation is that the small deficit from the experimental lower limit can be compensated by invoking small threshold effects at the GUT-scale or the TeV scale. But there is a possibility that this model would be constrained if the future measurements increase the existing lower limit by about one order or longer.
In the present model, however, the one-loop prediction of proton lifetime is nearly 40 times longer than the current limit which may be accessed by the search experiments of the next generation 4 . The two-loop and small threshold effects [22] may also bring this prediction closer to the current experimental limit without disturbing precision unification. Since the TeV scale spectrum in the present model is richer and also two of the mediating fermions for radiative seesaw are different from the two RH neutrinos of ref. [28], there would be a variety of other possible ways by which this model can be experimentally distinguished from others. Pending these and other related investigations [36], the present estimation shows that improvement on the existing proton lifetime measurement by at least upto one order [7] would clearly favor the present SO(10) based radiative seesaw model of precision unification.
(b)Accelerator tests. It has been shown that the color-octet fermion present in the TeV-scale spectrum would be pair produced at the LHC with nearly 1000, 15, and 2.5 number of events for its mass m F C = 2.0 TeV, 3.0 TeV, and 3.5 TeV, respectively, with 100fb −1 beam luminousity at energy √ s = 14 TeV [19]. Its relic density problem can be evaded either nonperturbatively, or by invoking second inflation [23], or even perturbatively, by making its lifetime shorter [20] through the introduction of the complete Higgs multiplet 10 H ⊂ SU(5) contained in 16 H ⊂ SO (10) at an appropriate mass scale without disturbing precision unification. In the third case, there is a clear possibility of the boosted production of CDM candidates F σ or F b with displaced vertices at LHC via F CFC pair production [36]. Alternatively, the octet fermion can be replaced by a pair of color octet scalars to achieve the same precision unification and their LHC signatures have been discussed in detail [37]. They can be pair produced copiously at LHC energies and will manifest themselves as resonances in multijet final states. Another specific distinguishing signal at LHC or Tevatron, but more prominent at ILC, would be the pair production and decay of heavy charged fermions contained in F φ and F χ of the non-standard spectrum of the extended SO(10) model [36].
Summary and conclusion. While discrete symmetries are externally imposed on model extensions to maintain stability of incorporated dark matter, a minimal non-SUSY SO(10) model naturally possess the stabilizing matter parity discrete symmetry, but it does not unify gauge couplings and neither does it predict prospective DM candidates. Although it predicts very attractive neutrino mass generation mechanisms, they involve high seesaw scales, 10 13 − 10 15 GeV, inaccessible for experimental tests in foreseeable future. Here the model is successfully extended to realize a low scale spectrum that achieves precision unification with experimentally testable proton lifetime, predicts an inert scalar doublet, and other potential fermionic DM candidates. With matter parity conservation, the type-I and type-II seesaw formulas are automatic consequences of the model, but mediated by the DM and the inert doublet, most interestingly, it also predicts the verifiable low-scale radiative seesaw formula of Ma type along with the natural emergence of its small quartic coupling. Moreover, the new contribution to neutrino mass dominates over the conventional ones making a major impact on neutrino physics and dark matter while opening up high potential as a theory of fermion masses in general. Our estimation in the SU(5) × Z 2 based radiative seesaw model reveals the proton lifetime for p → e + π 0 to be somewhat less than the current experimental lower limit, but in the present SO(10) model the lifetime turns out to be nearly 40 times longer which marks one of its clear distinguishing features.