$SU(5) \times U(1)_X$ Axion Model with Observable Proton Decay

We propose a $SU(5) \times U(1)_X \times U(1)_{PQ}$ model, where $U(1)_X$ is the generalization of the $B-L$ (baryon minus lepton number) gauge symmetry and $U(1)_{PQ}$ is the global Peccei-Quinn (PQ) symmetry. There are four fermions families in $\bf{{\overline 5}} + \bf{10}$ representations of $SU(5)$, a mirror family in $\bf{5}+\bf{{\overline {10}}}$ representations, and three $SU(5)$ singlet Majorana fermions. The $U(1)_X$ related anomalies all cancel in the presence of the Majorana neutrinos. The $SU(5)$ symmetry is broken at $M_{GUT} \simeq (6-9)\times 10^{15}$ GeV and the proton lifetime $\tau_p$ is estimated to be well within the expected sensitivity of the future Hyper-Kamiokande experiment, $\tau_p \lesssim 1.3 \times 10^{35}$ years. The $SU(5)$ breaking also triggers the breaking of the PQ symmetry, resulting in axion dark matter (DM), with the axion decay constant $f_a$ of order $M_{GUT}$ or somewhat larger. The CASPEr experiment can search for such an axion DM candidate. The Hubble parameter during inflation must be low, $H_{inf} \lesssim 10^9 $ GeV, in order to successfully resolve the axion domain wall, axion DM isocurvature and $SU(5)$ monopole problems. With the identification of the $U(1)_X$ breaking Higgs field with the inflaton field, we implement inflection-point inflation, which is capable of realizing the desired value for $H_{inf}$. The vectorlike fermions in the model are essential for achieving successful unification of the SM gauge couplings as well as the phenomenological viability of both axion DM and inflation scenario.


Introduction
A variety of well-established experimental results in particle physics and cosmology have exposed some of the inadequacies of the Standard Model (SM) of particle physics [2]. These include the confirmation of the existence of non-baryonic dark matter (DM), observation of tiny but non-zero masses for SM neutrinos, the observed asymmetry between the matter and antimatter abundance in the universe, the necessity of cosmic inflation in the very early stages of the universe's evolution, and the strong CP puzzle. The SM must be supplemented with new physics to account for these observations.
Among the various proposed extensions of the SM, the models based on grand unified theory (GUT) are attractive because they predict unification of the SM gauge interactions and also explain the quantization of the electric charges of the SM fermions [1]. An interesting grand unification scenario utilizes the anomaly free gauged U(1) X extension of the SM, where the U(1) X symmetry [3] is the generalization of the B − L (baryon minus lepton number) symmetry [4]. The generalized U(1) X charge of each particle is defined as a linear combination of its hypercharge (Q Y ) and B − L charge (Q B−L ), Q X = x H Q Y + Q B−L , where x H is a free parameter [5]. For x H = −4/5 [6], the SM quarks and leptons are unified in the 5 and 10 representations of SU (5). The three SM singlet Majorana neutrinos needed to cancel all the U(1) X related anomalies can explain the origin of observed neutrino masses and flavor mixings via the type-I seesaw mechanism [7]. The unification of the three SM gauge couplings can be achieved by adding components of vector-like quark pairs from the 5 ⊕5 and 10 ⊕10 representations of SU(5) [6].
In this article, we propose a model based on the symmetry SU(5) × U(1) X × U(1) P Q , where U(1) P Q is the global Peccei-Quinn (PQ) symmetry [8], which addresses all the inadequacies of the SM discussed above. The PQ symmetry solves the strong CP problem [9] and the associated axion from the PQ symmetry breaking is the DM candidate [10]. The SU(5) symmetry breaking also triggers the breaking of U(1) P Q , and so the DM physics is intimately connected to the physics of grand unification. In particular, the axion decay constant f a is comparable to the SU(5) GUT symmetry breaking scale, M GU T ∼ 10 15 − 10 16 GeV. To resolve the SU(5) GUT monopole problem [11] one may consider the low scale inflation scenario with H inf ≪ M GU T , where H inf is the value of the Hubble parameter during the inflation. See also Ref. [12]. However, in this case, the axion DM scenario suffers from the cosmological fatal axion domain wall problem and axion DM isocurvature problem (for a review see, for example, Ref. [13]). With the axion decay constant f a ≃ M GU T , the resolution of the axion domain wall and axion DM isocurvature problems require a low value for H inf 10 9 GeV [14]. Well-known inflationary scenarios with the Coleman-Weinberg or Higgs potential with minimal coupling to gravity [15], and a quartic potential with non-minimal coupling to gravity [16] predict a relatively large H inf ≃ 10 13−14 GeV [17]. With the identification of the U(1) X Higgs field with the inflaton field, we implement the so-called inflection-point inflation (IPI) scenario [18], which can realize H inf < 10 9 GeV. The new fermions in the model are key to achieving successful unification of the SM gauge couplings as well as the phenomenological viability of both the axion DM and the IPI inflation scenario. The Majorana fermions generate the observed baryon asymmetry via leptogenesis [19]. We identify sets of model parameters such that the new physics scenarios discussed above including proton decay are phenomenologically viable. Table 1: Particle content of SU(5) × U(1) X × U(1) P Q model. It includes four fermion families, ψ i 5(10) (i = 1, 2, 3, 4), one mirror family, ψ 5(10) , three Majorana fermions, (N c ) j (j = 1, 2, 3), and four complex scalars (Σ, S, H and Φ). All the fermions are in their left-handed spinor representation and "c" denotes charge conjugation.
In the following, we consider the spontaneous breaking of SU(5) and U(1) P Q symmetries, followed by the breaking of U(1) X and the electroweak symmetry. The Higgs potential for Σ and S fields is given by Here, the couplings parameters are all real and positive and we have neglected mixed terms between Σ and S fields with H/Φ, assuming that the associated couplings to be adequately small because H/Φ fields not essential for the breaking of SU(5) and P Q symmetries. The SU (5) and U(1) P Q symmetry breaking is accomplished by the Σ and S fields vacuum expectation values (VEVs), namely, Σ = v Σ /(2 √ 15) diag(−2, −2, −2, 3, 3) and S = v S / √ 2. Solving the stationary conditions for the potential in Eq. (1) we obtain Applying these results to evaluate the mass spectrum for the scalar and gauge fields we obtain 12 superheavy massive gauge bosons, 37 massive scalars and one massless scalar. The details about the scalar mass spectrum is presented in Appendix 1. For concreteness, let v Σ = √ 30κ 1 v S , with the coupling parameters κ 1,2 = λ 1 = λ S = 0.3, κ 3 = −0.011, λ 2 = −0.049, and λ 3 = λ 4 = 0.375 such the 35 scalar masses are given by where the numbers in the exponents are the degeneracy of each mass eigenvalue. The massless scalar field, which we identify to be the axion, is given by where χ S (χ Σ ) is the imaginary component of S (Σ) that acquires the VEV. Following SU(5)×U(1) P Q symmetry breaking, the residual symmetry is SU(3) c ×SU(2) L × U(1) Y ×U(1) X . In the following, we neglect the mixing between Φ and H Higgs fields which will be justified later. It allows us to independently examine the Φ and H sector Higgs potential. The VEV of Φ far exceeds the electroweak VEV of H, and so the U(1) X symmetry is primarily broken by Φ. Setting where v X denotes its VEV, Φ potential is given by The breaking of the U(1) X symmetry by the VEV of Φ also generates masses for the U(1) X gauge boson Z ′ and the real component φ, respectively, where g is the U(1) X gauge coupling. Finally, the electroweak symmetry gets broken after the charge neutral component of the SU(2) L doublet Higgs field in H field acquires its VEV, v H = 246 GeV.
Let us now consider fermion masses. We introduce the Yukawa interactions only for ψ 4 5 (10) and ψ 5 (10) , Because there is one copy of ψ 5 (10) , only one linear combination of the four ψ i 5(10) obtain a non-zero mass from the S and Σ VEVs. Here, without loss of generality we work in a basis where ψ 4 5(10) and ψ 5(10) are the massive states. The decomposition of the pairs under the SM gauge group and their masses will be discussed in Sec. 3.
The Yukawa interactions of the fermions with H are given by In the following analysis we assume Y i4 H1,H2 ≪ 1 (i = 1, 2, 3), so that ψ i 5,10 (i = 1, 2, 3) are identified with the SM fermions and the mixing between the SM fermions and ψ 4 5(10) is non-zero but negligibly small. This is crucial to ensure the decay of the exotic heavy fermions. The mass spectrum of ψ 4 5,10 will be discussed in Sec. 3. The Yukawa interactions involving the Majorana neutrinos are expressed as where we have used the mass basis for ψ i 5 and a flavor-diagonal basis for the Majorana neutrinos. After breaking of the U(1) X and the electroweak symmetry, the first and second terms in Eq. (10) generate the Dirac and Majorana type masses for the neutrinos

Gauge Coupling Unification
In this section we evaluate the renormalization group (RG) running of the SM gauge couplings including the contribution from the new fermions and scalars which have masses much smaller than the SU (5) The decompositions of ψ 4 5,10 under the SM gauge group are given by The SM decomposition of their partners, ψ 5,10 , are the conjugate of the representations shown in Eq. (13). Using the benchmark v Σ = √ 30κ 1 v S , we evaluate masses of the vector-like pairs within the multiplets. For O(1) Yukawa coupling values, we find that the pairs in ψ 4 5(10) − ψ 5(10) may have a large mass splitting between them. For example, if we fix the mass of D c (Q) to be O(1) TeV, the masses of the remaining components in the multiplet, without loss of generality, is approximately given by Y 4 10(5) × v Σ . The CMS collaboration for the LHC has set the lower limit of around 1500 GeV [26] at 95% confidence level for vector-like quarks with hypercharge (−2/3) and the vector-like leptons doublets with hypercharge (−1/2) in the mass range of 120 − 790 GeV [26] are excluded at 95% confidence level.
In the following analysis of the RG running of the SM gauge couplings, let us fix the Yukawa couplings such that the lepton doublet L has mass M L = 4.5 × 10 12 GeV, the quarks D c and Q have degenerate mass M Q = 5000 GeV, and U c and E c have GUT scale masses. We numerically solve the RG equations for SM gauge couplings listed in Appendix. 2. The left panel of Fig. 1 shows our results for the RG running of the SM gauge couplings as a function of the energy scale µ. The solid lines labeled by α i = g 2 i /4π (i = 1, 2, 3) denote the SM gauge couplings for U(1) Y , SU(2) L and SU(3) c , respectively. For comparison, in Fig. 1 we also show the RG running of the SM gauge couplings in the absence of the new fermions which are depicted by the dotted lines. In the former case, the SM gauge couplings successfully unify at around M GUT ≃ 7.53 × 10 15 GeV with the unified coupling value α GU T = α 1 = α 1 = α 3 ≃ 1/35.8. Using these values, the proton lifetime from its decay mediated by the SU(5) GUT gauge bosons can be approximated as [27] where m p = 0.983 GeV is the proton mass. This is consistent with the current experimental lower bound on proton lifetime given by the Super-Kamiokande with τ p (p → π 0 e + ) 4.0 × 10 34 yr [29]. Importantly, the predicted lifetime is within the expected sensitivity reach of future Hyper-Kamiokande, τ p 1.3 × 10 35 yr [30]. The color triplet scalar field contained in H can also mediate proton decay; the Super-Kamiokande experiments excludes the colored scalar mass lighter than O(10 11 ) GeV [27]. The validity of the proton lifetime estimate in Eq. (14) requires the colored Higgs mediated proton decay to be suppressed, particularly, the colored Higgs mass must to be greater than 10 11 GeV. Consider the quartic interactions of H with S/Σ fields, for instance, H † Σ † ΣH. Since both S and Σ have VEVs close to M GU T , consistency of proton lifetime estimate require these (positive) quartic couplings to be greater than O(10 −11 ).
In the right panel of Fig. 1  In the top left panel of Fig. 2 the blue (cyan) shaded region denote the range for the new quarks mass, M Q , and the new lepton mass, M L , to achieve the SM gauge couplings unification with an accuracy of 5% (1%) or less. We define the accuracy as a percentage difference between the energy scales where the SM gauge couplings α 1,2 and α 2,3 are unified (see, for example, the running of couplings in Fig. 1

Axion Dark Matter
The relic abundance of axion DM is given by [13] Ω a h 2 ≃ 0.12 θ a 3.40 × 10 −3 where f a = v P Q /N DM is the axion decay constant, N DW is the domain wall number and θ a is the so-called misalignment angle. The observed DM relic abundance is Ω a h 2 = 0.120 ± 0.0012 [31], and the axion decay constant is bounded from below by the measurement of the supernova SN 1987A pulse duration, f a 4 × 10 8 GeV [32]. The axion/DM field fluctuation during inflation generates isocurvature density perturbations in the DM power spectrum, P iso = H inf πθmfa 2 , which is severely constrained by the Planck measurements [33] where the adiabatic power spectrum P adi (k * ) ≃ 2.2 × 10 −9 with pivot scale k * = 0.05 Mpc −1 [31]. We obtain In our model, v P Q = v 2 Σ + v 2 S and N DW = 3 [34] such that f a = v P Q /N DM ≃ M GU T . From Eq. (15), θ a = 7.70 ×10 −3 is fixed to reproduce the observed DM in the universe. Together with Eq. (15), we obtain an upper bound H inf 5.73 × 10 8 GeV. Therefore, the value of the Hubble parameter during inflation must be relatively low for the viability of the axion DM scenario that we have considered. For f a = O(10 16 ) GeV and higher, the the axion mass is O (10 −9 ) eV, which can be searched by the CASPEr experiment [35].

Inflection-Point Inflation
The inflaton potential that exhibits an approximate inflection-point around φ = M is given by where V 0 = V (M), V n ≡ d n V /dφ n | φ=M , and φ = M is identified as the horizon exit scale corresponding to the pivot scale k * = 0.05 Mpc −1 used in Planck measurements [33]. Requiring the inflationary predictions to be consistent with the Planck measurements [33] of the curvature perturbation amplitude ∆ 2 R = 2.099×10 −9 and spectral index n s = 0.965, V 1,2,3 can be expressed in terms of V 0 , M and the number of e-folds during the inflation N as (see Ref. [18] for details) For the remainder of this article, we set the e-folding number N = 60 to solve the horizon problem of big bang cosmology.
We identify V (φ) in Eq. (18) with the RG improved U(1) X Higgs/inflaton potential where λ φ (φ) is determined by solving the following RG equations: Here, we have simplified the notation using Y i ≡ Y i M to denote the Majorana neutrino Yukawa couplings in Eq. (11), and the beta-function of λ φ is given by Using the RG improved inflaton potential together with the RG equation for λ φ , V 1,2,3 in Eq. (18) may be expressed as where the prime denotes derivatives with respect to φ. Approximate inflection-point conditions at M, , a choice which will be justified shortly. For simplicity, we also set Y 2 (M) = Y 3 (M). We later show that the inflection point conditions require Y 2,3 (M) and g(M) to be of the same order and λ φ (M) ∝ g(M) 6 . Using this we can approximate The Hubble parameter during inflation is given by Substituting H inf 5.73 × 10 8 GeV, the upper bound on Hubble parameter to solve the axion domain wall and isocurvature problems is expressed as For H inf 10 9 GeV, the inflationary prediction for the tensor-to-scalar ratio r is tiny (H inf = 2.47 × 10 14 GeV √ r).
To evaluate the masses of Z ′ gauge boson, Majorana neutrinos and inflaton, we now consider the low energy values of the relevant couplings.
Evaluating the other inflection-point condition, Mβ ′ λ φ (M) ≃ 16λ φ (M), by using the RG equations in Eqs. (21) and (27), we obtain We note that the contributions of the new fermions to the beta-function of g in Eq. (21) are key to obtaining λ φ (M) > 0 in Eq. (28), which is essential for the stability 6 of the U(1) X Higgs/inflaton potential. Equating the expressions for λ φ (M) in Eqs. (21) and (24), we find Since the beta-function of the quartic coupling in Eq. (22) is dominated by the gauge and Yukawa couplings, the RG equation for λ φ can be solved analytically, and its value for φ ≪ M can be estimated as [18]  The masses of the inflaton, Z ′ boson and Majorana neutrinos in Eqs. (7) and (11), evaluated at φ = v X , are given by where we have used g(v X ) ≃ g(M) and Y i (v X ) ≃ Y i (M). Note that the new particle spectrum is determined by v X and M in our model.

Thermal Leptogenesis and Reheating
To generate the observed baryon asymmetry we consider thermal leptogenesis [19], which is the one of the simplest realization of the scenario in models with type-I seesaw mechanism. Since the Majorana neutrinos have non-degenerate masses, a successful thermal leptogenesis requires the lightest Majorana neutrino mass, m N 1 > 10 9−10 GeV with reheat temperature T R > m N 1 [36]. To prevent the U(1) X gauge interactions [37] and Yukawa interactions [38] from keeping the Majorana neutrinos in thermal equilibrium with the SM particles and suppressing the generation of lepton asymmetry, we require these processes to decouple before the temperature of the thermal plasma drops to T ∼ m N 1 .
where f SM denote the SM fermions, is effectively a four-Fermi interaction. For T > m N 1 , the thermally-averaged cross section for this process is given by [39] σv ≃ 11 1280π The annihilation/creation rate of (N c ) 1 in the thermal plasma Γ(T ) = n eq (T ) σv , where n eq (T ) ≃ 2T 3 /π 2 is the equilibrium number density. This process decouples at T ∼ m N 1 if Γ/H| T =m N 1 < 1, where H(T ) ≃ πT 2 /M P is the corresponding value of the Hubble parameter. It leads to a lower bound on v X v X > 3.48 × 10 10 GeV m N 1 10 9 GeV The thermally averaged cross section for the process involving Yukawa interactions of N 1 R , particularly, N R 1 N 1 R ↔ φφ, with m N 1 > m φ is approximated as [40] σv ≃ 1 4π Requiring Γ/H < 1 at T = m N 1 to avoid the suppression of the generation of lepton asymmetry, we find v X > 5.95 × 10 10 GeV m N 1 10 9 GeV which is slightly stronger than the lower bound obtained for the Z ′ mediated process in Eq. (33). Let us fix M = 0.05M P to be our benchmark for consistency with the axion DM bound in Eq. (26) with F a ≃ M GU T . Together with m N 1 = 10 9 GeV = m Z ′ /10, we find v X ≃ 1.70 × 10 12 GeV which is consistent with the above bound on v X . With these values, the mass of the remaining Majorana neutrinos m N 2,3 ≃ 9.30 × 10 9 GeV, and the mass of the inflaton m φ ≃ 8.10 × 10 4 GeV. As we have discussed earlier, for successful thermal leptogenesis, the reheat temperature (T R ) must satisfy T R > m N 1 . Assuming an instantaneous decay of the inflaton field, the reheat temperature can be estimated as where g * ≃ 100 and Γ φ is the total decay width of the inflaton. To estimate Γ φ , we consider the following mixed quartic interaction between Φ and the SM doublet Higgs field H in the scalar potential: The decay width of φ is approximated as and the reheat temperature is given by Hence, T R > m N 1 can be achieved with λ ′ 9.86 × 10 −9 .

Summary
It is well-known that SM needs to be supplemented with new physics in order to address its inadequacies related to DM physics, neutrino masses and mixings, baryon asymmetry in the universe, cosmic inflation, and strong CP problem. We have proposed an extension of the SM which is based on SU(5) grand unification that accounts for all of the above inadequacies. Our model is based on SU(5)×U(1) X ×U(1) P Q symmetry, where the U(1) X gauge symmetry is the generalization of the B − L symmetry, and U(1) P Q is the global Peccei-Quinn (PQ) symmetry. It includes four fermion families in 5 + 10 representation of SU(5), a mirror family in 5 + 10 representations, and three SU(5) singlet three Majorana fermions. The U(1) X related anomalies cancel in the presence of the Majorana neutrinos. The scalar sector includes four complex scalars, Σ, S, H and Φ. The new fermions are essential for achieving a successful unification of the SM gauge couplings. We have shown that the SM gauge couplings unify at M GU T ≃ (6 − 9) × 10 15 GeV for a wide range of new fermion masses, and the proton lifetime τ p is estimated to be well within the expected sensitivity of the future Hyper-Kamiokande experiment, τ p 1.3 × 10 35 years. The new fermions also stabilize the SM Higgs potential at high energies. The SU(5) adjoint scalar Σ is also charged under the PQ symmetry, and hence the spontaneous breaking of the SU(5) also triggers the breaking of the PQ symmetry, resulting in axion dark matter. The axion decay constant f a is of the same order as the SU(5) symmetry breaking scale M GU T or somewhat greater. For f a ∼ 10 16 GeV and higher, the mass of the axion DM mass is O(10 −9 ) eV and smaller, which can be searched by the CASPEr experiment. The value of the Hubble parameter during inflation must be low, H inf 10 9 GeV, in order to successfully resolve the axion domain wall, axion DM isocurvature, and SU(5) monopole problems. With the identification of the U(1) X Higgs field with the inflaton field, we have implemented the low-scale inflection-point inflation which is capable of realizing the desired value for H inf . The new fermions are also essential for the phenomenological viability of both the axion DM and inflation scenarios. We have also shown that the inflaton decay after the end of inflation can reheat the universe to a sufficiently high temperature such that the Majorana fermions generate the observed baryon asymmetry in the universe via leptogenesis.

Acknowledgements
This work is supported in part by the United States Department of Energy grant DE-SC0012447 (N. Okada) and DE-SC0013880 (D. Raut and Q. Shafi).