The Minimal GUT with Inflaton and Dark Matter Unification

Giving up the solutions to the fine-tuning problems, we propose the non-supersymmetric flipped $SU(5)\times U(1)_X$ model based on the minimal particle content principle, which can be constructed from the four-dimensional $SO(10)$ models, five-dimensional orbifold $SO(10)$ models, and local F-theory $SO(10)$ models. To achieve gauge coupling unification, we introduce one pair of vector-like fermions, which form complete $SU(5)\times U(1)_X$ representation. Proton lifetime is around $5\times 10^{35}$ years, neutrino masses and mixing can be explained via seesaw mechanism, baryon asymmetry can be generated via leptogenesis, and vacuum stability problem can be solved as well. In particular, we propose that inflaton and dark matter particle can be unified to a real scalar field with $Z_2$ symmetry, which is not an axion and does not have the non-minimal coupling to gravity. Such kind of scenarios can be applied to the generic scalar dark matter models. Also, we find that the vector-like particle corrections to the $B_s^0$ masses can be about 6.6%, while their corrections to the $K^0$ and $B_d^0$ masses are negligible.


Introduction
It is well known that a Standard Model (SM) like Higgs boson (h) with mass m h = 125.09 ± 0.24 GeV was discovered at the LHC [1][2][3], and thus the SM particle content has been confirmed. Moreover, there are many possible directions for new physics beyond the SM: supersymmetry, extra dimensions, strong dynamics or say composite Higgs field, extra gauge symmetries, and Grand Unified Theory (GUT), etc. However, we do not have any new physics signal at the 13 TeV Large Hadron Collider (LHC) yet. Therefore, we may need a e-mail: hushan@itp.ac.cn to reconsider the principle for new physics beyond the SM, and then propose promising models.
First, let us briefly review the convincing evidence for new physics beyond the SM • Dark Matter (DM) is a necessary ingredient of cosmology, considering the cosmic microwave background (CMB) or the rotation curves of spiral galaxies, etc [4,5].
• The non-zero masses and mixing of neutrinos have been found from the atmospheric [9] and solar neutrino experiments [10], as well as the reactor anti-neutrino experiments [11], etc.
• The nearly scale-invariant, adiabatic, statistically isotropic, and Gaussian density fluctuations (see, e.g., [12]) point to cosmic inflation, which can solve the horizon and flatness problems of the Universe as well.
Second, there are two kinds of theoretical problems in the SM: fine-tuning problems and aesthetic problems. The finetuning problems are: (i) The cosmological constant problem: why is the cosmological constant so tiny? (ii) The gauge hierarchy problem: the SM Higgs boson mass square is not stable against quantum corrections and has quadratic divergences, while the electroweak scale is about 16 order smaller than the reduced Planck scale M Pl 2.43 × 10 18 GeV. (iii) The strong CP problem: the θ parameter of Quantum Chromodynamics (QCD) is smaller than 10 −10 from the measurements of the neutron electric dipole moment [13,14]. (iv) The SM fermion mass hierarchy problem: the electron mass is about 5 orders smaller than top quark mass. Also, the aesthetic problems are: (i) there is no explanation for the structure of gauge interactions; (ii) there is no explanation of fermion mass structures; (iii) there is no explanation for charge quantization; (iv) there is no realization of gauge coupling unification. The aesthetic problems can be solved in Grand Unified Theories (GUTs) if we can realize gauge coupling unification. In addition, the SM Higgs quartic coupling becomes negative around 10 9 GeV for central measured values of the SM parameters. Thus, the SM Higgs vacuum is not stable, which is called the stability problem [15][16][17]. Interestingly, the measured Higgs mass roughly corresponds to the minimal values of the Higgs quartic and top Yukawa coupling as well as the maximal values of the SM gauge couplings allowed by vacuum meta-stability [17]. In short, the SM vacuum might be meta-stable while not absolutely stable.
Neglecting the fine-tuning and aesthetic problems, Davoudiasl et al. proposed the New Minimal Standard Model (NMSM) to address the above new physics evidence based on the principle of the minimal particle content and most general renormalizable Lagrangian [18]. Dark energy is explained by a tiny cosmological constant, the dark matter particle is a real scalar with Z 2 symmetry, the inflaton is another real scalar, neutrino masses and mixing can be addressed via the seesaw mechanism [19][20][21][22][23], and baryon asymmetry is generated via leptogenesis [24]. Interestingly, inflation is still consistent with current observations if we consider polynomial inflation [25], and the NMSM is still fine via meta-stability due to the minimality principle. Later, Asaka, Blanchet and Shaposhnikov proposed the νMSM to explain baryon asymmetry, neutrino oscillations, and dark matter via sterile right-handed neutrinos with masses around a few KeV [26,27]. In 2015, Salvio proposed a simple SM completion [28]. Compared to the NMSM, the main differences are: (i) the dark matter candidate is the axion, and the strong CP problem is solved via the invisible axion model proposed by Kim, Shifman, Vainshtein, and Zakharov (KSVZ) [29,30]; (ii) Higgs field as the inflaton. On the other hand, the string landscape can explain the cosmological constant problem and gauge hierarchy problem [31][32][33][34][35][36], but it cannot explain the strong CP problem [37]. However, for the non-supersymmetric KSVZ model, at least it is not clear whether the string landscape can stabilize the axion. This is the reason why Barger et al. proposed the intermediate-scale supersymmetric KSVZ axion model [38]. Also, there exists a serious difficulty for Higgs inflation since the scale of the Higgs field during inflation is larger than that of the perturbative unitarity violation [39,40]. Recently, Ballesteros, Redondo, Ringwald and Tamarit proposed the SM Axion Seesaw Higgs portal inflation (SMASH) model [41,42] to explain the above new physics evidence and the strong CP problem, where the axion is a dark matter candidate, as in Ref. [28].
In this paper, we still neglect the fine-tuning problems, and we study the Minimal GUT which can solve all the aesthetic problems in the SM. We consider the non-supersymmetric flipped SU (5) × U (1) X models [43][44][45]. To achieve gauge coupling unification, we introduce one pair of vector-like fermions, which form a complete SU (5) × U (1) X representation. This kind of models can be constructed in the fourdimensional SO(10) models [46], five-dimensional orbifold SO(10) models [47], and local F-theory SO(10) models [48,49]. The doublet-triplet splitting problem can be solved at tree level, the proton lifetime is about 5 × 10 35 years, neutrino masses and mixing can be explained via the seesaw mechanism, baryon asymmetry can be generated via leptogenesis, and the stability problem can be solved as well. Especially, we for the first time show that inflaton and dark matter particle can be unified to a real scalar field with Z 2 symmetry, unlike all the previous models where such a scalar is either an axion or has non-minimal coupling to gravity (Ricci scalar R) [28,41,42,[50][51][52]. In other words, this is a brand new unification of the inflaton and dark matter particle. After inflation, the interaction between inflaton and Higgs field is reduced to that in the NMSM. Thus, such a kind of scenarios can be applied to the general scalar dark matter models. Furthermore, we find that the corrections to the B 0 s masses from vector-like particles might be about 6.6%, while their corrections to the K 0 and B 0 d masses are negligible.

Model building
We introduce three families of the SM fermions, two Higgs fields H and h, and one pair of vector-like particles (F x , F x ), whose quantum numbers under the SU (5) × U (1) X gauge group and SM particle contents are where i = 1, 2, 3, and and H are the left-handed quark and lepton doublets, right-handed uptype quarks, down-type quarks, charged leptons, neutrinos, and Higgs field, respectively.
To break the SU (5) × U (1) X gauge symmetry down to the SM gauge symmetry, we introduce the following Higgs potential at the GUT scale: where i, j, k, l, and m are SU (5) Lie algebra indices. After minimizing the potential, the field acquires a Vacuum Expectation Value (VEV) at < N c >= v GUT component, and then the SU (5) (2) will generate the GUT-scale mass to D φ but not the SM Higgs doublet H . Thus, we naturally obtain the doublet-triplet splitting at tree level, but we do need fine-tuning to keep the doublet light due to quantum corrections. The Yukawa coupling and vector-like mass terms are where M Pl is the reduced Planck scale. Once field develops a VEV, the N c i , N c x , and N x can obtain masses around 10 14 GeV times their corresponding Yukawa couplings. Assuming M V ≈ 1 TeV and μ i ≈ 0 TeV, we have the vector-like particles (Q x , Q c x ) and (D x , D c x ) at low energy without involving any more fine tuning. As shown previously, this particle content leads to gauge coupling unification [53][54][55][56][57][58]. The main difference is that these vector-like particles in our models form the complete GUT multiplets, which is an interesting point as well.

Gauge coupling unification
We study the gauge coupling unification by taking M V = 1 TeV and μ i = 0 in Eq. (3) and using two-loop Renormalization Group Equations (RGEs). The result is given in Fig. 1. Defining the gauge coupling unification condition as 3 )/2, we obtain α −1 GUT = 35.7 and the GUT scale M GUT = 2.2 × 10 16 GeV. The difference between α −1 GUT and α −1 2 /α −1 3 is about 1.0% or so. With the approximation formulas in Ref. [59], we obtain the proton lifetime for the decay channel p → e + π 0 via heavy gauge boson exchanges to be around 5 × 10 35 years.

Dark energy
Similar to the NMSM, we simply postulate a cosmological constant of the observed size,

Neutrino masses and mixing and baryon asymmetry
The neutrino masses and mixing can be explained via the seesaw mechanism [19][20][21][22][23] since the right-handed neutrinos . Also, the baryon asymmetry can be explained via thermal leptogenesis [24]. The right-handed neutrinos are in the thermal equilibrium in the early Universe, and the lepton asymmetry is generated from the CP violating decays of the lightest right-handed neutrino when it is out of thermal equilibrium. The nonperturbative sphaleron interactions violate B + L but preserve B − L, and then the baryon asymmetry is generated from the lepton asymmetry.

Dark matter and inflation
To unify the dark matter particle and the inflaton, we introduce a real scalar S with a Z 2 symmetry so that it is stable. The potential for S and φ is where m S is around the electroweak scale, and f is a mass parameter in the unit of the reduced Planck scale M Pl . Thus, the inflaton potential is given by V I (S), which is the αattractor model [60]. In terms of the well-known slow-roll parameters where X ≡ dX/dS, the scalar spectral index, the tensor-toscalar ratio, the running of the scalar spectral index, and the power spectrum are, respectively, In Fig. 2, we present the numerical results for r versus n s , where the inner and outer circles are 1σ and 2σ boundaries, respectively, from the Planck 2015 results [61] for TT, TE, and EE + lowP. Therefore, our model might be highly consistent with the experimental data. Because f is the only parameter which determines the inflationary observable n s , r , and α s , we present the slow-roll parameters , η, and ξ versus f in Fig. 3. Inflation ends when any slow-roll parameter violates the slow-roll condition. When f ≤ 1.0 M Pl and f ≥ 3.3 M Pl , η violates the slow-roll condition |η| < 1. When 1.0 M Pl < f ≤ 3.0 M Pl , ζ violates the slow-roll condition |ζ | < 1. When, finally, 3.0 M Pl < f ≤ 3.2 M Pl , violates the slow-roll condition < 1.
To have (n s , r ) within the 1σ and 2σ regions of the Planck 2015 results for TT, TE, and EE+ lowP in Fig. 2, we find that f should lie in the ranges 0 < f ≤ 13.4 M Pl for N = 60 and 0 < f ≤ 7.3 M Pl for N = 50, and in the ranges 0 < f ≤ 18.8 M Pl for N = 60 and 0 < f ≤ 11.2 M Pl for N = 50, respectively. The numerical values of α s are always very small, at the order of 10 −4 . Also, the minimum of the inflaton potential V I (S) is at φ = 0, and interestingly, the inflaton potential will not give any mass to S due to (d 2 V I (S)/d S 2 ) 1/2 | S=0 = 0. Thus, after inflation, S becomes a dark matter particle, and its Lagrangian is reduced to that of the NMSM since V I (S) is negligible at low energy. Therefore, S can be a viable dark matter candidate. The current viable parameter space is that the dark matter mass is close to 62.5 GeV via Higgs resonance for small k (k ∼ 0.06 or smaller), or the dark matter mass is larger than about 450 GeV for relatively large k ∼ 0.2 [63,64]. Let us give a benchmark point with f = 10.0 M Pl . We obtain n s = 0.964592, r = 0.0442495, α s = −0.0006154 and N = 60 for the initial Therefore, V I (S) can indeed be neglected at low energy.

Stability problem
We study the two-loop RGE running of the Higgs quartic coupling. Because it is very sensitive to the top quark mass, we consider the central value m t = 173.34 GeV and 1σ deviations of top quark mass [65]. The numerical results are given in Fig. 4. For comparison, we also present that in the SM by taking the central value of the top quark mass. In addition, we include the dark matter contribution from the k term in Eq. (5) by considering both the small k ∼ 0.06, and the relatively large k ∼ 0.2 for the viable dark matter parameter space [63]. We show numerically that the k term can indeed be neglected. Similarly, the Yukawa coupling λ in Eq.
(2) between the SM Higgs field and GUT Higgs field can also be neglected if such a coupling is not large, for example, 0.5, because of its short RGE running. Therefore, to evade the stability problem, we predict the top quark mass to be smaller than its one sigma upper bound, m t = 174.1 GeV. The key point is that the SM gauge couplings become stronger at high scale due to the extra vector-like particles. In our model, the correct values of the SM quark masses and CKM mixing can be generated through the mixings of vector-like quarks with SM quarks [69]. We assume that all the elements in up-type and down-type quark mass matrices are zero except the top quark mass. We can use bi-unitary transformation to diagonalize the mass matrices in up-type and down-type sectors, and define a general 5×5 non-unitary CKM matrix, following the approach in Ref. [66]. We define λ a qq ≡ V * aq V aq for mesons with down quarks. The correction to the mixing (M 12 ) qq , which is the 12 element of 2 × 2 mass matrix in the neutral meson oscillation system, is where qq stands for quarks participating in the box diagram leading to the neutral meson mixing [66]. S (x t ) and S (x U , x t ) are the IL functions defined as in Ref. [

Comments on reheating
The challenging question for our inflaton and dark matter unification scenario is reheating. There are two kinds of solutions: (i) Inflaton decay only occurs during the initial stage of field oscillations after inflation and then is kinematically forbidden at late time [68]. In this approach, we need to introduce two SM singlet fermions, and then it is not minimal. (ii) Z 2 symmetry is broken at high scale at a meta-stable vacuum and thus the inflaton can decay for reheating. After the metastable vacuum decays into the real vacuum, the Z 2 symmetry is restored, and then the inflaton is a dark matter candidate. Because the first solution has already been studied previously [68], we will not repeat it here. Thus, we shall briefly explain the idea for the second solution [69]. In this solution, we consider the following inflaton potential V I (S): where 0 < S a < v S < S b < S e . To have the continuous inflaton potential, we require at |S| = S a and |S| = S b . Thus, S = v S is a meta-stable vacuum, and the Z 2 symmetry is broken at this vacuum. With λ S > k, we have m S > 2m φ at the meta-stable vacuum, and S might decay into two Higgs particles. Thus, we can indeed realize the reheating. Moreover, we choose the proper parameters S , λ S , and v S so that the meta-stable vacuum can decay into the real vacuum with S = 0 just after reheating. Thus, the Z 2 symmetry will be restored, and S is a dark matter candidate as well. The detailed study will be given elsewhere [69].

Discussions and conclusion
We have proposed the non-supersymmetric minimal GUT with flipped SU (5) × U (1) X gauge symmetry and one pair of vector-like particles, which can incorporate all the convincing new physics beyond the SM based on the principle of the minimal particle content. The gauge coupling unification can be realized, the proton lifetime is about 5 × 10 35 years, and the doublet-triplet splitting problem at tree level as well as the stability problem can be solved. The possible signals from neutral meson mixing have been studied as well. Remarkably, we proposed a brand new scenario for the unification of inflaton and dark matter particle, which might be applied to the generic scalar dark matter models.