Fermionic spectral action and the origin of nonzero neutrino masses

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Physics Letters B

www.elsevier.com/locate/physletb

Fermionic spectral action and the origin of nonzero neutrino masses

Mairi Sakellariadou

^a,∗

, Andrzej Sitarz

^b,c,1

aTheoreticalParticlePhysicsandCosmologyGroup,DepartmentofPhysics,King’sCollegeLondon,UniversityofLondon,Strand,London,WC2R2LS,UK bInstituteofPhysics,JagiellonianUniversity,prof.StanisławaŁojasiewicza11,30-348Kraków,Poland

cInstituteofMathematicsofthePolishAcademyofSciences,´Sniadeckich8,00-656Warszawa,Poland

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

Articlehistory:

Received29March2019

Receivedinrevisedform27May2019 Accepted17June2019

Availableonline19June2019 Editor:A.Ringwald

Weproposethatthefermionicpartoftheactionintheframeworkofthenoncommutativedescriptionof theStandardModelisspectral,inananalogouswaytothebosonicpartoftheactionthatiscustomary consideredasbeingspectral.Wethendiscussthetermsthatappearintheasymptoticexpansionofthe fermionicspectralaction.

©^{2019PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Currentexperimentaldatahaveconfirmedneutrinooscillations, implying that at least two of the neutrinos have small albeit nonzeromasses[1]. Severalmodels havebeen consequentlypro- posedtoexplaintheoriginofsuchnonzeroneutrinomasses,con- sideringneutrinosasbeingeitherDiracfermionsorMajoranapar- ticles[2].

Inthe geometric interpretation ofthe Standard Modelof par- ticlephysics,proposed within the framework ofnoncommutative geometry, neutrinos were originally massless Majorana particles [3]. Yet the experimental confirmation that, beyond any doubt, neutrinosofdifferentflavoroscillate,hasenforcedtheintroduction oftheneutrinomassesintothemodel.MostoftheStandardModel phenomenology obtained through the noncommutative spectral geometryapproachisbased onconsideringeither DiracorMajo- ranamassiveneutrinosandemployingthesee-sawmechanism [4].

In this note we propose a consistent treatment of both the fermionic and the bosonic parts of the action [5], an approach which implies nontrivial corrections leading to nonminimal fermioncouplings. The idea behind the fermionic spectral action we propose, follows the customary bosonic spectral action ap- proach.Forthebosonicspectralaction,oneassumestheexistence ofsomeenergycut-off,andthusobtainsaneffectiveactionthat dependson the spectrum of the Dirac operator truncatedat . Onethencomputestheasymptoticexpansionoftheleadingterm

*

Correspondingauthor.

E-mailaddresses:mairi.sakellariadou@kcl.ac.uk(M. Sakellariadou), andrzej.sitarz@uj.edu.pl(A. Sitarz).

1 Partially supported by Polish National Science Centre (NCN) grant 2016/21/B/ST1/02438.

in.Weproposeasimilarapproachforthefermionicaction,usu- ally expressedas the expectationvalue of theDirac operator for a fermionic field , which depends only on the truncated Dirac operator (namely one considers only the terms witheigenvalues smallerthanthecutoff).

Thepurposeofthispaperistoinvestigatethequalitativecon- sequences of the spectral fermionic action we propose, leaving thedetailsofthemodel(includingtheLorentzianformulation[6], fermion doubling[7,8], various forms of thefermionic action for differentKO-dimensionsetc.)forafuturestudy.Thoughusually,it is argued that the bosonic spectral action offers a window into very high energies [9] and implies a natural framework for an early universe cosmology [10], we believe that correction terms tothefermionicspectral actioncanbeevencurrentlyobserved.A previously proposed toy model [11] suggestedthat thefermionic spectralactionmightberesponsibleforadditionalmassterms;in thefollowing wediscussindetailthe possiblenonminimalinter- actionterms.

2. Thefermionicaction

Thefermionicactionforthespectraltriple,whichgivesthedy- namics andinteractions betweenfermions andthe bosonic field, isusuallyformulated astheexpectationvalueoftheDiracopera- tor D_A thattakestheinner fluctuationsintoaccount,inthestate givenbyafermionicfield:

S_f

=  |

^DA

.

^(2.1)

Forthealmostcommutativegeometries,withaspecificKOdimen- sion2 (mod 8),which isused inthedescription ofthe Standard Model, one could consider another version ofthe action (on the totalHilbertspace),intermsoftheantisymmetricbilinearform:

https://doi.org/10.1016/j.physletb.2019.06.037

0370-2693/©^{2019PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

SJ,f

= 

J

 |

DA

,

(2.2) where J the real structure on the spectral triple defined by the choice of an algebra, a Hilbert space and a self-adjoint Dirac operator. This form may be restricted to the subspace of right- handedfermions,reducing theunnecessarydoublingofthespace offermions[7].

Inthe noncommutative geometryapproach there exists, how- ever,aremarkableandquiteunnaturaldifferencebetweenthetwo partsoftheaction.Whilethebosonicpartdependsonlyonthere- ducedspectrumoftheDiracoperator,thefermioniconeconsiders explicitlythe full spectrum. This hasbeen first observedin [12], whereitwaspostulatedthatalsothefermionicactionshouldhave asimilarform,yettheproblemwasnotdiscusseduntiltheanaly- sisof[11].

3. Thespectralfermionicaction

Weproposetowritethefermionicactionfunctional,usingsim- ilar cut-off regularization as is customary done for the bosonic case.Letusnotethatazeta-functionregularizationforthebosonic spectralaction hasbeenproposedin[13] in ordertoaddressthe issuesofrenormalizability andspectral dimensions.Certainly, us- ingthespectralactiongeneralizationleadstoissueswiththeinter- pretationoftheEuclideanformulationofspinorsandthereduction ofthemodelduetotheso-calledfermiondoubling(or,morecor- rectly,doubledoubling).

Weshallwork solelyintheEuclideansetup,leavingasidepo- tential problems and looking only for terms that could lead to newphysics. Our approach followsthe one used forthe bosonic spectralaction,wheretheleadingtermsintroduceonebyonethe geometry, the interactions, andthe coupling between them. The fermionicterm,by definitiongivesonlytheminimalcouplingbe- tween fermions, gauge fieldsand the Higgs. In what follows we willinvestigatethe type andformofnonminimalcouplings, mo- tivatedbythegeometricstructureoftheinteractions asdescribed bynoncommutativegeometry.

Let us propose (for the simplest Euclidean model) the cutoff fermionicaction

Sg,

=  |

^g

(

DA

)  ,

^(3.1)

fora suitable function g and taking g(x)=^g(_^x).Observe, that we take a function of DA andnot D²_A on purpose,as the spec- trum of D_A is not necessarily invariant with respect to change DA→ −^DA. Since any function can be splitted into the sum of an evenandan oddfunction,andan evenfunction canbe taken asafunctionofD²_A,andusing

g

(

x

) =

f

(

x²

) +

xh

(

x²

),

(3.2)

onerealizesthatEq. (3.1) includestwochoices:

S_e,

= |

^f



D²_A

 ,

^So,

=

¹

 |

^DAh_



D²_A

 ,

^(3.3)

where in principle, f,h are two arbitrary functions of the cut- offtype. Usingthe factthat DA commuteswithh_

D²_A wecan rewritethelatterexpressionas:

S_o,

=

¹

 |

^h



D²_A



D_A

,

^(3.4)

which wouldenable usto use the tools ofheat trace expansion [14,15] forcommutativeandalmostcommutativegeometries.

3.1. Commutativegeometries

LetM beaRiemannianspinmanifoldandL²(S)itsspinorbun- dle and a spinorfield. Defineby P_ an endomorphism ofthe bundle of spinors that locally, at each point x∈^{M, projects on} the spinor (x), (x)→ (^x)(^x)|(^x)^{, or} equivalently, using physicsnotation P_= |(^x)(^x)|^.

Let D beaDiracoperator thatliftsthetorsion-free Levi-Civita connectiontothespinorbundle.AlthoughonecanconsiderDirac operators that arise from connections with torsions, we concen- trate on theusually assumedcaseof vanishingtorsion. We shall interprettheabovefermionicactiontermsasarisingfromtheheat kernelexpansionofthetype:

Se,

=

^Tr



P_g_



D²



,

(3.5)

noting that P_ isa local operator (endomorphismof thebundle onwhichD acts).Followingthesameapproachasforthebosonic spectralaction,weusetheheattraceexpansion:

Tr



T e^{−t D2}

 = 

n=⁰ t^12(n−4)



M

a_n

(

x

,

T

),

(3.6)

where

a₀

(

T

,

x

) = (

⁴

π )

⁻²tr T

,

a₁

(

T

,

x

) = (

⁴

π )

⁻²tr



T



−

^R 6

+

^E

.

^(3.7)

For the fermionic spectral action, the coefficients of this expan- sionarethemomentsofthefunctions f andh [15] (seeEq. (3.2)).

Heretr denotesthelocaltraceoperationintheendomorphismsof the spinor bundle takenatpoint x.Note that trP= (^x)(x) andtherefore,fora4-dimensionalmanifoldweshallhavethefol- lowingleadingterms,whicharisefromthefactthat fortheDirac operator E=^R/4,withR thescalarRiccicurvature:



⁴



M

√

g

(

trP_

) = 

⁴



M

√

g

(

^x

)|(

^x

),



²



M

√

g tr



P_ R

12

= 

²



M

√

g R

12

(

^x

) |(

^x

) .

(3.8)

The leadingterm(in⁴),whichresemblesthecosmologicalcon- stant term, corresponds to the bare fermion mass term whereas the second term (in ²) comes from the even part of spectral theactionanddescribesthenon-minimal couplingoffermionsto gravitythroughthescalarcurvature[18,19].

Asfortheoddpartofthefermionicspectralaction,wepropose towriteitusingsimilarargumentsas:

S_o,

=

¹



^Tr



P_,Dh_



D²



,

(3.9)

where the local operator P,D is and endomorphism of the spinorbundleoftheform:

(

x

) →

^D

(

x

)(

^x

)|(

^x

).

^(3.10)

In a similar way as for the even case, observe that tr P_,D=

(^x)|^D(x)^with(,)thelocalscalarproductonthesectionsof thespinorbundle,withvaluesinC^∞(M).

Theoddcomponentofthefermionicspectralactionintroduces atleadingorder(in³)thefermiondynamics:



³



M

√

g

(

^x

)|

^D

(

x

),

^(3.11)

whilethenextordertermswouldcontribute (inthecaseofpure gravity)furthercouplingtoscalarcurvature R:





M

√

g R

12

(

^x

) |

^D

(

x

) .

^(3.12)

3.2.TheEinstein-Yang-Millssystem

Theabove discussedcaseextends tothe situationofa simple noncommutativemodificationofgeometry [5] whereoneconsid- ersthealgebraofMn(C)valuedfunctionsonthespinmanifold M andusesthe familyofDiracoperators constructedbygauge fluc- tuation of the Dirac operator D. Such family, which is obtained throughtheso-calledinternal fluctuationsofthemetric,isofthe form:

DA

=

^D

⊗

^id

+

^A

,

(3.13)

where A is a gauge potential ( A =

ia_i[^D,b_i]^{) for a}i,b_i ∈ C^∞(M)⊗^Mn(C). This includes, in particular, the case of Dirac operatorstwistedbyaconnectiononavectorbundle.

Ifthespectraltripleisrealandsatisfiestheorder-onecondition [20] oneshouldmodifytheabove familyby correcting A through (withasigndetermined by J D= ±^{D J ) J A J−1.Sincethe square} oftheDiracoperator contains thegauge curvature F , theformu- lae for the heat trace expansion are modified, E = (^R/12)+ ^{F ,} andconsequentlythefirstthreeleadingfermionicactiontermsare modifiedasfollows:



⁴



M

√

g

(

trP_

) = 

^4N



M

√

g

(

^x

) |(

^x

) ,



³



M

√

g

(

^x

)|

^DA

(

x

),



²



M

√

g

(

^x

) |



R

12

+

^F

(

x

) .

(3.14)

The only difference from the previous case is – apart from the multiplicityoffermionicfieldsthatcomesfromtherepresentation ofthe algebra MN(C) – the minimal couplingof fermions with gaugefieldsthatappearsinthesecondterm(in³)aswellasthe PauliinteractionLagrangianthatappearsinthethirdterm(in²), thatlocallylookslike



M

√

g

(

^x

)|

^F

(

x

) =



M

√

g

(

x

)

^†



γ

^μ

γ

^νF_μν

(

x

) 

(

x

),

(3.15)

andwhich canbe nontrivialeveninthe caseofelectrodynamics, whichin the casewhen J is presentcorresponds to the algebra C⊕ C^.

3.3.Thealmostcommutativegeometries

Let usnow consider the simplest noncommutative geometry, madefromtheproductofasmoothfour-dimensionalmanifold M (withafixedspinstructure),byadiscretefinite-dimensionalnon- commutative internal space F , defined in the language of finite spectraltriples.SinceeffectivelytheDiracoperatorofthisproduct geometryisofthesametypeasforanEinstein-Yang-Millssystem, onedoesnotexpectanysignificantqualitativedifferencesfromthe correspondingstudiesofcommutativegeometries.

Letusrecall the basic notions.We take asthe underlyingal- gebra of the model A=^C∞(M)⊗AF, which is represented on

the Hilbert space L²(S)⊗HF, and the Dirac operator is D = DM⊗^id+

γ

5⊗^DF,whereagainD denotesthestandardDiracop- eratoronthespinmanifoldM andDF istheDiracoperatorofthe finite spectral triple (A,H,^DF). We refer forthe details of con- structionofproductgeometriesandrelatedissuesto[3].

ThefamilyofDiracoperatorsDA arisessimilarlyasdiscussed intheprevious section,asfluctuationsoftheDiracoperator.They includeboth the classicalgauge fields,with theunitary group of innerautomorphismsofthealgebraAF aswellasthegaugefields related to discrete geometry, which are interpreted as the Higgs field.Moreprecisely,

DA

=

^D

⊗

^id

+ A + ( γ

⁵

⊗

¹

)

D_F

(

H

),

(3.16)

whereA aretheinner fluctuationsoftheDiracoperator D (con- taining,ifweassumetherealityofthespectraltriple,alsothereal partoffluctuation)andDF(H)areinnerfluctuationsoftheprod- uctgeometrywithrespecttothediscreteDiracoperator DF.Note thatbothAandD_F(H)are,fromthetechnicalpointofview,just matrix-valuedfunctionsonthemanifold M,whicharerepresented onL²(S)⊗HF.

Toobtain theleading terms inthe spectral actionwe use the heattraceasymptoticexpansionforthesquareoftheDiracopera- tor,

D²_A

= ∇

^

∇ −

^E

,

(3.17)

where∇ ^{isaconnectionon thespinorbundletensored with}HF andE istheendomorphismofthelatterbundle.

Inlocalcoordinatesoverthemanifold,with

γ

μ beingtheusual gammamatrices,wehave:

E

= −(

^DF

())

²

− 

μ<ν

γ

^μ

γ

^ν

F

_μν

+

ⁱ

γ

5

γ

^μ

M(

D_μ

(

H

)),

(3.18)

where F_{μν is} the curvature tensor of the gauge connections, (DF())² isthepotentialtermforthefields H andthelast term M(_Dμ(H) is the endomorphismof the bundle that dependson thecovariantderivativeoffieldsH .

We shall analyzethese terms inthe next section, in the par- ticular caseof the almost commutative geometry underlyingthe StandardModel.

4. TheapplicationtotheStandardModel

Letusbriefly recall thebasics ofthe StandardModeldescrip- tion within the framework of almost commutative geometry. To obtain the Standard Model the minimal choice ofthe algebra in the spectraltriple defining thediscrete internal space F is AF= C⊕ H ⊕^M3(C).Thisalgebraisrepresentedona16-dimensional Hilbertspacethatincludesallfermions(assumingDiracneutrinos) or 15-dimensionalifwe work with Majorananeutrinos only. For thedetails oftheactionin aconvenientbasis see[20] or [16,17]

for a mostrecent formulation andprinciples of constructing the Diracoperatorbothforthequarkandleptonicsector.

The discrete Dirac operator written in the basis of fermions, taken in the order(for leptons)

ν

R,eR,

ν

L,eL (not that asa rule eachfermionisdenotedcumulativelyforN generations)is

DF

=

⎛

⎜ ⎜

⎝

0 0

ϒ

^∗_ν 0

0 0 0

ϒ

^∗_e

ϒ

_ν 0 0 0

0

ϒ

e 0 0

⎞

⎟ ⎟

⎠ ,

^(4.1)

whereϒ_ν,ϒe areN×^{N massandmixingmatrices.Thefluctuated} discreteDiracoperator D_F,H is:

DF,H

=

⎛

⎜ ⎜

⎜ ⎜

⎝

0 0

ϒ

_ν^∗H⁰

ϒ

_ν^∗H⁻ 0 0

−ϒ

^∗eH⁻

ϒ

^∗_eH⁰

ϒ

νH⁰

−ϒ

eH⁻ 0 0

ϒ

νH⁻

ϒ

eH⁰ 0 0

⎞

⎟ ⎟

⎟ ⎟

⎠ ,

(4.2)

whereH=^H0+^{H−j denotesa}quaternionicfield(Higgsdoublet).

ThediscretepartoftheDiracoperatorhassuchformalsoforthe quarksector, ifwetake quarksintheorderq^u_R,q^d_R,q^u_L,q^d_L andthe massandmixingmatricesare,respectivelyϒu andϒu.

4.1. ThefermionicspectralactionfortheSM

Aswehavepreviously seen,thefirsttwoleading termsofthe fermionic spectral action are the bare mass term andthe usual fermionicaction.AstheStandardModelischiral,intheLorentzian version thebare mass termisnot possible asitis not gauge in- variant. In the Euclidean version, however, it can appear in the modelwiththefermiondoublingbutwillhavetovanishifonere- quiresthattheactiontermsarerestrictedtothephysicalspaceof fermions(byremovingthefermiondoubling[7]).

The next termyields the usual part ofaction, whichincludes thedynamicaltermforfermions,minimalcouplingtogaugefields and the coupling between the Higgs field and fermions, which givesthemasstermsinthebrokensymmetryphase.

Theonlypossiblecorrectionsandneweffectscanbe therefore visibleintheterm,whichisproportionalto².Ofcourse,weshall havetheresimilartermsasintheEinstein-Yang-Millssystem,that is the nonminimal coupling of fermions to gravity (through the scalarcurvature)andthePauli-typeinteractionterms(couplingto curvatureofconnections)[21].

However,we shalladditionallyhavethetermofthefermionic spectral actionthat containsthe squareofthe fluctuateddiscrete partofthe Diracoperator, DF(H)²,that contains the Higgsfield.

WediscussnowtwointerestingcasesofDiracandMajorananeu- trinos, concentrating ouranalysis onthe leptonic sector. Observe thatthesamecouldbe,ofcourse,usedforthespatialpartofthe Diracmanifold leadingtohigher-derivativetermsinthefermionic actionthathavebeenconsideredinsomemodels[22,23]

4.2. Diracneutrinos

WithintheStandardModel,thesquareofthefiniteDiracopera- torgivesthecorrectionstothefermionmasses.Usingthenotation wehaveintroduced previously,wecompute D_F()² restrictedto theleptonsector:

(

DF,H_l

)

²

=

⎛

⎜ ⎜

⎜ ⎝

ϒ

_ν^∗

ϒ

_ν

|

^H

|

²

+ ϒ

^∗_R

ϒ

^∗_R 0 0 0

0

ϒ

^∗_e

ϒ

e

|

^H

|

^{2 0 0}

0 0

ϒ

^∗_ν

ϒ

_ν

|

H

|

² 0

0 0 0

ϒ

^∗_e

ϒ

e

|

^H

|

²

⎞

⎟ ⎟

⎟ ⎠ ,

(4.3) whereϒe,ϒ_{ν are}themassandmixingmatrices,respectively.

Theorderofthecorrections,whencomparedtothemainmass termareof theorder1/andthereforewillbe negligiblewhen comparedtotheDiracmasstermscomputedinthevacuumexpec- tationvalueoftheHiggsfield:H=^Hv (thatis H⁰=^Hv,H⁻=^0).

4.3. Majorananeutrinos

InthenoncommutativedescriptionoftheStandardModelthat usesonlyleft-handedneutrinosthereisnoroomfortheneutrino

mass terms.The naturalinterpretation ofsuch modelis interms of Majorananeutrinos, asafter restrictingit to the physical sub- space (reducing the fermiondoubling) the neutrinospinor fields are their own antiparticles. Of course, there are known mecha- nisms to generate possible mass terms, yet all of them involve quadratic coupling to the Higgs. In view of the analysis of the fermionic spectral action we discussed previously, we argue be- lowthatonecanobtainsuchtermsfromthenextleadingtermin theheatkernelexpansion.

Observe,thatiftherearenoright-handedneutrinos,thefluctu- ationsofthediscreteDiracoperatorsontheleptonicsector,inthe basis(eR,

ν

L,eL),are:

D_F,H

=

⎛

⎜ ⎝

0

−ϒ

^∗_e^H−

ϒ

^∗_eH⁰

−ϒ

eH⁻ 0 0

ϒ

eH⁰ 0 0

⎞

⎟ ⎠ .

^(4.4)

Notethat bytakingthetermwiththesquare ofthefinitepartof theDiracoperatorwouldnotgive anythingnew.Indeed,comput- ing(DF,H)²attheHiggsvacuumexpectationvalueweobtain:

(

DF,H_v

)

²

=

⎛

⎜ ⎝

ϒ

_e^∗

ϒ

_e

|

^Hv

|

^{2 0 0}

0 0 0

0 0

ϒ

^∗_e

ϒ

_e

|

^Hv

|

²

⎞

⎟ ⎠ ,

which similarly as in the Dirac masses case can only add small correctionstothealreadynonvanishingmassofchargedleptons.

Toseethatsomeextratermsarepossibleweneedtogeneral- izetheformofthespectralactiontononscalarfunctions.Sofarwe have assumedthat thefunction f_,which we havetakento de- finetheevenpartofthespectralactionisscalar,thatisforevery operator T thatcommuteswithDA weassumethat f(DA)com- muteswithT aswell.However,thisisnotnecessaryandwemay considerotherfunctionsprovidedthefullgaugeinvariancewillbe preserved.

Leavingasidethequestionabouttheclassificationofsuchfunc- tions,forthespecificmodel,weobservetheexistenceofaparticu- larone.Let

τ

betheoperatormapping(

ν

L,e_L)to(e^c_L,−

ν

^c_L)where (

ν

^c_L,e^c_L) denotes the respectiveantiparticles. Taking as f τ_(DA)=

τ

f(D)

τ

westill obtaina selfadjoint operatorprovided that

τ

is selfadjoint(orantiselfadjoint,asisthecaseforchosen

τ

).

Theoperator

τ

couldbewrittenusingPaulimatricesasi

σ

2◦^{J ,} where J is therealityoperatorofthemodel(see[3,4])restricted to the leptonic sector. The action of thequaternionic partof the algebraontheantiparticle sectorinthenoncommutativedescrip- tionoftheStandardModelisthroughh for¯ h∈ H^{,wherequater-} nions are represented ascomplex matrices in M2(C) andh de-¯ notes complex conjugated matrix. Since for any quaternion we have

τ

h= ¯^h

τ

thenDF,H

τ

(e^c_L,

ν

_L^c)isinvariantundertheSU(2)part ofgaugetransformations.

Writing explicitlyinthe (e^c_R,

ν

^c_L,e^c_L)basis thematrixelements of DF,H

τ

:

D_F,H

τ =

⎛

⎜ ⎝

0

−ϒ

e^∗H⁰

−ϒ

e^∗H⁻

ϒ

_eH⁰ 0 0

−ϒ

eH⁻ 0 0

⎞

⎟ ⎠ .

^(4.5)

Then the terms in the fermionic spectral action, that arise from Tr(P_f τ_(D²) in the next-to-leading order, could be explicitly rewrittenas:



²

(ϒ

e

ϒ

e∗

) 

ν

^c_L

,

e^c_L

 

_H⁰ H⁻

 

H⁰

,

H⁻

  ν

^c_L

e^c_L



+

^h.c

.

(4.6) Aswe haveobservedbeforetheentireexpressionisgaugeinvari- ant andcan be identified asa Weinbergterm (sometimescalled

Weinbergoperator)[24],whichisusedtodescribeeffectivemech- anism of neutrino mass generation. As the operator is, in fact, non-renormalizable,oneoftenexplainsthephysics behindtheef- fectivetermasoriginatingfromyetunknown heavy intermediate particles.

Afterthe Higgsfield getsits vacuumexpectationvalue,which inour choiceof theparametrization is H⁰=^Hv,H⁻=^{0, a neu-} trinomassisgenerated,dependingonthescale,Higgsvacuum expectationvalue Hv, massesof charged leptons and the coeffi- cientsof the cutoff function f τ_. Recall, however, that the usual mass terms appear at order ³, compared to ² for the Wein- bergterm.Thereforeifoneassumestobemuchlargerthanthe Higgsvacuumvalue (in manymodels thisis around the scaleof GUT)thenthegeneratedneutrinomasseswillnecessarilybesmall, whichagreeswiththeexperiment.

5. ConclusionsandoutlookbeyondtheStandardModel

As we have shown, even the simplestmodel, which is based on the almost commutative geometry with the finite part de- scribedby aspectral triple,leads totheappearance ofcorrection terms that give some non-minimal interactions between gravity andfermions,gaugefieldsandfermionsaswellastheHiggsfield andfermions.It isinterestingthat noextraparticles arerequired toexplaintheappearanceofneutrinomasses. Ofcourse,wehave demonstratedonlythatthespectralactionforfermionsintheEu- clideanversion couldbe constructed andcomputed leavingaside the problemwhether any restriction could appear when consid- ering the Lorentzian version, in particular with respect to the fermiondoublingproblem[7].

It is also interesting that the next order corrections lead to termsthathavebeenconsideredinvariousmodels,includingPauli interactiontermsandnonminimalcouplingtogravity. Connecting theiroriginstothesamespectralactionprincipleasinthecaseof neutrinomassescouldhelptosetpossiblelimitsontheirobserva- tionalevidenceorsetconstraintsonthemodelsfromcosmological observationsinasimilarwayitcanbedoneforthebosonicaction [25,26].

The neutrinomass corrections are possible in both Dirac and Majorananeutrinomodels,andmayleadtosmallneutrinomasses.

The correction terms are nonrenormalizable (which is similar to higherordertermsfromthebosonicaction), yetcouldbetreated asaneffectivedescription.

Themodelallowsanextensionoftheassumedformofthecut- off function to include also a nonscalar part, which means that someinternalpermutationsoftheeigenspacesthatarewithinthe range of the spectral projection P_ are allowed. This point cer- tainlyrequiresfurtherstudies,asitisnecessarytounderstandthe allowedfreedominthechoiceofthefunction.Inparticular,toin- troduce the neutrino mixing matrix one needs to generalize the cutofffunctionfurther,byaddingamixingtothefunction(thatis modifying

τ

operatorsothatitisnotdiagonalforthethreefami- lies).

Acknowledgements

The authors would like to thank the refereefor remarks and comments. The research was initiated thank to support of the

Shorttermscientific missionsexchange,COSTActionMP1405QS- PACE. Publicationissupportedby theJohnTempleton Foundation Grant, ConceptualProblems in UnificationTheories” (No. 60671).

MS is supported inpart by the Scienceand Technology Facilities Council (STFC),UKundertheresearchgrantST/P000258/1.

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