Minimal Models for Axion and Neutrino

The PQ mechanism resolving the strong CP problem and the seesaw mechanism explaining the smallness of neutrino masses may be related in a way that the PQ symmetry breaking scale and the seesaw scale arise from a common origin. Depending on how the PQ symmetry and the seesaw mechanism are realized, one has different predictions on the color and electromagnetic anomalies which could be tested in the future axion dark matter search experiments. Motivated by this, we construct various PQ seesaw models which are minimally extended from the (non-) supersymmetric Standard Model and thus set up different benchmark points on the axion-photon-photon coupling in comparison with the standard KSVZ and DFSZ models.


Introduction
The existence of neutrino mass and dark matter is a clear sign of new physics beyond the standard model (SM). Another long-standing issue in SM is the strong CP problem [1] which is elegantly resolved by the Peccei-Quinn (PQ) mechanism [2]. It predicts a hypothetical particle called the axion as a pseudo-Nambu-Goldstone (NG) boson of an anomalous global symmetry U (1) PQ which is spontaneously broken at an intermediate scale v PQ ≈ 10 9-12 GeV [3]. The PQ symmetry is realized typically in the context of a heavy quark (KSVZ) model [4] or a two-Higgs-doublet (DFSZ) model [5].
The PQ symmetry breaking may be related to the seesaw mechanism explaining the smallness of the observed neutrino masses [6][7][8][9] identifying the PQ symmetry as the lepton number U (1) L [10,11]. Let us note that the seesaw mechanism realized at the intermediate scale v PQ can provide a natural way to explain the matter-antimatter asymmetry in the universe through leptogenesis [12]. An attractive feature of this scenario is that the axion is a good candidate of cold dark matter through its coherent production during the QCD phase transition [13]. As the axion is well-motivated dark matter candidate, serious efforts are being made to search for it by various experimental groups such as ADMX [14], CAPP [15] and IAXO [17]. The traditional KSVZ or DFSZ models have been considered as two major benchmarks in search for the axion dark matter. In the context of the PQ mechanism combined with the seesaw mechanism, however, the electromagnetic and color anomaly coefficients can take different values, and thus can have different predictions in the future axion search experiments. This motivates us to consider minimal extensions of the SM in which various seesaw models [6][7][8] are extended to realize the KSVZ or DFSZ axion, and compare their predictions with the conventional KSVZ and DFSZ models. This paper is organized as follows. We will first set up minimal extensions of the SM to combine the PQ and seesaw mechanisms in non-supersymmetric and supersymmetric theories in Sections 2 and 3, respectively. The corresponding model predictions are presented in Section 4, and then we conclude in Section 5.

Minimally extended standard model for the PQ and seesaw mechanism
A PQ seesaw model is characterized by how a global U (1) X symmetry, playing the role of the PQ symmetry and the lepton number, is implemented to act on a specific set of extra fermions carrying non-trivial X charges. Such an U (1) X symmetry is supposed to be broken spontaneously by the vacuum expectation value of a scalar field σ assuming a scalar potential: with v σ ∼ 10 9-12 GeV which sets the scales of the axion decay constant F a and the heavy seesaw particles. In the case of the type-I and type-II seesaw introducing a singlet fermion (righthanded neutrino) [6] and a Higgs triplet scalar [7] c iτ 2 with the charge conjugation ˜ c = C¯ T and the Pauli matrix τ 2 . The non-trivial X-charges are assigned as follows (1), the complex scalar field σ can be written as where a ≡ A σ is nothing but the KSVZ axion, and the real scalar ρ is supposed to get mass ∼ v σ which sets the axion and seesaw scales.

• DFSZ+type-III (DFSZ-III):
In a DFSZ axion model, the PQ symmetry is implemented by extending the Higgs sector with two , and a Higgs singlet σ , and allowing the scalar potential term which sets the PQ ( X ) charge relation of the two Higgs bosons: Then the Yukawa Lagrangian for DFSZ-III reads where one can choose s = 1 or 2 depending on which we categorize two different DFSZ models. As we again have two choices for the triplet mass operator with σ or σ * , there are four different DFSZ-III models. Eqs. (6), (7) give six X -charge relations to be satisfied by the eight fields (other than σ ). As will be discussed shortly, the orthogonality of the axion and the longitudinal degree of the Z boson gives another condition. Then, one finds that there is freedom to choose one of the three quark charges. Taking X Q L ≡ 0, we get the following X-charge assignment: where we have X 1 = X d and X 2 = −X u leading to the QCD anomaly c 3 = (X u + X d )N g = 6 with the number of the generation N g = 3.
After the breaking of SU (2) and v σ , of 1 , 2 and σ , the axion and the longitudinal degree of the Z boson denoted by a and G 0 , are given by [18]: where A 1 , A 2 and A σ are the phase fields of 1 , 2 and σ .
Then the orthogonality of a and G 0 is guaranteed by with the normalization X σ = 1 and x ≡ v 2 /v 1 .

Minimal supersymmetric PQ seesaw model
To implement the PQ symmetry in supersymmetric models, let us introduce two chiral superfields σ and σ having the opposite X charges, say, X σ = −Xσ ≡ +1, and its spontaneous breaking is assumed to occur by the typical superpotential: where σ = v σ / √ 2 and σ = vσ / √ 2 is implied in the notation.
Here Ŝ is a gauge singlet superfield and carry PQ charge zero.
The supersymmetric version of the KSVZ model introduces the heavy quark superpotential which defines the PQ charge relation: X + X c = −Xσ ≡ +1 leading to the QCD anomaly: c 3 = N + c as in the nonsupersymmetric case. Here W MSSM is the usual Minimal Supersymmetric Standard Model (MSSM) superpotential given by (13) which is separated from the PQ mechanism. The supersymmetric DFSZ model provides a natural framework to resolve the μ problem as well [19] (14) where M P is the reduced Planck mass and the right size of the μ term, μ = λ μ v 2 σ /2M P , arises after the PQ symmetry breaking. The usual PQ charges assignment consistent with the above superpotential iŝ σĤ uĤdQû cdcLlc where we have put X Q ≡ 0 as before and X u + X d = 2 follows from the charge normalization of X σ = +1. At this stage, there is arbitrariness in choosing the value of X L , but it will be fixed in seesaw extended PQ models which has no physical consequences. Note that the QCD anomaly of the supersymmetric DFSZ model is again given by c 3 = (X u + X d )N g = 6.
Now let us consider the seesaw extensions of the supersymmetric PQ models. As in the non-supersymmetric case, Type-I seesaw introducing right-handed (singlet) neutrinos does not change the results of the standard KSVZ and DFSZ models. Thus, we discuss the Type-II and -III extensions in order.
which set the PQ charges of the leptonic fields: • DFSZ+Type-II (sDFSZ-II): Similarly to the previous case, the superpotential for the DFSZ model combined with Type-II seesaw takes the form: which is invariant under the PQ symmetry with the charge assignment of (15) extended to the leptonic sector as follows: • KSVZ+Type-III (sKSVZ-III): In supersymmetric Type-III seesaw one introduces three triplet superfields (with Then the whole superpotential of the KSVZ model realized in Type-III seesaw is which defines the PQ charges of the leptonic fields as in the non-supersymmetric case: • DFSZ+Type-III (sDFSZ-III): Type-III seesaw introduces three triplet superfields (with Y = 0): The superpotential is which set the PQ charges of the leptonic fields: Ll cˆ

Model implications to the axion detection
To discuss the implications of the PQ seesaw models presented in the previous sections, let us first summarize some basic properties of the axion relevant for our discussion [3]. After the PQ symmetry breaking by a generic number of scalar fields φ having the PQ charge X φ and φ = v φ / √ 2, the following combination of the phase fields A φ defines the axion direction: Integrating out all the relevant PQ-charged fermions, the axion gets the effective axion-gluon-gluon and axion-photon-photon couplings through its color and electromagnetic anomalies, respectively: −L   where c aγ γ counts the electromagnetic anomaly normalized by the color anomaly, and c χ S B is the modified effect by the chiral symmetry breaking including the strange quark contribution. Each PQ seesaw model presented in the previous section gives a different prediction on the coefficient c aγ γ and thus on the future sensitivity of the axion search at ADMX or CAPP. Following Eq. (31), the electromagnetic anomaly of each model is given by where N DW = 1 and 6 are used for the KSVZ and DFSZ models, respectively.

Conclusion
We have considered minimal extensions of the SM combining the KSVZ or DFSZ axion with various seesaw models in the framework of the (non-) supersymmetric theories, which provides a popular solution to the strong CP problem as well as the smallness of neutrino masses. We have showed that depending on how to embed U (1) PQ in a seesaw model, the electromagnetic and color anomaly coefficients take different values, and thus each model has a different prediction on the axion-photon-photon coupling which could be tested in the future axion search experiments. This sets up various benchmark points for the minimal PQ seesaw models in comparison with the standard KSVZ and DFSZ models which are summarized in Eq. (31) and Fig. 1.