Minimal models for axion and neutrino

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Physics Letters B

www.elsevier.com/locate/physletb

Minimal models for axion and neutrino

Y.H. Ahn

^a,∗

, Eung Jin Chun

^b,∗

aCenterforTheoreticalPhysicsoftheUniverse,InstituteforBasicScience(IBS),Daejeon,34051,RepublicofKorea bSchoolofPhysics,KIAS,Seoul130-722,RepublicofKorea

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received17October2015

Receivedinrevisedform25November2015 Accepted27November2015

Availableonline30November2015 Editor:A.Ringwald

ThePQmechanismresolvingthestrongCPproblemandtheseesawmechanismexplainingthesmallness of neutrino masses may be related in away that the PQ symmetry breaking scale and the seesaw scalearisefromacommonorigin.DependingonhowthePQsymmetryandtheseesawmechanismare realized,onehasdifferentpredictionsonthecolorandelectromagneticanomalieswhichcouldbetested inthefutureaxiondarkmattersearchexperiments.Motivatedbythis,weconstructvariousPQseesaw modelswhichareminimallyextendedfromthe(non-)supersymmetricStandardModelandthussetup differentbenchmarkpointsontheaxion–photon–photoncouplingincomparisonwiththestandardKSVZ andDFSZmodels.

©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. 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) bosonof an anomalous global symmetry U(1)PQ which is spontaneously broken at an intermediate scale vPQ≈^{109–12 GeV[3].ThePQsymmetryisrealizedtypicallyinthe} contextofaheavyquark(KSVZ)model[4]oratwo-Higgs-doublet (DFSZ)model[5].

The PQ symmetry breaking may be related to the seesaw mechanism explaining the smallness of the observed neutrino masses [6–9] identifying the PQ symmetry as the lepton num- berU(1)L[10,11].Letusnotethattheseesawmechanismrealized at the intermediate scale v_PQ can provide a natural way to ex- plain the matter–antimatter asymmetry in the universe through leptogenesis[12].Anattractivefeatureofthisscenarioisthatthe axion isa good candidateofcold darkmatter through its coher- entproductionduringtheQCDphasetransition[13].Astheaxion is well-motivated dark matter candidate, serious efforts are be- ingmadetosearch foritbyvarious experimentalgroupssuch as ADMX[14],CAPP[15]andIAXO[17].ThetraditionalKSVZorDFSZ modelshavebeenconsideredastwomajorbenchmarksinsearch fortheaxiondarkmatter.

*

Correspondingauthor.

E-mailaddresses:yhahn@ibs.re.kr(Y.H. Ahn),ejchun@kias.re.kr(E.J. Chun).

In the context of the PQ mechanism combinedwith the see- saw mechanism,however,theelectromagneticandcoloranomaly coefficientscan takedifferent values,andthuscan havedifferent predictionsinthefutureaxionsearchexperiments.Thismotivates ustoconsiderminimalextensionsoftheSMinwhichvarioussee- sawmodels[6–8]areextendedtorealizetheKSVZorDFSZaxion, and compare their predictions with the conventional KSVZ and DFSZmodels.

Thispaperisorganizedasfollows.Wewillfirstsetupminimal extensionsoftheSMtocombinethePQ andseesaw mechanisms innon-supersymmetricandsupersymmetrictheoriesinSections2 and3,respectively.Thecorresponding modelpredictionsarepre- sentedinSection4,andthenweconcludeinSection5.

2. MinimallyextendedstandardmodelforthePQandseesaw 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,isimplementedtoactonaspecific setofextrafermions carrying non-trivial X charges. Such an U(1)X symmetry is sup- posed to be broken spontaneously by the vacuum expectation valueofascalarfield

σ

assumingascalarpotential:

V

( σ ) = λ

_σ

( | σ |

²

−

¹

2v²_σ

)

² (1)

with vσ ∼^{109–12GeV which sets the scales of the axion decay} constant Fa and the heavy seesaw particles. In the case of the type-I and type-II seesaw introducing a singlet fermion (right- handed neutrino) [6] and a Higgs triplet scalar [7] respectively, http://dx.doi.org/10.1016/j.physletb.2015.11.067

0370-2693/©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

their combinations withthe KSVZ and DFSZ axion models leads to the same results to the conventional ones. Thus, we consider thetype-IIIseesaw(byheavyleptontriplets)[8]implementingthe PQsymmetryinthemannerofKSVZorDFSZ.

• ^KSVZ+^type-III (KSVZ-III): In a KSVZ model combined with type-IIIseesaw(leptontripletswithzerohypercharge),weadd as usual an extra heavy quark field which transforms as (3,1,0)underSU(3)c×^SU(2)L×^U(1)Y,andthree(Majorana) leptontriplets whichtransformas(1,3,0). Theright-handed andleft-handedleptontripletsaredenotedby

=

NR

/ √

2 E⁺_R E⁻_R

−

^NR

/ √

2

,

^c

=

(

N_R

)

^c

/ √

2

(

E⁻_R

)

^c

(

E⁺_R

)

^c

−(

NR

)

^c

/ √

2

(2) where heavy neutrino NR, heavy charged leptons E^±_R, ^c≡ i

τ

2˜^ci

τ

2withthechargeconjugation˜^c=^C¯^T andthePauli matrix

τ

2.Thenon-trivialX-chargesareassignedasfollows

σ

L

R

L_L l_R

X

+

¹

+

¹₂

−

¹₂

∓

¹₂

∓

¹₂

∓

¹₂ ⁽³⁾

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

−

L^KSVZ-III_Yuk ^±

= −

L^SM_Yuk

+

Lh

σ

R

+

^LYD

˜ +

1

2Tr

[

^ch

σ ]

1

2Tr

[

^ch

σ

^∗

] +

^{h.c. (4)}

where =(φ⁺,φ⁰)^T and L=(

ν

L,L) stand for the Higgs doublet and the lepton doublet in the SM, respectively, and ˜ =ⁱ

τ

2^∗. Depending on the X-charge signs of the triplet fermiononecouples

σ

or

σ

^∗tothetripletasdenotedbyIII₊ or III₋,respectively. Note that we took the normalization of Xσ=^{1 inEq.(3)underwhichtheQCD anomalyisthenum-} beroftheheavyquarks:c3=^N.

After the U(1)X breaking by an appropriate scalar potential (1),thecomplexscalarfield

σ

canbewrittenas

σ = √

¹

2e^{i Aσ/vσ}

(

v_σ

+ ρ )

(5)

where a≡^{Aσ is nothing but the KSVZ axion, and the real} scalar

ρ

issupposed to getmass ∼^vσ which setsthe axion andseesawscales.

• ^DFSZ+^type-III(DFSZ-III):InaDFSZaxionmodel,thePQsym- metryisimplementedbyextendingtheHiggssectorwithtwo Higgsdoublets, i=(φ⁺_i ,φ_i⁰)^T withi=¹,2,andaHiggssin- glet

σ

,andallowingthescalarpotentialterm

V

(

1

,

2

, σ ) λ

_σ

^†₁

2

σ

²

+

h.c. (6) whichsetsthePQ(X)chargerelationofthetwoHiggsbosons:

2=^X1−^X2 againunderthenormalizationofXσ=^1.

ThentheYukawaLagrangianforDFSZ-IIIreads

−

L^DFSZ_Yuk ^s-III±

=

Q_LYu

˜

2uR

+

Q_LY_d

1dR

+

L Y

s

R

+

^{L Y}D

˜

2

+

1

2Tr

[

^ch

σ ]

1

2Tr

[

^ch

σ

^∗

] +

^h.c.

,

(7) where one can choose s=1 or 2 depending on which we categorize two differentDFSZmodels. As we againhavetwo

choicesforthetripletmass operatorwith

σ

or

σ

^∗,thereare fourdifferentDFSZ-IIImodels.

Eqs.(6),(7)give six X-charge relations tobesatisfiedby the eight fields (other than

σ

). As will be discussed shortly, the orthogonalityofthe axionandthelongitudinaldegree ofthe Z boson gives another condition. Then, one finds that there is freedom to choose one ofthe three quark charges. Taking XQ_L≡^{0,wegetthefollowingX-charge}assignment:

σ

1 2 QL uR dR LL lR

X 1 +^Xd −^Xu 0 −^Xu −^Xd ∓¹₂ ∓¹₂+^Xu ∓¹₂+^Xu−^Xs

(8) wherewehave X₁=^Xd and X₂= −^Xu leadingtotheQCD anomalyc3=(Xu+^Xd)Ng=^{6 withthenumberofthegener-} ation Ng=^3.

AfterthebreakingofSU(2)L×^U(1)Y×^U(1)X bythevacuum expectationvalues,v1,v2andvσ,of1,2and

σ

,theaxion andthelongitudinaldegreeofthe Z bosondenotedbyaand G⁰,aregivenby[18]:

a

∝

^Xdv1A1

−

^Xuv2A2

+

^Xσv_σA_σ

,

G⁰

∝

^v1A₁

+

^v2A₂

,

(9)

where A1, A2 and Aσ are thephasefieldsof1,2 and

σ

. ThentheorthogonalityofaandG⁰isguaranteedby

X_d

=

^2x

(

x

+

¹

/

x

)

^{and Xu}

=

²

/

x

(

x

+

¹

/

x

)

⁽¹⁰⁾

withthenormalizationXσ=^{1 andx}≡^v2/v1. 3. MinimalsupersymmetricPQseesawmodel

ToimplementthePQsymmetryinsupersymmetricmodels,let usintroduce two chiralsuperfields

σ

ˆ and

σ

ˆ¯ havingthe opposite X charges,say, Xσ= −^Xσ¯ ≡ +^1,andits spontaneousbreaking is assumedtooccurbythetypicalsuperpotential:

W_PQ

= λ

SS

ˆ ( σ ˆ σ ˆ¯ −

¹

2v_σv_σ¯

)

(11)

where

σ

=^vσ/√

2 and ¯

σ

=^vσ¯/√

2 isimpliedinthenotation.

Here Sˆ isagaugesingletsuperfieldandcarryPQchargezero.

ThesupersymmetricversionoftheKSVZmodelintroducesthe heavyquarksuperpotential

W_KSVZ

=

^WMSSM

+ ˆ

h

ˆ

^c

σ ˆ¯

(12) whichdefinesthePQchargerelation:X+^Xc= −^Xσ¯ ≡ +^{1 lead-} ingtotheQCDanomaly:c3=^N+^c asinthenonsupersymmetric case. Here WMSSMistheusual MinimalSupersymmetricStandard Model(MSSM)superpotentialgivenby

W_MSSM

= ˆ

^{Q Y}uu

ˆ

^cH

ˆ

_u

+ ˆ

^{Q Y}d_d

ˆ

^cH

ˆ

_d

+ ˆ

^LY

ˆ

^cH

ˆ

_d

+ μ

H

ˆ

_uH

ˆ

_d (13) whichisseparatedfromthePQmechanism.

ThesupersymmetricDFSZmodelprovidesanaturalframework toresolvethe

μ

problemaswell[19]byextendingtheHiggssec- tor

W_DFSZ

= ˆ

^{Q Y}u^u

ˆ

^cH

ˆ

_u

+ ˆ

^{Q Y}dd

ˆ

^cH

ˆ

_d

+ ˆ

LY

ˆ

^c_H

ˆ

_d

+ λ

_μ

σ ˆ

²

M_PH

ˆ

_uH

ˆ

_{d (14)}

where MP isthereducedPlanckmassandtherightsizeofthe

μ

term,

μ

=λμv²σ/2M_P,arisesafterthePQsymmetrybreaking.The

usual PQ chargesassignment consistent withtheabove superpo- tential is

ˆ

σ

H

ˆ

_u H

ˆ

_d Q

ˆ

u

ˆ

^c _d

ˆ

^c L

ˆ ˆ

_l^c X

+

¹

−

^Xu

−

^Xd 0

+

^Xu

+

^Xd

+

^XL

−

^XL

+

^Xd

(15) where we have put XQ ≡^{0 as before and X}u+^Xd=^{2 follows} fromthecharge normalizationof Xσ= +1.Atthisstage,thereis arbitrarinessin choosing the value of XL, but it will be fixed in seesawextendedPQmodels whichhasnophysicalconsequences.

Notethat theQCD anomalyofthesupersymmetricDFSZmodelis againgivenbyc3=(Xu+^Xd)Ng=^6.

Nowletusconsidertheseesawextensionsofthesupersymmet- ricPQmodels.Asinthenon-supersymmetriccase,Type-I seesaw introducingright-handed (singlet)neutrinos doesnot change the resultsof thestandard KSVZ andDFSZ models.Thus, we discuss theType-IIand-IIIextensionsinorder.

•^KSVZ+^Type-II(sKSVZ-II): Type-II seesaw introduces a Dirac pair of SU(2)L triplet superfields with the hypercharge Y =

±^1:ˆ =(ˆ⁺⁺,ˆ⁺,ˆ⁰)andˆ^c=(ˆ^c0,ˆ⁻,ˆ⁻⁻).Itscombi- nation withtheKSVZmodelcan berealizedby thesuperpo- tential:

W^sKSVZ-II±

=

^WKSVZ

+ ˆ

^LYνL

ˆ ˆ + λ

dH

ˆ

_dH

ˆ

_d

ˆ +

λ

_σ

σ ˆ¯ ˆ ˆ

^c

λ

_σ¯

σ ˆ ˆ ˆ

^{c (16)}

whichsetthePQchargesoftheleptonicfields:

L

ˆ ˆ

_l^c

ˆ ˆ

^c

X 0 0 0

±

^{1 (17)}

•^DFSZ+^Type-II(sDFSZ-II): Similarly to the previous case, the superpotentialfortheDFSZmodelcombinedwithType-IIsee- sawtakestheform:

W^sDFSZ-II±

=

^WDFSZ

+ ˆ

^LYνL

ˆ ˆ + λ

_dH

ˆ

_dH

ˆ

_d

ˆ +

λ

_σ

σ ˆ¯ ˆ ˆ

^c

λ

_σ¯

σ ˆ ˆ ˆ

^{c (18)}

which is invariant under the PQ symmetry with the charge assignmentof(15)extendedtotheleptonicsectorasfollows:

L

ˆ ˆ

l^c

ˆ ˆ

^c

X

−

X_d

+

2X_d

+

2X_d

±

1

−

2X_d (19)

•^KSVZ+^Type-III(sKSVZ-III):InsupersymmetricType-IIIseesaw one introduces threetriplet superfields(with Y=^{0) denoted} by

ˆ =

_N

ˆ

^c

/ √

2 _E

ˆ

E

ˆ

^c

− ˆ

^Nc

/ √

2

(20) Then thewholesuperpotential oftheKSVZmodelrealizedin Type-IIIseesawis

W^sKSVZ-III±

=

^WKSVZ

+ ˆ

^LYD

ˆ

H

ˆ

_u

+

1

2

λ

_σ

σ ˆ¯

Tr

[ ˆ ˆ ]

1

2

λ

_σ¯

σ ˆ

Tr

[ ˆ ] ˆ

⁽²¹⁾

which definesthe PQcharges oftheleptonic fields asinthe non-supersymmetriccase:

ˆ

L

ˆ

_l^c

ˆ

X

∓

¹₂

±

¹₂

±

¹₂ ⁽²²⁾

• ^DFSZ+^Type-III(sDFSZ-III): Type-III seesaw introduces three tripletsuperfields(withY=^0):

ˆ =

N

ˆ

^c

/ √

2 E

ˆ

E

ˆ

^c

− ˆ

^Nc

/ √

2

(23) Thesuperpotentialis

W^sDFSZ-III±

=

^WDFSZ

+ ˆ

^LYD

ˆ

H

ˆ

_u

+

1

2

λ

_σ

σ ˆ¯

Tr

[ ˆ ] ˆ

1

2

λ

_σ¯

σ ˆ

Tr

[ ˆ ] ˆ

⁽²⁴⁾

whichsetthePQchargesoftheleptonicfields:

L

ˆ ˆ

_l^c

ˆ

X

∓

¹₂

+

Xu

±

¹₂

−

Xu

+

X_d

±

¹₂ ⁽²⁵⁾

4. Modelimplicationstotheaxiondetection

TodiscusstheimplicationsofthePQseesawmodelspresented inthe previous sections,letusfirst summarizesome basicprop- erties of the axion relevant for our discussion [3]. After the PQ symmetrybreaking byageneric numberofscalarfieldsφ having thePQcharge Xφ andφ=^vφ/√

2,thefollowingcombinationof thephasefields Aφdefinestheaxiondirection:

a

=

φ

XφvφAφ

/

vPQ with vPQ

=

φ

X²_φv²_φ

.

(26)

IntegratingoutalltherelevantPQ-chargedfermions,theaxiongets the effective axion–gluon–gluon and axion–photon–photon cou- plings through its color and electromagnetic anomalies, respec- tively:

−

L

^a Fa

g₃² 32

π

^2G

a

μνG

˜

_a^μν

+ ˜

^caγ γ a Fa

e² 32

π

^2FμνF

˜

μν (27)

where the axion decay constant Fa is defined by Fa≡^vPQ/c3, and ˜caγ γ is the ‘modified’ electromagnetic anomaly normalized by the color anomaly c₃ of the PQ symmetry. Below the QCD scale QCD∼^{200 MeV,the} axion–gluon–gluon anomaly coupling inducestheaxionpotential

V

(

a

) =

^ma²F_a²

1

−

^{cos a} Fa

(28) wheretheaxionmassiscalculatedtobe

m_a

√

z 1

+

^z

m_πf_π

F_a

≈

^{6 μeV}

10¹²GeV F_a

(29) withz≡^mu/md≈⁰.5,mπ=135 MeV and fπ=^{92 MeV.}

UnderthePQchargenormalizationofXσ= +^{1 (andX}σ¯ = −¹⁾ in the non-supersymmetric (supersymmetric) axion models dis- cussed in the previous section, the color anomaly c3 counts the number of distinct vacua developed in the axion potential (28) whichsetstheaxionicdomainwallnumberN_DW= |^c3|^{.Then the} axion–photon–photoncouplingconstantisgivenby

˜

caγ γ

=

^caγ γ

−

^cχS B (30)

with caγ γ

≡

^2Tr

[

^{X Q}em²

]

c3

and c_χS B

≡

² 3

4

+

¹

.

05z 1

+

¹

.

05z

≈

¹

.

98 wherecaγ γ countstheelectromagneticanomalynormalizedbythe coloranomaly,andcχS B isthemodified effectbythechiralsym- metrybreakingincludingthestrangequarkcontribution.

Fig. 1.Theaxion–photon–photoncoupling|gaγ γ|asafunctionofaxionmassma invariousPQseesawmodels.Thefuturesensitivities oftheADMXandCAPPex- perimentsare shown inthe cyan and redthick-dashed lines,respectively. (For interpretationofthereferencestocolorinthisfigure,thereaderisreferredtothe webversionofthisarticle.)

EachPQseesawmodelpresentedintheprevioussection gives adifferentpredictiononthe coefficientc_aγ γ andthus onthefu- ture sensitivityof the axion search at ADMXor CAPP. Following Eq.(31),theelectromagneticanomalyofeachmodelisgivenby

c_{aγ γ}

=

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

±

^2Ng

= ±

^{6 KSVZ-III}± 8

3

±

¹

=

¹¹₃

,

⁵₃ DFSZ1-III_±

1

3

±

¹

=

⁵₃

, −

²₃ ^DFSZ2-III_±

±

10 sKSVZ-II_±

8

3

−

²₃

±

¹⁰₃

=

¹⁶₃

,−

⁴₃ ^sDFSZ-II±

±

2Ng

= ±

6 sKSVZ-III_±

8

3

−

²₃

±

¹

=

³

,

1 sDFSZ-III_±

(31)

where NDW=1 and 6 are used for the KSVZ and DFSZ models, respectively.

In Fig. 1, we plot the axion–photon–photon coupling gaγ γ ≡

˜

caγ γ

α

em/2

π

Faasafunctionofaxionmassma,andcomparethem withtheconventional KSVZ(caγ γ =^{0) andDFSZ(c}aγ γ =⁸/3, or 1/3) predictions.Theexperimentssuch asADMX[14],CAPP[15], CAST[16],IAXO[17],are projected toprobe someregions ofthe parameterspaceoftheaxioncouplingtophotonsanditsmass.In Fig. 1thecyan- (red-) thickdashedboundaryindicates thefuture sensitivityoftheaxiondarkmatter searchby ADMX(CAPP)[21].

The current ADMX results [20] exclude only a limited region of KSVZtypemodelsandDFSZ₂-III₋,sDFSZ-II_±modelsoverthemass range ofma=³.3–3.69 μeV. Solar axion search experimentslike CASTandIAXOarealsosensitivetothePQaxions.CASTprobesthe axion massrangeof ma≈⁰.1–1 eV for gaγ γ⁹×^{10−11GeV−1,} while IAXO would have sensitivity to much larger axion masses compared to CAST if g_aγ γ ⁹×^{10−12GeV−1. Most recently,} CAST has improved the limit on the axion–photon–photon cou- pling to gaγ γ <1.47×^{10−10GeV−1 at 95% C.L. [16]. This may} excludethemodelsabovetheKSVZlineoverthemassrangema 0.06–0.4 eV,whichcanbeseenbyconsideringthemassvaluesat

gaγ γ =¹.47×^{10−10GeV−1 in thevarious modelslike sKSVZ-II}− (ma⁰.06 eV), sKSVZ-II₊, sKSVZ-III₋, KSVZ-III₋ (ma⁰.1 eV), sKSVZ-III₊, KSVZ-III₊ (m_a⁰.2 eV), sDFSZ-II_± (m_a⁰.24 eV), DFSZ2-III₋(ma⁰.3 eV),andKSVZ(ma⁰.4 eV).

5. Conclusion

We have considered minimal extensions of the SM combin- ing the KSVZ or DFSZ axion withvarious seesaw models in the frameworkofthe(non-)supersymmetrictheories,whichprovides apopularsolutiontothestrongCPproblemaswellasthesmall- nessofneutrinomasses.Wehaveshowedthatdependingonhow toembedU(1)PQinaseesawmodel,theelectromagneticandcolor anomaly coefficients take different values, and thus each model has a different prediction on the axion–photon–photon coupling whichcouldbetestedinthefutureaxionsearchexperiments.This sets upvariousbenchmarkpointsfortheminimalPQseesawmod- elsincomparisonwiththestandardKSVZandDFSZmodelswhich aresummarizedinEq.(31)andFig. 1.

Acknowledgement

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

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