Type II Dirac seesaw with observable ΔNeff in the light of W-mass anomaly

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Type II Dirac seesaw with observable N _eff in the light of W-mass anomaly

Debasish Borah

^a

, Satyabrata Mahapatra

^b

, Dibyendu Nanda

^c

, Narendra Sahu

^b,∗

aDepartmentofPhysics,IndianInstituteofTechnologyGuwahati,Assam781039,India

bDepartmentofPhysics,IndianInstituteofTechnologyHyderabad,Kandi,Sangareddy502285,Telangana,India cSchoolofPhysics,KoreaInstituteforAdvancedStudy,Seoul02455,RepublicofKorea

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

Articlehistory:

Received26April2022 Accepted8July2022 Availableonline14July2022 Editor:G.F.Giudice

WeproposeatypeIIseesawmodelforlightDiracneutrinostoprovideanexplanationfortherecently reported anomalyin Wboson mass by the CDFcollaboration with 7σ statistical significance. Inthe minimalmodel,therequiredenhancementinWbosonmassisobtainedattreelevelduetothevacuum expectationvalueofarealscalartriplet,whichalsoplaysaroleingeneratinglightDiracneutrinomass.

Dependinguponthecouplingsandmassesofnewlyintroducedparticles,wecanhavethermallyornon- thermallygeneratedrelativisticdegreesoffreedomNeffintheformofrighthandedneutrinoswhich can be observed at future cosmology experiments. Extending the model to a radiative Dirac seesaw scenariocanalsoaccommodatedarkmatterandleptonanomalousmagneticmoment.

©^{2022TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

The CDF collaboration has provided an updated measurement of the W boson mass MW =⁸⁰⁴³³.5±⁹.4 MeV [1] using the data corresponding to8.8fb⁻¹ integratedluminosity collected at theCDF-IIdetectorofFermilabTevatroncollider.Thisnewlymea- sured value has 7

σ

deviation from the standard model (SM) expectation(MW =⁸⁰³⁵⁷±^{6 MeV). Thishas ledto several dis-} cussions on possible implications and interpretations in the last week relatedto effectivefield theory [2,3], electroweakprecision parameters [4–7],beyondstandardmodel(BSM)physicslikedark matter(DM)[8–13],additionalscalarfields[14–24],supersymme- try[25–29] andseveralothers[30–42].Assumingthisanomaly to beoriginatingfrombeyondstandardmodel(BSM)physics,herewe consider a seesaw model forDirac neutrinoswhere a real scalar tripletplaysanon-trivialroleingeneratingtherequiredenhance- ment inW bosonmass aswell aslight neutrino masses. Due to theexistenceofadditionallightspeciesintheformofrightchiral parts of light Diracneutrinos, we can get enhancement ineffec- tiverelativisticdegreesoffreedomN_eff dependinguponYukawa couplings and masses of additional particles including those in- volvingthetriplet scalar.Weshowthat suchenhanced Neff can not onlybeconstrainedbyexistingdatafromthePlanckcollabo-

*

Correspondingauthor.

E-mailaddresses:dborah@iitg.ac.in(D. Borah),ph18resch11001@iith.ac.in (S. Mahapatra),dnanda@kias.re.kr(D. Nanda),nsahu@phy.iith.ac.in(N. Sahu).

ration,butalsoremainswithinreachofnextgenerationcosmology experiments.AfterdiscussingtheminimalmodeloftreelevelDirac seesaw,weconsideraradiativeversionofitwhichcanaccommo- datedarkmatteraswellasleptonanomalousmagneticmoments, whicharealsosignaturesofBSMphysics.

2. TreelevelDiracseesaw

WhilemostoftheneutrinomassmodelsinvoketheMajorana nature of neutrinos, the possibility of Dirac seesaw is relatively lessexploredyetequallyappealing.DifferentBSMframeworksfor originof light Diracneutrino mass can be found in [43–74] and referencestherein. Here we consider a type II seesaw realisation ofthe dimension sixoperator forlight Diracneutrinomass [61].

The particle content of the minimal model is shownin Table 1.

A vector-like fermion doublet and a real scalar triplet are introducedtorealisethetypeIIDiracseesaw.¹Asoftlybrokendis- crete Z2 symmetryisintroducedtoforbiddirectcouplingofright handed neutrino

ν

R to the SM lepton and Higgs in the form of LH˜

ν

R. An overall globallepton number symmetry isassumed as inconventionalDirac seesawmodels tokeep leptonnumbervio- latingMajoranamasstermsawayfromtheLagrangian.

1 Thisisnotauniquechoiceinvolvingarealscalartriplet.Forexample,onecan considervector-likefermiontriplet insteadoffermiondoublettootorealisethe sameseesawrealisationofdimensionsixoperator.Suchchoicesdonotchangethe analysissignificantlyandwesticktotheminimalchoiceoffermiondoublet.

https://doi.org/10.1016/j.physletb.2022.137297

0370-2693/©^{2022TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

Fig. 1.Origin of tree level Dirac Neutrino Mass.

TherelevantYukawaLagrangiancanbewrittenas

−

L

⊇

^M

+

^yLL

R

+

^yνR

LH

˜ ν

R

+

^h

.

c (1) where=(ψ⁰,ψ⁻)^T isthevector-likefermiondoublet.

Scalarpotentialoftheminimalmodelcanbewrittenas

V

(

H

, ) = − μ

²_HH^†H

+ λ

H

H^†H

2

−

M²Tr

²

+ λ

Tr

⁴

+ λ

(

Tr

²

)

²

+ λ

H

H^†H

Tr

²

+ λ

_H

H^†

²H

+ ζ

H^†

H

.

(2)

Here weconsider M²<0 suchthat theneutralcomponentof acquires only an induced vacuum expectation value (VEV) after electroweaksymmetrybreaking.Aftersymmetrybreaking,neutri- noswillacquireaDiracmassattreelevelfromthediagramshown inFig.1.Thiscanbeestimatedas

m_ν

=

^yLyνR

H M

.

(3)

3. Wbosonmass

Any newphysicswhich couldbe responsible fortheW boson massanomaly,canbeparametrised byobliqueparametersS,T,U [75,76]. Considering the U parameter to be suppressed, one can parametrise any BSM physics contribution to W boson mass in termsofS,T parameters.Taking

α

EW,GF,mZasinputparameters, the requiredfitting of S,T parameters in view ofthe recent W- massanomaly hasbeendiscussedin[3].It shouldbe notedthat, changeinS,T parametersduetoBSMphysicswillalsochangethe weakmixingangleθW,whichispreciselymeasured.Themassof WbosonmW andsin²_θW(mZ)_{M S} canbeexpressedintermsofelec- troweakprecisionparametersnamely,SandTas[77],

m_W

=

⁸⁰

.

357 GeV

(

1

−

⁰

.

0036S

+

⁰

.

0056T

) ,

sin²_θ

W

(

m_Z

)

_MS

=

⁰

.

23124

(

1

−

⁰

.

0157S

+

⁰

.

0112T

) ,

(4) which implies that the compatibility of the enhanced W boson mass withprecision measurements of θW requiresboth S andT parameters to be non-zero, also seen from the fits presented in [3].However,inthecaseofrealscalartriplet,wewillnotgetany correctiontoSparameter,evenatone-looplevel[78].Ontheother hand,Tparameter can getcontributionfrom theinduced VEVof theneutralcomponentoftriplet.TherequiredlimitonT,inorder toexplaintheenhancedWbosonmass,willbe

v

= ζ

v² 2

√

2M²

.

(7)

Since isarealscalartripletwithhyperchargezero,it willcon- tributeonlytoWmassandnottoZmassattreelevel.Therefore, itwillchangeonlytheTparameterthroughitscontributiontoW- massandcanbeexpressedas,

T

=

^v2

α

ζ

²

M⁴

,

(8)

where

α

is the fine structure constant. It should be noted that, thevector-likefermiondoublet,inspiteofhavingSU(2)L gauge interactions,cannotgiverisetotherequiredenhancementviara- diativecorrectionsduetodegeneratemassesofitscomponents.

4. ObservableN_eff

Thermalisationofthe

ν

Rs givesextracontributionto thetotal radiationenergydensityoftheuniverse N_eff.Thequantity N_eff is definedasthe contributionofnon-photon componentstothera- diationenergydensitynormalisedby thecontributionofa single activeneutrinospecies(

νL)[79] i.e.

N_eff

≡

rad

−

γ

_νL

^,

⁽⁹⁾

which,duringtheeraofrecombination(z∼^{1100),isrestrictedby} Planck2018data[80] tobe

N_eff

=

²

.

99⁺₋⁰₀^._.³⁴₃₃ (10)

at2

σ

CLincluding baryon acousticoscillation (BAO)data. At 1

σ

CL it becomes more stringent to Neff=².99±⁰.17. Both these bounds are consistent with the standard model (SM) prediction N_eff^SM=³.045 [79,81,82].Upcoming cosmicmicrowavebackground (CMB)experimentslikeSPT-3G[83] andCMB-S4[84] willbeable toconstrainN_eff upto amuchhigheraccuracy,havingthepoten- tialtoverifyBSMscenarioslike ourswhichlead toenhancement inNeff.SomerecentstudiesonlightDiracneutrinosandenhanced N_eff can be found in [72,85–94]. For thermalised

ν

R, the en- hancementtoN_eff canbeexpressedas,

N_eff

=

N_eff

−

N_eff^SM

=

N_νR

T_νR

T_νL

4

=

N_νR

⎛

⎝

^g∗

T_ν^dec

L

g_∗

T_ν^dec_R

⎞

⎠

4/3

.

(11)

Fig. 2.ContributiontoNeffasafunctionoftheYukawainteractionyνR forthree benchmarkvaluesofMψ.

This can be computed simply by assuming instantaneous decou- pling of

ν

R’s and finding the corresponding decoupling temper- ature. This also agrees to a great accuracy with more detailed analysis involving Boltzmann equations [88,94]. The correspond- ing resultsare showninFig.2forbenchmark choicesof mass and Yukawa couplings which dictate the thermalisation as well as decoupling temperature of

ν

R. Clearly, the entire parameter spaceremainswithin thereachoffutureCMBexperiments.How- ever,

ν

R canalsobeproducedthroughfreeze-inmechanismifthe corresponding interactions are not sufficientto keep themin the thermalplasma.Fornon-thermalproductionof

ν

R,wecancalcu- late the enhancement in N_eff by solving the Boltzmannequation which tracksthenon-thermal productionof

ν

R fromother parti- cles [91]. The corresponding Boltzmann equation can be written, forourmodel,as

dY_νR

dx

=

²

β(

x

)

H s^1/3x

^E

^Y_ψ^eq

,

(12) where xisdefinedasM_ψ/T,s istheentropydensityoftheuni- verse, and β(T)=

1+^{T dg}_3g^s/_s^dT

. The 2 factor in the right hand side ofthe Boltzmannequation isdueto theDiracnature of

ν

R. Thecollisionterm<E>canbewrittenas

E

=

^y

2νR

(

M²_ψ

−

^m2_h

)

² 32

π

M²_ψ

1

−

^m

2 h

M²_ψ

.

(13)

Here,weconsiderthatthe

ν

R producedfromthedecayofψ which was in thermalequilibriumwiththe SM particles owingto their gauge interactions.Onecanwritethecorrespondingenhancement toN_eff as

N_eff

=

^3s4/3YνR

ρ

νL

T=TCMB

,

(14)

wherethenumericalvalueofYνR atT=^TCMB isfoundbysolving equation (12). Theresults are shownin Fig.3 forchosen bench- markparameters. As expected, withtherise inYukawacoupling, the non-thermal contribution to

ν

R density and hence Neff in- creases.AscanbenoticedwhilecomparingwithFig.2,theYukawa coupling remains suppressed in this case for

ν

R to be in non- thermalregime.

InFig.4,weshowthescanintheplaneofζandMwhichsat- isfy enhanced W-masscriteriareportedby theCDFcollaboration.

TherequiredenhancementinW-massconstrainstheVEVof to

Fig. 3.Non-thermalcontributiontoNefffromthedecayoffermiondoubletasa functionoftheYukawainteractionyνR.

Fig. 4.AllowedparameterspaceinζvsMplanefromtheTparameterconstraint showninequation(5).HerewehavefixedMψat250GeVandyLat10⁻⁸.

lieinfewGeVregimewhichcan betranslatedintoconstraintsin ζ,M plane followingEq. (7) as showninFig.4. Incolour code, weshowtheYukawacouplingstrength of

ν

R consideringittobe thermallygeneratedproviding alarge contributionto N_eff within reachoffutureCMBexperiments,asshowninFig.2.Wealsotake intoaccounttheconstraintonthescaleofneutrinomass.Togen- eratethe correctneutrino mass,the product of yLyνR hasto be O(10⁻¹⁰) assuming Mψ and^H=^{v havesimilar order. In case} ofverysmall y_L (say y_L ∼ ^{10−8), y}νR hasto belarge enoughto generatethecorrectneutrinomass.Such interactions can leadto the thermalisationof

ν

R in the early universe whoseeffects can be seen as the extra radiationenergy density atthe time of re- combination. On the other hand, if we consider sizable yL this makes yνR automatically small from the requirement of correct neutrinomass. In thiscase, although

ν

R can not be thermalised dueto such feeble interactions, they can still be produced from thenon-thermaldecayofψ asthey werepresentinthe thermal plasmaduetotheirgaugeinteractions.

5. RadiativeDiracseesaw

TheparticlecontentofthemodelisshowninTable2.Therel- evantLagrangiancanbewrittenas:

−

L

⊇

^M

+

^MNN N

+

^Y

H N

+

^YNlL

η ˜

1N_R

Fig. 5.Origin of Scotogenic Dirac Neutrino Mass.

+

^YR

L

η ˜

2

ν

R

+

^Yψl

L

η

1l_R

+

^h

.

c (15) One of the discrete symmetriesnamely Z₂ willbe broken while theotherwillremainunbrokenensuringthestabilityofdarkmat- ter.Aftertheelectroweaksymmetrybreaking,lightDiracneutrino massesariseatone-looplevelasshowninFig.5.Thecorrespond- ingone-loop contributioncanbeestimatedfollowing[45,54].The analysis for W boson mass correction from triplet VEV and en- hancementofN_eff fromthermalornon-thermal

ν

R remainsmore or lesssame as in the minimal model andhence we only com- mentonadditionalphenomenology ofthe radiativemodelbelow.

ItshouldbenotedthatsimilartothetreelevelDiracseesawmodel discussed earlier, here also we consider a global lepton number symmetryinordertokeeptheMajoranamasstermsaway.

After electroweakphase transition,theVEVofSM Higgsleads tomixingbetweentheneutralcomponentofdoubletψ⁰(as(^T= (ψ⁰,ψ⁻))andN givingrisetothewellmotivatedsinglet-doublet DM Model which has been studied extensively in the literature [95–115]. Asclear fromtheLagrangian inEq. (15),herewe con- sider a Diracversion ofthe singlet-doubletDM model.The mass termsforthesefieldscanthenbewrittentogetherasfollows:

−

L^VF_mass

=

M

ψ

⁰

ψ

⁰

+

M

ψ

⁺

ψ

⁻

+

MNN N

+ √

^{Y v}

2

ψ

⁰N

+ √

^{Y v} 2 N

ψ

⁰

=

N

ψ

⁰

MN Y v

/ √

2 Y v

/ √

2 M

N

ψ

⁰

+

^M

ψ

⁺

ψ

⁻

.

(16) The unphysicalbasis,

N ψ⁰T

is relatedtophysical basis, ψ1 ψ2

T

throughthefollowingunitarytransformation:

N

ψ

⁰

=

U

ψ

1

ψ

2

=

cos

θ −

^sin

θ

sin

θ

cos

θ

ψ

1

ψ

2

,

(17)

wherethemixingangleisgivenby:

tan 2

θ = −

√

2Y v M

−

MN

.

(18)

Themasseigenvaluesofthephysicalstatesψ1andψ2 arerespec- tivelygivenby:

Y

= −

Msin 2

θ

√

2v

,

M

=

^Mψ1sin²

θ +

^Mψ2cos²

θ,

(21) whereM=^Mψ2−^Mψ1

Thus the interactionterms of thedark sector fermions inthe massbasisare:

L^DM_int

= [

ⁱ

γ

^μ

(∂

_μ

−

^ig

σ

^a

2 W^a_μ

−

^igY 2B_μ

)]

+

^N

(

i

γ

^μ

∂

μ

)

N

− (

Y

H N

+

^h

.

c

.)

L^DM_int

=

^gZ

sin²

θ ψ

1

γ

^μZ_μ

ψ

1

+

^cos2

θ ψ

2

γ

^μZ_μ

ψ

2

+

^sin

θ

cos

θ (ψ

1

γ

^μZ_μ

ψ

2

+ ψ

2

γ

^μZ_μ

ψ

1

)

+

^gW

(

sin

θ ψ

1

γ

^μW_μ⁺

ψ

⁻

+

^cos

θ ψ

2

γ

^μW_μ⁺

ψ

⁻

) +

^gW

(

sin

θ ψ

⁺

γ

^μW_μ⁻

ψ

1

+

^cos

θ ψ

⁺

γ

^μW_μ⁻

ψ

2

)

−

^gZc_2w

ψ

⁺

γ

^μZ_μ

ψ

⁻

−

^e0

ψ

⁺

γ

^μA_μ

ψ

⁻

− √

^Y 2h

sin 2

θ (ψ

1

ψ

1

− ψ

2

ψ

2

) +

^{cos 2}

θ (ψ

1

ψ

2

+ ψ

2

ψ

1

)

(22) where g_Z =^e0/2s_wc_w, g_W =^e0/√

2s_w and s_w, c_w,c_2w denote sinθW andcosθW andcos 2θW respectively.

Theimportantparameterswhichdecidetherelicabundanceof ψ1 aremass ofDM(M_ψ1), themass splitting(M) betweenthe DMandthenext-to-lighteststableparticle(NLSP)andthesinglet- doubletmixinganglesinθ.HerewehaveusedmicrOmega[116]

to calculate the relic density of DM. In this scenario the an- nihilation cross-section increases with sinθ, as it enhances the SU(2)componentandhenceresultsinsmallerrelicdensity.When the mass-splitting is small it leads to effective co-annihilation and hence reduces the relic density contribution (due to less Boltzmann suppression). In this case the effect of sinθ on relic abundance is quite negligible whereas for large M, asthe co- annihilationbecomessuppressed,theeffectofsinθ orrelicabun- danceis clearly visible. Forsmall sinθ, the effectiveannihilation cross-sectionissmallleadingtoalargerelicabundance,whilefor large sinθ therelic abundanceis smallprovided that the M is largeenoughtoreducetheco-annihilationcontributions.

InFig.6,wehaveshownthepointssatisfyingcorrectrelicden- sityintheplaneof Mψ1 versus M.We canseethat,forawide rangeof singlet-doublet mixing(sinθ) which is indicated by the colour bar,we cangetcorrectrelicabundance.Thisresultcanbe explainedby understanding the interplaybetweensinθ and M in deciding the effective annihilation cross-section of the DM. If we dividethe plane of M_ψ1 versus m into two regions:(I)the

Fig. 6.DMRelicdensity(fromPLANCK)allowedparameterspaceshownintheplane ofMψ1 vsM.

bottom portionwithsmallm,whereM decreases withlarger mass of ψ1, (II) the top portion with larger mass splitting m, wherem increasesslowlywithlargerDMmassM_ψ1.Inthefor- mercase, foragivenrangeofsinθ,theannihilationcross-section decreasesasmassofDMincreases.Therefore,moreco-annihilation contribution is needed for compensation,which requires M to decrease. This also implies that the region below this is under abundantassmallmass-splittingimplylargeco-annihilation,while theregionabove,isoverabundant aslargeMlead tosmallco- annihilation for a given mass ofDM. Now in region (II), as M is large, co-annihilation contribution is much smaller here and hence theannihilation processes effectivelydecide the relicden- sity. However asthe Yukawa coupling Y ∝Msinθ, for a given sinθ,larger M leads to larger Y andtherefore larger annihila- tioncross-sectionleading tounderabundance,whichcanonlybe brought to correct ballpark by having a larger DM mass.That is thereason,incase-(II),theregionabovetheshownallowedregion ofcorrectrelicdensityisunderabundant, whiletheregionbelow itisoverabundant.

Now imposing the constraints fromDM direct search experi- mentsontopoftherelicdensityallowed parameterspace(Fig.6) intheM versus M_ψ1 plane, weobtainFig.7,whichshowsthat the parameterspaceiscruciallytamed downascomparedto the parameterspacesatisfyingcorrectrelicdensity.Asinthisscenario, dueto thesinglet-doubletmixing, theDM ψ1 canscatter offthe target nucleusat terrestrialdirect search experiments,via Zand Higgsmediatedprocesses,thecross-sectionfor Z-bosonmediated DM-nucleonscatteringisgivenby [117,118]

σ

_SI^Z

₌

^G

2 Fsin⁴

θ

π

A²

μ

_r²

Z fp

+ (

A

−

^Z

)

fn

₂

^{2 (23)}

and similarly the spin-independent DM-nucleon cross-section throughHiggsmediationisgivenby

σ

_SI^h

₌

⁴

π

A²

μ

²_r^Y

2sin²2

θ

M_h⁴

_mp v

f_{T u}^p

+

^f_{T d}^p

+

^f_{T s}^p

+

² 9f_{T G}^p

+

^mn

v

f_{T u}ⁿ

+

^f_{T d}ⁿ

+

^fT sⁿ

+

² 9f_{T G}ⁿ

2

(24)

Itclearly showsthatifsinθ islarge,thenthe interactionstrength is large andhence the DM-nucleon cross-section becomes large.

Fig. 7.DMRelicdensity(fromPLANCK)+Directsearch(fromXENON1T)allowed parameterspaceshownintheplaneofMψ1 vsM.

Fig. 8.Lepton(g−²)in scotogenic Dirac model.

Thusthedirectsearchexperimentsconstraintssinθ toagreatex- tent.Thisstringentupperlimitonsinθissinθ≤0.05.Asaresult, only smallMregionbecomesviable,ascorrectrelicdensitycan beachieveddominantlythroughco-annihilationprocesses.

Recently, the E989 experiment at Fermilab has measured the muonanomalous magnetic momentaμ =(g−²)_μ/2 reporting a 4.2

σ

observed excess of aμ=²⁵¹(59)×^{10−11 [119]. On the} otherhand,themeasurementofthefinestructureconstantusing Cesiumatoms[120] hasled to a discrepancyinelectron anoma- lousmagneticmomentae innegativedirectionwith2.4

σ

statisti- calsignificance.Inourmodel,theone-loop contributiontolepton anomalous magneticmoment(g−²)l canarise withthe Z2-odd particlesintheloopasshowninFig.8.Thiscontributionisgiven by [121,122]:

a_l

= −

^ml

16

π

²M²_η+sin 2

θ (

Y_NlY_ψl

)

×

M_ψ

1F

_M2

ψ1

M²_η+

−

^Mψ₂F

^Mψ²2

M²_η+

−

^m

2 l

8

π

²M²_η+

(

Y_Nl² cos²

θ +

^Y_ψ²_l^sin2

θ )

G

_M²

ψ₁ M_η²₊

−

^ml2

8

π

²M²_η+

(

Y_ψ²_lcos²

θ +

^Y2_Nl^sin2

θ )

G

_M²

ψ₂ M_η²₊

(25) WheretheloopfunctionsF andG aregivenby

Fig. 9.Lepton(g−²)in radiative Dirac seesaw model.

F

(

x

) =

¹

−

^x2

+

^2xlogx 2

(

1

−

x

)

³

G

(

x

) =

¹

−

^6x

+

^3x2

+

^2x3

−

^6x2logx

12

(

1

−

^x

)

^{4 (26)}

Wecalculatedtheanomalousmagneticmomentforbothelectron andmuonwiththeparameterspaceconsistentwithrelicanddi- rectsearchconstraintsforDMandtheresultisshowninFig.9.We find that this radiative Dirac seesaw model with singlet-doublet DMcangiverisetobothelectronandmuon(g−²)incertainre- gionofparameterspaceonlywhentherelevantcouplingsY_ψl and YNl are large approaching theperturbativity limit. The blackand cyan lines show the correct ball park for aμ and a_e respec- tively.Itisworthmentioningherethattherelativesignofaeand aμ canbeeasily achievedby appropriatelychoosingthesignof therelevantYukawacouplings.

Here it is worth mentioning that the presence of the inert scalardoublets intheradiativeDiracseesaw modelcanalsocon- tributeto thecorrection toW bosonmass.However ifthe mass- splittingbetweenthechargedandtheneutralcomponentsofthe doublet is within 10GeV andifthe massof thedoublet is cho- sen beyond 800 GeVor so, then it can not explain the W-mass anomaly asin such a case the S andT parameters are found to be of the order O(10⁻⁵) and O(10⁻³) respectively. Similar con- clusionwasalsogivenin [123].Forsimplicity,wehaveconsidered a parameter space for the inertdoublet scalar such that its co- annihilations withthe DMare alsonegligible implyingthat inert doubletcomponentsremainmuchheaviercomparedtoDM.Given that DM mass lies up to a few hundred GeV and inert doublet mass remainsheavier, thedominantcorrection toW boson mass in oursetup is comingfromthe triplet scalarVEV only.In addi- tion, the presence of the additional singlet-doublet fermion also doesnotaffecttheS andTparametersignificantly toexplain the CDF-II resultsasthe mass-splittingbetweenthecharged andthe neutral componentsremain extremelysmallas aconsequence of theupperlimitonmixinganglesinθfromDMdirectsearchexper- iments.²Thus,thesinglet-doubletDiracfermionDMwithmassup to afewhundredGeVwithdesiredrelicanddirectdetectioncross section automatically keepsthe contribution of Z2-odd particle’s radiative contribution to W-mass suppressed. We find that after imposition oftherelevantDMconstraints,theallowed parameter

2 SimilarconclusionsareobtainedevenforMajoranasinglet-doubletDM[115].

parameteratoneloopremainssmallTη<0.01.Insuchacase,we could get their maximum contribution to S parameter to be ap- proximately Sη≤⁰.003,which stillkeepsθW awayfromthe LEP estimate. Considering the possibility of both scalar doublets and tripletvia radiative andtreelevelcontribution precision parame- tersrespectively,shouldbeabletoreducethistension.

6. Conclusion

Wehaveproposeda Diracseesaw mechanismforlight neutri- noswherearealscalartripletplaysanon-trivialrole.Inthemin- imal model with tree level seesaw, vector-like fermion doublets arealsointroducedtoallowtripletcouplingstoleptonsleadingto the seesaw mechanism. The induced VEV of the neutral compo- nentofthetripletnotonlytakespartinneutrinomassgeneration butalso provides the necessary enhancement in W boson mass, inview ofthe recentlyreported resultsby theCDF collaboration ofFermilab.Depending upontriplet andvector-likefermiondou- blet couplingsto righthanded neutrinos, we findthat additional effectiverelativistic degreesof freedom N_eff can be generatedin theearlyuniverseeitherthermallyornon-thermally.WhilePlanck 2018 data already constrains such enhancement in N_eff, future CMBexperimentswillbeabletoprovemostpartoftheparameter space, providing a complementary probe of the model. We then discuss a radiative version of the same Dirac seesaw model af- terincludingadditionalscalardoubletsandfermionsingletswhich alsoleadsto therealisationofsinglet-doubletDiracfermiondark matterphenomenology.Themodelcanalsoaccommodateanoma- lous magnetic moments of electron and muon provided the re- spective Yukawa couplings are of order one in order to remain consistent with other phenomenological requirements. The pro- posedscenarios provide interesting phenomenology and comple- mentaryprobesofrealscalartripletoriginofW-massanomalyin the context of tree level andradiative Dirac seesaw. Apart from therich phenomenology provided by theradiative model, italso hasthepotentialtoreducethetensionbetweenWmassandpre- cisionmeasurementofθW whichexistinthepuretripletoriginof Wmassanomaly.Incorporatingcomparablecontributionofscalar doublets at one loop and scalar triplet at tree level to W mass should be able to relax this condition. A detailed exploration of suchapossibilityalongwiththechangeinDM,g-2resultsisleft forfuturestudies.

Declarationofcompetinginterest

The authors declare the following financial interests/personal relationshipswhichmaybeconsideredaspotentialcompetingin- terests:

Narendra Sahureports financial support,equipment, drugs,or supplies,andtravelwere providedbyIndiaDepartmentofAtomic Energy.

Acknowledgements

DN would liketo thankSanjoy Mandal forusefuldiscussions.

The workofDN issupported byNationalResearchFoundation of Korea (NRF)’sgrants, grantsno.2019R1A2C300500913.SM would liketothankPurusottamGhoshforusefuldiscussion.NSacknowl- edgesthesupportfromDepartmentofAtomicEnergy(DAE)- Board ofResearchinNuclear Sciences(BRNS),GovernmentofIndia(Ref.

Number:58/14/15/2021- BRNS/37220).

References

[1]T.Aaltonen,etal.,CDF,Science376(2022)170.

[2]J.Fan,L.Li,T.Liu,K.-F.Lyu,arXiv:2204.04805,2022.

[3]E.Bagnaschi, J.Ellis, M.Madigan, K. Mimasu,V.Sanz,T.You,arXiv:2204. 05260,2022.

[4]J.deBlas,M.Pierini,L.Reina,L.Silvestrini,arXiv:2204.04204,2022.

[5]A.Strumia,arXiv:2204.04191,2022.

[6]P.Asadi,C.Cesarotti,K.Fraser,S.Homiller,A.Parikh,arXiv:2204.05283,2022.

[7]C.-T.Lu,L.Wu,Y.Wu,B.Zhu,arXiv:2204.03796,2022.

[8]Y.-Z.Fan,T.-P.Tang,Y.-L.S.Tsai,L.Wu,arXiv:2204.03693,2022.

[9]C.-R.Zhu,M.-Y.Cui,Z.-Q.Xia,Z.-H.Yu,X.Huang,Q.Yuan,Y.Z.Fan, arXiv:

2204.03767,2022.

[10]B.-Y.Zhu,S.Li,J.-G.Cheng,R.-L.Li,Y.-F.Liang,arXiv:2204.04688,2022.

[11]J.Kawamura,S.Okawa,Y.Omura,arXiv:2204.07022,2022.

[12]K.I.Nagao,T.Nomura,H.Okada,arXiv:2204.07411,2022.

[13]X.Liu,S.-Y.Guo,B.Zhu,Y.Li,arXiv:2204.04834,2022.

[14]K.Sakurai,F.Takahashi,W.Yin,arXiv:2204.04770,2022.

[15]G.Cacciapaglia,F.Sannino,arXiv:2204.04514,2022.

[16]H.Song,W.Su,M.Zhang,arXiv:2204.05085,2022.

[17]H.Bahl,J.Braathen,G.Weiglein,arXiv:2204.05269,2022.

[18]Y.Cheng,X.-G.He,Z.-L.Huang,M.-W.Li,arXiv:2204.05031,2022.

[19]K.S.Babu,S.Jana,V.P.K.,arXiv:2204.05303,2022.

[20]Y.Heo,D.-W.Jung,J.S.Lee,arXiv:2204.05728,2022.

[21]Y.H.Ahn,S.K.Kang,R.Ramos,arXiv:2204.06485,2022.

[22]M.-D.Zheng,F.-Z.Chen,H.-H.Zhang,arXiv:2204.06541,2022.

[23]P.F.Perez,H.H.Patel,A.D.Plascencia,arXiv:2204.07144,2022.

[24]S.Kanemura,K.Yagyu,arXiv:2204.07511,2022.

[25]X.K.Du,Z.Li,F.Wang,Y.K.Zhang,arXiv:2204.04286,2022.

[26]T.-P.Tang, M.Abdughani, L. Feng,Y.-L.S.Tsai, Y.-Z.Fan, arXiv:2204.04356, 2022.

[27]J.M.Yang,Y.Zhang,arXiv:2204.04202,2022.

[28]P.Athron,M.Bach,D.H.J.Jacob,W.Kotlarski,D.Stöckinger,A.Voigt,arXiv:

2204.05285,2022.

[29]A.Ghoshal,N.Okada,S.Okada,D.Raut,Q.Shafi,A.Thapa,arXiv:2204.07138, 2022.

[30]G.-W.Yuan,L.Zu,L.Feng,Y.-F.Cai,Y.-Z.Fan,arXiv:2204.04183,2022.

[31]P.Athron,A.Fowlie,C.-T.Lu,L.Wu,Y.Wu,B.Zhu,arXiv:2204.03996,2022.

[32]M. Blennow, P. Coloma, E.Fernández-Martínez, M. González-López, arXiv:

2204.04559,2022.

[33]J.J.Heckman,arXiv:2204.05302,2022.

[34]H.M.Lee,K.Yamashita,arXiv:2204.05024,2022.

[35]L.DiLuzio,R.Gröber,P.Paradisi,arXiv:2204.05284,2022.

[36]A.Paul,M.Valli,arXiv:2204.05267,2022.

[37]T.Biekötter,S.Heinemeyer,G.Weiglein,arXiv:2204.05975,2022.

[38]R.Balkin,E.Madge,T.Menzo,G.Perez,Y.Soreq,J.Zupan,arXiv:2204.05992, 2022.

[39]K.Cheung,W.-Y.Keung,P.-Y.Tseng,arXiv:2204.05942,2022.

[40]X.K.Du,Z.Li,F.Wang,Y.K.Zhang,arXiv:2204.05760,2022.

[41]M.Endo,S.Mishima,arXiv:2204.05965,2022.

[42]A.Crivellin,M.Kirk,T.Kitahara,F.Mescia,arXiv:2204.05962,2022.

[43]K.S.Babu,X.G.He,Mod.Phys.Lett.A4(1989)61.

[44]J.T.Peltoniemi,D.Tommasini,J.W.F.Valle,Phys.Lett.B298(1993)383.

[45]Y.Farzan,E.Ma,Phys.Rev.D86(2012)033007,arXiv:1204.4890.

[46]S.CentellesChuliá,E.Ma,R.Srivastava,J.W.F.Valle,Phys.Lett.B767(2017) 209,arXiv:1606.04543.

[47]E.Ma,N.Pollard,R.Srivastava,M.Zakeri,Phys.Lett.B750(2015)135,arXiv:

1507.03943.

[48]M. Reig,J.W.F. Valle,C.A.Vaquera-Araujo, Phys. Rev.D94(2016) 033012, arXiv:1606.08499.

[49]C.Bonilla,J.W.F.Valle,Phys.Lett.B762(2016)162,arXiv:1605.08362.

[50]C.Bonilla,E.Ma,E.Peinado,J.W.F.Valle,Phys.Lett.B762(2016)214,arXiv:

1607.03931.

[51]E.Ma,O.Popov,Phys.Lett.B764(2017)142,arXiv:1609.02538.

[52]E.Ma,U.Sarkar,Phys.Lett.B776(2018)54,arXiv:1707.07698.

[53]D.Borah,Phys.Rev.D94(2016)075024,arXiv:1607.00244.

[54]D.Borah,A.Dasgupta,J.Cosmol.Astropart.Phys.1612(2016)034,arXiv:1608. 03872.

[55]D.Borah,A.Dasgupta,J.HighEnergyPhys.01(2017)072,arXiv:1609.04236.

[56]D.Borah,A.Dasgupta,J.Cosmol.Astropart.Phys.1706(2017)003,arXiv:1702. 02877.

[57]S.CentellesChuliá, R.Srivastava, J.W.F.Valle, Phys. Lett.B773(2017)26, arXiv:1706.00210.

[58]C.Bonilla,J.M.Lamprea,E.Peinado,J.W.F.Valle,Phys.Lett.B779(2018)257, arXiv:1710.06498.

[59]D.Borah,B.Karmakar,Phys.Lett.B780(2018)461,arXiv:1712.06407.

[60]S.CentellesChuliá,R.Srivastava,J.W.F. Valle,Phys.Lett.B781(2018)122, arXiv:1802.05722.

[61]S.CentellesChuliá,R.Srivastava,J.W.F.Valle,arXiv:1804.03181,2018.

[62]D.Borah,B.Karmakar,D.Nanda,J.Cosmol.Astropart.Phys.1807(2018)039, arXiv:1805.11115.

[63]D.Borah,B.Karmakar,Phys.Lett.B789(2019)59,arXiv:1806.10685.

[64]S.CentellesChuliá,R.Cepedello,E.Peinado,R.Srivastava,J.HighEnergyPhys.

10(2019)093,arXiv:1907.08630.

[65]S.Jana,V.P.K.,S.Saad,arXiv:1910.09537,2019.

[66]D.Borah,D.Nanda,A.K.Saha,arXiv:1904.04840,2019.

[67]A.Dasgupta,S.K.Kang,O.Popov,Phys.Rev.D100(2019)075030,arXiv:1903. 12558.

[68]E.Ma,arXiv:1907.04665,2019.

[69]E.Ma,arXiv:arXiv:1905.01535,2019.

[70]S.Saad,Nucl.Phys.B943(2019)114636,arXiv:1902.07259.

[71]S.Jana,V.P.K.,S.Saad,Eur.Phys.J.C79(2019)916,arXiv:1904.07407.

[72]D.Nanda,D.Borah,arXiv:1911.04703,2019.

[73]T.A.Chowdhury,M.Ehsanuzzaman,S.Saad,arXiv:2203.14983,2022.

[74]N.Narendra, N. Sahoo,N. Sahu,Nucl.Phys. B936(2018) 76, arXiv:1712. 02960.

[75]M.E.Peskin,T.Takeuchi,Phys.Rev.Lett.65(1990)964.

[76]M.E.Peskin,T.Takeuchi,Phys.Rev.D46(1992)381.

[77]K.S.Kumar,S.Mantry,W.J.Marciano,P.A.Souder,Annu.Rev.Nucl.Part.Sci.

63(2013)237,arXiv:1302.6263.

[78]J.R.Forshaw,D.A.Ross,B.E.White,J.HighEnergyPhys.10(2001)007,arXiv:

hep-ph/0107232.

[79]G.Mangano,G.Miele,S.Pastor,T.Pinto,O.Pisanti,P.D.Serpico,Nucl.Phys.B 729(2005)221,arXiv:hep-ph/0506164.

[80]N.Aghanim,etal.,Planck,arXiv:1807.06209,2018.

[81]E.Grohs,G.M.Fuller,C.T.Kishimoto,M.W.Paris,A.Vlasenko,Phys.Rev.D93 (2016)083522,arXiv:1512.02205.

[82]P.F.deSalas,S.Pastor,J.Cosmol.Astropart.Phys.1607(2016)051,arXiv:1606. 06986.

[83]J.Avva, etal., SPT-3G,J. Phys. Conf.Ser.1468 (2020)012008, arXiv:1911.

08047.

[84]K.N.Abazajian,etal.,CMB-S4,arXiv:1610.02743,2016.

[85]K.N.Abazajian,J.Heeck,Phys.Rev.D100(2019)075027,arXiv:1908.03286.

[86]P.FileviezPérez,C.Murgui,A.D.Plascencia,Phys.Rev.D100(2019)035041, arXiv:1905.06344.

[87]C.Han,M.López-Ibáñez,B.Peng,J.M.Yang,arXiv:2001.04078,2020.

[88]X.Luo,W.Rodejohann,X.-J.Xu,J.Cosmol.Astropart.Phys. 06(2020)058, arXiv:2001.04078.

[89]D. Borah, A. Dasgupta, C. Majumdar, D. Nanda, Phys. Rev. D 102(2020) 035025,arXiv:2005.02343.

[90]P.Adshead,Y.Cui,A.J.Long,M.Shamma,arXiv:2009.07852,2020.

[91]X.Luo,W.Rodejohann,X.-J.Xu,arXiv:2011.13059,2020.

[92]D.Mahanta,D.Borah,arXiv:2101.02092,2021.

[93]Y.Du,J.-H.Yu,arXiv:2101.10475,2021.

[94]A.Biswas,D.Borah,D.Nanda,J.Cosmol.Astropart.Phys.10(2021)002,arXiv:

2103.05648.

[95]R.Mahbubani, L. Senatore,Phys. Rev.D73 (2006) 043510, arXiv:hep-ph/ 0510064.

[96]F.D’Eramo,Phys.Rev.D76(2007)083522,arXiv:0705.4493.

[97]R.Enberg,P.J.Fox,L.J.Hall,A.Y.Papaioannou,M.Papucci,J.HighEnergyPhys.

11(2007)014,arXiv:0706.0918.

[98]T. Cohen, J. Kearney, A. Pierce, D. Tucker-Smith, Phys. Rev. D 85 (2012) 075003,arXiv:1109.2604.

[99]C.Cheung,D.Sanford,J.Cosmol.Astropart.Phys.1402(2014)011,arXiv:1311.

5896.

[100]D.Restrepo,A.Rivera,M.Sánchez-Peláez,O.Zapata,W.Tangarife,Phys.Rev.

D92(2015)013005,arXiv:1504.07892.

[101]L.Calibbi,A.Mariotti, P.Tziveloglou, J. HighEnergy Phys. 10(2015) 116, arXiv:1505.03867.

[102]G.Cynolter,J.Kovács,E.Lendvai,Mod.Phys.Lett.A31(2016)1650013,arXiv:

1509.05323.

[103]S.Bhattacharya,N.Sahoo,N.Sahu,Phys.Rev.D93(2016)115040,arXiv:1510. 02760.