1 Introduction

KANAZAWA-18-05

September, 2018

An extension of the SM based on effective Peccei-Quinn Symmetry

Daijiro Suematsu111e-mail:  suematsu@hep.s.kanazawa-u.ac.jp

Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan

Abstract
Peccei-Quinn (PQ) mechanism based on a chiral global symmetry is considered to be a simple and elegant solution for strong problem. Fact that the mechanism could be experimentally examined through the axion search makes it much more interesting and recently it causes a lot of attention again. However, it is also known that the mechanism is annoyed by two serious problems, that is, a domain wall problem and goodness of global symmetry. Any global symmetry is considered not to be exact due to the quantum effect of gravity. In this paper, we consider a solution to these problems, in which quark mass hierarchy and mixing, neutrino mass generation and existence of dark matter are closely related. In our solution, PQ symmetry is assumed to be induced through symmetry breaking at an intermediate scale of a local U(1) symmetry, and a global U(1) symmetry which plays a role of Froggatt-Nielsen symmetry . In the lepton sector, a remnant of the PQ symmetry controls neutrino mass generation and dark matter existence.

## 1 Introduction

Strong problem is one of serious problems in the standard model (SM), which is suggested by an experimental bound of the electric dipole moment of a neutron [1]. The bound requires a fine tuning of for a parameter in QCD. Invisible axion scenario based on a chiral global symmetry, which is called Peccei-Quinn (PQ) symmetry [2, 3], is known to give a simple and elegant solution to it. Since it predicts the existence of a light and very weakly interacting pseudoscalar [4, 5], this solution could be examined experimentally. Moreover, it is known to present a good candidate for cold dark matter (DM) under a suitable condition [6]. Its experimental search is proceeded now.

On the other hand, the scenario has two fatal problems generally. The first one is known as a domain wall problem [7]. Although PQ symmetry is explicitly broken to its subgroup through the QCD instanton effect, the is also spontaneously broken to its subgroup when PQ symmetry is spontaneously broken by a vacuum expectation value (VEV) of scalar fields and quark condensates. This brings about degenerate vacua, which are separated by topological defects called domain wall. Since the energy density of domain walls dominates cosmological energy density of the Universe inevitably, the Universe is over-closed contradicting to the observations. It is known that the domain wall problem could be escaped for a non-degenerate vacuum which has [8], even if the cosmological inflation occurs before the PQ symmetry breaking.

The second one is related to goodness of the PQ symmetry. The PQ symmetry is a global symmetry, which is used to be considered broken by the gravitational effect [9]. If this breaking effect due to the gravity is larger than the QCD instanton effect, the PQ mechanism cannot solve the strong problem. In order to escape this dangerous situation, such symmetry breaking operators caused by the gravity should be forbidden up to dimension ten [10]. There, the PQ symmetry is considered to be realized as an accidental symmetry induced by some gauge symmetry or a discrete symmetry, which satisfies such a constraint on its goodness. In such a direction, several works has been done by now [11].

In this paper, we propose a model which can escape these two problems in invisible axion models [4, 5]. Although the SM has been confirmed by the discovery of the Higgs scalar [12], it cannot explain several experimental and observational data such as quark mass hierarchy and CKM mixing, neutrino masses and their large mixing [13], and also the existence of DM [14]. In the present model construction, we take account of these problems also.aaaModel construction to explain these problems including the strong CP problem has been done in various articles [15, 16, 17, 18].. For this purpose, we impose on the model, where is a gauge symmetry but is a global symmetry whose charge is flavor dependent. Then, the latter could play a role of Froggatt-Nielsen symmetry [19]. This symmetry is assumed to be spontaneously broken to PQ symmetry at some intermediate scale. We require that guarantees the goodness of to be kept up to a consistent level required by the strong problem. After the spontaneous breaking of , both a non-degenerate QCD vacuum and Yukawa couplings with desirable flavor structure are induced in a quark sector [18]. In a leptonic sector, the scotogenic model [20] which connects the neutrino mass generation and the existence of DM is brought about as a low energy effective model.

The remaining parts are organized as follows. In the next section, we present a model by fixing the symmetry and the field contents in the model. We discuss features of the model such as the symmetry breaking, the domain wall number, the goodness of the PQ symmetry and so on. In section 3, phenomenological features of this model are discussed, such as quark mass hierarchy and CKM mixing, neutrino mass generation, leptogenesis, DM abundance and so on. We summarize the paper in section 4.

## 2 A model with U(1)g×U(1)FN

We start presenting a brief review of QCD vacuum degeneracy in the PQ mechanism [1]. If the anomaly takes a value for the PQ charge assignment of colored contents in the model, the transformation of the colored fermions shifts a parameter as [2]

 ¯θ→¯θ+2πN, (1)

where is a coefficient of an effective term induced by instantons and it is defined as . stands for a quark mass matrix. If the PQ symmetry is spontaneously broken by a VEV of a scalar field , behaves a dynamical variable corresponding to a pseudo Nambu-Goldstone boson associated with this breaking, which is called axion [3, 4, 5]. Since a period of is and potential for can be represented by using a QCD scale as

 V(¯θ)=Λ4QCD(1−cos¯θ), (2)

this potential for has -fold degenerate minima. The axion is fixed as which is defined at a region . This requires that axion decay constant should be identified as .

Each degenerate vacuum is separated by potential barriers called domain wall [7]. It can be identified with a topological defect which is produced through the spontaneous breaking of . is called domain wall number and it is written as for definiteness in the following part. In case, the walls are produced although the vacuum is unique. They have a string at its boundary which is generated due to the breaking of . This type of domain wall can quickly disappear as studied in [21]. On the other hand, in case, each string has domain walls and they generate complex networks of strings and walls. Since these networks are stable, they dominate the energy density of the Universe to over-close it. Thus, if inflation does not occur after the breaking, the present Universe cannot be realized unless is satisfied. Inflation could make the present Universe to be covered with a unique vacuum if inflation occurs after the PQ symmetry breaking. Thus, low scale inflation could give a solution in the case. However, we focus on a case in the present study.

Here two points on the domain wall problem should be remarked. First, a non-degenerate vacuum can be realized even for the case with . As an interesting example, we may consider a case where the VEV of the scalar does not break spontaneously. Since two vacua could be identified each other by this unbroken symmetry, is realized just as in the case. Second, we should notice that there are two estimations for axion relic density by taking account of the decay of domain walls in the case [22], which give different conclusions. One of them suggests that the domain wall problem might not be solved even in the case unless the axion decay constant is less than a certain limit. Another one claims that the axion produced through the domain wall decay is subdominant in comparison with the one due to axion misalignment. In the following discussion, we assume that the axion energy density coming from the domain wall decay is subdominant and could be a solution for the strong problem.

Now, we try to construct a model so as to escape the domain wall problem by [17, 18] and to guarantee the goodness of global symmetry at a required level by gauge symmetry. A framework to keep the goodness of the PQ symmetry has been proposed in [10]. We would like to follow a similar scenario to it.

We impose on the model above an intermediate scale and introduce new fields with the charge of this symmetry. They are two SM singlet complex scalars , and also six types of color triplet fermions , which are assumed to have no charge of and their subscripts and represent their chirality. The charge of these fields are given in Table 1. In this charge assignment, each VEV of and induces the symmetry breaking

 U(1)g×U(1)FN⟨σ⟩⟶U(1)PQ⟨S⟩⟶Z2, (3)

where we assume . The charge is defined as a linear combination where and are the charges of and , respectively. As we find it later, is not be broken through quark condensate either.

We have to address various anomalies associated to the introduction of new fields, first of all. All of the gauge anomaly for , and are easily found to be cancelled within these field contents. On the other hand, the QCD anomaly for is not cancelled but it is calculated as in this extra fermion sector. Since plays its role as a global symmetry after the first step of the symmetry breaking in eq. (3) and anomaly takes the same value as , the strong problem is expected to be solved by the PQ mechanism based on an axion caused in the spontaneous symmetry breaking of due to a VEV of . In order to escape the domain wall problem, the total anomaly including contribution from quark sector should be or .bbbIf and the quark condensates do not break a subgroup of , two vacua can be identified by this symmetry and then the model with can be considered to have . This suggests that the corresponding anomaly of the quark sector should take a value among , , and . As we see it later, this value is closely related to the quark mass hierarchy and the CKM mixing. Three examples (i) (iii) of the charge assignment for the quark sector is presented in Table 2. In these cases, can be realized.

Next, we move to the problem on the goodness of this and the mass generation of extra colored fermions. It is easy to see that a lowest order term in the potential of and , which is invariant but violating, is

 gM7plσ∗2S9+h.c., (4)

where is a constant and violation is considered to be induced by the gravitational effect so that the operator is suppressed by Planck mass . If the PQ mechanism works well in this model, the contribution to the axion mass from eq. (4) should be less than the one coming from the potential (2) due to the QCD instanton effect [10]. Since the latter is given as [3], this condition gives the constraint on such as

 ⟨σ⟩ <∼ 6×1012(1011 GeV⟨S⟩)92 GeV. (5)

It should be consistent with our assumption for the symmetry breaking pattern (3) within the astrophysical and cosmological constraints on the axion decay constant which is [1]. It requires that the VEV of should satisfy

 109 GeV <∼ ⟨S⟩ <∼ 2×1011 GeV, (6)

for the case. It suggests that the axion seems to be difficult to be a dominant component of DM since have to be rather small in this scenario. From these discussions, we find that the axion in this model is characterized by a lower mass bound such as  eV and a coupling with photon such as [23] where for (i), for (ii), and 6 for (iii).

The extra colored fields can get their mass only through the VEVs of and . It is crucial for the consistency of the model what scale of masses they can have. The following operators are invariant under ,

 σ¯Q(1)LQ(1)R,σ¯Q(2)LQ(2)R. (7)

On the other hand, could be generated as a invariant operator after the breaking at . These operators give masses to these extra colored fermions through and . However, since they have no hypercharge, they cannot couple with ordinary quarks and then have no decay modes to be stable.cccIt may be possible to assume that these fermions have hypercharge and couple with ordinary quarks to have decay modes. However, in that case, we have to introduce a lot of fields to cancel the gauge anomaly. We do not consider such a possibility here. If they are in thermal equilibrium during the history of the Universe, we have to note that several contradictions such as the existence of fractionally charged hadrons and their over-abundant contribution to the energy density could appear [23]. The most strong constraint on their abundance comes from searches of fractionally charged particles, which requires for the abundance of and ordinary nucleons [24]. This constraint could be satisfied even if is in the thermal equilibrium, as long as reheating temperature is assumed to be much lower than the mass of which is the lightest extra colored fermion. Since is assumed not to be restored after the reheating, these extra colored fermions are not produced in the thermal bath through the reheating process and the model can escape this problem. In fact, we can confirm that the mass derived by an coupling could satisfy the above constraint for parameters used in the following study and the reheating temperature such as  GeV. Such a low reheating temperature could cause a problem if we consider the thermal leptogenesis due to the decay of thermal right-handed neutrinos. We will come back this point later.

Now, we couple this model with the SM including a lepton sector. Since the axion could not be a dominant component of DM in this scenario as discussed above, we need to prepare a candidate for DM. For this purpose, the leptonic sector is extended by an additional doublet scalar and three right-handed neutrinos so as to realize the scotogenic model [20, 25, 26]. An example of the charge assignment for the leptonic sector is shown in Table 3. After the symmetry breaking due to , invariant operators are considered to be generated in both Yukawa couplings and a scalar potential of an effective theory at energy regions below . An interesting point is that nonrenormalizable Yukawa couplings are controlled by the charge of each quark and lepton [16, 18]. In fact, if we define

 nuij=12(XuRj−XqLi),ndij=12(XdRj−XqLi),nNij=12(XNRi+XNRj), nνij=12(XNRj−XℓLi−1),neij=12(XeRj−XℓLi), (8)

quark Yukawa couplings are written as

 −Lqy = 3∑i=1,j[yuij(SM∗)|nuij|¯qLiϕuRj+ydij(SM∗)|ndij|¯qLi~ϕdRj], (9)

where and . On the other hand, Yukawa couplings relevant to neutrino mass generation are written as

 −Lℓy=3∑i,j=1[hνij(SM∗)|nνij|¯ℓLiηNRj + heij(SM∗)|neij|¯ℓLi~ϕeRj (10) + hNij(SM∗)|nNij|M∗¯NcRiNRj+h.c.].

The third term related to the mass of right-handed neutrinos should satisfy , since renormalizable one is forbidden by . In these formulas (9) and (10), should be replaced by for . The scalar potential at energy regions lower than is written as

 V1 = m2SS†S+κ1(S†S)2+κ2(S†S)(ϕ†ϕ)+κ3(S†S)(η†η) (11) + m2ηη†η+m2ϕϕ†ϕ+λ1(ϕ†ϕ)2+λ2(η†η)2+λ3(ϕ†ϕ)(η†η)+λ4(ϕ†η)(η†ϕ) + λ52[SM∗(η†ϕ)2 +h.c.],

where is taken to be real. On the other hand, the scalar potential for the light scalars and after gets the VEV can be expressed as

 V2 = ~m2ηη†η+~m2ϕϕ†ϕ+~λ1(ϕ†ϕ)2+~λ2(η†η)2 (12) + ~λ3(ϕ†ϕ)(η†η)+λ4(ϕ†η)(η†ϕ)+~λ52[(η†ϕ)2 +h.c.],

which is found to coincide with the scalar potential of the scotogenic model.

In eqs. (11) and (12), scalar masses and couplings are shifted from ones at higher energy regions due to the symmetry breaking effect by and , respectively [17]. The shift of parameters in (11) can be summarized as

 κ1=¯κ1−ξ2S4ξσ,κ2=¯κ2−ξSξϕ2ξσ,κ3=¯κ3−ξSξη2ξσ, λ1=¯λ1−ξ2ϕ4ξσ,λ2=¯λ2−ξ2η4ξσ,λ3=¯λ3−ξϕξη2ξσ, m2S=¯m2S+ξS⟨σ⟩2,m2ϕ=¯m2ϕ+ξϕ⟨σ⟩2,m2η=¯m2η+ξη⟨σ⟩2, (13)

where over-lined parameters correspond to the ones before the symmetry breaking and represents a coupling constant for an operator in the potential at energy scales larger than .

On the other hand, the shift of parameters in (12) can be given as

 ~λ1=λ1−κ224κ1,~λ2=λ2−κ234κ1,~λ3=λ3−κ2κ32κ1, ~λ5=λ5⟨S⟩M∗,~m2ϕ=m2ϕ+κ2⟨S⟩2,~m2η=m2η+κ3⟨S⟩2. (14)

The parameters in eq. (14) should satisfy conditions for a vacuum defined in to be stable, which are written as

 ~λ1,2>0,~λ3>−2√~λ1~λ2,~λ3+λ4−|~λ5|>−2√~λ1~λ2. (15)

In these equations, the lowest dimension operators invariant under are listed for nonrenormalizable ones. There could be violating contributions to them which are induced by the gravity effect. However, since they are suppressed by a factor at least, their effect can be safely neglected under the condition (5). These formulas show that Yukawa couplings for the quarks and the leptons have a suppression by powers of after the PQ symmetry breaking due to . Neutrino Yukawa couplings in the leptonic sector are also found to reduce to the ones in the scotogenic model. Moreover, term in eq. (12) could be small so as to cause small mass difference between neutral components of the extra doublet scalar . One should remind that it is a crucial element of the neutrino mass generation in the original scotogenic model.

## 3 Phenomenological features of the model

### 3.1 Quark mass hierarchy and CKM mixing

After the PQ symmetry breaking due to , eq. (9) induces Yukawa couplings for quarks with a suppression factor where and is determined by the PQ charge of quarks just like Froggatt-Nielsen mechanism [16, 18].ddd In the different context, flavor structure of quarks and leptons have been extensively studied using flavons resulting from various types of flavor symmetry [27, 28, 29]. Elements of quark mass matrices derived from these are represented as

 mfij=yfijϵ|nfij|⟨ϕ⟩, (16)

where a superscript stands for up and down sector and then . If we define the quark mass eigenstates as and using the unitary matrices and , they satisfy the condition

 (Uf†)αiyfijϵ|nfij|Vfjβ=mfα⟨ϕ⟩ δαβ, (17)

where represents a mass eigenvalue in the -sector. The CKM matrix is expressed as . If we use the PQ charge of quarks given in Table 2, the quark mass matrices defined by and can be written for each example as

 (i) Mu=⎛⎜ ⎜⎝yu11 ϵ4yu12 ϵ3yu13 ϵ2yu21 ϵ3yu22 ϵ2yu23 ϵyu31 ϵ2yu32 ϵyu33⎞⎟ ⎟⎠⟨ϕ⟩,Md=⎛⎜ ⎜⎝yd11 ϵ3yd12 ϵ2yd13 ϵ3yd21 ϵ4yd22 ϵ3yd23 ϵ2yd31 ϵ5yd32 ϵ4yd33 ϵ⎞⎟ ⎟⎠⟨~ϕ⟩, (ii) Mu=⎛⎜ ⎜⎝yu11 ϵ4yu12 ϵ2yu13 ϵ4yu21 ϵ7yu22 ϵyu23 ϵyu31 ϵ8yu32 ϵ2yu33⎞⎟ ⎟⎠⟨ϕ⟩,Md=⎛⎜ ⎜⎝yd11 ϵ6yd12 ϵ5yd13 ϵ5yd21 ϵ3yd22 ϵ2yd23 ϵ2yd31 ϵ2yd32 ϵyd33 ϵ⎞⎟ ⎟⎠⟨~ϕ⟩, (iii) Mu=⎛⎜ ⎜⎝yu11 ϵ4yu12 ϵ2yu13 ϵ4yu21 ϵ7yu22 ϵyu23 ϵyu31 ϵ8yu32 ϵ2yu33⎞⎟ ⎟⎠⟨ϕ⟩,Md=⎛⎜ ⎜⎝yd11 ϵ5yd12 ϵ5yd13 ϵ5yd21 ϵ2yd22 ϵ2yd23 ϵ2yd31 ϵyd32 ϵyd33 ϵ⎞⎟ ⎟⎠⟨~ϕ⟩,

While flavor dependent PQ charge of quarks could bring about these mass matrices, it can also cause flavor changing neutral processes with axion emission [27, 16], which can be severely constrained through experiments. The strongest constraint on due to such processes is known to come from , whose experimental bound is given as [30]. Since the axion is introduced in the effective theory through the replacement , eq. (9) gives axion-quark interaction terms

 inuijmuij afa ¯uLiuRj+indijmdij afa ¯dLidRj+h.c., (19)

where is given in eq. (16). If we focus our attention to the down-sector and use the quark mass eigenstates defined above, corresponding terms in eq. (19) can be rewritten as

 i⟨ϕ⟩fa[(Ud†)αinuijydijϵnuijVdjβ−(Vd†)αinuijy∗djiϵnuijUdiβ]a ¯dαdβ (20) + i⟨ϕ⟩fa[(Ud†)αinuijydijϵnuijVdjβ+(Vd†)αinuijy∗djiϵnuijUdiβ]a ¯dαγ5dβ ≡ iSαβ a ¯dαdβ+iAαβ a ¯dαγ5dβ.

If we apply eqs. (8) and (17) to eq. (20), the coupling constants and are found to be expressed as

 Sαβ=mα−mβ2faX+αβ,Aαβ=mα+mβ2faX−αβ,

where is defined by

 X±αβ=(Vd†)αiX(dRi)(Vd)iβ±(Ud†)αiX(dLi)(Ud)iβ. (21)

Since the decay width of can be estimated by using this as [16, 31]

 Γ=|X+ds|2128πm3Kf2a(1−m2πm2K)3, (22)

we obtain the strong constrain on by applying the experimental bound to this formula as

 fa>2.4×1011 |X+ds| GeV. (23)

On the other hand, since the condition (6) should be satisfied, eq. (23) requires . The PQ charge of quarks is required not only to reproduce the quark mass eigenvalues and the CKM mixing but also to satisfy this constraint.

We examine these issues in the examples shown in Table 2. Since these realize , the axion decay constant satisfies . In order to study features of the examples quantitatively, we need to fix a value of and coupling constants . Needless to say, the validity of the scenario is determined through how good predictions can be derived for less number of independent coupling constants without serious fine tuning. The results in each example are ordered for a typical parameter set. In this analysis, the phase of is not taken into account, for simplicity.

In example (i), we assume and the coupling constants are fixed as

 yu11=yu23=yu32=yu33=1,yu13=yu22=yu31=0.1,yu12=yu21=0.7, yd21=yd22=yd31=yd32=1,yd11=yd13=yd23=0.1,yd12=0.022,yd33=0.3,

where the number of independent parameters can be identified as six. For this parameter set, the quark mass eigenvalues and the CKM matrix are obtained as

 mu≃2.6 MeV,mc≃1.1 GeV,mt≃174 GeV,md≃6.7 MeV,ms≃92 MeV,mb≃4.2 GeV,VCKM≃⎛⎜⎝0.97−0.23−0.00520.230.97−0.0180.00920.0171.0⎞⎟⎠.

In this case, eq. (23) requires  GeV.

In example (ii), is assumed and are fixed as

 yu11=yu13=yu21=yu31=yu32=yu33=1,yu22=yu23=0.1,yu12=0.32, yd11=yd21=yd31=1,yd22=0.1,yd23=−0.03,yd32=yd33=0.26.yd12=yd13=60,

where the number of independent parameters can be identified as seven. For this parameter set, we obtain

 mu≃4.0 MeV,mc≃1.3 GeV,mt≃174 GeV,md≃3.9 MeV,ms≃93 MeV,mb≃4.6 GeV,VCKM≃⎛⎜⎝0.970.240.0042−0.240.97−0.0056−0.00540.00431.0⎞⎟⎠.

Eq. (23) requires  GeV.

In example (iii), if we assume the same values for and as the ones in the example (ii), and are taken as

 yd11=yd21=yd31=ϵ,yd22=0.1,yd23=−0.03,yd32=yd33=0.26.yd12=yd13=60.

Since and take the same form as the ones of the example (ii), the quark mass eigenvalues and the CKM matrix take the same values as the ones in the example (ii). The number of independent parameters used here can be identified as eight. The bound on is estimated as  GeV, which is one order of magnitude smaller than the previous two examples.

These examples show that the constraint on coming from the flavor dependent PQ charge assignment could be much stronger than the astrophysical constraint as suggested in [16]. However, it could be consistent with the cosmological upper bound of even if the realization of realistic values for the quark mass eigenvalues and the CKM mixing is imposed. On the other hand, the consistency of this constraint with the upper bound of imposed by the goodness of the PQ symmetry could depend largely on the PQ charge assignment. In fact, although the consistency is complete in the example (iii), the situation is marginal in the examples (i) and (ii). In the example (iii), the scenario is found to work well even if serious fine tuning of the coupling constants is not adopted. The obtained results seem to be rather good compared with the data listed in [13] although the number of independent parameters are smaller than the number of physical observables in the quark sector.

### 3.2 Leptonic sector

In this model, the neutrino mass generation is forbidden at tree-level due to even after the breaking of , since is assumed to have no VEV, However, since both right-handed neutrino masses and mass difference between the neutral components of are induced after the breaking of as found form eqs. (10) and (12), small neutrino masses are generated radiatively in the same way as the original scotogenic model through a one-loop diagram which is shown in Fig. 1. If we apply the PQ charge given in Table 3 to eq. (10), the Dirac mass matrix for charged leptons which is defined by and the the Majorana mass matrix for right-handed neutrinos are expressed as

 (24)

In the mass matrix , we take account that allowed operators start from the nonrenormalizable ones.

This right-handed neutrino mass matrix suggests that three mass eigenvalues tend to take the same order values. If we assume values of the Yukawa coupling constants appropriately, the eigenvalues of can be fixed, for example, aseeeIn this choice, we refer to the previous work [17].

 M1≃1.0×108 GeV,M2≃4.2×108 GeV,M3≃1.9×109 GeV, (25)

where we assume  GeV. The neutrino mass generated through a one-loop diagram can be approximately written as

 (Mν)ij=3∑k=1~hνik