SMART U(1)$_X$ $-$ Standard Model with Axion, Right handed neutrinos, Two Higgs doublets and U(1)$_X$ gauge symmetry

To address five fundamental shortcomings of the Standard Model (SM) of particle physics and cosmology, we propose a SMART U(1)$_X$ model which is a $U(1)_X \times U(1)_{PQ}$ extension of the SM. The $U(1)_X$ gauge symmetry is a generalization of the well-known $U(1)_{B-L}$ symmetry and $U(1)_{PQ}$ is the global Peccie-Quinn (PQ) symmetry. Three right handed neutrinos are added to cancel $U(1)_X$ related anomalies, and they play a crucial role in understanding the observed neutrino oscillations and explaining the observed baryon asymmetry in the universe via leptogenesis. The PQ symmetry helps resolve the strong CP problem and also provides axion as a compelling dark matter (DM) candidate. The $U(1)_X$ gauge symmetry enables us to implement the inflection-point inflation scenario with $H_{inf} \lesssim 2 \times 10^{7}$ GeV, where $H_{inf}$ is the value of Hubble parameter during inflation. This allows us to overcome a potential axion domain wall problem as well as the axion isocurvature problem. The SMART U(1)$_X$ model can be merged with $SU(5)$ as we briefly show.


Introduction
A variety of cosmological and particle physics observations have demonstrated some shortcomings of the Standard Model (SM) of particle physics. Cosmological and astrophysical observations strongly support, indeed require, the existence of dark matter (DM) which accounts for about 25% of the total (critical) energy density in the universe [1]. However, no viable nonbaryonic DM candidate exists in the SM. Experimental observations of neutrino oscillations and flavor mixings indicate that neutrinos have a tiny but non-zero mass [2]. But neutrinos are massless to all orders in perturbation theory in the SM. Cosmological and astrophysical observations have also confirmed that ordinary baryonic matter dominates over the anti-baryons in the universe [2]. The SM fails to generate this so-called baryon asymmetry in the universe (BAU) [3]. Experimental measurement of the electric dipole moment of neutron require that the effective dimensionless parameterθ QCD of the SM must be tiny,θ QCD ≃ 0.7×10 −11 [4]. This is the so-called strong CP problem [5] of the SM. Finally, according to the current cosmological paradigm, the universe experienced cosmic inflation, a period of accelerated expansion in the early stages of its evolution. Inflation solves two major problems of standard big bang cosmology [6], namely the origin of the observed spatial flatness of the universe and the observed uniformity of the cosmic microwave background radiation with δT /T ≃ 10 −5 [7]. Moreover, the primordial density fluctuations generated during inflation can seed these tiny fluctuations which are essential to reproduce the observed large scale structures of the universe. The SM needs to be extended to accommodate a realistic inflationary scenario.
It is clear that these five fundamental shortcomings of the SM are at the frontiers of high energy physics and cosmology research and crucial to understand the origin and evolution of our universe. Equivalently, it implies that the SM is at best an effective theory description of nature and needs to be supplemented with new physics beyond the SM. In this paper, we propose a gauged U(1) X extension of the SM which addresses all these shortcomings and also offer some testable predictions 4 .
We augment the SM with U(1) X × U(1) P Q symmetry, where U(1) P Q is the well-known global Peccie-Quinn (PQ) symmetry [9], and U(1) X [10] is a generalization of the well-known U(1) B−L gauge symmetry [11]. The generalized U(1) X charge of each field is defined as a linear combination of its hypercharge and B − L charge, and is determined by a single free parameter x H [12]. The B − L charges for the particles are reproduced in the limit x H → 0. To cancel the U(1) X associated anomalies, three generations of SM singlet Majorana right handed neutrinos (RHNs) are added. These SM singlet RHNs explain the origin of observed neutrino masses via type-I seesaw mechanism [13]. Furthermore, the RHNs can generate the observed BAU via leptogenesis [14]. In addition to a SM singlet Higgs field which breaks U(1) X symmetry, the model also contains a SM and U(1) X singlet Higgs field which breaks U(1) P Q symmetry and a pair of SM doublet Higgs fields which are crucial to implement the PQ symmetry. In this regard, our model is a U(1) X extension of the well-known Dine-Fishler-Serednicki-Zhitnitsky (DFSZ) model [15,16].
The PQ symmetry solves the strong CP problem and also provides axion as a compelling DM candidate [17]. The axion models can potentially encounter two major cosmological problems, namely, the axion domain wall problem and the isocurvature problem. For a review, see, for example, Ref. [18]. The topological defects (strings and domain walls) associated with PQ symmetry breaking can potentially dominate the energy density of the universe which is inconsistent with the cosmological observation. In addition, if inflation takes place after PQ symmetry breaking, it can induce isocurvature fluctuations which are severely constrained by observations [1]. Both of these problems can be resolved in an inflation scenario 5 with a Hubble parameter during inflation H inf 2 × 10 7 GeV. Such a low value for the Hubble parameter cannot be realized in a simple inflation scenario based, say, on a Coleman-Weinberg or Higgs potential with a minimal coupling to gravity [20], or a quartic potential with non-minimal coupling to gravity [21]; both scenarios predict H inf ≃ 10 13−14 GeV [22]. This leads us to consider the inflection-point inflation (IPI) scenario [23] (see also Ref. [24]). To realize this, it is crucial that the inflaton field which drives inflation has both gauge and Yukawa interactions. The Higgs field which breaks the U(1) X gauge symmetry is a unique candidate in our model for the inflaton field. We will refer to this model as SMART U(1) X : SM with Axion, Right handed neutrinos, Two Higgs doublets and U(1) X gauge symmetry.
We also consider a merger of the SMART U(1) X model with grand unification. It was previously demonstrated in Ref. [25] that the SM quark and lepton representations in a U(1) X extended SM can be embedded inside SU(5) for a fixed x H = −4/5. Therefore this grand unified theory (GUT) scenario, the U(1) X symmetry explains the origin of charge quantization. By incorporating the PQ symmetry, the model we consider is SU(5) × U(1) X × U(1) P Q . With the addition of suitable new vector-like fermions, the three SM gauge couplings successfully unify at M GU T ≃ 9.8 × 10 15 GeV. These new fermions are also essential for a successful implementation of the IPI scenario, which plays an important role in solving both the axion domain wall and isocurvature problems. In addition, we show that the SM Higgs potential can also be stabilized in the presence of the new vector-like fermions.
The recently proposed Trans-Planckian Censorship Conjecture (TCC) [26], when applied to slow-roll inflation, significantly lowers the bound on the value of Hubble parameter during inflation, H inf 1 GeV [27]. An IPI inflation scenario consistent with TCC was recently examined by the authors of this paper [28] in a model whose particle content matches that of the SMART U(1) X model with x H = 0. Therefore, the model in Ref. [28] is the B − L version of the SMART U(1) X . This paper is organized as follows: In Sec. 2 we describe the SMART U(1) X model. We discuss the axion DM scenario in Sec. 3 and show that both the axion domain wall and isocurvature problems can be solved if H inf 2 × 10 7 GeV. An IPI scenario with this constraint is discussed in Sec. 4, where we consider reheating after inflation and identify the model parameters required for successful leptogenesis. In Sec. 5 we consider the merger of SU(5) with the SMART U(1) X . Our results are summarized in Sec. 6.

SMART U(1) X
The particle content of our model is listed in Table 1, where all fermion fields are left-handed. The U(1) X charge of each particle is defined as are their SM hypercharge and B − L (baryon minus lepton) number, respectively. We fix x Φ = 1 without loss of generality so that the U(1) X charges of all the particles are uniquely determined by a single free parameter x H . The B − L charges are reproduced in the limit x H → 0. In addition to three generations of SM quarks and leptons (i = 1, 2, 3), there are three SM singlet Majorana neutrinos, (N c ) i , which cancel all the U(1) X related gauge and mixed gauge-gravitational anomalies. The scalar sector includes four complex scalar fields, namely the two SM doublet Higgs fields, H u,d , and two SM singlet Higgs fields, Φ and S. The gauge U(1) X and global U(1) P Q invariant Higgs potential is given by where the couplings parameters are all chosen to be positive, the dot represents contraction of SU(2) indices by epsilon tensor, and the vacuum expectation values (VEVs) of Φ and S are given by Φ = v X / √ 2 and S = v P Q / √ 2, respectively. The mixed quartic interactions includes terms such as ( For simplicity, we assume these mixed quartic couplings to be adequately small, and they do not alter our results. We also assume that both U(1) X and U(1) P Q symmetry are broken at a scale much higher than the electroweak scale as well as Λ s .
We parameterize S and Φ in terms of real fields as follows: Note that the Higgs field Φ is charged under both U(1) X and U(1) P Q , whereas S is charged only under U(1) P Q . After S and Φ fields acquire their VEVs, the would-be Nambu-Goldstone (NG) boson χ(x) is absorbed by the U(1) X gauge boson (Z ′ ) and a(x) is identified with the axion. After the U(1) X × U(1) P Q symmetry breaking, the masses of φ, s and Z ′ gauge boson are respectively given by where g is the U(1) X gauge coupling.
Our assumption of negligible mixed quartic couplings allows us to separately analyze the SM doublet Higgs potential at low energies from the singlet Higgs sector. In Eq. (2.1), the PQ symmetry breaking Higgs field S generates the mixing mass term between the two doublet Higgs fields, Because of their charge assignments, H u (H d ) only couple with up-type (down-type) SM fermions (see Eqs. (2.4) and (2.5) below). The doublet Higgs potential at low energies in this case is the same as that of the type-II two Higgs doublet extension of the SM. Since the two Higgs doublet model is well studied in the literature [29], we will skip the detailed phenomenology of the Higgs potential at low energies.
In addition to the Yukawa interactions for the SM quarks, the Lagrangian includes the following new Yukawa interactions involving the Majorana neutrinos: where Y D (Y i ) is the Dirac (Majorana) Yukawa coupling, and we have chosen a flavor-diagonal basis for Y i . After the U(1) X and electroweak symmetry breakings, the Dirac and the Majorana masses for the neutrinos are generated, where H 0 u = v u / √ 2 is the VEV of the charge neutral component of H u .

Axion Dark Matter
As discussed before, the axion DM scenario possibly suffers from two major cosmological problems, namely, the axion domain wall problem and the axion DM isocurvature problem. This domain wall problem can be solved if inflation takes places after the PQ symmetry breaking, or equivalently, H inf < F a = v P Q /N DW , where F a is the axion decay constant and N DW is the domain wall number. In our case, N DW = 12. The measurement of supernova SN 1987A pulse duration provides a model-independent constraint on the axion decay constant F a 4 × 10 8 GeV [30]. We will shortly show that the resolution of the isocurvature problem leads to an even stronger constraint on H inf than the one required to solve the axion domain wall problem,

Relic Abundance of Axion Dark Matter
The PQ symmetry breaking produces cosmic strings, which can efficiently decay into axions [18]. Because inflation takes place after the PQ symmetry breaking in our setup, the axions produced from cosmic string decay contribute negligibly to the relic abundance. Although QCD interactions can also produce axions in the thermal plasma, their relic abundance from such processes was shown to be negligible compared to the observed abundance of DM in Ref. [31]. At the end of QCD phase transition, the coherently oscillating axion field behaves like cold DM and contribute dominantly to the relic abundance of the axion [18], . (3.1) Here, the misalignment angle θ m is the initial displacement of axion field from the potential minima at the onset of oscillations, and δθ m = H inf /(2πF a ) is the fluctuation of θ m generated by inflation. A natural choice for the misalignment angle is θ a ≃ 1, and we have used H inf < F a to obtain the final expression in Eq. (3.1). Requiring that the axion fully account for the DM in the universe, Ω a h 2 = 0.120 ± 0.0012 [1], the axion decay constant F a is determined as a function of θ m ,

Axion Dark Matter Isocurvature Fluctuations
If inflation takes place after the PQ symmetry breaking, it induces isocurvature fluctuation in the axion DM power spectrum [1], which is constrained by Planck measurements [1], namely where P adi (k * ) ≃ 2.2 × 10 −9 [1] is the adiabatic power spectrum and k * = 0.
Together with Eqs. (3.2), we find that δθ 2 m /θ 2 m ≃ 10 −11 , or equivalently, H inf /F a 3.0×10 −5 θ m . This shows that the axion DM isocurvature constraint imposes a much stronger restriction on H inf than the one required to solve the axion domain wall problem, H inf < F a .
Let us set θ m to be O(1) and F a = 7.11 × 10 11 GeV such that the axion saturates the observed DM abundance. We therefore obtain an upper bound on the Hubble parameter during inflation, H inf < 2.08 × 10 7 GeV. In the next section, we discuss an IPI scenario with the Hubble parameter value of this magnitude which also solves both the axion domain wall and axion DM isocurvature problems.

Inflection-Point Inflation
Let us begin by highlighting the key results of the inflection-point inflation (IPI) scenario driven by a real scalar field φ. See Ref. [23] for details. An inflaton potential which exhibits an approximate inflection-point around φ = M can be expressed as where V 0 = V (M), V n ≡ d n V /dφ n | φ=M , and we identify φ = M to be the horizon exit scale that corresponds to the pivot scale k * = 0.05 Mpc −1 used in Planck measurements [32].
With the potential in Eq. (4.1), the inflationary slow-roll parameters are given by where M P = 2.43 × 10 18 GeV is the reduced Planck mass. In terms of the slow-roll parameters, the inflationary predictions for the scalar spectral index (n s ), the tensor-to-scalar ratio (r), and the running of the spectral index (α) are respectively given by The amplitude of the curvature perturbation ∆ 2 R is expressed as The central values from the Planck 2018 results [32], ∆ 2 R = 2.195 × 10 −9 and n s = 0.9649, can be used to express V 1,2 in terms of V 0 and M: The following expression for the number of e-folds has been derived in Ref. [23]: Combining Eqs. (4.5) and (4.6) we obtain In the following analysis, we set N = 60 to solve the horizon problem of big bang cosmology. Using Eqs. (4.3), (4.5), and (4.7), the prediction for the running of spectral index is obtained as α = −2.74×10 −3 . This is consistent with the Planck 2018 measurements, α = −0.0045±0.0067, and can be tested in the future [33]. Next we identify the inflaton field used in the IPI analysis with the real component of the U(1) X Higgs field, φ = √ 2Re[Φ]. The inflaton potential in Eq. (4.1) is identified with the renormalization group improved U(1) X Higgs potential, To obtain the second expression, we have assumed that during inflation φ ≫ v P Q and λ φ (φ) is determined by solving the the following renormalization group equations (RGEs): Here, the beta-function of λ φ and is given by (4.10) The renormalization group improved Higgs potential together with the RGE for λ φ can be used to express V 1,2,3 in Eq. (4.1) as where the prime denotes derivatives with respect to φ. To realize an approximate inflectionpoint at M, we impose V 1 /M 3 ≃ 0 and V 2 /M 2 ≃ 0, which yields the following relations: (4.14) For the benchmark values θ m ≈ 1 and F a = 7.11 × 10 11 GeV, we obtain an upper bound M < 0.11 M P . Therefore, the tensor-to-scalar ratio is predicted to be be tiny, r < 6.55 × 10 −15 , well below the reach of any foreseeable experiments.

Low Energy Predictions
In the following analysis, we will estimate the low energy values for the gauge coupling  be [23] λ where we have used Eq. (4.12) to obtain the final expression. The masses of the particles in Eq. (2.3), evaluated at φ = v X , are then given by In Fig. 1, we plot the RG running of the inflaton quartic coupling (left) and the corresponding RG improved effective inflaton potential (right) which exhibits an approximate inflection-

Thermal Leptogenesis and Reheating
Leptogenesis [14] is a relatively simple mechanism to generate the observed baryon asymmetry in a model with the type-I seesaw mechanism. If Majorana neutrinos are non-degenerate in mass, a successful thermal leptogenesis scenario requires the lightest Majorana neutrino mass (m N 1 ) to be heavier than 10 9−10 GeV and the reheat temperature T R > m N 1 [35]. In our setup, the U(1) X gauge interactions [36] and Yukawa interactions [37] of the Majorana neutrinos can keep these neutrinos in thermal equilibrium with the SM particles. These processes will suppress the generation of lepton asymmetry until they freeze out. By requiring these processes to decouple before the temperature of the thermal plasma drops to T ∼ m N 1 , we now derive the conditions necessary to prevent such a suppression.
We first consider the Z ′ mediated process, where f SM are the SM fermions. Since m Z ′ > m N 1 , the Z ′ mediated process is effectively a four-Fermi interaction, and its thermal-averaged cross section for T m N 1 can be approximated as [23] σv ≃ F (x H ) 768π where F (x H ) = 13 + 16x H + 10x 2 H . The process decouples at T ∼ m N 1 , if Γ/H| T =m N 1 < 1, where Γ(T ) = n eq (T ) σv is the annihilation/creation rate of (N c ) 1 with an equilibrium number density n eq (T ) ≃ 2T 3 /π 2 , and H(T ) ≃ πT 2 /M P is the corresponding value of the Hubble parameter. This leads to a lower bound v X > 2.17 × 10 10 GeV (F (x H )) 1/4 m N 1 10 9 GeV (4.21) Because we require M < 0.11M P to solve the axion domain wall and isocurvature problems, it follows from Eq. (4.19) that m N 1 > m φ . In this case, another process, N R 1,2 N 1,2 R ↔ φφ, can suppress the generation of lepton asymmetry as investigated in Ref. [37]. The thermal-averaged cross section of this process is roughly given by [38] σv ≃ 1 4π Requiring Γ/H < 1 at T = m N 1 to avoid the suppression of the generation of lepton asymmetry, we obtain v X > 7.92 × 10 10 GeV m N 1 10 9 GeV For the remainder of this section, let us fix x H = 0, M = 0.05M P and m N 1 = 10 9 GeV to be our benchmark values, and we find v X ≃ 1.93 × 10 12 GeV, m φ ≃ 9.30 × 10 4 GeV and m N 2,3 ≃ 9.32 × 10 9 GeV from Eq. (4.19). This value of v X is consistent with Eq. (4.23), which is stronger than condition in Eq. (4.21) for x H = 0. The observed baryon asymmetry therefore can be produced by thermal leptogenesis if the reheat temperature T R > m N 1 .
The SM particles produced from the decay of the inflation reheat the universe, thereby connecting inflation to the standard hot big bang cosmology. To estimate the reheat temperature, we assume an instantaneous inflaton decay, which yields the standard formula where g * ≃ 100 and Γ φ is the total decay width of the inflaton. To estimate the inflaton decay width, we consider the following interactions between Φ and the SM doublet Higgs fields, The decay width of φ can be approximated as where we have neglected the mass of Higgs bosons in the final state. The reheat temperature is given by (4.27) Thus, an adequate reheat temperature for successful thermal leptogenesis, T R > m N 1 can be achieved with λ ′ 2.31 × 10 −9 . Such a small value of λ ′ has a negligible effect on the effective potential of the inflaton and can be safely ignored as far as the inflationary phase is concerned.
The presence of the new quarks can also stabilize the SM Higgs potential 7 at high energies [40]. We shall work in the so-called alignment limit for the two Higgs doublet model with tan β = v u /v d ≫ 1, such that H u ⊃ h sin β and the quartic self-coupling λ u can be effectively identified with the SM Higgs (h) and its quartic self-coupling (λ h ), respectively [29]. We fixed the mixed quartic coupling between H u and H d to be negligible in our analysis. Following Ref. [40], we numerically solve the RGEs. We set two vector-like quark pairs and the new Higgs doublet mass to be 2.5 TeV.
In the left panel of Fig. 2, we plot the RG running of the SM gauge couplings as a function of the energy scale µ. The diagonal solid lines labeled α i = g 2 i /4π (i = 1, 2, 3) depict the SM gauge couplings for U(1) Y , SU(2) L and SU(3) c , respectively. The SM gauge couplings successfully unify at around M GUT ≃ 9.8 × 10 15 GeV, with α GU T ≃ 1/35. With these values, the gauge boson mediated proton lifetime 8 is estimated to be [42]  where m p = 0.983 GeV is the proton mass. This is consistent with the experimental lower bound obtained by the Super-Kamiokande, τ p (p → π 0 e + ) 10 34 yr [43]. The predicted lifetime is close to the sensitivity limit of future Hyper-Kamiokande, τ p 1.3 × 10 35 yr [44]. In the right panel, the solid (dotted) curve depicts the RG running of the SM Higgs quartic coupling with (without) the new vector-like quarks and a the new Higgs doublet contributions, and the horizontal dashed line depicts λ h = 0. In the presence of the new vector-like fields, λ h (µ) > 0, and thus the SM Higgs potential is stabilized.
Next we study the IPI scenario. Note that in the SMART U(1) X case and we have λ φ (M) < 0 for −0.87 x H −0.69 from Eq. (4.16), which implies that the inflaton potential is unstable.
With the inclusion of the new vector-like quarks (D + D c and Q + Q c ), which is essential for the unification of the SM gauge couplings, we now show that the inflaton potential remains stable for any value of x H . As in the SMART U(1) X case, we fix Y 1 (M)/g(M) = √ 2/5 and Y 2,3 (M) = Y (M), such that the conditions β λ φ (M) = 0 leads to Y (M) ≃ 2.63 g(M), which is the same relation as before. However, in the presence of the new vector-like quarks and the extra Higgs doublet, which are also charged under the U(1) X symmetry, the U(1) X gauge coupling RG equation in Eq. (4.9) is modified to This leads to a new relation between the quartic and gauge couplings at the inflation scale φ = M, which is positive for any value of x H . Therefore the inflaton potential is stable and we can realize a successful IPI scenario for x H = −4/5.
To discuss reheating and leptogenesis, we next consider the low energy predictions for the masses and couplings. Since the quartic coupling at low energies only depends on λ φ (M), its low energy value is still determined by Eq. , we obtain v X ≃ 1.68 × 10 12 GeV, and the particle masses are given by m φ ≃ 8.23 × 10 4 GeV, and m N 2,3 ≃ 9.30 × 10 9 GeV. The conditions for preventing a suppression of lepton asymmetry from the gauge (Yukwa) interactions of the lightest Majorana neutrino leads to a lower bound of v X > 2.46 (7.92) × 10 10 GeV. This is consistent with the v X value obtained for the benchmark. The observed baryon asymmetry can be generated by thermal leptogenesis for T R ≃ 10 10 > m N 1 GeV and this can be realized with λ ′ 2.31 × 10 −6 as shown in Eq. (4.27).

Summary
We have proposed a SMART U(1) X model which is an anomaly free U(1) X extension of the SM supplemented by a U(1) P Q symmetry. The U(1) X charge of each particle is defined as a linear combination of its hypercharge and B − L charge and is determined by a single free parameter x H . Three right handed neutrinos (RNHs) are added to cancel all U(1) X related anomalies. These neutrinos, as is well-known, also explain the origin of the observed neutrino oscillations via the type-I seesaw mechanism and generate the observed baryon asymmetry via leptogenesis. The U(1) P Q symmetry solves the strong CP problem and also provides the axion as a compelling dark matter (DM) candidate. By identifying a U(1) X breaking Higgs field as the inflaton, we have implemented a low-scale inflection-point inflation (IPI) scenario in our model to realize a Hubble parameter during inflation H inf 2 × 10 7 GeV. The U(1) X gauge symmetry is crucial for the implementation of the IPI scenario. This low-scale inflation is a key for resolving the axion domain wall and the axion DM isocurvature problems. After the end of inflation, we have shown that the inflaton decay adequately reheats the universe to allow for a successful implementation of the leptogenesis scenario.
We have also examined another possibility to merge the SMART U(1) X with SU(5) grand unification. For x H = −4/5, all the SM quarks and leptons are embedded within the SU(5) × U(1) X × U(1) P Q representations. We extended the particle content of the SMART U(1) X with a pair of new vector-like fermions and a new Higgs doublet with O(1) TeV mass to realize successful unification of the three SM gauge couplings at M GU T = 9.8 × 10 15 GeV. This leads to proton lifetime estimate of 2.6 × 10 35 yr, which is close to the sensitivity limit of the future Hyper-Kamiokande experiment. These new fermions also help stabilize the SM Higgs potential as well as the inflaton potential. The latter is crucial for successful implementation of the IPI scenario which is a key for solving the two cosmological problems encountered in the axion DM scenario. The leptogenesis scenario is also implementable in the SU(5) model.