Minimal radiative Dirac neutrino mass models

Julian Calle 1,∗ Diego Restrepo 1,† Carlos E. Yaguna 2,‡ and Oscar Zapata 1§1 Instituto de Fısica, Universidad de Antioquia,

Calle 70 # 52-21, Apartado Aereo 1226, Medellın, Colombia and2 Escuela de Fısica, Universidad Pedagogica y Tecnologica de Colombia,

Avenida Central del Norte # 39-115, Tunja, Colombia

Neutrinos may be Dirac particles whose masses arise radiatively at one-loop, naturally explainingtheir small values. In this work we show that all the one-loop realizations of the dimension-fiveoperator to effectively generate Dirac neutrino masses can be implemented by using a single localsymmetry: U(1)B−L. Since this symmetry is anomalous, new chiral fermions, charged under B−L,are required. The minimal model consistent with neutrino data includes three chiral fermions, twoof them with the same lepton number. The next minimal models contain five chiral fermions andtheir B−L charges can be fixed by requiring a dark matter candidate in the spectrum. We list thefull particle content as well as the relevant Lagrangian terms for each of these models. They arenew and simple models that can simultaneously accommodate Dirac neutrino masses (at one-loop)and dark matter without invoking any discrete symmetries.

I. INTRODUCTION

The interpretation of neutrino experimental data in terms of neutrino oscillations is compatible with both Ma-jorana or Dirac neutrino masses [1]. The former possibility has received the most attention but, given the lack ofsignals in neutrinoless double beta decay experiments [2–7], the latter cannot be dismissed. If neutrinos are Diracparticles, the Standard Model (SM) particle content must be extended with right-handed neutrinos, and some sym-metry must be imposed to prevent their Majorana mass terms. At least a Z3 symmetry is required to guaranteethe neutrino Diracness through

Lν = yD (νR)†L ·H + h.c. , (1)

with1 L ·H = εabLaHb, where L is the lepton doublet, H is the SM Higgs doublet with hypercharge Y = 1, and yD

is the matrix of neutrino Yukawa couplings. To be compatible with neutrino oscillation data [9], yD should be atleast of order 2× 3. A possible assignment for the set of SM fields that transform non-trivially under Z3 is: L ∼ ω,

(eR)† ∼ ω2 and (νR)

† ∼ ω2, with ω3 = 1. At this level, the neutrino mass problem is not longer a phenomenologicalissue but a theoretical one, in which it is necessary to explain the smallness of the Yukawa couplings in yD, whichmust be of order 10−11.

To do so, we assume that the symmetry allows for the 5-dimensional operator with total lepton number conser-vation [10]

L5 =h

Λ(νR)

†L ·H S∗ + h.c. , (2)

where Λ is the new physics scale, and that this operator is first realized at one-loop level [11] (see [11–19] for thetree-level realizations).

Regarding the symmetry, we follow the usual approach of promoting baryon number (B) minus lepton number(L) from an accidental global symmetry of the SM, to a local Abelian symmetry, U(1)B−L, which is spontaneouslybroken. One of the main novelties of our work is that we do not impose any other symmetries, discrete or otherwise.Thus, the charges of the right-handed neutrinos under U(1)B−L should be such that the tree-level Dirac mass

∗ julian.callem@udea.edu.co† restrepo@udea.edu.co‡ carlos.yaguna@uptc.edu.co§ oalberto.zapata@udea.edu.co1 Throughout the text we will follow the convention of defining only left-handed Weyl spinors [8], and we will use the SU(2) metric to

build scalar products.

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term (1) is forbidden. This requirement automatically excludes the usual assignment where the three right-handedneutrinos have B − L charges equal to −1.

The classification of all the topologies at one-loop level that realize the effective operator (2) has been presentedin [11]. There, in addition to the U(1)B−L symmetry, at least one additional Z2 symmetry was imposed to avoidthe Majorana mass terms for the right-handed neutrinos, and a further Z ′2 was required to avoid i) the appearanceof tree-level realizations in the cases when U(1)B−L is not able to do it, and ii) to have a dark matter candidatein the particle spectrum –one of the new particles needed to realize (2). Here, we focus instead on the simplestrealizations of each topology that can be realized with a single symmetry, U(1)B−L. This same symmetry would beresponsible for the stability of possible dark matter candidates appearing in the different realizations. Let us stressthat until now in the literature there is not a simple realization of operator (2) at one-loop invoking only a singlesymmetry. There were some efforts in this direction but either some Majorana terms were left out, which wouldneed to be forbidden with an extra Z2-symmetry [20, 21], or the found models require many extra fields [20].

Using only U(1)B−L, we find, for each topology, the realization with the minimum number of fields. The minimalmodel requires three chiral fields, two of them sharing the same lepton number, so that the spectrum contains twomassive Dirac neutrinos. The next minimal models include five chiral fields. Interestingly, their B − L charges arefixed once the requirement to have a dark matter particle is imposed. Hence, these new models may account fordark matter and Dirac neutrino masses (at one-loop) without invoking any discrete symmetries.

The rest of the paper is organized as follows. In the next section we introduce the notation and derive theconditions necessary to realize the different topologies that give rise to one-loop Dirac neutrino masses. Our mainresults are presented in section III. There, the particle content of the minimal models, with three and five chiralfields, is spelled out. Finally, in section IV our conclusions are drawn.

II. GENERAL SETUP

We use the notation for the topologies defined in [11], which are displayed in figure 1. There the flux of leptonnumber is illustrated by the wide colored arrows. The green arrow represents the flux of the doublet lepton number,L(Li) = −1, the yellow is for L(νRβ) = −ν with β at least 1, 2, the blue for L(S) = s, and the red for some internalcirculating L charge associated with a chiral fermion, which we choose as a free parameter in our setup.

We do not consider the T1-1 or T4 topologies of [11] because the former is already included in topology T3-1 whenthe X3 scalar field is decoupled, whereas the latter requires further symmetries to forbid the tree-level contribution2.In [11] it was assumed that the internal fermion lines were already vectorlike fermions, which allow them to use theν = 1 solution for anomaly cancellation conditions, but they needed to impose additional Z2 symmetries to forbidthe tree-level contribution to neutrino masses and to stabilize the dark matter. Here, we instead assume chiral fieldsfor the fermions that are singlets under the SM gauge group. This allow us to search for new minimal realizationsof the topologies with a single extra symmetry beyond the SM. It is clear that our solution with three chiral states,corresponding to the three right handed neutrinos, can be easily extrapolated to all the solutions found in [11],with the advantage that not further discrete symmetries are imposed. We explicitly illustrate this for the case oftopology T3-1-A, whose minimal solution involves just a Dirac fermion.

In our setup, the only extra symmetry beyond the SM, U(1)B−L, must forbid the tree-level terms

Lν = yDβi (νRβ)†Li ·H +MR

βγ (νRβ)†

(νRγ)†S + h.c , (3)

as well as allowing for the dimension five operator (2). This is accomplished by the L charge assignments in Table I.There, ν is the lepton number of the left-handed antineutrino, which is common to at least the two right-handedneutrinos required to explain the neutrino oscillation data. As already mentioned, the solution with ν = 1, studiedin [11], is no longer considered in this work.

To find the new possible solutions, we explore the anomaly cancellation conditions with five chiral fields bychecking that their charges do not generate direct or induced Majorana mass terms for them, that is, the fermion loop

mediators are Dirac-like fields. In addition to the chiral fields for β = 1, 2, we introduce (νRk)†

with L(νRk) = −νk,

2 The exception cases are the IV and V solutions of T4-3-I. However, they require mixing between the charged leptons and the newfermion fields, thus leading to charged lepton flavor violation processes which are quite constrained.

3

L νR

X2X3

S

H

X4 X1 L νR

X2X3

H

S

X4 X1

T1-3-D T1-3-E

L X1 νR

X2X3

H S

T3-1-A

L X1 νR

X2X4

H SX3

L X1 νR

X2X4

S HX3

T1-2-A T1-2-B

FIG. 1. Topologies (in notation of [11]) leading to one-loop Dirac neutrino masses.

Fields H Li (νRβ)† S

L 0 −1 ν 6= 1 ν − 1 6= −2ν

TABLE I. General assignment of lepton number for external legs of the one-loop topologies in figure 1

and a heavy Dirac fermion field with Weyl components ψL and (ψR)†, such that L(ψL) = l and L(ψR) = −r

respectively. The linear and cubic anomaly cancellation conditions are [22, 23]

2ν + νk + l + r = 3 , 2ν3 + ν3k + r3 + l3 = 3 . (4)

From the linear equation

νk = 3− 2ν − l − r . (5)

To further proceed, we separate the topologies in two types: The set (A) with T1-3-D and T3-1-A, correspondingto the ones with the yellow ν flux in figure 1, and where S is in a vertex only involving scalars; and the set (B)

4

with T1-3-E and T1-2-(A/B), with the blue s flux in figure 1, and where S is in a Yukawa-type vertex. For eachcase we have:

(A) The heavy Dirac fermion has a vectorlike mass, such that

l = −r . (6)

Replacing back in eq. (5) we obtain

νk =3− 2ν . (7)

Solving the cubic equation for ν gives rise to two different roots: 1 and 4. Thus, we choose ν = 4 which leadsto νk = −5.

On the other hand, the fact that the new fermion field ψ does not contribute to the anomalies implies that theonly possible realization within the T1-3-D topology are the solutions I and II with ψL = Li and ψR = eRi,since in these cases the corresponding contributions to the anomalies are already taken into account.

(B) The two SM-singlet chiral fields can acquire a Dirac mass (after the spontaneous symmetry breaking (SSB)of U(1)B−L) through

Lψ = hS (ψR)†ψLS + h.c. . (8)

If we choose r as the free charge circulating in the loop, and since from Table I we have s = ν − 1, then fromthe condition in eq. (8): r + l + s = 0, we get

l =1− ν − r , (9)

and replacing back in eq. (5)

νk =2− ν . (10)

Using (9) and (10) in the cubic condition for anomaly cancellation, eq. (4), we end up with the one-parametersolution

ν =r2 − r + 2

3− r . (11)

If νk = 0 we would have a solution with four chiral fields when ν = 2, however the required r charge isirrational and will not be further considered here.

To label the solutions we use the conventions of [11], as Ta-n-b-I α, where “a” refers to the topology itself, “n”indicates the different choices of the fermion and scalar lines in a given topology, “b” denotes the field assignmentsfor external fields, “I” denotes the simplest solution with standard model singlet scalars or fermions, and α is theparameter which fixes the hypercharges of the Xi fields inside the loop.

It is worth mentioning that when both ψL and ψR are SM singlets (in the T3-1-A-I with α = 0, T1-3-E-I withα = 0, T1-2-A-I with α = 0 and T1-2-B-II with α = +1 models), the condition r 6= 1 or l 6= −1 must be imposedin order to avoid the tree-level realization of (2) that involves a SM singlet fermion mediator –the so-called type I

Dirac seesaw. In this way one of the two required terms in that realization (ψ†RLi ·H and ν†RβψLS∗) is forbidden.

As usual for scotogenic models, we demand the lightest neutral particle running in the loop to be stable. Forthe case of scalar DM, the stability is guaranteed if there is no linear in the scalar loop mediators in the scalarpotential neither Yukawa interactions with two SM fermions. The minimal DM scenarios that may arise are then thesinglet [24–26], doublet [27, 28] and singlet-doublet [29–32] scalar DM. It is worth mentioning that since Majoranamass terms for the fermion loop mediators are not allowed, the fermion DM candidate is either singlet [33] orsinglet-doublet Dirac DM [34].

The wanted solutions must satisfy the constraints regarding DM stability and Diracness of light neutrinos, andguarantee that the direct or induced fermion mass terms between the right handed neutrinos and ψL or ψR areforbidden. The reason to exclude this kind of mixings is that in such a case the DM would not be stable, because

5

Fields (νRi)† (νRj)

† (νRk)† ψL (ψR)† S

L(A) +4 +4 −5 −r r +3

(B) +8

5+

8

5+

2

5

7

5−10

5+

3

5

TABLE II. Solutions for Dirac neutrino masses with Dirac loop mediators for i 6= j 6= k.

the fermion loop mediator would decay into particles of the visible sector. For instance, the induced (through S)or direct mixing between (νRk)† with either ψL or (ψR)† leads to the decay into S and νRk for the induced mixing,and into Z ′µ and νRk for the direct mixing.

The solutions of the two sets are displayed in Table II. Solution (A) is the well known one studied in [15] fortree-level realization of the 5-dimensional operator. In this solution r is quite free, in fact r = ±1/2,±1/3, . . . Thesolution (B) was obtained after exploring all the solutions of eq. (11) for rational values of |r| ≤ 10, and with boththe numerator and denominator less or equal than 10. Since the anomaly cancellation conditions are invariantunder the exchange of r with l, a second solution for (B) exists with the charges of ψL and (ψR)† exchanged.

Higher SU(2)L fermion representations, as required in T1-2 topologies, need to be introduced as vectorlike fermionsto not spoil the anomaly cancellation conditions of the standard model. We will denote vectorlike doublet Weylfermions fields with Y = −1 (Y = +1) as ΨL,R (ΥL,R).

Regarding the scalars circulating in the loop, we will use σ and η to represent SU(2)L scalar singlets and scalardoublets respectively, and we will denote their non-zero lepton number with the same symbols.

A final comment is in order. Because U(1)B−L is promoted to a gauge symmetry, the vacuum expectation value

of S, 〈S〉 = vS/√

2, induces a non-zero mass to the associated gauge boson ZBL. The expression for its mass canbe cast as MZBL

= gBLvS |s|, where gBL is the B −L gauge coupling and s is the B −L charge of S. On the otherhand, since ZBL couples to all the SM fermions (they have non zero B − L charges) it can be produced in hadronand lepton colliders leading to observable signatures. Indeed, from the non-observation of any of such signatures inthe LEP and LHC data there exist constraints on its mass and gauge coupling [35–39] (see e.g. [20, 21] for specificanalysis in B − L scotogenic Dirac models).

III. SOLUTIONS

The new solutions correspond to the case in which some of the Xi fermion fields in figure 1 can be chosen aschiral fields. We explore the solutions with the minimal number of fermion fields beyond the stadard model. We areinterested, therefore, in the solutions in which at least two right handed neutrinos have the same U(1)B−L charge,because in such a case both of them can couple to the same set of extra chiral fermions. All the solutions for theB − L charges presented below have been choosen in such a way the DM particle does not decay.

A. Chiral T1-3-D-I (α = −2)

We will start our analysis with the case in which an internal fermion line in figure 1 can be interpreted as astandard model field. Therefore, only the right handed neutrinos contribute to the anomaly cancellation conditionsof the SM with U(1)B−L. There are three well known solutions with three chiral fields [15, 40]. However, solution(A) in Table II, is the only one that satisfies our constraints.

In fact, with the additions of two charged scalars, σ±1,2, which are singlet under SU(2)L, we can build the Dirac

version of the Zee mechanism to generate neutrino masses [41]. The couplings required to build the diagramdisplayed in figure 2 are

L ⊂[fijLi · Lj σ+

1 + hije (eRi)†Lj · H + hiβR eRi νRβσ

+2 + h.c

]+ V (σ±1 , σ

±2 , H) (12)

where Li are the SM lepton doublets, H = (H+, H0)T , H = iσ2H∗ and V (σ±1 , σ

±2 , H) is the scalar potential.

6

L νR

σ−2σ−

1

S

H

L e−R

Chiral T1-3-D-I (α = −2)

FIG. 2. Dirac Zee model

Fields(νRi)

† (νRj)† (νRk)†

X1 X2 X3 X4S

T1-3-D-I e−Rlσ−2 σ−1 Ll

L +4 +4 −5 −1 −5 −2 −1 3

TABLE III. Chiral T1-3-D-I (α = −2): Solutions for the Dirac Zee model with i 6= j 6= k (i, j, k, l = 1, 2, 3).

Therefore, ψL (ψR) in solution (A) of Table II corresponds to three lepton doblets Li (right-handed electronse−Ri) with l = −1 (r = 1) as the usual lepton number. Since we use one set of charged scalar fields, σ±1,2, the modelhas two massless chiral fields, one of them, νRk, contributing to effective number of relativistic degrees of freedom,Neff [21, 40, 42]. From the Lepton number flux in figure 2, we have

L(σ+1 ) =− 2 , L(σ+

2 ) =− 5 . (13)

The full solution is presented in Table III. This is by far the minimal model for Dirac neutrino masses with a gaugedU(1)B−L. The Dirac Zee model with ν = νk = 1 and extra discrete symmetries has been studied in [41].

It is worth noticing that the restrictions from Neff are expected to be stronger in our model because of thelarger lepton number assignment for the right-handed neutrinos. However, since they do not couple directly to anySM particles, we can simply assume that their interaction with the extra gauge boson and scalars are sufficientlysuppressed that they decouple early enough from the thermal bath. On the other hand, it is clear that there isnot a DM candidate in this model. Indeed, this is just a specific example of models with one-loop Dirac neutrinomasses but without a DM candidate that can be obtained within our setup.

Regarding νRk, it can give rise to either a third Dirac neutrino mass if we extend the scalar sector with S′ andσ′±2 of L charges −6 and +8 respectively, or a Majorana dark matter candidate if we extend the scalar sector witha S′ of L charge +10 [40]. In both cases we end up with a physical Goldstone boson (GB) which could contributeto Neff through interactions with the Higgs [43]. The conditions that GB decouples from the bath in the earlyuniverse are analyzed in [40] and require couplings of GB with SM Higgs not larger than 10−3. This discussion canbe easily extended to the other one-loop realizations below.

We have implemented the model with three non-zero Dirac neutrino masses in SARAH [44]. We use the methodin [41, 45] to express f13, f23, h11, h22, hR33, hR31, hR32 and hR23 as a function of the neutrino masses and mixings,by using fR12, hR12, hR13, hR21 as free parameters. As an example of the consistency of the model, we show in figure 3the observable Br(µ+ → e+γ) < 4.2 × 10−13 [46] as function a f12. The other parameters were fixed as θ = 0.1,Mσ1

= 500 GeV, Mσ2= 750 GeV, hR12, hR13, hR21 = 10−4, where θ is the mixing angle between the charged mass

eigenstates σ1 and σ2. We can see that the value for the parameter f12 is restricted to values lower than 0.02.

7

10−6 10−5 10−4 10−3 10−2

f12

10−28

10−26

10−24

10−22

10−20

10−18

10−16

10−14

10−12

Br(µ

+→

e+γ

)

FIG. 3. Relationship between Br(µ+ → e+γ) and the parameter f12

B. Chiral T1-3-E-I (α = 0)

We consider now the topologies where two SM-singlet chiral fields acquire a Dirac mass after the SSB of U(1)B−Lfrom the term Lψ in eq. (8). For them we will use the solution (B). The relevant terms in the Lagrangian include

L ⊃ Lψ + Lσψ + Lηψ + V (σa, ηa, S,H), (14)

where Lψ was given in eq. (8),

Lσψ =hβa1 (νRβ)†ψL σ

∗a + h.c

Lηψ =h′ia1 (ψR)

†Li · ηa + h.c , (15)

and V (σa, ηa, S,H) is the scalar potential.After the spontaneous breaking of the U(1)B−L symmetry, this topology is reduced to the well known Dirac

radiative seesaw model, but with different Lepton number assignments. In fact, the model with ν = νk = r = 1 andtwo extra Z2 discrete symmetries was first introduced in [47, 48], while the case with r 6= 1 which requires only oneextra Z2 symmetry was studied in [49]. The minimal set of fermion fields is achieved when the two non-zero Diracneutrino masses are generated with two set of SM singlet and doublet scalars: ηa , σa [49].

From the figure 4, the charges of the scalars are

η = 1− r , σ = 1− r . (16)

The solution compatible with the fields in figure 4 is displayed in Table IV. The phenomenology of the radiativeseesaw model with ν = 1 has been already studied in the literature [20, 21, 47–49], where either singlet Dirac (ψ)or singlet-doublet scalar (σa, ηa) DM is realized. Because of the similar charges associated to solution (B), we donot expect significant differences with respect to those works.

C. Chiral T1-2-A-I α = 0

The solution compatible with the fields in figure 5 (left), requires at least the following terms in the Lagrangian

L ⊃Lψ + Lσψ +[MΨ(ΨR) ·ΨL + hia2 (ΨR) · Li σa + y1 (ψR)

†ΨL ·H + h.c

]+ V (σa, S,H) , (17)

where ΨL =(Ψ0L, Ψ−L

)T, (ΨR) =

((Ψ−R)†, −(Ψ0

R)†)T

and V (σa, S,H) is the scalar potential. It follows that thismodel requires a set of at least two SM-singlet scalars to generate a rank-2 neutrino mass matrix, and allows foreither singlet scalar (σa) or singlet-doublet Dirac (ψ,Ψ) DM.

8

L νR

ση

H

S

ψR ψL

FIG. 4. T1-3-E-I (α = 0)

L σ νR

ψL

ΨL

H SψR

ΨR

L η νR

ΥR

S HψL

ψR

ΥL

FIG. 5. Chiral T1-2-A-I (α = 0) Chiral T1-2-B-II (α = 0)

From the figure 5 (left)

σ = 1− r . (18)

The corresponding charges for the solution (B) are shown in Table IV.

D. Chiral T1-2-B-II α = 0

The solution compatible with the fields in figure 5 (right), requires at least the following terms in the Lagrangian

L ⊃Lψ + Lηψ +[MΥ(ΥR) ·ΥL + h′

aβ2 (ΥL) · ηa νRβ + y′1ΥR ·H ψL + h.c

]+ V (ηa, S,H) , (19)

where ΥL =(Υ+L , Υ0

L

)T, (ΥR) =

((Υ0

R)†, −(Υ+R)†)T

and V (ηa, S,H) is the scalar potential. It follows that thismodel requires a set of at least two scalar doublets to generate a rank-2 neutrino mass matrix, and allows for eitherdoublet scalar (ηa) or singlet-doublet Dirac (Υ, ψ) DM.

From the figure

η = 1− r . (20)

The solutions compatible with the fields in figure 5 (right) correspond to the ones displayed in Table IV.

The phenomenological analysis of the chiral realizations of topologies T1-2 will be done elsewhere. It is worthnoting, however, that a model independent analysis of the effect of the right handed neutrinos on Neff, can be inferredfrom [42]. The ratio MZBL

/(νgBL), where gBL is the U(1)B−L gauge coupling, must be larger than 7–8 TeV inorder to be in agreement with the cosmological constraints.

9

Fields(νRi)

† (νRj)† (νRk)†

X1 X2 X3 X4

S

T1-3-E-I ψL ηa σa ψR

L

8

5

8

5

2

5

7

5

15

5

15

5

10

5

3

5

T1-2-A-I σa ψR ψL (ΨR) ΨL

L15

5

10

5

7

5−10

5

10

5

T1-2-B-II ηa (ΥR) ΥL ψR ψL

L15

5−7

5

7

5

10

5

7

5

TABLE IV. Chiral solutions for Dirac neutrino masses with the minimal T1 topologies with i 6= j 6= k and α = 0.

L ψR νR

ση

H S

ψL

FIG. 6. T3-1-A-I (α = 0): B − L flux in the Dirac radiative seesaw

E. vectorlike solutions

In [11], all the obtained solutions for the topologies in figure 1 have internal vectorlike fermions. We want tostress that it is possible to realize all of them with a single extra symmetry U(1)B−L. For that, we can use thesolution (A) with a proper choice of the circulating free charge r.

For example, in the simplest case of just one circulating SM-singlet Dirac fermion line as in T3-1-A, we have thediagram displayed in figure 6.

From the flux of lepton number in figure 6, we have.

η =1− r , σ =ν − r . (21)

The corresponding charges for solution (A) are shown in Table V.Since the charges of the right-handed neutrinos are now bigger than those in solution (B) used in section III B,

the constrains from Neff are expected to be stronger. In fact, a recent detailed phenomenological analysis of the

Fields(νRi)

† (νRj)† (νRk)†

X1 X2 X3 X4

S

T3-1-A-I (ψR)† ψL ηa σa −

L +4 +4 −5 r −r 1− r 4− r − 3

TABLE V. T3-1-A-I (α = 0): Solutions for Dirac radiative seesaw model with i 6= j 6= k (i, j, k = 1, 2, 3).

10

Dirac radiative seesaw [21] includes the case of a right-handed neutrino with ν = 4 and can be fully applied here.In particular, the restrictions from Neff for both scalar (σa, ηa) and Dirac fermion (ψ) dark matter cases are moreimportant than the restriction from ZBL searches at the LHC. We remit the reader there for further details.

It is clear that after SSB of U(1)B−L both solutions T1-3-E-I (α = 0) in section III B and T3-1-A-I (α = 0)reduce to the Dirac radiative seesaw. Moreover, we can add extra circulating charges in the loop. In fact, when allthe circulating particles in the loop are color octets [49], we can have a bound state dark matter candidate formedby two Dirac color-octet fermions [50]. With our solutions, the extra Z2 symmetry in [49] is not longer required.

IV. CONCLUSIONS

We found new and simple models for Dirac neutrino masses within an extension of the Standard Model by anspontaneously broken U(1)B−L gauge symmetry. Specifically, we studied the minimal chiral realizations, at one-loop, of the dimension-5 total lepton number conserving operator that gives rise to Dirac neutrino masses withoutimposing extra symmetries. The minimal models contain three or five chiral fields, two of them with the samecharges under B − L. In the latter case, their charges can be fixed by the requirement to have a dark matterparticle in the spectrum. The full particle content as well as the relevant Lagrangian terms were given for eachof these models. We also showed that known solutions with vectorlike fermions can be obtained with just thesingle symmetry U(1)B−L. These new models, therefore, can simultaneously accommodate one-loop Dirac neutrinomasses and dark matter without invoking any discrete symmetries.

NOTE: During the completion of this work, a related study [51] has appeared. They also present the solution insection III E (Table V) by fixing r = 1/2.

ACKNOWLEDGMENTS

Work supported by Sostenibilidad-UdeA, and by COLCIENCIAS through the Grants 111565842691 and111577657253. D.R thanks Martin Hirsch for illuminating discussions. O.Z. acknowledges the ICTP Simonsassociates program and the kind hospitality of the Abdus Salam ICTP where part of this work was done.

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