Minimal Models for Axion and Neutrino

# Minimal Models for Axion and Neutrino

## Abstract

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.

34

## I 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 srongCP () which is elegantly resolved by the Peccei-Quinn (PQ) mechanism Peccei:1977hh (). It predicts a hypothetical particle called the axion as a pseudo-Nambu-Goldstone (NG) boson of an anomalous global symmetry which is spontaneously broken at an intermediate scale GeV kim-carosi (). The PQ symmetry is realized typically in the context of a heavy quark (KSVZ) model KSVZ () or a two-Higgs-doublet (DFSZ) model DFSZ ().

The PQ symmetry breaking may be related to the seesaw mechanism explaining the smallness of the observed neutrino masses Minkowski:1977sc (); Konetschny:1977bn (); Foot:1988aq (); Kang:1980kn () identifying the PQ symmetry as the lepton number  Cheng:1995fd (); Mohapatra:1982tc (). Let us note that the seesaw mechanism realized at the intermediate scale can provide a natural way to explain the matter-antimatter asymmetry in the universe through leptogenesis Fukugita:1986hr (). 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 Sikivie:2006ni (). 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 Asztalos:2003px (), CAPP CAPP () and IAXO IAXO1 (). 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 moviates us to consider minimal extensions of the SM in which various seesaw models Minkowski:1977sc (); Konetschny:1977bn (); Foot:1988aq () 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 II and III, respectively. The corresponding model predictions are presented in Section IV, and then we conclude in Section V.

## Ii Minimally Extended Standard Model for the PQ and Seesaw Mechanism

A PQ seesaw model is characterized by how a global 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 charges. Such an symmetry is supposed to be broken spontaneously by the vacuum expectation value of a scalar field assuming a scalar potential:

 V(σ)=λσ(|σ|2−12v2σ)2 (1)

with GeV which sets the scales of the axion decay constant and the heavy seesaw particles. In the case of the type-I and type-II seesaw introducing a singlet fermion (right-handed neutrino) Minkowski:1977sc () and a Higgs triplet scalar Konetschny:1977bn () respectively, their combinations with the KSVZ and DFSZ axion models leads to the same results to the conventional ones. Thus, we consider the type-III seesaw (by heavy lepton triplets) Foot:1988aq () implementing the PQ symmetry in the manner of KSVZ or DFSZ.

• KSVZ+type-III (KSVZ-III) : In a KSVZ model combined with type-III seesaw (lepton triplets with zero hypercharge), we add as usual an extra heavy quark field which transforms as (3,1,0) under , and three (Majorana) lepton triplets which transform as (1,3,0). The right-handed and left-handed lepton triplets are denoted by

 Σ=(NR/√2E+RE−R−NR/√2),Σc=((NR)c/√2(E−R)c(E+R)c−(NR)c/√2) (2)

where heavy neutrino , heavy charged leptons , with the charge conjugation and the Pauli matrix . The non-trivial X-charges are assigned as follows

 X σ ΨL ΨR Σ LL lR +1 +12 −12 ∓12 ∓12 ∓12
(3)

compatible with the Yukawa Lagrangian for the KSVZ-III model,

 −LKSVZ−III±Yuk = −LSMYuk+¯¯¯¯ΨLhΨσΨR+¯¯¯¯LYD~ΦΣ+{12Tr[¯¯¯¯¯¯ΣchΣσΣ]12Tr[¯ΣchΣσ∗Σ]+h.c. (4)

where and stand for the Higgs doublet and the lepton doublet in the SM, respectively, and . Depending on the -charge signs of the triplet fermion one couples or to the triplet as denoted by III or III, respectively. Note that we took the normalization of in Eq. (3) under which the QCD anomaly is the number of the heavy quarks: .
After the breaking by an appropriate scalar potential (1), the complex scalar field can be written as

 σ=1√2eiAσ/vσ(vσ+ρ) (5)

where is nothing but the KSVZ axion, and the real scalar is supposed to get mass 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 Higgs doublets, with , and a Higgs singlet , and allowing the scalar potential term

 V(Φ1,Φ2,σ)∋λΦσΦ†1Φ2σ2+h.c. (6)

which sets the PQ () charge relation of the two Higgs bosons: again under the normalization of .
Then the Yukawa Lagrangian for DFSZ-III reads

 −LDFSZs−III±Yuk = ¯¯¯¯QLYu~Φ2uR+¯¯¯¯QLYdΦ1dR+¯¯¯¯LYℓΦsℓR (7) + ¯¯¯¯LYDΣ~Φ2+{12Tr[¯¯¯¯¯¯ΣchΣσΣ]12Tr[¯¯¯¯¯¯ΣchΣσ∗Σ]+h.c.,

where one can choose or depending on which we categorize two different DFSZ models. As we again have two choices for the triplet mass operator with or , there are four different DFSZ-III models.
Eqs. (6,7) give six -charge relations to be satisfied by the eight fields (other than ). As will be discussed shortly, the orthogonality of the axion and the longitudinal degree of the boson gives another condition. Then, one finds that there is freedom to choose one of the three quark charges. Taking , we get the following X-charge assignment:

 X σ Φ1 Φ2 QL uR dR Σ LL lR 1 +Xd −Xu 0 −Xu −Xd ∓12 ∓12+Xu ∓12+Xu−XΦs
(8)

where we have and leading to the QCD anomaly with the number of the generation .
After the breaking of by the vacuum expectation values, and , of and , the axion and the longitudinal degree of the boson denoted by and , are given by Chun:1995hc () :

 a ∝ Xdv1A1−Xuv2A2+XσvσAσ, G0 ∝ v1A1+v2A2, (9)

where and are the phase fields of and . Then the orthogonality of and is guaranteed by

 Xd=2x(x+1/x)andXu=2/x(x+1/x) (10)

with the normalization and .

## Iii Minimal Supersymmetric PQ Seesaw Model

To implement the PQ symmetry in supersymmetric models, let us introduce two chiral superfields and having the opposite X charges, say, , and its spontaneous breaking is assumed to occur by the typical superpotential:

 WPQ=λS^S(^σ^¯σ−12vσv¯σ) (11)

where and is implied in the notation. Here is a gauge singlet superfield and carry PQ charge zero.

The supersymmetric version of the KSVZ model introduces the heavy quark superpotential

 WKSVZ=WMSSM+^ΨhΨ^Ψc^¯σ (12)

which defines the PQ charge relation: leading to the QCD anomaly: as in the nonsupersymmetric case. Here is the usual Minimal Supersymmetric Standard Model (MSSM) superpotential given by

 WMSSM=^QYu^uc^Hu+^QYd^dc^Hd+^LYℓ^ℓc^Hd+μ^Hu^Hd (13)

which is separated from the PQ mechanism.

The supersymmetric DFSZ model provides a natural framework to resolve the problem as well kim-nilles () by extending the Higgs sector

 WDFSZ=^QYu^uc^Hu+^QYd^dc^Hd+^LYℓ^ℓc^Hd+λμ^σ2MP^Hu^Hd (14)

where is the reduced Planck mass and the right size of the term, , arises after the PQ symmetry breaking. The usual PQ charges assignment consistent with the above superpoential is

 X ^σ ^Hu ^Hd ^Q ^uc ^dc ^L ^lc +1 −Xu −Xd 0 +Xu +Xd +XL −XL+Xd
(15)

where we have put as before and follows from the charge normalization of . At this stage, there is arbitrariness in choosing the value of , 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 .

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.

• KSVZ+Type-II (sKSVZ-II): Type-II seesaw introduces a Dirac pair of triplet superfields with the hypercharge : and . Its combination with the KSVZ model can be realized by the superpotential:

 WsKSVZ−II± = WKSVZ+^LYν^L^Δ+λd^Hd^Hd^Δ+{λσ^¯σ^Δ^Δcλ¯σ^σ^Δ^Δc (16)

which set the PQ charges of the leptonic fields:

(17)
• DFSZ+Type-II (sDFSZ-II): Similarly to the previous case, the superpotential for the DFSZ model combined with Type-II seesaw takes the form:

 WsDFSZ−II± = WDFSZ+^LYν^L^Δ+λd^Hd^Hd^Δ+{λσ^¯σ^Δ^Δcλ¯σ^σ^Δ^Δc (18)

which is invariant under the PQ symmetry with the charge assignment of (15) extended to the leptonic sector as follows:

 X ^L ^lc ^Δ ^Δc −Xd +2Xd +2Xd ±1−2Xd
(19)
• KSVZ+Type-III (sKSVZ-III): In supersymmetric Type-III seesaw one introduces three triplet superfields (with ) denoted by

 ^Σ=(^Nc/√2^E^Ec−^Nc/√2) (20)

Then the whole superpotential of the KSVZ model realized in Type-III seesaw is

 WsKSVZ−III± = WKSVZ+^LYD^Σ^Hu+{12λσ^¯σTr[^Σ^Σ]12λ¯σ^σTr[^Σ^Σ] (21)

which defines the PQ charges of the leptonic fields as in the non-supersymmetric case:

 X ^L ^lc ^Σ ∓12 ±12 ±12
(22)
• DFSZ+Type-III (sDFSZ-III): Type-III seesaw introduces three triplet superfields (with ):

 ^Σ=(^Nc/√2^E^Ec−^Nc/√2) (23)

The superpotential is

 WsDFSZ−III± = WDFSZ+^LYD^Σ^Hu+{12λσ^¯σTr[^Σ^Σ]12λ¯σ^σTr[^Σ^Σ] (24)

which set the PQ charges of the leptonic fields:

 X ^L ^lc ^Σ ∓12+Xu ±12−Xu+Xd ±12
(25)

## Iv 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 kim-carosi (). After the PQ symmetry breaking by a generic number of scalar fields having the PQ charge and , the following combination of the phase fields defines the axion direction:

 a=∑ϕXϕvϕAϕ/vPQwithvPQ=√∑ϕX2ϕv2ϕ. (26)

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∋aFag2332π2Gaμν~Gμνa+~caγγaFae232π2Fμν~Fμν (27)

where the axion decay constant is defined by , and is the ‘modified’ electromagnetic anomaly normalized by the color anomaly of the PQ symmetry. Below the QCD scale MeV, the axion-gluon-gluon anomaly coupling induces the axion potential

 V(a)=m2aF2a(1−cosaFa) (28)

where the axion mass is calculated to be

 ma≃√z1+zmπfπFa≈6μeV(1012GeVFa) (29)

with , MeV and MeV.

Under the PQ charge normalization of (and ) in the non-supersymmetric (supersymmetric) axion models discussed in the previous section, the color anomaly counts the number of distinct vacua developed in the axion potential (28) which sets the axionic domain wall number . Then the axion-photon-photon coupling constant is given by

 ~caγγ=caγγ−cχSB (30)
 withcaγγ≡2Tr[XQ2em]c3andcχSB≡234+1.05z1+1.05z≈1.98

where counts the electromagnetic anomaly normalized by the color anomaly, and is the modified effect by the chiral symmetry breaking including the strange quark contribution.

Each PQ seesaw model presented in the previous section give a different prediction on the coefficient and thus on the future sensitivity of the axion search at ADMX or CAPP. Following Eq. (30), the electromagnetic anomaly of each model is given by

 caγγ=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩±2Ng=±6KSVZ-III±83±1=113,53DFSZ1-III±13±1=53,−23DFSZ2-III±±10sKSVZ-II±83−23±103=163,−43sDFSZ-II±±2Ng=±6sKSVZ-III±83−23±1=3,1sDFSZ-III± (30)

where and are used for the KSVZ and DFSZ models, respectively.

In FIG.  I, we plot the axion-photon-photon coupling as a function of axion mass , and compare them with the conventional KSVZ () and DFSZ (, or ) predictions. The experiments such as ADMX Asztalos:2003px (), CAPP CAPP (), CAST Arik:2015cjv (), IAXO IAXO1 (), are projected to probe some regions of the parameter space of the axion coupling to photons and its mass. In Fig. 1. the cyan- (red-) thick dashed boundary indicates the future sensitivity of the axion dark matter search by ADMX (CAPP) expAxion (). The current ADMX results Hoskins:2011iv () excludes only a limited region of KSVZ type models and DFSZ-III, sDFSZ-II models over the mass range of eV. Solar axion search experiments like CAST and IAXO are also sensitive to the PQ axions. CAST probes the axion mass range of eV for , while IAXO would have sensitivity to much larger axion masses compared to CAST if . Most recently, CAST has improved the limit on the axion-photon-photon coupling to GeV at 95 C.L. Arik:2015cjv (). This may exclude the models above the KSVZ line over the mass range eV, which can be seen by considering the mass values at GeV in the various models like sKSVZ-II ( eV), sKSVZ-II, sKSVZ-III, KSVZ-III ( eV), sKSVZ-III, KSVZ-III ( eV), sDFSZ-II ( eV), DFSZ-III ( eV), and KSVZ ( eV).

## V 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 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. (30) and FIG. I.

###### Acknowledgements.
Y.H.Ahn is supported by IBS under the project code, IBS-R018-D1.

### Footnotes

1. E-mail : yhahn@ibs.re.kr
2. Email: ejchun@kias.re.kr
3. preprint: KIAS-P15031
4. preprint: CTPU-15-12

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