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Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

A simple motivated completion of the standard model below the Planck scale: Axions and right-handed neutrinos

Alberto Salvio

DepartamentodeFísicaTeórica,UniversidadAutónomadeMadridandInstitutodeFísicaTeóricaIFT-UAM/CSIC,Madrid,Spain

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

Articlehistory:

Received22January2015

Receivedinrevisedform5March2015 Accepted6March2015

Availableonline9March2015 Editor: M.Trodden

Keywords:

Higgs Vacuumstability Axions

Right-handedneutrinos Cosmicinflation

We study a simple Standard Model (SM) extension, which includes three families of right-handed neutrinoswithgenericnon-trivialflavorstructureandaneconomicimplementationoftheinvisibleaxion idea. Wefindthatinsomeregionsoftheparameter spacethismodel accountsforallexperimentally confirmedpiecesofevidenceforphysicsbeyondtheSM:itexplainsneutrinomasses(viathetype-Isee- saw mechanism),darkmatter, baryonasymmetry(throughleptogenesis),solvesthestrong CPproblem andhasastableelectroweakvacuum.ThelastpropertymayallowustoidentifytheHiggsfieldwiththe inflaton.

©2015TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Although no unambiguous signal of physics beyond the SM (BSM)hasappearedsofarattheLHC, thereisnodoubt thatthe SM has to be extended. Neutrino oscillations, which lead to the existenceofsmall(left-handed)neutrinomasses,andtheobserva- tionalevidence fordarkmatter (DM)is enough tostate that the SMisincomplete.

Other unsatisfactory features of the SM are an insufficient baryonasymmetry oftheuniverse,thestrongCP,gauge hierarchy andcosmologicalconstantproblems.

Moreover,precision calculations[1,2] indicatethat theSM po- tentialdevelopsaninstability atascaleoftheorderof1010 GeV, forcentralmeasuredvaluesoftheSMparameters.Thisisnotpar- ticularly worrisomeperse because theprobability oftunnelingto the absolute minimum, where life is impossible, is spectacularly small[2].However, it maylead to some issues during the expo- nentialexpansionoftheearlyuniverse(inflation)[3–5].Moreover, the (absolute) stability up to the Planck scale MPl may lead to thepossibilityofHiggsinflation[6–9],linkingparticlephysicsand cosmology: this is interesting because it provides us with rela- tions betweenparticle physics andcosmologicalobservables.The presenceofsuch an instability intheSM isnot firmly confirmed because ofnon-negligible uncertainties on the top mass andthe

E-mailaddress:alberto.salvio@uam.es.

QCDgaugecoupling;but,ifconfirmed,itwouldsuggest thatright- handed neutrinos (at scales suitable forthe see-saw mechanism andthermalleptogenesis)andthephysics ofthe QCDaxion may berelevantfortheissueoftheelectroweak(EW)vacuuminstabil- ityandthereforeinflation.

Theaimofthispaperistoidentifyasimpleandwell-motivatedmodel where thefollowingsignalsofBSM physicscanallbeaddressedand whichaddstotheSMonlyright-handedneutrinosandtheextrafields neededtoimplementtheaxionidea:

1. Smallneutrinomasses.Weadoptperhapsthe simplestexpla- nation: the type-Isee-sawmechanism based on right-handed neutrinos. The addition of right-handed neutrinos also sym- metrizes the field content ofthe SM givingto each SM left- handedparticlearight-handedcounterpart.

2. Darkmatter. As a DMcandidate we consider theaxion [10], a lightspin-0particlewhoseexistenceisimpliedbythespon- taneous symmetry breaking ofa U(1) symmetry, the Peccei–

Quinn (PQ) symmetry [11] that explains whystrong interac- tionsdonotviolateCP.Inparticular,weconsidertheinvisible axion model proposed by Kim, Shifman, Vainshtein and Za- kharov(KSVZ)[12],whichhasasimplestructureandasmall numberoffreeparameters.

3. Baryonasymmetry. In order to explain such asymmetry we makeuseof(thermal)leptogenesis[13],whichisimplemented with the same right-handed neutrinos that allow the light neutrinostohavemasses.

http://dx.doi.org/10.1016/j.physletb.2015.03.015

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

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4. Inflationandvacuuminstability.Aswestatedbefore,thein- flatoncould beidentifiedwiththeHiggsbosonprovidedthat theEW vacuumisstable,1 takingintoaccountenergies upto the Planckscale.Wethereforelookforregions oftheparam- eter space where the EW vacuum is stable, even for central valuesoftheSMobservables.

5. StrongCPproblem.The solutionweconsider isthefirst and mostfamousone:thePQsymmetry,thesamesymmetrylead- ingtotheaxionDMcandidateabove.

It is important to note that the first two points represent a proof of BSM physics, while the others are indications, although veryplausible ones. The spirithere issimilar to the one of[15], focusingonproblems1,2and3andaddingtotheSM fieldcon- tentonlyright-handedneutrinoswithmassesbelowtheEWscale.

Inthis case, indeed, the right-handed neutrinos can significantly contributetodarkmatter[16](seealso[17]forareview andfur- therreferences) andneutrinooscillationsprovidea mechanismto generatebaryon asymmetry through a different version oflepto- genesis[18].Moreover,it ispossibleto extendthisframework to includethe axion idea andto look forsimultaneous solutions of problems1,2,3,4and5.Herethereis noclaimthat thesimple modelwestudyistheonlyoneabletoaddressalltheseissues.

Inthe list above we did not include the gauge hierarchyand thecosmologicalconstantproblemsbecausetheycanbothbe ad- dressedwithanthropicarguments2[19].Ontheotherhand,there seemstobenoanthropicsolutiontothestrongCPproblem;thus technicalnaturalnessappears to be the only possibleway toex- plainthesmallvalueoftheQCDθ angle.

Letussummarizenowthecontentsofthearticle.InSection2 we define the model. In Section 3 we discuss the observational constraintson its parameters. The theoretical ingredients for the extrapolation up to MPl are provided in Section 4. In Section 5 weinvestigate whetherthemodelcan havea stableEW vacuum takingintoaccount energiesup to MPl.Oneofthe conditionsfor stabilityisthattheHiggsquarticcouplingremainsalwayspositive.

Thereis,however,anotherconditiontobefulfilledtoensureasta- blevacuum. This section contains thecentral new resultsofthis paper.Finally,inSection6weprovideourconclusions.

2. Themodel

WeconsiderthemodelwithLagrangian:

L

=

Lgravity

+

LSM

+

LN

+

Laxion

,

(1)

whererepeatedindicesunderstandasummation.Thegaugegroup ofthemodelistheSMone:

GSM

=

SU

(

3

)

c

×

SU

(

2

)

L

×

U

(

1

)

Y

.

Lgravity are the terms in the Lagrangian, which include the pure gravitationalpartandthepossiblenon-minimalcouplingbetween gravity and the other fields. In particular, the term proportional to |H|2R, where H is the Higgs doublet and R is the Ricci scalar, plays an importantrole in Higgsinflation [6]. LSM is the

1 Byaddingnon-renormalizableoperatorswithindependentcoefficientsonemay entertheregionofmetastability[14],wedonotconsider thispossibilityinthe presentpaper.

2 Thegaugehierarchyproblemcanofcoursebesolvedinatechnicallynatural way(e.g.withSUSY,compositeHiggs,etc.)inmodelsthatexplainsomeoftheis- suesmentionedabove[20];also,SUSYlargeextradimensions[21]offersapossible waytoaddressthecosmologicalconstantproblem;butthisisdoneatthepriceof introducingmanymorefieldsthanthoseofthemodelstudiedhereandsometimes thenecessityofanultravioletcompletionatmuchsmallerenergies.

SM Lagrangian (minimallycoupled to gravity). LN is the part of the Lagrangian that depends on the right-handed neutrinos Ni (i=1,2,3):

LN

=

iNi

 ∂

Ni

+



1

2NiMi jNj

+

Yi jLiH Nj

+

h.c.



,

(2)

where Mi j and Yi j are the elements of the Majorana mass ma- trix M andthe neutrinoYukawa couplingmatrix Y , respectively.

Thanks to the complex Autonne–Takagi factorization, we take M realanddiagonalwithoutlossofgenerality:

M

=

diag

(

M1

,

M2

,

M3

),

where the Mi (i=1,2,3) are mass parameters, the Majorana massesofthethreeright-handedneutrinos.

Finally,LaxionrepresentstheadditionaltermsintheLagrangian duetothechosen axionmodel.As statedintheintroduction,we consider the first invisible axion model (the KSVZ model3). The fieldsofthismodelthatare notcontainedintheSM arethefol- lowing.

AnextraDiracfermion. (InWeylnotation)itisapairoftwo- componentfermionsq1andq2 inthefollowingrepresentation ofGSM

q1

∼ (

3

,

1

)

0

,

q2

∼ (¯

3

,

1

)

0

.

(3) NamelytheyformacoloredDiracfermionwithnointeractions withthegaugefieldsofSU(2)L×U(1)Y.

Anextracomplexscalar. Thisscalar A ischargedunderU(1)PQ andneutralunderGSM.

TheLagrangianofthisaxionmodelis Laxion

=

i



2 j=1

qj

/

Dqj

+ |∂

μA

|

2

− (

y q2Aq1

+

h.c.

) − 

V

(

H

,

A

)

andtheclassicalpotentialofthefullmodelis

V

(

H

,

A

) = λ

H

(|

H

|

2

v2

)

2

+ 

V

(

H

,

A

),

(4) where



V

(

H

,

A

) ≡ λ

A

( |

A

|

2

fa2

)

2

+ λ

H A

( |

H

|

2

v2

)( |

A

|

2

fa2

).

Theparameters v, faandy canbetakenrealandpositivewithout lossofgenerality.ThePQsymmetryactsonq1,q2andA asfollows q1

eiα/2q1

,

q2

eiα/2q2

,

A

eiαA

,

(5) which forbidsan explicitmassterm Mqq1q2+h.c. The SM fields andtheright-handedneutrinosare insteadneutralunderU(1)PQ. Moreover,thereistheaccidentalsymmetry

q1

→ −

q1

,

q2

q2

,

A

→ −

A

.

(6) Thismodelhastheadvantageofbeingsimpleandhaving(inad- dition to the SM andtype-I see-saw parameters) only three real parameters:λH A, λA and y; itis themostgeneralonegiven the fieldcontentandsymmetriesdescribedabove.Inparticular,notice thatλH Aistheonlytree-levelcouplingbetweentheaxionandSM sectors.

TheEWsymmetrybreakingistriggeredby thevacuumexpec- tationvalue (VEV) v174 GeV of theneutralcomponent H0 of

3 Forarecentinterestingworkwhereanotheraxionmodelandscalargenerations ofneutrinomassesareconsidered,see[22].

(3)

the Higgsdoublet. After that the neutrinosacquire a Diracmass matrix

mD

=

v Y

,

(7)

whichcanbeparameterizedas mD

= 

mD1

,

mD2

,

mD3

 ,

(8)

where mDi (i=1,2,3) are column vectors. Integrating out the heavyneutrinos Ni,onethen obtainsthefollowinglight neutrino Majoranamassmatrix

mν

=

mD1mTD1

M1

+

mD2mTD2

M2

+

mD3mTD3

M3

.

(9)

By means of a unitary (Autonne–Takagi) redefinition of the left- handed SM neutrinos we can diagonalizemν to obtain themass eigenvalues m1, m2 and m3 (the left-handed neutrino Majorana masses). Calling Uν the unitary matrix that implements such transformation, also known as the Pontecorvo–Maki–Nakagawa–

Sakata (PMNS)matrix,thatisUν mν U νT =diag(m1,m2,m3),wecan parameterize=V ν P12,where

Vν

=



c

12c13 s12c13 s13eiδ

s12c23c12s13s23eiδ c12c23s12s13s23eiδ c13s23 s12s23c12s13c23eiδc12s23s12s13c23eiδ c13c23

 ,

withsi jsini j),ci jcosi j);θi j aretheneutrinomixingangles and P12 is a diagonal matrix that contains two extra phases, in additiontotheone,δ,containedinV ν :

P12

=

e

iβ1 0 0

0 eiβ2 0

0 0 1

⎠ .

(10)

Eveninthemostgeneralcaseofthreeright-handedneutrinos,it is possibleto express Y in terms oflow-energy observables, the heavymassesM1,M2 andM3 andextraparameters[23]:

Y

=

UνDmR DM

v

,

(11)

where

Dm

diag

(

m1

,

m2

,

m3

),

DM

diag

(

M1

,

M2

,

M3

)

andR isagenericcomplexorthogonalmatrix,whichcontainsthe extra parameters. This is usefulfor usbecause the observational constraintsarenotdirectlyon Y ,butthey areratheronthelow- energyquantitiesmi,Uν andon Mi(seeSection3).Onecanshow that thesimplerandrealistic caseoftwo right-handedneutrinos [24]belowMPlcanberecoveredbysettingm1=0 and

R

=

cos z0

sin z0 10

ξ

sin z

ξ

cos z 0

⎠ ,

wherez isacomplexparameterandξ= ±1.

ThePQsymmetryisbrokenbothspontaneouslyandbyanoma- lies.Thespontaneous symmetrybreaking isinducedby faA, leadingtothefollowingDiracmassof{q1,q2}:

Mq

=

y fa

.

Moreover, A contains a (classically) massless particle, the axion, which acquiresa small massthanks to the quantum breakingof

thePQsymmetry,andamassiveparticlewithsquaredmass M2A

=

fa2



4

λ

A

+

O



v2 fa2



.

(12)

Aswewillreviewbelow,theobservationalboundsimplythatthe correctionsO

v2/fa2

areverysmallandwillbeneglectedinthe following.

3. Observationalconstraints

We now discuss the observational constraints, which we will takeintoaccountintherestofthepaper.

As far as the neutrino massesmi (i=1,2,3) are concerned, datafromatmosphericandsolarneutrinostellusrespectively[25]

(seealso[26–28]forpreviousdeterminations)



m221

=

7

.

50+00..1917

×

105eV2

,



m23l

=

2

.

457+00..047047

×

103eV2

,

wherem2i jm2im2j andm23l≡ m231fornormalorderingand

m23l≡ m232forinvertedordering.

As far as the mixing angles and phases of the PMNS matrix are concerned, the mostrecent central valuesandcorresponding uncertainties can also be found in [25]: for anyordering of the neutrinomassesthe3

σ

rangesare

0

.

270

s212

0

.

344

,

0

.

385

s223

0

.

644

,

0

.

0188

s213

0

.

0251

,

(13)

while δ spans the whole range from 0 to 2

π

at 3

σ

level (for examplefornormalorderingwehaveδ/0=306+3970,while,forin- vertedordering,δ/0=254+6362).Currentlynosignificantconstraints areknownforβ1 andβ2.

Wenowturntotherequirementstohavesuccessfulleptogen- esis[13]:neutrinosshouldbelighterthan0.15 eVandthelightest right-handedneutrinoMajoranamassMl hastofulfill[29]

Ml



1

.

7

×

107GeV

.

(14)

Inorderto beconservativewe havereportedtheweakestbound, but depending on the assumptions one can have stronger con- ditions.4 Notice, however, that the mechanisms of [15] and[18]

discussed in the introduction can evade these bounds and use right-handedneutrinomassesbelowtheEWscale;aswewillsee, it is less challenging to achieve vacuumstability in thiscase. In other models,iftheHiggsfield acquiresalargeVEVduring infla- tion,[30] argued thatthe subsequentHiggsrelaxationtotheEW vacuumcangeneratethebaryonasymmetry.

Regardingtheaxionsector,inordertoaccountforDMthrough themisalignment mechanism[31] (withanorderone initialmis- alignmentangle)andtoeludeaxiondetectiononeobtainsrespec- tively an upper(seee.g. [32])andlower bound(see e.g. [33]) on theorderofmagnitudeofthescaleofPQsymmetrybreaking fa:

108GeV



fa



1012GeV

.

(15)

Theupperboundisobtainedbyrequiringthattheaxionfieldtakes a value oforder fa atearly times,whichiswhat we expect,but is not necessarily the case; also the precise value of the lower boundismodeldependent.Therefore(15)shouldnot betakenas

4 Forexampleifthe initialabundanceofright-handedneutrinosat T Ml is zerothentheboundisMl2.4×109GeV[29].

(4)

sharpbounds, but it certainly gives a plausible range of fa. An- othersource ofuncertaintyisintroducedifone insteadconsiders light right-handed neutrinos [16,17], which can then contribute to dark matter, as mentioned in the introduction; in this case, indeed, the upper bound becomes stronger as it is obtained by requiringtheaxion contributionnot toexceed theobserveddark matterabundance.In anycase,(15)ensures that fa v andthe termsO

v2/fa2

in (12)can be neglected. Moreover, notice that boundson fa canonly constraintheratio MA/

λA asitisclear from(12).WhenMA v andMq v (whichweassume)theEW constraintsarefulfilled.

Inaddition to contributingto darkmatter, the axion also un- avoidablymanifest itself asdark radiation asit is alsothermally produced[34–36].Thispopulationofhotaxionscontributestothe effectivenumberofrelativisticspecies,butthesizeofthiscontri- butioniscurrentlywellwithintheobservationalbounds[36].

Finally,ofcourse wealsohaveconstraintson theSM parame- ters.AfterthediscoveryoftheHiggsbosonattheLHC[37,38]the last SM parameter, the Higgs mass, hasbeen determined within small uncertainties, and there are no free SM parameters any- more.WetakethevaluesanduncertaintiesoftheSMmassesand couplingsgivenin[2](seealsothereferencestherein).Thedeter- minations of[2]are not significantly affected by the presenceof theextraheavydegreesoffreedom.

4. RGEanalysisandthresholds

Sincewe wanttostudythepredictionsofthismodelatener- giesmuch above the EW scale, up to the Planck scale, we need the complete set of renormalizationgroup equations (RGEs). We adoptthe MS renormalizationschemetodefine therenormalized couplings and the corresponding RGEs. Moreover, for a generic renormalizedcouplingg wewritetheRGEsas

dg

d

τ = β

g

,

(16)

whered/d

τ

≡ ¯

μ

2d/d

μ

¯2 and

μ

¯ isthe MS renormalizationenergy scale.Theβ-functionsβgcanalsobeexpandedinloopsas

β

g

= β

(g1)

(

4

π )

2

+ β

(g2)

(

4

π )

4

+ . . . ,

(17)

whereβ(gn)/(4

π

)2n isthen-loopcontribution.

Letusstartfromenergiesmuchabove MA,MqandMi j.Inthis casethe1-loopRGEsare(see[39–42]forpreviousdeterminations ofsometermsintheseRGEs)

β

(1)

g12

=

41g41

10

, β

(1)

g22

= −

19g42

6

, β

(1)

g23

= −

19g34 3

, β

(1)

y2t

=

y2t



9

2yt2

8g23

9g22 4

17g21

20

+

Tr

(

YY

)

 ,

β

λ(1)

H

=



12

λ

H

+

6 y2t

9g21 10

9g22

2

+

2 Tr

(

YY

)

 λ

H

3 y4t

+

9g42 16

+

27g41

400

+

9g22g12 40

+ λ

2H A

2

Tr

((

YY

)

2

), β

λ(1)

H A

=



3 y2t

9g21 20

9g22

4

+

6

λ

H

 λ

H A

+ 

4

λ

A

+

Tr

(

YY

) +

3 y2



λ

H A

+

2

λ

2H A

, β

λ(1)

A

= λ

2H A

+

10

λ

2A

+

6 y2

λ

A

3 y4

,

β

Y(1)

=

Y



3 2y2t

9

40g21

9 8g22

+

3

4YY

+

1 2Tr

(

YY

)

 , β

(1)

y2

=

y2

(

4 y2

8g23

),

whereg3,g2andg1=√

5/3gY arethegaugecouplingsofSU(3)c, SU(2)L andU(1)Y respectivelyand yt isthetopYukawacoupling.

The explicitformofthecomplete setofthe RGEsabovewas not explicitly presented before, but the RGEs fora generic quantum field theory (without gravity)were computedup to 2-loop order in[43](seealso[44]foracomputerimplementationofthem).

Next,weconsiderwhathappensingoingfromenergiesabove MAtoenergiesbelowMA:asdiscussedin[45,41]onehastotake intoaccount ascalarthresholdeffect:inthelow energyeffective fieldtheorybelowMAonehastheeffectiveHiggsquarticcoupling

λ = λ

H

λ

2H A 4

λ

A

.

(18)

This isthe result ofintegrating out the massivescalar degree of freedomattree-level. Thereasonwhythisshiftoccursisbecause setting the heavy scalar to zero is not a consistent truncation, namelyitisnotconsistentwiththeequationsofmotion.Inprac- ticeoneshoulddothefollowing:belowMAtheRGEsaretheones given above with βλH A and βλA removed and λH replaced by λ. AboveMAoneshouldincludeβλH A andβλA andfindλH usingthe fullRGEsandtheboundaryconditionin(18)at

μ

¯=MA.

As far as the newfermions are concerned, following [46] we adopt the approximation in which the new Yukawa couplings run only above the corresponding mass thresholds; this is im- plemented technically by substituting Yi jYi jθ (

μ

¯ Mj) and yyθ (

μ

¯ −Mq) on the right-hand side of the RGEs. The situa- tionis differentfromthescalar one,assettingthefermionfields tozerobelowtheirmassthresholdisconsistent.

Finally notice that, the SM parameters can run in an energy rangebiggerthantheoneof Y ,λA,λH A and y.Therefore,wein- cludeforthemthe2-loopRGE contribution;we donot,however, showexplicitlythe2-looppartbecauseofitscomplexity.

5. Stabilityanalysis

Since we use the 1-loop RGEsof the non-SMparameters, we approximatetheColemanandWeinberg[47] effectivepotentialof themodelwithitsRG-improvedtree-levelpotential:wesubstitute thebarecouplingsintheclassicalpotentialwiththecorresponding runningones.

Theconditionsthatensuretheabsolutestabilityofthevacuum H0=v and A= fahavebeenstudiedin[41]:theyare

I. λH(

μ

¯)>0 andλA(

μ

¯)>0;

II. c4λH(

μ

¯A(

μ

¯)− λ2H A(

μ

¯)>0.

Notice that once λH >0 and c>0 are fulfilledthen λA>0 is fulfilledtoo.ThefactthattheMS couplingsaregaugeinvariant,as proved in [48,2],guarantees that our results willnot be affected byanygaugedependence.

ThefirstconditionλH>0,atthelevelofapproximationweare using,maylead tothe possibilityofHiggs inflation [7–9].There- fore having absolute stability may also allow us to identify the inflaton withtheHiggs field. However, one should keep inmind that perturbative unitarity5 is violated above some high energy scale [50,51].Once the backgroundfields aretaken intoaccount,

5 Thisunitarityproblemcanbesolvedbyaddinganextrarealscalarfield[49,41].

Theextensionofthepresentanalysistoincludesuchscalarisbeyondthescopeof thispaper.

(5)

Fig. 1. Phasediagramofthemodel,showingtheregionwithabsolutestabilityupto thePlanckscale.TheregionwhereconditionIIisfulfilledisinsidetheregionwith λH¯)>0.WesetthecentralvaluesoftheSMparametersattheEWscaleand thelow-energyneutrinoparameters;however,wecheckedthatvariationsofm2i andθi j(within5σ aroundtheircentralvalues)andvariationsofδ,β1andβ2have anegligibleeffectonthisplot.Moreover,wesetthelightestneutrinomassm1=0, M2=1014GeV,M3>MPlandz=0.Switchingthesignofξdoesnotchangethe plot.Theaxiondecayconstantissettofa=1011GeV andλA(MA)=0.05.

however, the authors of [52] find that such energy is paramet- rically higher than all relevant scales during the history of the Universe.Neverthelesssomeextra assumptionsonthe underlying ultravioletcompletionarenecessary[51,52,8].

The question ofthe stability ofthe EW vacuumhas beenad- dressed previously in other economic extensions of the SM. The SM extended only by adding a single right-handed neutrino or three right-handed neutrinos with degenerate masses was stud- iedin[46,40].Extensionswithasingletscalarwereconsidered in [53,41,54]andotherswithoneright-handedneutrinoandan ex- trarealscalar werestudiedin[55].However,we donot knowof anypreviousworkthataccountedforall problemslistedinthein- troduction.6

InFigs. 1 and2weshowregionsoftheparameterspacewhere thestability conditionsare fulfilledforall valuesof

μ

¯ upto MPl andotherswheretheyarenot.Thevaluesoftheparametersused inthat plotcanalsoexplainneutrinomasses,darkmatter,baryon asymmetry andthe strongCP problem(throughthemechanisms discussed in the introduction), fulfilling all bounds of Section 3.

Moreover, the regions where λH >0 all the way up to MPl cor- respondtothepossibilityofHiggsinflation.InFig. 2 weseethat increasing y(Mq)shrinkstheregionwhereconditionIIforstability isfulfilled:thisisbecause y contributespositively (negatively)to therunningofλH A(λA),whichthenincreases(decreases)andthis makesitmoredifficulttosatisfythatcondition.Wealsoobserved

6 AfterpostingthisarticleonthearXivourattentionwasdrawntotheinterest- ingRef.[56].Theauthorsdiscussamodelverysimilartooursandanticipatethat allthoseproblems(withtheexceptionoftheoriginofinflation)canbesolved:in thatworkthePQsymmetryisanextensionoftheSMleptonnumber.Thisallows torelatethescales faandMi[57].However,anexplicitanalysiswasnotpresented in[56].Aswewillseenow,suchananalysishereleadstoregionswherethesi- multaneoussolutionsoccurandotherswheretheydonot.

Fig. 2. The same as inFig. 1, but with a different value of y.

Fig. 3. RGevolutionofthequarticcouplingsλH,λAandthecombinationofquartics

cdefinedinconditionIIforstability.Theverticalsolidlineindicatestheposition ofthescalarthreshold,MA.Thestripesontherightindicatetheregionpresumably dominatedbyPlanckphysics.Thevaluesoftheparametersarethesameusedin Fig. 1.

thatchangingthevalueofλA(MA)and fa changesthelocationof thatregion,sothatthesizeoftheparameterspacethatiscompat- ible withabsolutestability islarger. NoticethatFigs. 1 and2also indicate that lighter right-handedneutrino massesfavor the sta- bilityconditions.Thiscanbe qualitativelyunderstood:smaller Mi generically correspond to smaller Yi j, Eq.(11), andto a reduced destabilizingeffectinconditionsIandIIbecauseofthewayY ap- pearsinβλ(1)

H andβλ(1)

H A.

InFigs. 3 and4weshowtheevolutionofthequarticcoupling combinationsrelevantforthestabilityanalysisasafunctionofthe renormalization scale.The parameters are chosen ina waycom- patible with theregions of, respectively, Figs. 1 and 2, where all stability conditions are fulfilled. There are no Landau poles be- low thePlanckscaleandthecouplingsremainperturbative when thestabilityconditionsarefulfilled.Theregionwithstripesonthe rightcorrespondsto theregimewherePlanckphysicsisexpected

(6)

Fig. 4. ThesameplotasinFig. 3,butwiththevaluesoftheparametersusedin Fig. 2.

tobedominant;thebehaviorofthecurvesthereisthuspresum- ablyunreliable.

Atthesametime, itisimportantto noticethatthere arealso regionsoftheparameterspace, wheretheresultsonthestability analysisobtainedin the SM are not significantly changed by the additionof Ni,qj and A.InthelimitλH A0 theaxionsectoris decoupledfromtherest,and,iftheneutrinoYukawacouplingsare smallenough,onerecovers theSM resultsataverygoodlevelof accuracy.

6. Conclusions

Inthispaperwe havefoundregionsoftheparameterspaceof asimplebutwell-motivatedmodelthatcanaccountforallexper- imentally confirmed signals of physics beyond the SM: neutrino oscillations (through the addition of three right-handed neutri- nos),darkmatter(duetotheaxion),baryonasymmetry(generated bythermal leptogenesis),inflation (which could be drivenby the HiggsfieldsincetheEWvacuumcanbeanabsoluteminimumfor energiesuptothePlanckscale)andthestrongCPproblemthatis automaticallysolvedbythePQsymmetryleadingtotheaxion.

Thismodelisan extensionofthe SM,whichonly addstothe SMthreeright-handedneutrinosaswellthescalarfieldandextra coloredfermionof thesimple invisibleaxion model proposed by KSVZ.

We have found that there are values of the parameters such that the important features listed above are all present together withperturbativity(alwaysuptothePlanckscale).

An importantextension for the presentwork may be the in- clusionofquantumgravity, whichhasbeencompletely neglected here.Somestepsinthisdirectionhavebeentakenin[42].Butthe role ofgravitational quantum effects inthe stability issueof the SMisstillunclear.

Acknowledgements

I thank J. Alberto Casas, Michele Frigerio, Thomas Hambye, Michele Maltoni, Mikhail Shaposhnikov and Cédric Weiland for very useful discussions. This work has been supported by the Spanish Ministry of Economy and Competitiveness under grant FPA2012-32828, Consolider-CPAN (CSD2007-00042), the grant SEV-2012-0249 of the “Centro de Excelencia SeveroOchoa” Pro- gramme and the grant HEPHACOS-S2009/ESP1473 from the C.A.

deMadrid.

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