One-loop Dirac neutrino mass and mixed axion/WIMP dark matter

Cristian D. R. Carvajal1, ∗ and Oscar Zapata1, †

1 Instituto de Fısica, Universidad de Antioquia, Calle 70 No. 52-21, Medellın, Colombia.(Dated: April 4, 2019)

We consider the Peccei-Quinn (PQ) mechanism as the one behind the Dirac neutrino masses when

these are generated through the d = 5 effective operator LHNRφ at one loop level, with φ beinga Standard Model singlet scalar. In this setup, the PQ symmetry guarantees that the one-looprealization of such an effective operator gives the leading contribution to the Dirac neutrino massesby forbidding the contributions arising from its tree level realizations. All the mediators in the one-loop neutrino mass diagrams can be stabilized by a remnant ZN symmetry from the PQ symmetrybreaking, thus forming a dark sector besides the axion sector and leading to mixed axion/WIMPdark matter scenarios.

I. INTRODUCTION

In the last decades the neutrino oscillations have been firmly established thanks to a great dedicated experimentalprogram [1], leading to a concise and a clear understanding of the neutrino oscillation pattern [2]. However, thiswisdom has not shed light on the underlying mechanism behind the neutrino masses and the properties of neutrinosunder the particle-antiparticle conjugation operation. This adds to the negative results regarding the Majorana natureof the neutrinos from neutrinoless double-beta decay experiments [3–8]. Hence, a growing interest in Dirac neutrinosmass models has recently appeared, especially in models that contain a weakly interacting massive particle (WIMP)[9] as the dark matter candidate that is not trivially connected with the neutrino mass generation mechanism [10–24].

It worth noting here that the main motivation for considering WIMPs is that they lie at the scale which at thephysics beyond the Standard Model (SM) is expected to appear, thus leading to signals in dedicated experimentslooking for dark matter (DM). However, the WIMP paradigm is not free of theoretical and experimental challenges[25], such as the lack of signals in those experiments so far, which have led to severe constraints over the parameterspace of WIMP models [25–28]. Therefore, it would be relevant to consider approaches beyond the WIMP paradigmsuch as the multi-component DM [29, 30], where the DM of the Universe is composed by, for instance, WIMPs andQCD axions [31–36].

Within the framework of the SM it is not possible to form a Dirac mass term for the neutrinos because of the lackof the right-handed partners of active neutrinos νL. Once they are added, Dirac neutrino masses are generated viathe d = 4 operator

O4 = y LHNR + h.c., (1)

where NR are right-handed neutrinos, L = (νL, `L)T is the left-handed lepton doublet and H = (H+, H0)T is theHiggs doublet. Under the lepton number conservation (to protect the Diracness of neutrinos) this operator leads tosub-eV neutrino masses for |y| . 10−13, thus leaving unsettled the explanation of the smallness of the neutrino massscale. From this, it would be reasonable to forbid O4 through a certain symmetry X while generating Dirac neutrinomasses via higher dimensional operators, either at tree level or loop level. On those lines, if the Higgs doublet is theonly scalar in the particle spectrum the lowest dimensional operator would be of dimension d = 6, while if a scalarSM singlet φ charged under X is added then a d = 5 operator O5 would arise [14, 21, 24, 37].

O5 = y′LHNRφ+ h.c.. (2)

It follows that after the spontaneous breaking of X (due to the vacuum expectation value of φ) an effective O4-likeoperator will arise but with the benefit that O4 itself is forbidden. Additionally, the X symmetry may also may serveas the stabilizing symmetry of possible WIMP candidates arising from the one-loop (ultraviolet) realizations of O5

[13–19].In this work, we associate the Peccei-Quinn (PQ) mechanism [38] to scotogenic models with one-loop Dirac neutrino

masses1 through identification of φ inO5 as the scalar field hosting the QCD axion and the PQ symmetry as responsible

∗ cdavid.ruiz@udea.edu.co† oalberto.zapata@udea.edu.co1 See Refs. [33, 35, 39–54] for scenarios where the PQ mechanism is deeply related to the neutrino mass generation (at tree or loop level).

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for the absence of O4 and the stability of the possible WIMP candidates. It follows that after the spontaneouselectroweak and PQ symmetry breaking Dirac neutrino masses will be generated via the O5 operator, and imposingthat H does not carry a PQ charge (while L and/or NR do) the contribution to the neutrino masses from the O4

operator is automatically forbidden, and under some PQ charge assignments it is also possible to make the one looprealization of O5 the main source of neutrino masses. That is, the contributions of the three Dirac seesaw mechanismsare no longer present and the smallness of neutrino masses is related to the radiative character besides the large masssuppression coming from the loop mediators. Furthermore, we will show that the same PQ charge assignments leadto a residual ZN discrete symmetry after the PQ symmetry breaking that guarantees the stability of the lightest ofthe mediators in the one-loop neutrino mass diagrams, thus leading naturally to axion/WIMP DM scenarios.

II. FRAMEWORK

The apparent non-observation of CP violation originating from the θ term in the QCD Lagrangian is a strongtheoretical motivation for going beyond the SM, since it can be dynamically explained through the PQ mechanism[38]. Such a mechanism requires one to extend the SM gauge group with an anomalous global symmetry, U(1)PQ,which is spontaneously broken at some high scale [55–58] by the vacuum expectation value vS of the scalar S thathosts the QCD axion a(x) [59, 60],

S =1√2

[ρ(x) + vS ]eia(x)/vS . (3)

Here ρ(x) is the radial part that will gain a mass of order of the PQ symmetry breaking scale, which is constrained byseveral astrophysical phenomena such as supernova cooling [61] and black hole superradiance [62] to vS ∼ [109, 1017]GeV. As canonical axion model we consider a hadronic KSVZ type model [55, 56], that is, we add to the SM theaxionic field S and two chiral singlet quarks DL and DR, all of them charged under the PQ symmetry and interactingthrough the Yukawa term yDSDLDR.

In the present framework S additionally plays the role of φ in the O5 operator, with both L and NR charged underthe PQ symmetry whereas H does not in such a way O5 is allowed and O4 is banned. Contrary to the originalKSVZ model, where all the SM fermions are neutral under the PQ symmetry, in this framework the leptons do havePQ charges but the quarks remain neutral. The lepton number L conservation is further imposed with the usualassignment for the SM fields (1 for the leptons and 0 for the rest), 1 for NR and 0 for S, in order to prevent theappearance of the Majorana mass term NRNR and the d = 5 Weinberg operator LHLH, along their induced partnersNRNRS and LHLHS, and thus protecting the Dirac nature of neutrinos. However, it should be noted that byallowing large values for the PQ charges and not imposing L conservation is possible to obtain a consistent modelwhere the Diracness of neutrinos is guaranteed [47] (see Sec. IV for a specific example).

More specifically, the S, H and NR fields transform under the PQ symmetry as

S → eiXSξS, H → H, NR → eiXNRξNR, with XS 6= 0. (4)

It follows that the PQ charge of the lepton doublet must be XL = XNR+ XS in order to simultaneously make the

O5 operator PQ-invariant and O4 forbidden. Regarding the PQ charges of DR and DL, they are related throughthe Yukawa term yDSDLDR so XDL

= XDR+ XS . This, in turn, implies that the color anomaly coefficient C of the

resulting axion models is set by C = |XS |/2 = M , with M a positive integer2, i.e. the PQ charge of S must be even.On the other hand, we assume XDR

6= 0 to avoid the mixing term qLHDR (the SM quark sector is not charged underthe U(1)PQ). All in all, these L and PQ charge assignments automatically forbid O4 and the Majorana mass termsfor NR and νL, and leave invariant O5. However, they do not necessarily prevent the contributions to the neutrinomass matrix arising from the tree-level realizations of O5.

In the presence of right-handed neutrinos and a scalar singlet, the tree-level realizations of O5 demand the intro-duction of either a SM singlet vector-like fermion, an extra Higgs doublet or a vector-like lepton doublet as mediatorfields, which lead to the three Dirac seesaw mechanisms [14, 21, 24, 37]: type I, type II and type III, respectively.On the other hand, all possible realizations -considering new fields (fermions and scalars) transforming as singlets,doublets or triplets under SU(2)L, and color singlets, with arbitrary values of the hypercharge but fixed up to a

2 The color anomaly is given by C =∑i |XDL

−XDR|T (Ci), where the sum is over all the irreducible SU(3)C × U(1)EM representations

and T (Ci) is the index of the SU(3)C representation of the DL,R fields (T (3) = 1/2) [63]. Note that the condition of C to be an integeris mandatory to guarantee the periodicity of the axion potential, 2πfa = 2πvS/C, and the interpretation of the axion as the phase ofthe field S (implying a periodicity of 2πvS) [64, 65]. Notice here that vS is related to the axion decay constant fa via fa = vS/C, wherethe color anomaly also sets the number of vacua of the S (the so called domain wall number [64]).

3

S H

NR LX1

X2

X3

X4

(a) T1-1-A model

SH

NR LX1

X2

X3

X4

(b) T1-1-B model

NR X1 L

X2

S X3 H

X4

(c) T1-2-A model

NR X1 L

X2

SX3H

X4

(d) T1-2-B model

NR X1

X2

S X3 L

X4

H

(e) T1-3-D model

NR X1

X2

S

X3

X4

H L

(f) T1-3-E model

S H

X1

X2 X3

NR L

(g) T3-1-A model

NR X3 S

X1

X4X2

L H

(h) T4-3-I model

FIG. 1: One-loop diagrams with T1, T3 and T4 topologies leading to Dirac neutrino masses.

free parameter α3- of O5 leading to Dirac neutrino masses at one-loop level were studied in Ref. [21]. There it wasobtained that a finite list of sets of mediator fields exists, which give rise to neutrino masses at one-loop level, classifiedaccording to the topology of the diagram. In that analysis, a Z2 symmetry was used to forbid O4, with both NR andS odd and the SM sector even under this Z2, and the new fermions were assumed to be vector-like to ensure anomalycancellation of the SM gauge symmetries. We will use their results regarding the renormalizable and genuine one-loopmodels to study the necessary conditions to implement the PQ mechanism as the responsible behind the one-loopDirac neutrino masses4.

At one-loop level, there are three different topologies that lead to renormalizable and genuine one-loop realiza-tions [21]: the T1 (box-like), T3 (triangle-like) and T4 (penguin-like) topologies. For T1 and T4 topologies, thereexist different possibilities of assigning the fields in O5 to the four external legs, hence several one-loop diagramsappear, see Fig. 1. The mediator fields in all the one-loop diagrams are denoted by Xi, where i runs from 1 to 3(4)for the T3 topology (T1 and T4 topologies) and denotes the number of mediators. The Lorentz nature, scalar orfermion, is shown in the corresponding one-loop diagram by a dashed or solid line, respectively.

In Table I we show the L and PQ charges of the mediator fields that are consistent with our assumptions: i)axionic one-loop realizations of the O5 operator and prohibition of the O4 operator, ii) lepton number conservation,iii) vanishing contributions of the tree-level realizations of O5 to the neutrino mass matrix, and iv) existence ofa remnant ZN symmetry after the PQ breaking (see next Section). For completeness purposes we display in theAppendix VI the sets of quantum numbers of the mediator fields for each model [21]. It turns out that for α = 0, 1or 2 (the values leading to either Y = 0 singlet fermions, or Y = ±1 doublet fermions or Y = ±1 doublet scalars) oneor two Dirac seesaw mechanisms arise, implying that the one-loop contribution is subdominant (the exceptional caseis the T4-3-I-V model since all the mediators fields are SU(2)L triplets).

3 The hypercharge of the mediators is set to Y = α or Y = α ± 1 so that the possible values for α are 0,1,2 in order to have SU(2)Lmultiplets featuring a neutral particle.

4 We will closely follow the notation introduced in Ref. [21] for the topologies, fields, and genuine models.

4

Model Symmetry X1 X2 X3 X4

T1-(1/2)-AU(1)L δ − 1 δ δ δ

U(1)PQ XX4 − (XNR + XS) XX4 −XS XX4 XX4

T1-(1/2)-BU(1)L δ − 1 δ δ δ

U(1)PQ XX4 − (XNR + XS) XX4 −XS XX4 −XS XX4

T1-3-DU(1)L δ δ + 1 δ + 1 δ

U(1)PQ XX4 XNR + XX4 XNR + XX4 + XS XX4

T1-3-EU(1)L δ δ + 1 δ + 1 δ

U(1)PQ XX4 + XS XNR + XX4 + XS XNR + XX4 + XS XX4

T3-1-AU(1)L δ δ + 1 δ + 1

U(1)PQ XX1 XNR + XX1 XNR + XX1 + XS

T4-3-IU(1)L −1 δ − 1 δ δ

U(1)PQ −(XNR + XS) XX4 − (XNR + XS) XX4 −XS XX4

TABLE I: L and PQ charges of the mediator fields Xi of T1, T3 and T4 models. δ, XX1and XX4

are free realparameters whereas XS and XNR

denote the PQ charges of S and NR, respectively. The Lorentz nature of Xi isinferred from the corresponding one-loop diagram (see text for details).

• T1 models. The six T1 models have four mediators, where the T1-1-A and T1-1-B models have three scalarmediators, T1-3-D and T1-3-E models have two scalars, whereas there is one single scalar mediator in T1-2-A andT1-2-B models. The T1 models differ on the number of scalar and fermion mediator fields, or on the exchange ofthe S and H. Note that models T1-1-A and T1-2-A share the same L and PQ charges, and this is the same forT1-1-B and T1-2-B models. This is because the charge assignment is independent of the SU(2)L transformationproperties of the fields. Each T1 model has four sets of possible electroweak charges for the mediator fields, allof them featuring at least one of the mediators of the Dirac seesaw mechanisms when α = 0, 1 or 2 (see TableIV).

• T3 models. There is one type of model with a T3 topology, which involves three mediators with two of themscalars. The T3 model has four sets of possible electroweak charges for the mediator fields, with at least one ofthe mediators of the Dirac seesaw mechanisms when α = 0, 1 or 2 (see Table V).

• T4 models. The T4-3-I models have four mediators with two fermions, with the particular feature that X1

is not running inside the loop. This in turn implies that all its quantum numbers are fixed: it transforms asa SU(3)L triplet with zero hypercharge and the L and PQ charges are settled by the lepton doublet charges:L(X1) = −1 and XX1

= −XL = −(XNR+ XS). On the other hand, the T4-3-I-V model does not contain any

mediator field of the Dirac seesaw mechanisms, however, it requires the mixing between the neutral and chargedleptons with the neutral and charged components of X1, which induce charged lepton flavor violating processesat tree level.

In the next Section we will present the conditions on the charges of the mediator fields that must be fulfilled toguarantee, at the same time, the presence of a WIMP candidate in the spectrum and the absence of tree-level Diracseesaw mechanisms.

III. RADIATIVE NEUTRINO MASSES AND AXION/WIMP DARK MATTER

It turns out that the same values for α that lead to tree-level Dirac seesaw mechanisms are the ones that allowus to have an electrically neutral and PQ charged particle in the spectrum, thus opening the possibility of havingmulticomponent DM scenarios comprising such particle (a WIMP) and the axion. Concretely, the existence of aneutral particle in the spectrum is guaranteed by setting Q = T3 +Y/2 = 0 such that Y = −2T3 for at least one of themediators. This implies that α can only take the values 0, 1 or 2 (depending upon the SU(2)L representation), whichare precisely the same values that lead to the tree-level realizations of O5. Thus, the presence of a WIMP candidatein the models automatically implies that the one-loop contribution is the main contribution to the neutrino masses.

Regarding the stability of the WIMP particle, all the mediator fields (except X1 in T4 models5) can be made part

5 It is worth mentioning that in the T4 models the X1 field is not running inside the loop which makes it different from the other mediatorfields in what concerns DM.

5

of a dark (WIMP) sector by demanding that a residual ZN symmetry is generated after the U(1)PQ breaking, in thesame line of the scotogenic models. This is achieved when XS is an even integer N (to preserve the periodicity of theaxion potential) and the PQ charges of the loop mediators are XXi

6= 0 mod(XS).On the other hand, the conditions over the PQ charges of the mediator fields that avoid tree-level Dirac seesaw

mechanisms (and hence guarantee a WIMP particle in the spectrum) lead to PQ charges XXi of the form of (seeTable I)

XX4, XX4

±XS , XX4±XNR

, or XX4±XNR

±XS ; (for T1 and T4 models),

XX1 , XX1 ±XNR, or XX1 ±XNR

±XS ; (for T3 models). (5)

Consequently, in order to ensure a non-trivial charge for these fields under the remnant ZN (N = XS), the XX4charge

(XX1for T3-1-A model) must be an integer non-multiple of XS as long as NR transforms trivially, XNR

= 0 mod(XS)(this the simplest choice, however, see Sec. IV for other possible ZN assignment). Note further that if XNR

6= 0 and2XNR

6= XS the L conservation would not be mandatory since the Diracness of neutrinos would be guarantee for theZN properties of the lepton doublet, L∼ωN/2 (see Ref. [66] for details and and Sec. IV for an example).

To illustrate a specific charge assignment, let us assume XS = 2 and XNR= 0, which entails that the PQ charge of

the lepton doublet and singlet are both 2 (this is because of the Yukawa terms LHNRS and LH`R). Thus, if all themediators running inside the loop have odd PQ charges then a remnant Z2 symmetry would appear in such a waythat they are Z2 odd while the rest of the model particles are Z2 even.

The stability of electrically neutral particles naturally leads to WIMP scenarios. Among the possible scenariosthat may arise (see Tables IV, V and VI) we highlight the ultraviolet fermionic realizations of the Higgs portal [67]:the singlet (via the axion interactions), singlet-doublet and doublet-triplet Dirac DM [68–71]; and the renormalizablescalar DM models: the singlet scalar DM model [72–74], inert doublet model (IDM) [75, 76] and inert triplet model[77–79], along with the interplays singlet-doublet and doublet triplet [80–84].

Regarding the heavy quark D, since the SM quarks and H are not charged under the PQ symmetry, DR cannotcouple to the SM quark doublet through qLHDR as long as XDR

6= 0. However, it may interact with the quark singletvia the DLdRS term, which entails that XDL

6= XS to avoid the constraints due to the mixing the SM quarks. Withrespect to the interactions with the dark sector, if there exists a Y = 0 singlet scalar ϕ or a second Higgs doublet H2

in the spectrum then the terms DLdRϕ and qLH2DR would appear. The first term would require that XDL= Xϕ

while the second one XDR= −XH2

, so D must be also part of the dark sector in order for such terms be allowed.It follows that D can only decay into dark scalars (ϕ and/or H2) and SM quarks, leading to SUSY-like signals suchas jets plus missing energy at the LHC if D is not heavy enough (see e.g. Ref. [34] for an explicit analysis). In theopposite case, the Lagrangian would be invariant under the exchange (DL, DR) → (−DL,−DR) making it a stablecolored heavy quark, thus bringing some cosmological problems [85]. Therefore, the instability of the heavy quark Dimposes additional constraints over the viable models since it demands the existence of either a Y = 0 singlet scalaror an extra Higgs doublet in the dark sector. From Tables IV, V and VI, we obtain that the following models do notcontain such fields: T1-2-A-IV, T1-3-D-IV and T4-3-I-V.

On the other hand, since the PQ mechanism is at work suitable parameters for the QCD axion can be consideredin order to have the axion as a second DM candidate. In the framework of a broad class of inflationary scenariosQCD axions may be non-thermally produced in the early Universe through the vacuum-realignment mechanism andsurviving as cold DM matter. The axion contribution to energy density depends on the order in which cosmologicalevents take place, especially whether the breaking of the PQ symmetry occurred before or after inflation. For thecase of reheating temperatures lower than fa, the axion relic density due to the misalignment population is fixed bythe initial field displacement, the so-called misalignment angle θa, and fa [86, 87],

Ωah2 ≈ 0.18θ2a

(fa

1012 GeV

)1.19

. (6)

Thus the allowed window of axion parameters is

109 GeV . fa . θ−2a 1012 GeV, (7)

6 θ2a µeV . ma . 6 meV (8)

where ma ≈ 6µ eV(1012 GeV/fa) [59, 60] (see Refs. [88, 89] for the detailed study of the window for preferred axionmodels and Refs. [90–94] for the current status of experimental searches). For this case the axion population fromthe decay of topological defects such as string axions and domain walls is diluted by inflation.

It follows that in this framework, the DM of the Universe comprises the WIMP and the axion, which behave as twocompletely independent DM particles, without affecting the standard relic density calculations and the corresponding

6

L `R NR S ψ ϕ H2 DL DR

U(1)L 1 1 1 0 1 0 0 0 0

U(1)PQ 2 2 0 2 3 3 1 1 −1

Z2 + + + + − − − − −

TABLE II: L and PQ charges for model T3-1-A-I with α = 0 and δ = −1, XS = 2, Xψ = 1 and XNR= 0. The

charges under the remnant Z2 symmetry are also shown.

experimental bounds [31–36, 95]. Therefore, at the electroweak scale the models presented in this work are up to someextent similar to the renormalizable WIMP DM models mentioned above, with the main difference being that theDM is complemented by axions. And since there exist two contributions to the total DM relic abundance, we expectthat all the constraints over such WIMP scenarios would be substantially weakened, thus resulting in larger portionsof the parameter space that are still allowed within those scenarios (see Refs. [33, 34, 36, 95] for specific scenarios).In addition to this, since the WIMP may couple to the right-handed neutrino through Yukawa interactions furtherWIMP annihilation channels may appear and thus changing drastically the expected DM phenomenology, with thebonus that they are rather unconstrained because such interactions do not induce lepton flavor violation processes.

IV. A CASE STUDY

In this section we study a specific model resulting from the analysis made in previous sections, the T3-1-A-I modelwith α = 0 and a residual Z2 symmetry. Thus, the model contains the following fields: an SU(2)L doublet scalar X3,and two SM singlets, one being a Dirac fermion X1 while the other one is a complex scalar X2. We add a secondsinglet fermion in order to obtain two massive neutrinos. In what follows we rename these fields as X1 → ψc, X2 → ϕ∗

and X3 → H2, with ϕ = (ϕR + iϕI)/√

2, H2 = (H+, (H0 + iA0)/√

2))T and H2 = iσ2H∗2 . Moreover, H1 will denote

the SM Higgs doublet. The U(1)PQ and U(1)L charges are given in Table II.The more general Lagrangian invariant under GSM × U(1)L × U(1)PQ symmetry comprises the interaction terms

−L ⊃ VH1,S + Vϕ + VH2+ V1 + L1, where

VH1,S = −µ21|H1|2 + λ1|H1|4 + µ2

S |S|2 + λS |S|4 + λ6|H1|2|S|2, (9)

Vϕ = µ2ϕ|ϕ|2 + λϕ|ϕ|4 + λ7|H1|2|ϕ|2 + λ8|ϕ|2|S|2, (10)

VH2= µ2

2|H2|2 + λ2|H2|4 + λ3|H1|2|H2|2 + λ4|H†1H2|2 + λ9|H2|2|S|2, (11)

V1 = λ10|H2|2|ϕ|2 + λ11

[ϕ∗SH†2H1 + h.c.

], (12)

L1 = yiβψiH†2Lβ + hβiϕ

∗NRβψi + yQSDLDR + fβqLβH2DR + h.c.. (13)

Here some comments are in order. The λ6 term mixes the real component of the axion with the Higgs field, so we canneglect the λ6 without lost of generality. λ11 is responsible for the mixing between the neutral components of ϕ andH2 that leads to Dirac neutrino masses (notice that λ11 plays, to some extent, the role of the λ5 term in the scotogenicmodel [96]). Since the SM quarks are not charged under the PQ symmetry, D does not couple to SM particles throughDLdRϕ but does through qLH2DR. Note that, under the Z2 remnant symmetry6 the fields running in the loop ofdiagram T3-1-A in Fig. 1 are odd. Therefore, the lightest of them can be considered as the DM candidate (if itis neutral). This model at low energies, i.e., without the axion, is identical to the one presented in Ref. [10], sinceinstead of considering an ultraviolet completion realization it introduced a soft breaking term that allows the mixingbetween the neutral components of the singlet and doublet (in a similar way as λ11 does).

Let us consider the limit λ6 → 0 in order to avoid the mixing between the scalar S and the SM Higgs boson, and

the limit λ8,9 → 0 with the aim to avoid a large fine-tuning in the masses of ϕ and H0. In the basis(H0, ϕR

)Tand

(A0, ϕI)T the mass matrices for the Z2-odd neutral scalars are given by

MR,I =

(µ22 + 1

2 (λ3 + λ4)v2 12λ11vSv

12λ11vSv µ2

ϕ + 12λ7v

2

), sin(2θR,I) =

λ11vvSm2SR,I2

−m2SR,I1

, (14)

6 In fact, a larger dark U(1) symmetry is obtained, which contains the remnant Z2 as a subgroup.

7

H±

γ

l−β l−β′ψ

FIG. 2: Feynman diagram for the process `β → `β′γ.

where mSRj(mSIj

) are the two mass eigenvalues of MR (MI). The mass of the charged scalar H+ is given by

m2H+ = µ2

2 + 12λ3v

2, as in the IDM.Neutrino masses are generated at one-loop through the Feynman diagram corresponding to the T3-1-A model in

Fig. 1. The effective mass matrix is given by

(Mν

)ββ′

=1

64π2

λ11vSv

m2SR2−m2

SR1

∑i

hβiyiβ′mψi

[F

(m2SR2

m2ψi

)− F

(m2SR1

m2ψi

)]+ (R→ I), (15)

where F (x) = x ln(x)/(x−1), and R→ I means the exchange of the CP-even eigenstates (SRj) with the correspondingCP-odd eigenstates (SIj). It follows that the neutrino masses are suppressed by several factors: the loop factor, theYukawa couplings, the mass of the heaviest loop mediator and λ11. It turns out that the resulting suppression isenough to counteract the contribution coming from vS (see Eq. (17)). A simple numerical estimate for the effectivemass matrix can be calculated considering the limit7 λ3,4,7 → 0, µ2

2 → µ2ϕ and µ2

ϕ λ11vSv. In this case,

(Mν

)ββ′≈ λ11vSv

32π2

∑i

hβiyiβ′mψi

µ2ϕ −m2

ψi

[1−

m2ψi

µ2ϕ −m2

ψi

log

(µ2ϕ

m2ψi

)]. (16)

If µ2ϕ m2

ψ, that is considering fermion DM, it reduces to

(Mν

)ββ′≈ λ11vSv

32π2

∑i

hβiyiβ′mψi

µ2ϕ

∼ 0.05 eV×(hβiyiβ′

10−4

)(λ11

10−10

)(mψi

102 GeV

)(5× 103 GeV

µϕ

)2(vS

109 GeV

). (17)

The linear dependence of the neutrino masses with vS arises because the loop mediators are not lying at the PQ scale,a feature that is also shared with other radiative neutrino mass models e.g., [48]. Note that the smallness of λ11 isnatural in the ’t Hooft sense [97] since λ11 = 0 leads to an extra symmetry (the charge of S is no longer connectedwith the charges of the other particles). Furthermore, the smallness of λ11 is a necessary condition in order to have theloop scalar mediators at or below the TeV mass scale. This is also in consonance with the requirement of demanding atiny value for the scalar coupling between the Higgs boson H1 and the axion field S (these conditions are unavoidablein axion models as long as light loop mediators are required [33]).

On the other hand, the Yukawa interactions in Eq. (13) also lead to lepton flavor violation (LFV) processes atone-loop mediated by the charged scalar H± and ψi (see Fig. 2). The decay rate for this process is given by

Γ(`β → `β′γ

)=e2m5

β

16π

1

162π4m4H±

∑i

∣∣yiβy∗iβ′ ∣∣2(2t2i + 5ti − 1

12(ti − 1)3− t2i log ti

2(ti − 1)4

)2

. (18)

where ti ≡ m2ψi/m2

H± . Considering the limit of the heavy scalar, mH± mψi , it follows that∣∣∣∣∣yiµy∗ie(

5× 103 GeV

mH±

)2∣∣∣∣∣ . 0.1, (19)

7 The limits λ3,4,7 → 0 are motivated by the naturalness of the scalar potential since they avoid the imposition of large cancellationsamong the scalar potential parameters.

8

where the bound BR(µ→ eγ) < 5.7× 10−13 [98] has been used.Regarding WIMP DM, this model features either singlet fermion or singlet/doublet scalar candidates8. Such a

WIMP-like setup is similar to the model studied in Ref. [10] with the difference that in the present case the DM isadditionally composed by axions. Therefore, all the constraints that may exist on the model with only WIMPs wouldbe substantially weakened. Indeed, when the WIMP particle is mainly doublet (an IDM-like scenario) most of theintermediate DM mass range can be free of DM constraints [33, 34] (see Ref. [36] for the case of singlet scalar DM). Inthe case of fermion DM, in order to account for the total relic abundance the ψψ annihilation must be driven by hβiYukawa terms, since in this case there are not constraints coming from lepton flavor violating processes, so they canbe large enough [10]. In contrast, ψψ annihilation via yiβ Yukawa terms is not a viable option without some detailedfine-tuning of parameters. All in all, this shows the impact of considering mixed axion/WIMP DM scenarios.

Finally, it is worth mentioning that with a suitable PQ charge assignment (such as the one displayed in Table III)and without imposing lepton number conservation it is possible to obtain a consistent model where the Diracness ofneutrinos is guaranteed. In this case, a remnant Z4 symmetry is obtained from the PQ symmetry breaking which isresponsible for the stability the WIMP candidate.

L `R NR S ψ ϕ H2 DL DR

U(1)PQ 8 8 4 4 7 3 −1 5 1

Z4 1 1 1 1 ω3 ω3 ω3 ω ω

TABLE III: Alternative PQ charges for model T3-1-A-I that guarantee the Diracness of neutrinos without furtherassuming the lepton number symmetry. The charges under the remnant Z4 symmetry are also shown, where ω4 = 1.

V. SUMMARY

In this work we have studied the one-loop realizations of the d = 5 operator LHNRS that leads to Dirac neutrinomasses, with S being a singlet scalar field that hosts the QCD axion. As usual, the axion arises from the breaking ofthe Peccei-Quinn symmetry, which in our setup we used to not only solve the strong CP problem, but also to forbid

the operator LHNR (which generates Dirac neutrino masses at tree level) and the tree-level realizations of LHNRS.Thus, the neutrino masses are directly correlated to the axion mass (via the PQ symmetry breaking scale vs) andtheir smallness is due to the radiative character besides the mass suppression of the loop mediators. Furthermore,the PQ symmetry breaking leaves a residual ZN symmetry that allow us to guarantee the stability of the lightest ofthe mediators in the one-loop neutrino mass diagrams (as happens in the scotogenic models), thus leading naturallyto multicomponent DM scenarios with axions and WIMPs. We have illustrated our proposal by considering a specificmodel, where simple numerical estimates allow us to show the effectiveness of the scheme regarding neutrino masses,DM and lepton flavor violating processes.

VI. ACKNOWLEDGMENTS

We are thankful to Diego Restrepo for enlightening discussions. Work supported by Sostenibilidad-UdeA andthe UdeA/CODI Grant 2017-16286, and by COLCIENCIAS through the Grants 111565842691 and 111577657253.C. D. R. C. acknowledges the financial support given by the Departamento Administrativo de Ciencia, Tecnologıae Innovacion - COLCIENCIAS (doctoral scholarship 727-2015). O.Z. acknowledges the ICTP Simons associatesprogram and the kind hospitality of the Abdus Salam ICTP where part of this work was done.

APPENDIX

Below we display the sets of quantum numbers of the mediator fields for each model [21].

8 A detailed DM phenomenological study of this model will be done elsewhere.

9

Model Sol. XF1 XS

2 XS3 XS

4 α

T1-1-A

I 1α 1α 1α 2α−1 0, 2

II 2α 2α 2α 1α−1 ±1

III 2α 2α 2α 3α−1 ±1

IV 3α 3α 3α 2α−1 0, 2

T1-1-B

I 1α 1α 2α−1 2α−1 0, 2

II 2α 2α 1α−1 1α−1 ±1

III 2α 2α 3α−1 3α−1 ±1

IV 3α 3α 2α−1 2α−1 0, 2

(a)

Model Sol. XS1 XF

2 XF3 XF

4 α

T1-2-A

I 1α 1α 1α 2α−1 0, 2

II 2α 2α 2α 1α−1 ±1

II 2α 2α 2α 3α−1 ±1

IV 3α 3α 3α 2α−1 0, 2

T1-2-B

I 1α 1α 2α−1 2α−1 0, 2

II 2α 2α 1α−1 1α−1 ±1

III 2α 2α 3α−1 3α−1 ±1

IV 3α 3α 2α−1 2α−1 0, 2

(b)

Model Sol. XF1 XS

2 XS3 XF

4 α

T1-3-D

I 1α 1α 1α 2α+1 0

II 2α 2α 2α 1α+1 ±1

III 2α 2α 2α 3α+1 ±1

IV 3α 3α 3α 2α+1 0

T1-3-E

I 1α 1α 2α−1 1α 0, 2

II 2α 2α 1α−1 2α ±1

III 2α 2α 3α−1 2α ±1

IV 3α 3α 2α−1 3α 0, 2

(c)

TABLE IV: Possible quantum numbers for the mediators in the T1 topologies.

Model Sol. XF1 XS

2 XS3 α

T3-1-A

I 1α 1α 2α−1 0, 2

II 2α 2α 1α−1 ±1

III 2α 2α 3α−1 ±1

IV 3α 3α 2α−1 0, 2

TABLE V: Possible quantum numbers for the mediators in the T3 topology.

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