A minimal non-supersymmetric SO(10) model with Peccei--Quinn symmetry
Sofiane M. Boucenna Tommy Ohlsson Marcus Pernow
 
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        #1⟨ #1⟩boucenna@kth.seDepartment of Physics,
        	School of Engineering Sciences,
        	KTH Royal Institute of Technology,
        	AlbaNova University Center,
        	Roslagstullsbacken 21,
        	SE–106 91 Stockholm,
        	SwedenThe Oskar Klein Centre for Cosmoparticle Physics,
        	AlbaNova University Center,
        	Roslagstullsbacken 21,
        	SE–106 91 Stockholm,
        	Swedentohlsson@kth.seDepartment of Physics,
        	School of Engineering Sciences,
        	KTH Royal Institute of Technology,
        	AlbaNova University Center,
        	Roslagstullsbacken 21,
        	SE–106 91 Stockholm,
        	SwedenThe Oskar Klein Centre for Cosmoparticle Physics,
        	AlbaNova University Center,
        	Roslagstullsbacken 21,
        	SE–106 91 Stockholm,
        	SwedenUniversity of Iceland, 
        	Science Institute, 
        	Dunhaga 3, 
        	IS–107 Reykjavik, 
        	Icelandpernow@kth.seDepartment of Physics,
        	School of Engineering Sciences,
        	KTH Royal Institute of Technology,
        	AlbaNova University Center,
        	Roslagstullsbacken 21,
        	SE–106 91 Stockholm,
        	SwedenThe Oskar Klein Centre for Cosmoparticle Physics,
        	AlbaNova University Center,
        	Roslagstullsbacken 21,
        	SE–106 91 Stockholm,
        	Sweden
        We present a minimal non-supersymmetric SO(10) GUT breaking directly to the Standard Model gauge group. Precise gauge coupling unification is achieved due to the presence of two color-octet scalars, one of which is accessible to LHC searches. Proton lifetime is predicted to be below 4.5× 10^34 years, which is within the projected five-year sensitivity of the proposed Hyper-Kamiokande experiment. We find that the Standard Model observables are reproduced to a reasonable accuracy in a numerical fit, which also predicts the unknown neutrino parameters. Finally, the two scalar representations stabilize the electroweak vacuum and the dark matter is comprised of axions.
        A minimal non-supersymmetric SO(10) model with Peccei–Quinn symmetry
            Marcus Pernow
            Received: date / Accepted: date
        ====================================================================
        
        
        
        § INTRODUCTION
        
        
        Non-supersymmetric SO(10) grand unified theories (GUTs) <cit.> provide an appealing framework for physics beyond the Standard Model (SM). In addition to providing a unified description of the SM gauge group, they can naturally describe neutrino masses and the baryon asymmetry <cit.>, provide a dark matter candidate in the form of axions <cit.> or weakly interacting massive particles <cit.>. 
        
        In this letter, we will present an SO(10) GUT, which addresses the unification of the SM  and the problem of dark matter in a minimal way. Contrary to  common practice, we will not invoke an intermediate breaking step between the electroweak scale and the GUT scale (compare e.g. Refs. <cit.>). Indeed, we will break SO(10) directly to the SM gauge group.[For previous attempts at constructing non-supersymmetric SO(10) GUTs with direct breaking to the SM, see e.g. Refs. <cit.>.] Fermion observables are obtained via the actions of the 10+126 Higgs representations, which have been previously shown to be viable <cit.> and contain the necessary ingredients to implement leptogenesis. We will consider a global Peccei–Quinn (PQ) symmetry <cit.> to solve the strong CP problem and provide a dark matter candidate <cit.>. The PQ symmetry is broken at the GUT scale for the sake of minimality, since the lack of intermediate steps means that there is a priori no reason for the PQ fields to take vacuum expectation values (vevs) at any other scale.
        
        This letter is organized as follows: First, in Sec. <ref>, we describe our model and outline its most salient features. Then, in Sec. <ref>, we analyze the constraints on gauge coupling unification and the predictions for proton lifetime. In Sec. <ref>, we perform global fits to the parameters of our model and analyze the results and predictions of the model. Next, in Sec. <ref>, we briefly discuss how axion dark matter fits in our model and implications on inflation. Finally, in Sec. <ref>, we summarize and present our conclusions. 
         
        
        
        
        § DESCRIPTION OF THE MODEL
        
        
        We consider a GUT  based on the symmetry SO(10)×U(1)_PQ, where U(1)_PQ is the global PQ symmetry. The fermions are in the spinorial  16_F representation and the Yukawa interactions are obtained via the complexified 10_H and 126_H Higgs representations. The breaking of  SO(10)×U(1)_PQ is achieved using the 210_H, 126_H, and 45_H representations and proceeds in one step, directly to the SM symmetry. The electroweak gauge group is broken via the SU(2)_L doublets in the 10_H and 126_H representations. Schematically, it follows that
        
            SO(10)×U(1)_PQ   ⟶SU(3)_C×SU(2)_L ×U(1)_Y 
               ⟶SU(3)_C×U(1)_Q  .
        
        The PQ charges are such that
        
            16_F → e^ iα16_F  , 
               10_H → e^-2 iα10_H ,  126_H → e^-2 iα126_H , 
               210_H →210_H ,  45_H → e^4 iα45_H ,
        
        where α is a real number, and the Lagrangian of the Yukawa interactions reads
        
            ℒ_Y= 16_F (Y_1010_H + Y_126126_H) 16_F ,
        
        where the Yukawa couplings Y_10 and Y_126 are 3 × 3 matrices in flavor space. Although the 10_H is complexified to allow for two different vevs (see below) as phenomenologically required, the PQ symmetry forbids the Yukawa interactions with 10_H^*, thus retaining minimality of the model <cit.>. After the breaking of SO(10), the Yukawa couplings of the SM are formed by combinations of Y_10 and Y_126:
        
            Y_u   =v̅^u_10 Y_10 + v̅^u_126Y_126 , 
            
            Y_d   =v̅^d_10 Y_10 + v̅^d_126Y_126 , 
            
            Y_ν   =v̅^u_10 Y_10 - 3v̅^u_126Y_126 , 
              
            Y_ℓ   =v̅^d_10 Y_10 - 3v̅^d_126Y_126 ,
        
        where Y_u, Y_d, Y_ν, and Y_ℓ are Yukawa couplings for the up-type quarks, down-type quarks, neutrinos, and charged leptons, respectively. The quantities v^u_10, v^d_10, v^u_126, and v^d_126 are the vevs of the SU(2)_L Higgs doublets in the 10_H and 126_H, respectively, and we define x̅≡ x/v_SM with v_SM≃ 174.1 GeV. These vevs satisfy (v̅^u_10)^2 + (v̅^d_10)^2 + (v̅^u_126)^2 + (v̅^d_126)^2 = 1, and we assume that only one physical combination of these SU(2)_L Higgs doublets survives at low energy, corresponding to the SM Higgs doublet. Neutrino masses are generated using the seesaw mechanism <cit.> with the right-handed neutrinos obtaining masses from the SM singlet σ contained in the 126_H, namely
        
            M_R=v^σ Y_126 .
        
        
        § GAUGE COUPLING UNIFICATION AND PROTON DECAY
        
        
        Precise gauge coupling unification can be achieved in our model after direct breaking of SO(10) to the SM due to the presence of two color-octet scalar multiplets from the 210_H representation used in the breaking, namely S_1≡(8,1,1) and S_2≡(8,3,0). We assume the existence of a fine-tuning of the parameters of the scalar potential  leading to the required splitting of the 210_H.[Note that this may require introducing a new scalar representation to the model in order to have extra couplings. However, these new representations will be integrated out at the GUT scale and will have no impact on the model other than facilitating the splitting of the 210_H.]  Similar constructions of extra multiplets surviving at a lower scale to facilitate gauge coupling unification have been previously studied in SU(5)-based models in e.g. Refs. <cit.>. The representations S_1 and S_2 with masses M_1 and M_2, respectively, satisfying M_1 ≤ M_2 ≤ M_GUT, alter the renormalization group (RG) running such that gauge coupling unification is obtained if
        
            M_1   ≲ 5.92× 10^10 GeV , 
            
            M_2   ≈(M_1/GeV)^0.330×1.65× 10^7  GeV , 
            
            M_GUT   ≈(M_1/GeV)^-0.0447×7.34× 10^15 GeV .
        
        Equations (<ref>)–(<ref>) were obtained by solving the renormalization group equations (RGEs) at one-loop order, neglecting threshold effects.
        
        The proton lifetime τ_p is directly related to the GUT scale,  M_GUT. The most constraining decay channel is via the dimension-six operator for p→ e^+π^0 and an approximate relation for τ_p is given by <cit.>
            τ_p≡τ(p→ e^+π^0)≃4/πf_π^2/m_p1/α_H^2 A_R^21/F_qM_GUT^4/α(M_GUT)^2 ,
        
        where f_π≈ 139 MeV is the pion decay constant, m_p ≈ 938.3 MeV is the proton mass, α_H ≈ 0.012 GeV^3 is the hadronic matrix element, A_R ≈ 2.726 is a renormalization factor, and F_q ≈ 7.6 is a quark-mixing factor. Using these input parameter values, Eq. (<ref>) simplifies to
        
            τ_p ≈ (3.22 GeV^-5) ×M_GUT^4/α(M_GUT)^2 .
        
        
        In Fig. <ref>, we display the relationship between M_1 and τ_p  together with the current bound from Super-Kamiokande, τ_p > 1.6× 10^34 yr<cit.>, the projected five-year discovery potential of Hyper-Kamiokande, τ_p > 5.5× 10^34 yr<cit.>, as well as the current LHC lower bound  on the mass of the color-octet S_1, M_1>3.1 TeV<cit.>.[Note that the color-octet representation tested in Ref. <cit.>  is not the same as ours. However, a similar bound is expected to hold in our case.] The current constraint on τ_p from Super-Kamiokande gives the bound M_1 ≲ 1.99 × 10^6 GeV. Note that if Hyper-Kamiokande does not observe proton decay after five years, this would, together with the bound from LHC, strongly disfavor our model.
        
        
        
        Figure <ref> shows the evolution of the inverse gauge couplings with energy for the extreme cases M_1=3.10  × 10^3  GeV and M_1=1.99× 10^6 GeV.  Within these bounds, the variation of M_GUT and the gauge couplings at M_GUT is shown to be small, which is also evident from the small exponent in Eq. (<ref>). Finally, note that there are several other possible choices of sub-representations of the 210_H, which could also provide gauge coupling unification. We motivate our choice by selecting the minimal combination giving the lowest possible τ_p, thus being directly testable.
        
        
        
        § NUMERICAL FIT OF SM OBSERVABLES
        
        
         §.§ Numerical Procedure
        
        
        To make sure that the SM can be correctly reproduced in our model, we fit its 19 observables (5 lepton mass parameters, 6 quark masses, 3 leptonic mixing angles, 4 quark mixing parameters, and the Higgs mass) to the input parameters in Eqs. (<ref>)–(<ref>). The latter are conveniently parametrized as <cit.>
            Y_u   =r(H+sF)  , 
            
            Y_d   =H+F  , 
            
            Y_ν   =r(H-3sF) , 
            
            Y_ℓ   =H-3F  , 
            
            M_R   =t F  ,
        
        where H≡v̅^d_10  Y_10, F≡v̅^d_126 Y_126, r≡ v^u_10/v^d_10, s≡ v^u_126/(r v^d_126), and t ≡ v^σ/v̅^d_126. The notation of x̅≡ x/v_SM is the same as the one introduced in Eqs. (<ref>)–(<ref>). This parametrization of the vevs in ratios r and s gives enough freedom to satisfy the electroweak constraints.
        The 19 free parameters are then: 3 in H (after choosing a basis in which H is real and diagonal), 12 in F (complex symmetric), 1 in r (real), 2 in s (complex), and 1 in t (real). Since we have a one-step breaking, we also require v^σ∼ M_GUT, corresponding to t > M_GUT. However, since we neglect threshold effects <cit.> and higher-dimensional operators <cit.> which may impact this relation, we allow for a slightly broader variation as t> 0.1 M_GUT. In principle, one should also sample the value of the Higgs quartic coupling λ at M_GUT. However, it was observed to be consistently close to zero and small variations had a negligible effect on the predictions of the fit. Therefore, it was set to zero and not sampled. After obtaining the best-fit, we adjust the value of λ to ensure the stability of the vacuum, as explained in Sec. <ref>. The final χ^2 value and predictions are calculated with the adjusted value of λ.
        
        The input data at M_Z used in the fits (see Tab. <ref>) are obtained from the following sources: The values of the quark and charged lepton masses are taken from Tab. 3 of Ref. <cit.> and λ is computed from the values of the Higgs mass and vev therein. The quark-mixing parameters are computed from the ICHEP 2016 update by the CKMFitter Group <cit.>. For the leptonic mixing angles and neutrino mass-squared differences, we use Ref. <cit.> for both normal and inverted neutrino mass ordering. Similarly to  Ref. <cit.>, in order to improve the efficiency of the numerically challenging fits, we artificially enlarge the errors known to better than 5 % to a minimum of 5 % deviation from the central values.
        
        
         
        
        To fit the SO(10) parameters to the SM observables, we apply the following procedure: The 19 free parameters are sampled and transformed to the SM Yukawa and right-handed neutrino mass matrices, using Eqs. (<ref>)–(<ref>). Next, they are evolved down from M_GUT to the electroweak scale M_Z, using the RGEs at one-loop order <cit.>. The right-handed neutrinos are integrated out at each of their respective mass scales, resulting in the effective dimension-five operator for neutrino masses <cit.> which we then also run down to M_Z. The RGEs for the gauge couplings are properly modified at each mass scale M_1 and M_2 of the color-octet scalars S_1 and S_2. In order to gain in predictivity, we assume that the couplings between the scalars S_1 and S_2 and the Higgs play a negligible role in the running of λ. The RG running of the Higgs boson coupling is therefore dominated by the gauge, the top Yukawa, and the neutrino Yukawa couplings. At M_Z, the observables of the SM are calculated and compared to data using a standard χ^2 estimator  (note that due to the non-linearity of the problem, it is non-trivial to interpret the χ^2 function statistically, as noted, e.g., in Refs. <cit.>. Instead, it should be interpreted only as an indication of how easy it is to fit the data and is most useful as a comparison).
        
        In order to minimize the χ^2 function, we link the code performing the procedure described above to the sampler  from the  package <cit.>. The best-fit parameter values returned from this program are then further improved using the basin-hopping algorithm <cit.> from the  library <cit.>. To maximize the chance that the actual best-fit parameter values are found, we repeat this procedure multiple times.
        
        
        
        
         §.§ Results and Predictions
        
        
        For normal neutrino mass ordering, we find that the best-fit parameter values result in χ^2 ≃ 21.0. The predicted values and pulls are displayed in Tab. <ref>. The largest contribution to the χ^2 function originates from the leptonic mixing parameter sin^2θ_23^ℓ, for which we obtain 0.445, which is in the first octant, compared to the 1σ range (0.517,0.577) which lies in the second octant. However, note that the octant of θ_23^ℓ is still largely uncertain and values in the lower octant are still allowed by global neutrino oscillation fits <cit.>. Such a tension has also been observed in previous fits <cit.>. The other two contributions to the χ^2 function that are larger than unity stem from the down-quark mass m_d, which is found to be 2.70 MeV, while the 5 % range is (2.75,3.05) MeV, and the muon mass m_μ, which is found to be 0.111 GeV, while the 5 % range is (0.0978,0.108) GeV. The best-fit parameter values for normal ordering are determined to be
        
            H    = 
            [  -9.07178× 10^-7                0                0;                0  -6.85850× 10^-5                0;                0                0 -6.51835 × 10^-3 ] ,
            
            F   =
            [ 4.46924× 10^-6 + 3.11197× 10^-16i  5.65215× 10^-6  +8.15408× 10^-6i  -1.26725× 10^-4 -1.39399× 10^-5i;  5.65215× 10^-6  +8.15408× 10^-6i   1.52388× 10^-4 +6.77586× 10^-5i -1.79155× 10^-4 - 3.35227× 10^-4i;  -1.26725× 10^-4 -1.39399× 10^-5i -1.79155× 10^-4 - 3.35227× 10^-4i  8.32359× 10^-4 + 2.83292× 10^-4i ] ,
            
            r    = 59.8611,    s = 0.379855 -0.0631434i,    t = 2.02909× 10^15 GeV.
        
         
        
        The best-fit parameter values  allow us to make  predictions on the unknown neutrino parameters: (i) The sum of the light neutrino masses is Σ m_ν≃ 6.25×10^-2 eV, which is below the upper limit from cosmological observations <cit.>, (ii) the effective double beta-decay neutrino mass is predicted to be very difficult to observe <cit.> with m_ee≃ 3.33×10^-3 eV, (iii) the leptonic CP-violating phase δ_CP^ℓ≃ 0.187 is smaller than the value favored by global fits <cit.> (however, note that this phase is not yet directly measured), and finally, (iv) the neutrino mass spectrum is m_ν≃{3.54× 10^-3,9.55× 10^-3, 4.94 × 10^-2} eV for the light neutrinos and M_N ≃{2.09× 10^10, 3.88× 10^11,2.08× 10^12} GeV for the right-handed neutrinos.
        
        It is well known that inverted neutrino mass ordering is much more difficult to fit in SO(10) models than normal ordering (see e.g. Refs. <cit.>). Indeed, the best-fit for inverted ordering has χ^2 ≃ 918. It is interesting to note that global fits to neutrino oscillation data are also disfavoring inverted ordering <cit.>. As noted in other studies, it was also not possible to accommodate thermal leptogenesis (we find that χ^ 2 ≃ 413 in the case of normal ordering) via the decay of the lightest right-handed neutrino, due to constraints imposed by the SO(10) symmetry. Following Ref. <cit.>, we have calculated the leptogenesis prediction by solving the Boltzmann equations for the decay of the lightest right-handed neutrinos including washout effects (but not the flavor effects). However, a more precise treatment of leptogenesis may result in better fits, e.g., taking into account the contributions of the SU(2)_L triplet in the 126_H<cit.> (which would also contribute to neutrino masses via type-II seesaw <cit.>), other scalars in the 126_H <cit.>, or the strong thermal leptogenesis solution <cit.>.
         
         
        Our model also allows for a solution to the instability problem of the electroweak vacuum. This is directly associated to the Higgs quartic coupling λ (see e.g. Ref. <cit.>) and can be solved by adding appropriate new physics not far from the TeV scale. In our case, this is readily provided by the color-octet scalar S_2, and to a lesser extent the other color-octet scalar S_1 since S_2 has the largest effect on the RG running of the gauge couplings. As noted earlier, the fits were performed with λ(M_GUT)=0. However, we   observed that this did not fully prevent λ from becoming negative, albeit in a small energy region. To rectify this, we compensate by shifting λ(M_GUT) up by a small amount (0.005 for the best-fit) such that it remains positive throughout the whole energy range between M_Z and M_GUT. We verify that this small shift has a negligible effect on the observables in the fit. In Fig. <ref>, we present the RG running of λ in our model at one-loop order, taking into account all relevant contributions. Although the contributions of S_1 and S_2 are somewhat counter-balanced by that of the right-handed neutrinos, the total effect remains positive.
        
        
        
        § AXION DARK MATTER AND INFLATION
        
        
        Our model provides invisible axions as a solution to the dark matter problem via the Dine–Fischler–Srednicki–Zhitnitsky mechanism <cit.>. Since the U(1)_PQ symmetry breaks at M_GUT, the axion decay constant is f_A ∼ M_GUT≈ 4.51× 10^15 GeV. For this value, the upper bound on the isocurvature  fluctuations <cit.> constrains the inflation energy to be smaller than about 6.8× 10^8 GeV<cit.>, implying that the PQ symmetry is broken before inflation and the correct abundance of axion dark matter is fixed anthropically by tuning the value of the misalignment angle to be about 3.9 × 10^-3.
        
        
        
        
        § SUMMARY AND CONCLUSIONS
        
        
        Non-super­symmetric GUTs based on SO(10) gauge symmetry provide a promising framework for new physics. We have investigated a minimal non-super­symmetric SO(10)×U(1)_PQ model, which breaks directly to the SM. Gauge coupling unification is achieved by splitting a representation contributing to the breaking of SO(10), namely the 210_H, such that two color-octet scalar representations have intermediate masses. We have determined the proton lifetime to be below 4.5× 10^34 years, in reach of the sensitivity of the proposed Hyper-Kamiokande experiment. In particular, we have observed that the non-observation of proton decay at Hyper-Kamiokande after five years, combined with the lower bound on the mass from LHC searches, would strongly disfavor our model in its minimal realization. Furthermore, the two color-octet scalars help stabilize the electroweak vacuum. We have performed numerical fits to the para­meters of the model and found a reasonable agreement with data in the case of normal neutrino mass ordering. We have predicted the unknown neutrino parameters, and in particular, the leptonic CP-violating phase δ_CP^ℓ≃ 0.187. The PQ symmetry solves the strong CP problem and provides a dark matter candidate in the form of axions produced via the misalignment mechanism in the anthropic window. Although we have not discussed details of the inflationary scenario in this model, we have concluded that its scale should be below 6.8× 10^8 GeV. 
        It would be worthwhile to investigate baryogenesis and inflation in our model in more detail to have a model addressing all the shortcomings of the SM in a minimal SO(10) GUT (as in the model based on SU(5) ×U(1)_PQ presented in Ref. <cit.>).
        
        
        
        S.M.B. thanks the “Roland Gustafssons Stiftelse för teoretisk fysik” for partial financial support. T.O. acknowledges support by the Swedish Research Council (Vetenskapsrådet) through contract No. 2017-03934 and the KTH Royal Institute of Technology for a sabbatical period at the University of Iceland. M.P. thanks “Stiftelsen Olle Engkvist Byggmästare” for financial support through contract No. 2017/85 (179) as well as “Roland Gustafssons Stiftelse för teoretisk fysik”. Numerical computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC Center for High Performance Computing (PDC-HPC) at KTH Royal Institute of Technology in Stockholm, Sweden under project numbers PDC-2018-49 and SNIC 2018/3-559.
        apsrev4-1
