A minimal non-supersymmetric SO(10) model with Peccei--Quinn symmetry

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 $4.5\times 10^{34}$ 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.


I. INTRODUCTION
Non-supersymmetric SO(10) grand unified theories (GUTs) [1,2] 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 [3], provide a dark matter candidate in the form of axions [4][5][6][7] or weakly interacting massive particles [8][9][10].
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. [11][12][13]). Indeed, we will break SO(10) directly to the SM gauge group. 1 Fermion observables are obtained via the actions of the 10 + 126 Higgs representations, which have been previously shown to be viable [15] and contain the necessary ingredients to implement leptogenesis. We will consider a global Peccei-Quinn (PQ) symmetry [16] to solve the strong CP problem and provide a dark matter candidate [16][17][18][19].
This letter is organized as follows: First, in Sec. II, we describe our model and outline its most salient features. Then, in Sec. III, we analyze the constraints on gauge coupling unification and the predictions for proton lifetime. In Sec. IV, we perform global fits to the parameters of our model and analyze the results and predictions of the model. Next, in Sec. V, we briefly discuss how axion dark matter fits in our model and implications on inflation. Finally, in Sec. VI, we summarize and present our conclusions. * boucenna@kth.se † tohlsson@kth.se ‡ pernow@kth.se 1 For previous attempts at constructing non-supersymmetric SO(10) GUTs with direct breaking to the SM, see e.g. Refs. [8,14].

II. 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 where α is a real number, and the Lagrangian of the Yukawa interactions reads 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 [6,20]. After the breaking of SO(10), the Yukawa couplings of the SM are formed by combinations of Y 10 and Y 126 : These vevs satisfy (v u 10 ) 2 + (v d 10 ) 2 + (v u 126 ) 2 + (v 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 [21][22][23][24][25][26] with the right-handed neutrinos obtaining masses from the SM singlet σ contained in the 126 H , namely

III. 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 . 2 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 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 [27,28] 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. (12) simplifies to In Fig. 1, 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 [29,30], the projected five-year discovery potential of Hyper-Kamiokande, τ p > 5.5×10 34 yr [31], as well as the current LHC lower bound on the mass of the color-octet S 1 , M 1 > 3.1 TeV [32]. 3 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, rule out our model. Figure 2 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. (11). 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.

A. 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. (4)- (8). The latter are conveniently parametrized as [12,13,15,[33][34][35][36] where 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 [37,38] and higher-dimensional operators [39,40] which may impact this relation, we allow for a slightly broader variation as t > 0.1M GUT . In principle, one should also fit the value of the Higgs quartic coupling λ at M GUT . However, it was observed to be consistently very small and was therefore set to zero in the fits.
The input data used in the fits (see Tab. I) are obtained from the following sources: The values of the quark and charged lepton masses are taken from Tab. 3 of Ref. [41] 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 [42]. For the leptonic mixing angles and neutrino mass-squared differences, we use Ref. [43] for both normal and inverted neutrino mass ordering. Finally, Y ∆B is computed from Ω B h 2 reported by the Planck Collaboration [44]. In order to improve the efficiency of the (numerically challenging) fits, we artificially enlarge the error bars of the parameters known to great accuracy using 5 % deviation from the central value as a minimum error. 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. (14)- (18). Next, they are evolved down from M GUT to the electroweak scale M Z , using the renormalization group equations (RGEs) at one-loop order [45][46][47][48]. The righthanded neutrinos are integrated out at each of their respective mass scales, resulting in the effective dimensionfive operator for neutrino masses [49,50] 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 . At M Z , the observables of the SM are calculated and compared to data using a standard χ 2 test function (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. [41,51]. 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 Diver from the ScannerBit package [52]. The best-fit parameter values returned from this program are then further improved using the basin-hopping algorithm [53] from the Scipy library [54]. To maximize the chance that the actual best-fit parameter values are found, we repeat this procedure multiple times.

B. Results and Predictions
For normal neutrino mass ordering, we find that the best-fit parameter values result in χ 2 21.0. The largest contribution to the χ 2 function originates from the leptonic mixing parameter sin 2 θ 23 , 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 is still largely uncertain and values in the lower octant are still allowed by global neutrino oscillation fits [43]. Such a tension has also been observed in previous fits [15]. 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 The best-fit parameter values allow us to make predictions on the unknown neutrino parameters: (i) The sum of the light neutrino masses is Σm ν 6.25 × 10 −2 eV, which is below the upper limit from cosmological observations [30], (ii) the effective double beta-decay neutrino mass is predicted to be very difficult to observe [55,56] with m ee 3.33×10 −3 eV, (iii) the leptonic CP-violating phase δ CP 0.187 is smaller than the value favored by global fits [43] (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. [15,36]). 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 [43]. 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. 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 [57,58] (which would also contribute to neutrino masses via type-II seesaw [25,59,60]), other scalars in the 126 H [3], or the strong thermal leptogenesis solution [61,62].
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. [63]) 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 1 , and to a lesser extent the other color-octet scalar S 2 . As noted earlier, the fits were performed with λ = 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 no effect on the other observables in the fits. In Fig. 3, 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.

V. 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 [64,65]. 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 [66][67][68][69][70][71] constrains the inflation energy to be smaller than about 6.8 × 10 8 GeV [72], 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 .

VI. 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-super-symmetric 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 coloroctet 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 rule out 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 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. [73]).