Axion model with the ${\rm SU}(6)$ unification

We propose an ${\rm SU}(6)$ unified theory that solves the strong ${\it CP}$ problem with a high-quality axion. This is achieved by an automatic global Peccei-Quinn ${\rm U}(1)_{\rm PQ}$ symmetry and the gauged discrete $\mathbb{Z}_{4{\cal R}}$ symmetry. With the axion mass predictions of $m_a \sim {\cal O}(10^{-3} ) - {\cal O}(0.1) \,{\rm eV}$, as well as a universal axion-photon coupling in the unification models, the QCD axion can be probed in the upcoming experiments of IAXO. An intermediate gauge symmetry breaking scale characterized by the axion decay constant is obtained by the Peccei-Quinn quality argument, which is likely to achieve a successful leptogenesis through the type-I seesaw mechanism. The Higgs sector at the electroweak scale is a type-II two-Higgs-doublet model. The gauge coupling unification is possible with the supersymmetric extension.


I. INTRODUCTION
Grand unified theories (GUTs) [1,2] were proposed to unify all fundamental interactions. In the original Georgi-Glashow SU(5) GUT [1], it was assumed that there are no unobserved fermions beyond the Standard Model (SM). Aside from the aesthetic aspect of achieving the gauge coupling unification in its supersymmetric (SUSY) version, it is pragmatic to envision that a successful GUT could address as many physical issues beyond the SM as possible. One leading puzzle of the SM is the strong CP problem, which predicts the neutron electric dipole moment due to the nontrivial topological term of θ α3c 8π G a µνG µν a in the quantum chromodynamics (QCD) sector. However, the experimental upper limit of |d n | < 3.0 · 10 −26 e cm [3] leads to an extremely tiny upper bound of |θ| 10 −10 , whereθ is shifted from the θ parameter by the phase of the quark mass matrix M q asθ = θ + arg det M q . Axion field, which transforms under a global U(1) Peccei-Quinn (PQ) symmetry [4][5][6], is the most appealing candidate to solve the strong CP problem since its first proposal. Two widely studied benchmark scenarios of the invisible axion models are the Kim-Shifman-Vainshtein-Zakharov model [7,8] and the Dine-Fischler-Srednicki-Zhitnitsky model [9,10].

II. THE MINIMAL MODEL
The minimal setup of an anomaly-free SUSY SU(6) model is made up of three generational chiral superfields of6 ρ=I ,II F and 15 F that contain all necessary fermions. Two6 ρ F form an SU(2) F doublet, which is free from the Witten anomaly [81]. The SU(6) undergoes two stages of symmetry breaking [65,66,82,83]   Collectively, we tabulate the chiral superfields with their PQ and Z 4R charges in Table I. A discrete and gauged Z 4R symmetry with proper charge assignments is necessary to avoid the dangerous dimension-three PQbreaking operators. The SU(2) F is also gauged in order to avoid the constraint from the gravity. Indeed, the mixed gauge anomalies cancel as follows with n g = 3. Accordingly, the superpotential includes the following terms Schematically, we denote the Higgs VEVs and their hierarchies as follows with tan β being the ratio between two EW Higgs VEVs. After the first-stage symmetry breaking in Eq. (2) With the electric charge of −1/3, the D quarks can form d = 4 operators from the Yukawa couplings in Eq. (4), and the D-hadron lifetime [84][85][86] (10 11 ) GeV through the PQ quality analysis below. This satisfies the cosmological constraint of τ Q 10 −2 sec [87,88] from the big bang nucleosynthesis. In addition, the Yukawa couplings of6 , which satisfy the Davidson-Ibarra bound [89][90][91] of M N 10 9 GeV for a successful leptogenesis. Two SU(2) L EW Higgs doublets come from the6 I H and the 15 H and lead to the type-II 2HDM. Besides, a type-I seesaw mechanism can also be realized with the Yukawa couplings of6 All relevant Yukawa couplings in two stages of symmetry breaking will be explicitly given in Eqs. (B6) and (B8). Another remarkable feature is that the tree-level µ term of µH u H d ⊂ µ6 H I 15 H is impossible in the superpotential (4) due to the SU(6) gauge symmetry.

III. THE SU(6) AXION
The physical axion mainly comes from the . It can be obtained from the orthogonality between the U(1) PQ current and the U(1) N current, and we arrive at The physical PQ charges of (q 3 , q 6 ) for the (a 3 , a 6 ) fields are linear combinations of the U(1) PQ charges and the U(1) N charges [28] q ≡ c 1 PQ + c 2 N .
The coefficient of c 1 is determined by matching [92] the global anomaly factors of which leads to c 1 = 1. By denoting the overall size of the Higgs VEVs and their ratio as we find the physical PQ charges and axion decay constant of The physical axion becomes a phys = cos φ a 3 + sin φ a 6 . In addition, the electromagnetic (EM) anomaly factor is with C f being the the fermion representations under the SU(3) c , and (PQ f , q f ) being the PQ and EM charges of fermions.

IV. THE HIGH-QUALITY AXION
The leading PQ-breaking (with ∆PQ = −12) operator that is invariant under the SU (6), SU(2) F , and the discrete Z 4R symmetries in Table I with (I , J , K) being the SU(3) L indices. In order not to reintroduce further PQ-breaking operators, it is reasonable to expect the SUSY-breaking scale to be lower than v 331 . The total axion effective potential and the induced effectiveθ due to the Eq. (12) are with δ = Arg(k) and M pl = 1.22 × 10 19 GeV. According to the PQ quality requirement [31][32][33], the contribution from the V H H PQ to the energy density should be 10 −10 times less than that of the QCD axion potential, and we find In Fig. 1, we display upper limits to f a . Within the reasonable parameter choices of (tan β , tan φ , |k sin(δ)|), we consider the high-quality axion window of 10 8 GeV f a 10 10 GeV , ⇒ 10 9 GeV v 331 10 11 GeV , by using the relation in Eq. (10b). The corresponding axion masses [5,94] are   This is distinguishable from the axion masses of m a ∼ O(10 −9 ) eV in the SU (5) GUT [17,19,21,22]. The effective axion-photon couplings are parametrized by By using the QCD and EM anomaly factors in Eqs. (8b) and (11), we find that C aγγ = 0.75. Thus, the axionphoton couplings from the SU(6) GUT confirm the universal predictions as those in the SU(5) and SO (10) GUTs [21,22,28,100]. In Fig. 2, we present the benchmark models for the SU(6) axion in the (m a , |g aγγ |) plane. The high-quality axion (solid line) with mass range in Eq. (16) can be probed in the future IAXO [95][96][97] experiment.

V. THE AXION DOMAIN WALL
The V H H PQ term in Eq. (13) plays a role as a biased term [87,[101][102][103][104][105] to avoid the axion domain wall formation, which can be possible due to the periodicity of the effective potential term of V QCD . One thus requires domain walls to decay before the domination epoch [106] t dec < t form , t dec ∼ 10 −66 sec

VI. THE GAUGE COUPLING UNIFICATION
We briefly present the gauge coupling evolutions in terms of the one-loop renormalization group equations (RGEs). The gauge couplings are (α 3c , α 3L , α N ) for the G 331 symmetry, and (α 3c , α 2L , α Y ) for the G SM symmetry. The U(1) N coupling should be normalized by α 1 = 4 3 α N for the unification. The most general one-loop RGE solution for the gauge coupling α i of the gauge symmetry G i is with the one-loop β coefficients in the SUSY extension being Explicitly, the one-loop β coefficients for the SUSY SU (6) can be obtained by the spectrum in Table III and the branching rules in Eq. (B4) as follows To evaluate the RGEs, the tree-level matching conditions at the v 331 scale are The one-loop results are displayed in Fig. 3, with an intermediate scale of v 331 = 10 10 GeV and the latest electroweak precision measurements of (α 3c , α em , sin 2 θ W ) at the Z pole [109] as the inputs. A unification is indicated with Λ GUT ∼ 10 16 GeV for the SUSY SU (6). Notice that the intermediate G 331 -breaking scale usually requires the two-loop RGE analysis as well as the one-loop matching conditions with mass threshold effects [110,111]. Recent studies on the SO(10) and E 6 reveal the strong correlation to the proton lifetime predictions with these effects [112][113][114][115][116]. Thus, we defer the study of the proton lifetime predictions with the current constraint of τ p 2.4 × 10 34 yrs from the Super-Kamionkande [117,118] to future work.

VII. CONCLUSIONS
We have put forth a SUSY SU(6) model for the strong CP problem, by utilizing the emergent global DRS symmetry in Eq. (1). Historically, the emergent global symmetry was first mentioned in the study of strongly coupled theories. Its emergence and breaking are independent of the dynamical aspects of the gauge theories. A high-quality axion with its decay constant is found to be constrained from the PQ-quality requirement to the dimension-six PQ-breaking operator, and is most likely to be directly probed in the future IAXO searches. The type-I seesaw mechanism and a successful leptogenesis is also achievable at the corresponding PQ symmetrybreaking scale. Our work manifests that three seemingly unrelated issues of the strong CP problem, the neutrino mass origin, and the EW Higgs sector are coherently unified in the SU(6) framework. Though the current study of the SU(6) requires additional ingredients of discrete symmetries for suitable PQ-breaking operators, it is natural to extend SU(6) to higher unified groups, which enjoy the emergent global DRS symmetry in general. Historically, the nonminimal GUTs with gauge symmetries of SU(N ≥ 7) were considered to unify three generations of SM fermions [119][120][121][122]. It is therefore appealing to look for realistic GUTs that unify both the PQ quality problem and the flavor puzzle.
We prove that the non-SUSY SU(6) model cannot lead to a high-quality axion with any discrete gauge symmetry of Z k . To see this, we assign the most general Z k charges for fermions and Higgs fields in Table II, which are subject to the gauge anomaly cancellation of In addition, at least two of the following terms are necessary to reproduce the SM-like Higgs boson Yukawa couplings that are consistent with the current LHC results In this section, we list the SUSY SU(6) spectrum and the Yukawa couplings following the breaking pattern of SU(6) → G 331 → G SM . The U(1) N charge for the SU (6)  fundamental representation at the first-stage symmetry breaking is defined as follows Afterwards, the U(1) Y charge for the SU(3) L fundamental representation is given by and the electric charges are quantized by  Table III, with their representations under the G 331 and G SM . The first-stage branching rules of superfields containing Higgs arē The scalar components of (1 ,3 , − 1 3 ) H II ⊂6 H II and the (1 , 6 , + 2 3 ) H ⊂ 21 H will develop VEVs of ∼ v 331 for the symmetry breaking of G 331 → G SM . We can determine the global U(1) PQ [SU(6)] 2 anomaly factor from the fermions in Table III and Eq. (B4) as follows with n g = 3. The related trace invariants are T (6) = T (6) = 1 2 , T (15) = 2, and T (21) = 4. After the second-stage symmetry breaking, we find the following mass terms from the Yukawa couplings Y N6

[I F6
II] The (1 , 2 , −1) H I ⊂6 H I gives masses to down-type quarks and charged leptons, and (1 , 2 , +1) H ⊂ 15 H gives masses to up-type quarks. This justifies the lowenergy effective theory at the EW scale is the type-II 2HDM. The neutrino masses are realized through the type-I seesaw mechanism By combining the fermions in Table III