# Right-handed Neutrino Dark Matter, Neutrino Masses, and

## Transcript Of Right-handed Neutrino Dark Matter, Neutrino Masses, and

arXiv:2007.07920v1 [hep-ph] 15 Jul 2020

IIPDM-2020

Right-handed Neutrino Dark Matter, Neutrino Masses, and non-Standard Cosmology in a 2HDM

G. Arcadi,a,b S. Profumo,c F. S. Queiroz,d C. Siqueirad

aDipartimento di Matematica e Fisica Universitá di Roma Tre, Via Della Vasca Navale 84 00146 Roma, Italy bINFN Sezione Roma Tre, Via Della Vasca Navale 84 00146 Roma, Italy cUCSC dInternational Institute of Physics, Universidade Federal do Rio Grande do Norte, Campus Universitario, Lagoa Nova, Natal-RN 59078-970, Brazil E-mail: [email protected], [email protected], [email protected], [email protected]

Abstract. We explore the dark matter phenomenology of a weak-scale right-handed

neutrino in the context of a Two Higgs Doublet Model. The expected signal at direct detection experiments is diﬀerent from the usual spin-independent and spin-dependent classiﬁcation since the scattering with quarks depends on the dark matter spin. The dark matter relic density is set by thermal freeze-out and in the presence of nonstandard cosmology, where an Abelian gauge symmetry is key for the dark matter production mechanism. We show that such symmetry allows us to simultaneously address neutrino masses and the ﬂavor problem present in general Two Higgs Doublet Model constructions. Lastly, we outline the region of parameter space that obeys collider, perturbative unitarity and direct detection constraints.

Contents

1 Introduction

2

2 Right-Handed Neutrino Dark Matter in a 2HDM augmented by a

gauge symmetry

4

2.1 Yukawa Lagrangian

5

2.2 Gauge Anomalies

5

2.3 Scalar Potential

6

2.4 Relevant Interactions

8

3 Dark Matter Phenomenology

10

3.1 Relic Density - Standard Cosmology

10

3.2 Relic Density - Early Matter Domination

10

3.3 Direct Detection

12

3.4 Indirect Detection

13

3.5 Perturbative Unitarity

14

3.6 Collider bounds

14

3.7 Atomic Parity violation

14

3.8 Constraints on extra scalars

16

4 Results

16

5 Conclusions

17

A Anomaly Freedom

18

–1–

1 Introduction

There is ample evidence that the dark matter (DM) accounts for about 27% of the energy budget of our universe, i.e. ΩDM h2 = 0.12, as measured with very high precision by the PLANCK Collaboration [1]. Current observations, however, do not shed light on the microscopic nature of the DM, nor to they allow to discriminate between astrophysical or particle physics solutions to the DM puzzle. Assuming a particle nature for the DM component of the Universe, it is well known that the Standard Model (SM) of particle physics cannot provide a viable candidate, which should thus be one (or more) new exotic particles. Among the many possible options, WIMPs (Weakly Interacting Massive Particles) have been regarded as one of the most promising ones, since their abundance can be elegantly accommodated, through the thermal freeze-out mechanism, by requiring DM masses in the GeV-TeV range and interactions with the SM states of strength similar to weak interactions. Attempts to directly or indirectly detect WIMPs, however, have been up to now unsuccessful [2]. It should be pointed out, however, that only recently have we started to probe the “natural” parameter space of WIMPs [3].

Further evidence for new physics beyond the SM is provided by the experimental evidence of non-zero neutrino masses. As right-handed neutrinos are absent in the SM, at least in its minimal incarnation, the Higgs ﬁeld cannot generate a Dirac mass term for the neutrinos. If copies of right-handed neutrinos are included in the matter content of the SM, the Yukawa coupling needed to explain neutrino masses around 0.1 eV would be extremely, and unnaturally, suppressed. Being SM singlets, righthanded neutrinos can have as well a Majorana mass term without conﬂicting with gauge invariance. Given this peculiar feature of neutrino masses, New Physics beyond the SM is typically invoked for their origin. Neutrino oscillation experiments, see e.g. [4–11], have measured with great precision mass diﬀerences and mixing of the three light SM neutrinos. However, their individual masses are yet unknown and two mass orderings, normal and inverse, are allowed. Useful insight has been given, in this direction, by cosmology. Indeed measures of the Cosmic Microwave Background (CMB), constrain the sum of the neutrino masses through its eﬀect on structure growth that comes in terms of the early Integrated Sachs Wolfe eﬀect and gravitational lensing of the CMB [12]. In summary, they impose i νi < 0.12 eV [1]. A natural question arises at this point: can the same new physics beyond the SM be responsible, at the same time, of the generation of neutrino masses and of the DM component of the Universe? The answer is yes, and this has been driving a multitude of studies in the literature in the context of neutrino masses generated at tree-level [13–27].

As mentioned, the simplest way to generate light neutrino masses is to extend the SM with three right-handed neutrinos described by the following Lagrangian:

L ⊃ yabLa ΦNbR + M2a NacRNaR,

(1.1)

where Φ is the SM Higgs doublet, and M a Majorana bare mass term (matrix) for the right-handed neutrinos.

–2–

After spontaneous electroweak (EW) symmetry breaking, the ﬁrst term leads to a Dirac mass which mixes the left-handed and right-handed neutrinos. Three light neutrino masses are, at this point, generated through the so called (Type-I) seesaw mechanism. Since neutrino masses are given by mν mTDM −1mD, with mD M the Dirac mass term, it can be easily argued that there are multiple choices for the Yukawa couplings yab and the Majorana M mass that are consistent with the measured oscillation pattern and astrophysical constraints [28]. In particular, values of order one for yab would imply a very high scale M ∼ 1012 − 1015 GeV for the Majorana mass term, hardly accessible to experimental tests. Furthermore, in principle one cannot accommodate a viable DM candidate in this scenario.

A common origin for DM and neutrino masses in the context of the seesaw mechanism can be achieved in a somewhat orthogonal regime, with GeV-scale or lighter Majorana masses and comparatively small Yukawa couplings. The key aspect in this regime is the fact that the tree-level seesaw mechanism requires the presence of only two right-handed neutrinos to generate a pattern of masses and mixing of the light neutrinos compatible with laboratory tests. This leaves the freedom of assuming that the remaining right-handed neutrino be the DM candidate. In absence of additional ad-hoc symmetries, the latter would be allowed to decay into SM states. Nevertheless, it is possible to achieve a cosmologically stable state in the O(1 − 50) keV mass range and very suppressed Yukawa couplings. This kind of candidate is usually referred to as sterile neutrino DM. The minimal model, which we just summarized, which accommodates both DM and neutrino masses is referred to νM SM [29]. In this kind of scenario the DM has exceedingly suppressed interactions to be produced according the conventional freeze-out paradigm. The correct relic density can nevertheless be achieved through the so-called Dodelson-Widrow (also dubbed non-resonant) mechanism [30], consisting of production via active-sterile neutrino oscillations. A keV scale mass sterile neutrino is produced, through this mechanism, at temperatures of the order of 150 MeV. An approximate expression for its relic density is [31–35]:

ΩDM h2 ∼ 0.1

sin2 θi 3 × 10−9

MN 1.8 3 keV

(1.2)

where sin2 θi ∼ a ya2bv2/M 2, with v the SM Higgs vev. It is clear that, in this setup, the right-handed neutrino is unstable due to N → ννν, and to the loop-suppressed mode N → νγ decay. The decay width is controlled by the mixing angle and the right-handed neutrino mass, and can be observationally tested using line searches from N decay, producing a line at around half the mass of the sterile neutrino, thus typically in X-rays. Searches of X-ray signals, combined with bounds from structure formation substantially exclude the parameter space corresponding to the non-resonant production mechanism for DM illustrated above (see e.g. [36, 37] for some reviews). Bounds from X-rays can be overcome by relying on the the so-called Shi-Fuller [38] (resonant production) mechanism, i.e. enhanced DM production in presence of lepton asymmetry, hence requiring much smaller mixing angles to comply with the correct relic density. Tensions with structure formation are nevertheless still present [39]; no conclusive assessment can be made due to the intrinsic uncertainties of these limits. It

–3–

is also worth pointing out that the correct relic density through resonant production implies, in the minimal model illustrated above, very tight requirements on the parameters of the new neutrino sector [40]. Alternatively, additional physics can be invoked for the production of sterile neutrinos in the early universe, see e.g. [41, 42].

In our work, we are interested in the possibility of having a weak-scale, thermally produced right-handed neutrino1. To achieve this goal, extra interactions and/or symmetries beyond those of the SM must be invoked [44–46]. More concretely, here we consider a Two Higgs Doublet Model (2HDM) augmented by a spontaneously broken, new Abelian gauge symmetry. This scenario aims at addressing, at the same time, the problems of ﬂavor, dark matter and neutrino masses [47, 48]. However, as will be shown in the following, the correct DM relic density can be hardly achieved, in the standard thermal scenario, without tension with experimental constraints. To overcome this problem, we consider the possibility of a non-standard cosmological history, represented by a phase of early matter domination. In summary, our work will extend previous studies in the following aspects:

(i) We consider a 2HDM augmented by a spontaneously broken additional Abelian gauge symmetry;

(ii) We consider a viable solution to the ﬂavor problem;

(iii) We address and solve the issue of neutrino masses;

(iv) We accommodate a thermal right-handed neutrino dark matter;

(v) We explore the same setup in the context of an early matter domination period in the universe.

The paper is structured as follows: in the following Section 2, we introduce the 2HDM augmented by an additional gauge symmetry, including the relevant interactions; in Section 3, we explore the DM phenomenology, in Section 4, we discuss our results, and ﬁnally in Section 5, we present our conclusions.

2 Right-Handed Neutrino Dark Matter in a 2HDM augmented by a gauge symmetry

Two-Higgs doublet models (2HDM) are theoretically and experimentally appealing extensions of the SM [49–51]. One of the key features of 2HDM is that they do not aﬀect the ρ parameter. Moreover, 2HDM oﬀer a rich environment for new physics in the sector of Higgs physics [52–57], collider searches [58], and ﬂavor physics [59– 61]. Over the years, extensions to the original proposal have appeared that include the introduction of additional new gauge symmetries [62–76]. Later, these new gauge symmetries were used to simultaneously solve the ﬂavor problem in 2HDM and the issue

1In principle, this particle can be light, although this possibility is tightly constrained by CMB, direct and collider searches, for example, please see [43].

–4–

of neutrino masses and mixing [48, 77–79], as well as the dark matter puzzle [14, 80–83]. It is clear that 2HDM augmented by gauge symmetries are gaining interest. Motivated by this, here we propose a new 2HDM that accommodates a thermal right-handed neutrino dark matter by adding a B-L gauge symmetry. The anomaly cancellation of a new B-L symmetry can be easily performed, but there are additional requirements to be considered when this symmetry is embedded in the context of a 2HDM. To understand this fact we start our discussion with the Yukawa Lagrangian. While very appealing, the BL group is not the only viable option [47]. We will thus adopt in the following a more general notation so that one can straightforwardly extend our results to the case in which the SM group is augmented by a generic U (1)X symmetry.

2.1 Yukawa Lagrangian

We will work in the context of type-I 2HDM, where only one of the scalar doublets contributes to fermion masses. This setup naturally arises via the introduction of a new gauge symmetry under which the scalar doublets Φ1 and Φ2 transform diﬀerently. In this way, the Yukawa Lagrangian reads,

− LY1 = yadbQ¯aΦ2dbR + yaubQ¯aΦ2ubR + yaebL¯aΦ2ebR + h.c.,

(2.1)

−LY2 ⊃ yabL¯aΦ2NbR + yaMb (NaR)cΦsNbR + h.c. ,

(2.2)

Another Higgs ﬁeld Φs, singlet under the SM group but with charge QXs under the new gauge group (in the case of B − L, Q(B−L)s = −2), is introduced to spontaneously break the U (1)X symmetry. The corresponding vacuum expectation value (vev) will be indicated as vs in the following. The DM candidate is represented by the neutrino mass eigenstate N1, which is assumed to be odd with respect to a Z2 symmetry to ensure stability. The other two neutrinos are responsible for the generation of active

neutrino masses and mixing through type-I seesaw [44, 45, 83, 84]:

(ν N )

0 mD mTD MR

ν. N

(2.3)

Taking MR mD we get mν = −mTD M1R mD and mN = MR, as long as MR mD,

where mD = y√v2 and MR = yM√vs . Using the Casas-Ibarra parametrization [85] one

22

22

can straightforwardly reproduce current neutrino data (see e.g. [86]). We emphasize

that the right-handed neutrino N1R is decoupled by construction from this seesaw

mechanism as it is odd under a Z2 symmetry.

2.2 Gauge Anomalies

In order for the new U (1)X symmetry to be anomaly free, the following anomaly cancellation conditions must hold:

[SU (3)c]2 U (1)X → QuX + QdX − 2QqX = 0. [SU (2)L]2 U (1)X → QlX = −3QqX .

(2.4) (2.5)

–5–

[U (1)Y ]2 U (1)X → 6QeX + 8QuX + 2QdX − 3QlX − QqX = 0. U (1)Y [U (1)X ]2 → −(QeX )2 + 2(QuX )2 − (QdX )2 + (QlX )2 − (QqX )2 = 0.

[U (1)X ]3 → (QeX )3 + 3(QuX )3 + 3(QdX )3 − 2(QlX )3 − 6(QqX )3 = 0.

(2.6) (2.7) (2.8)

where l, q, e, u, d stand, respectively, for the lepton and quark doublets, right-handed

charged leptons, and the up-type and down-type right-handed quarks. QX2 is, instead, the charge of the Φ2 ﬁeld. In Appendix A, we describe how to ﬁnd the above relations. By using them, it is possible to ﬁnd several U (1)X anomaly free models [47], including the B − L approached here.

Taking QuX and QdX as free parameters, the anomaly conditions are satisﬁed if the SM spectrum is augmented with three right-handed neutrinos with charge QnX = −(QuX + 2QdX). Using the Yukawa Lagrangian in Eq.(2.1), we obtain QdX − QqX + QX2 = 0, QuX − QqX − QX2 = 0, QeX − QlX + QX2 = 0, which, combined with the anomaly conditions, allow us to express all the charges as function of QuX and QdX:

q

QuX + QdX

QX =

, 2

l

3 QuX + QdX

QX = −

2

,

QeX = − 2QuX + QdX ,

QuX − QdX

QX2 =

.

2

(2.9)

In Appendix A, we describe how to ﬁnd the conditions for anomaly freedom. By using them, it is possible to ﬁnd several U (1)X anomaly free models [47], including the B − L approached here.

2.3 Scalar Potential

The scalar potential in the presence of two scalar doublets, transforming diﬀerently under U (1)X, and a scalar singlet reads:

V (Φ1, Φ2) = m211Φ†1Φ1 + m222Φ†2Φ2 + m2sΦ†sΦs + λ21 Φ†1Φ1 + λ3 Φ†1Φ1 Φ†2Φ2 + λ4 Φ†1Φ2 Φ†2Φ1 +

2 + λ2 2

2

Φ†2Φ2 +

(2.10)

+ λ2s Φ†sΦs 2 + µ1Φ†1Φ1Φ†sΦs + µ2Φ†2Φ2Φ†sΦs + µΦ†1Φ2Φs + h.c. .

Notice that, given the fact that the two Higgs doublets have diﬀerent charged under the U (1)X group, only the tri-scalar operator Φ†1Φ2Φs + h.c. is allowed by the symmetries of the system.

The two doublets and the singlet can be decomposed as:

Φ=

φ+i √ ,

i (vi + ρi + iηi) / 2

(2.11)

–6–

Fields U (1)X U (1)B−L

2HDM with right-handed neutrino dark matter

uR dR

QL

LL

eR

NR

QuX QdX (QuX +2 QdX ) −3(QuX2+QdX ) −(2QuX + QdX ) −(QuX + 2QdX )

1/3 1/3 1/3

−1

−1

−1

Fields U (1)X U (1)B−L

Φ2

(QuX −QdX ) 2

0

Φ1

+ 5QuX 2

7QdX 2

2

Table 1. Field context of our 2HDM-U (1)X model where the lightest right-handed neutrino is dark matter, whose stability is protected by a Z2 symmetry.

1 Φs = √ (vs + ρs + iηs) .

2

From the terms of Eqs.(2.2)-(2.10) we can determine the following relations: QXs = QX1 − QX2 and −QlX − QeX + QnX = 0, leading to QX1 = 5Q2uX + 7Q2dX . The values of the U (1)X charges, both in the general case and for X = B − L, the speciﬁc case of study of this work, for the ﬁeld content of our model are summarized in Table 1.

After EW symmetry breaking and assuming that CP is preserved by the scalar potential, the CP-even and CP-odd components of the Higgs ﬁelds will mix, eventually leading to three CP-even mass eigenstates, which we indicate with the standard notation h, H and hs, two charged states H± and one CP-odd state, A. Throughout this study, we will identify h with the 125 GeV SM-like Higgs and hs with a mostly singlet-like Higgs, with negligible mixing with the other CP-even scalars. In such a case the mass of the latter is simply given by:

m2hs λsvs.

(2.12)

The other relevant parameter for DM phenomenology is the mass of the charged Higgs,

given by [47, 81]:

m2H ±

√ 2µvs − λ4v1v2 v2 . 2v1v2

(2.13)

In order to ensure negligible mixing between the Higgs doublets and singlet the µ parameter should be small but still satisfy:

λ4v1v2 µ> √ .

2vs

(2.14)

Unless diﬀerently stated, we will assume, for our analysis, the assignations λ4 = 0.1, λs = 0.1 and µ = 35 GeV.

–7–

2.4 Relevant Interactions

Let’s start with the interactions of the gauge bosons. The new U (1)X and the hypercharge gauge boson are described by the following Lagrangian:

L = − 1 Bˆ Bˆµν +

Xˆ Bˆµν − 1 Xˆ Xˆ µν.

gauge

4 µν

2 cosθW µν

4 µν

(2.15)

As we added a U (1)X gauge group a kinetic mixing between the two Abelian groups is, in general, allowed by the gauge symmetries. Assuming it to be zero at the

tree level, we can nevertheless not avoid its generation at the one-loop level since the

SM fermion are charged under both the two gauge group. In such case the induced kinetic mixing parameter can be estimated as | | ∼ g gX/(16π2) Q2X ln M 2/µ2 [87, 88] where, again, gX is the gauge coupling of the U (1)X and QX is the charge of the SM fermion. This expression roughly gives ∼ 10−2gX; consequently we can assume that kinetic mixing gives a subdominant contribution to DM phenomenology

and, hence, neglect it in numerical computation.

The Lagrangian responsible for the mass of the gauge bosons is:

L = (Dµφ1)†(Dµφ1) + (Dµφ2)†(Dµφ2) + (Dµφs)†(Dµφs) = + 41 g2v2W − µWµ+ + 18 gZ2 v2Z0 µZµ0 − 41 gZ (GX1v12 + GX2v22)Z0 µXµ + 18 (v12G2X1 + v22G2X2 v22 + vs2Q2Xs gX2 )XµXµ

where we have used the following expression for the covariant derivative:

(2.16)

Dµ = ∂µ + igT aWµa + ig Q2Y Bˆµ + igX Q2X Xˆµ,

(2.17)

while:

G = QYi + g Q . Xi cos θW X Xi

(2.18)

From the equation above we straightforwardly recover the SM value for the mass

of the W boson:

m2W = 14 g2v2,

(2.19)

the masses of the remaining gauge bosons are obtained upon diagonalization of the following mass matrix:

M=

gZ2 v2 −gZ (GX1 v12 + GX2 v22)

−gZ (GX1 v12 + GX2 v22) v12G2X1 + v2G2X2 v22 + qX2 gX2 vs2

(2.20)

which leads to the following rotation:

Zµ

cos ξ − sin ξ

Z µ = sin ξ cos ξ

Z0µ Xµ ,

(2.21)

–8–

where the mixing ξ is deﬁned in general as

2gZ (GX1 v12 + GX2 v22)

tan 2ξ =

m2 − m2

.

Z0

X

(2.22)

However, in this work we will mostly consider the regime m2X m2Z02 (we will further comment on this choice in the next section) and very small mixing angle. In

such a case we can write

m2Z m2Z0 = 14 gZ2 v2,

2

2 vs2 2 2 gX2 v2 cos2 β sin2 β

2

mZ mX = 4 gX QXs +

4 (QX1 − QX2) ,

(2.23)

and

sin ξ

GX1 v12 + GX2 v22 m2Z

m2Z = m2

Z

gX Q cos2 β + Q sin2 β + tan θ .

gZ X1

X2

W

(2.24)

Finally, we write the Lagrangian describing the neutral current interactions as follows:

L = − eJµ A − g cos ξJµ Z − sin ξ

NC

em µ 2 cos θW

NC µ

eJ µ +

g Jµ

em Z 2 cos θW NC

Zµ+

1 + 4 gX sin ξ

QRXf + QLXf ψ¯f γµψf + QRXf − QLXf ψ¯f γµγ5ψf Zµ+

1 − 4 gX cos ξ

QRXf + QLXf ψ¯f γµψf − QLXf − QRXf ψ¯f γµγ5ψf Zµ+

− 14 QN1gX cos ξ cos ξN1γµγ5N1Zµ + 41 QN1gX sin ξN1γµγ5N1Zµ,

(2.25)

where we have again adopted a general notation in terms of generic charges for the DM and the SM fermions under the new gauge symmetry (conditions for an anomaly free symmetry are, of course, automatically assumed).

Besides the neutral current there are other interactions that might be relevant for our phenomenology such Z W +W −, Z W +H−, HZ Z , hZ Z , HZZ . The presence of such interactions is an additional feature that distinguishes our work from previous studies in the literature [84, 90, 91]. In particular, the Z W +H− is proportional to ggXv/2 cos ξ, therefore it cannot be neglected. The details expression for the aforementioned couplings can be found e.g. in [81].

2In the opposite regime the same model can be used to interpret the recent XENON1T anomaly [89].

–9–

IIPDM-2020

Right-handed Neutrino Dark Matter, Neutrino Masses, and non-Standard Cosmology in a 2HDM

G. Arcadi,a,b S. Profumo,c F. S. Queiroz,d C. Siqueirad

aDipartimento di Matematica e Fisica Universitá di Roma Tre, Via Della Vasca Navale 84 00146 Roma, Italy bINFN Sezione Roma Tre, Via Della Vasca Navale 84 00146 Roma, Italy cUCSC dInternational Institute of Physics, Universidade Federal do Rio Grande do Norte, Campus Universitario, Lagoa Nova, Natal-RN 59078-970, Brazil E-mail: [email protected], [email protected], [email protected], [email protected]

Abstract. We explore the dark matter phenomenology of a weak-scale right-handed

neutrino in the context of a Two Higgs Doublet Model. The expected signal at direct detection experiments is diﬀerent from the usual spin-independent and spin-dependent classiﬁcation since the scattering with quarks depends on the dark matter spin. The dark matter relic density is set by thermal freeze-out and in the presence of nonstandard cosmology, where an Abelian gauge symmetry is key for the dark matter production mechanism. We show that such symmetry allows us to simultaneously address neutrino masses and the ﬂavor problem present in general Two Higgs Doublet Model constructions. Lastly, we outline the region of parameter space that obeys collider, perturbative unitarity and direct detection constraints.

Contents

1 Introduction

2

2 Right-Handed Neutrino Dark Matter in a 2HDM augmented by a

gauge symmetry

4

2.1 Yukawa Lagrangian

5

2.2 Gauge Anomalies

5

2.3 Scalar Potential

6

2.4 Relevant Interactions

8

3 Dark Matter Phenomenology

10

3.1 Relic Density - Standard Cosmology

10

3.2 Relic Density - Early Matter Domination

10

3.3 Direct Detection

12

3.4 Indirect Detection

13

3.5 Perturbative Unitarity

14

3.6 Collider bounds

14

3.7 Atomic Parity violation

14

3.8 Constraints on extra scalars

16

4 Results

16

5 Conclusions

17

A Anomaly Freedom

18

–1–

1 Introduction

There is ample evidence that the dark matter (DM) accounts for about 27% of the energy budget of our universe, i.e. ΩDM h2 = 0.12, as measured with very high precision by the PLANCK Collaboration [1]. Current observations, however, do not shed light on the microscopic nature of the DM, nor to they allow to discriminate between astrophysical or particle physics solutions to the DM puzzle. Assuming a particle nature for the DM component of the Universe, it is well known that the Standard Model (SM) of particle physics cannot provide a viable candidate, which should thus be one (or more) new exotic particles. Among the many possible options, WIMPs (Weakly Interacting Massive Particles) have been regarded as one of the most promising ones, since their abundance can be elegantly accommodated, through the thermal freeze-out mechanism, by requiring DM masses in the GeV-TeV range and interactions with the SM states of strength similar to weak interactions. Attempts to directly or indirectly detect WIMPs, however, have been up to now unsuccessful [2]. It should be pointed out, however, that only recently have we started to probe the “natural” parameter space of WIMPs [3].

Further evidence for new physics beyond the SM is provided by the experimental evidence of non-zero neutrino masses. As right-handed neutrinos are absent in the SM, at least in its minimal incarnation, the Higgs ﬁeld cannot generate a Dirac mass term for the neutrinos. If copies of right-handed neutrinos are included in the matter content of the SM, the Yukawa coupling needed to explain neutrino masses around 0.1 eV would be extremely, and unnaturally, suppressed. Being SM singlets, righthanded neutrinos can have as well a Majorana mass term without conﬂicting with gauge invariance. Given this peculiar feature of neutrino masses, New Physics beyond the SM is typically invoked for their origin. Neutrino oscillation experiments, see e.g. [4–11], have measured with great precision mass diﬀerences and mixing of the three light SM neutrinos. However, their individual masses are yet unknown and two mass orderings, normal and inverse, are allowed. Useful insight has been given, in this direction, by cosmology. Indeed measures of the Cosmic Microwave Background (CMB), constrain the sum of the neutrino masses through its eﬀect on structure growth that comes in terms of the early Integrated Sachs Wolfe eﬀect and gravitational lensing of the CMB [12]. In summary, they impose i νi < 0.12 eV [1]. A natural question arises at this point: can the same new physics beyond the SM be responsible, at the same time, of the generation of neutrino masses and of the DM component of the Universe? The answer is yes, and this has been driving a multitude of studies in the literature in the context of neutrino masses generated at tree-level [13–27].

As mentioned, the simplest way to generate light neutrino masses is to extend the SM with three right-handed neutrinos described by the following Lagrangian:

L ⊃ yabLa ΦNbR + M2a NacRNaR,

(1.1)

where Φ is the SM Higgs doublet, and M a Majorana bare mass term (matrix) for the right-handed neutrinos.

–2–

After spontaneous electroweak (EW) symmetry breaking, the ﬁrst term leads to a Dirac mass which mixes the left-handed and right-handed neutrinos. Three light neutrino masses are, at this point, generated through the so called (Type-I) seesaw mechanism. Since neutrino masses are given by mν mTDM −1mD, with mD M the Dirac mass term, it can be easily argued that there are multiple choices for the Yukawa couplings yab and the Majorana M mass that are consistent with the measured oscillation pattern and astrophysical constraints [28]. In particular, values of order one for yab would imply a very high scale M ∼ 1012 − 1015 GeV for the Majorana mass term, hardly accessible to experimental tests. Furthermore, in principle one cannot accommodate a viable DM candidate in this scenario.

A common origin for DM and neutrino masses in the context of the seesaw mechanism can be achieved in a somewhat orthogonal regime, with GeV-scale or lighter Majorana masses and comparatively small Yukawa couplings. The key aspect in this regime is the fact that the tree-level seesaw mechanism requires the presence of only two right-handed neutrinos to generate a pattern of masses and mixing of the light neutrinos compatible with laboratory tests. This leaves the freedom of assuming that the remaining right-handed neutrino be the DM candidate. In absence of additional ad-hoc symmetries, the latter would be allowed to decay into SM states. Nevertheless, it is possible to achieve a cosmologically stable state in the O(1 − 50) keV mass range and very suppressed Yukawa couplings. This kind of candidate is usually referred to as sterile neutrino DM. The minimal model, which we just summarized, which accommodates both DM and neutrino masses is referred to νM SM [29]. In this kind of scenario the DM has exceedingly suppressed interactions to be produced according the conventional freeze-out paradigm. The correct relic density can nevertheless be achieved through the so-called Dodelson-Widrow (also dubbed non-resonant) mechanism [30], consisting of production via active-sterile neutrino oscillations. A keV scale mass sterile neutrino is produced, through this mechanism, at temperatures of the order of 150 MeV. An approximate expression for its relic density is [31–35]:

ΩDM h2 ∼ 0.1

sin2 θi 3 × 10−9

MN 1.8 3 keV

(1.2)

where sin2 θi ∼ a ya2bv2/M 2, with v the SM Higgs vev. It is clear that, in this setup, the right-handed neutrino is unstable due to N → ννν, and to the loop-suppressed mode N → νγ decay. The decay width is controlled by the mixing angle and the right-handed neutrino mass, and can be observationally tested using line searches from N decay, producing a line at around half the mass of the sterile neutrino, thus typically in X-rays. Searches of X-ray signals, combined with bounds from structure formation substantially exclude the parameter space corresponding to the non-resonant production mechanism for DM illustrated above (see e.g. [36, 37] for some reviews). Bounds from X-rays can be overcome by relying on the the so-called Shi-Fuller [38] (resonant production) mechanism, i.e. enhanced DM production in presence of lepton asymmetry, hence requiring much smaller mixing angles to comply with the correct relic density. Tensions with structure formation are nevertheless still present [39]; no conclusive assessment can be made due to the intrinsic uncertainties of these limits. It

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is also worth pointing out that the correct relic density through resonant production implies, in the minimal model illustrated above, very tight requirements on the parameters of the new neutrino sector [40]. Alternatively, additional physics can be invoked for the production of sterile neutrinos in the early universe, see e.g. [41, 42].

In our work, we are interested in the possibility of having a weak-scale, thermally produced right-handed neutrino1. To achieve this goal, extra interactions and/or symmetries beyond those of the SM must be invoked [44–46]. More concretely, here we consider a Two Higgs Doublet Model (2HDM) augmented by a spontaneously broken, new Abelian gauge symmetry. This scenario aims at addressing, at the same time, the problems of ﬂavor, dark matter and neutrino masses [47, 48]. However, as will be shown in the following, the correct DM relic density can be hardly achieved, in the standard thermal scenario, without tension with experimental constraints. To overcome this problem, we consider the possibility of a non-standard cosmological history, represented by a phase of early matter domination. In summary, our work will extend previous studies in the following aspects:

(i) We consider a 2HDM augmented by a spontaneously broken additional Abelian gauge symmetry;

(ii) We consider a viable solution to the ﬂavor problem;

(iii) We address and solve the issue of neutrino masses;

(iv) We accommodate a thermal right-handed neutrino dark matter;

(v) We explore the same setup in the context of an early matter domination period in the universe.

The paper is structured as follows: in the following Section 2, we introduce the 2HDM augmented by an additional gauge symmetry, including the relevant interactions; in Section 3, we explore the DM phenomenology, in Section 4, we discuss our results, and ﬁnally in Section 5, we present our conclusions.

2 Right-Handed Neutrino Dark Matter in a 2HDM augmented by a gauge symmetry

Two-Higgs doublet models (2HDM) are theoretically and experimentally appealing extensions of the SM [49–51]. One of the key features of 2HDM is that they do not aﬀect the ρ parameter. Moreover, 2HDM oﬀer a rich environment for new physics in the sector of Higgs physics [52–57], collider searches [58], and ﬂavor physics [59– 61]. Over the years, extensions to the original proposal have appeared that include the introduction of additional new gauge symmetries [62–76]. Later, these new gauge symmetries were used to simultaneously solve the ﬂavor problem in 2HDM and the issue

1In principle, this particle can be light, although this possibility is tightly constrained by CMB, direct and collider searches, for example, please see [43].

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of neutrino masses and mixing [48, 77–79], as well as the dark matter puzzle [14, 80–83]. It is clear that 2HDM augmented by gauge symmetries are gaining interest. Motivated by this, here we propose a new 2HDM that accommodates a thermal right-handed neutrino dark matter by adding a B-L gauge symmetry. The anomaly cancellation of a new B-L symmetry can be easily performed, but there are additional requirements to be considered when this symmetry is embedded in the context of a 2HDM. To understand this fact we start our discussion with the Yukawa Lagrangian. While very appealing, the BL group is not the only viable option [47]. We will thus adopt in the following a more general notation so that one can straightforwardly extend our results to the case in which the SM group is augmented by a generic U (1)X symmetry.

2.1 Yukawa Lagrangian

We will work in the context of type-I 2HDM, where only one of the scalar doublets contributes to fermion masses. This setup naturally arises via the introduction of a new gauge symmetry under which the scalar doublets Φ1 and Φ2 transform diﬀerently. In this way, the Yukawa Lagrangian reads,

− LY1 = yadbQ¯aΦ2dbR + yaubQ¯aΦ2ubR + yaebL¯aΦ2ebR + h.c.,

(2.1)

−LY2 ⊃ yabL¯aΦ2NbR + yaMb (NaR)cΦsNbR + h.c. ,

(2.2)

Another Higgs ﬁeld Φs, singlet under the SM group but with charge QXs under the new gauge group (in the case of B − L, Q(B−L)s = −2), is introduced to spontaneously break the U (1)X symmetry. The corresponding vacuum expectation value (vev) will be indicated as vs in the following. The DM candidate is represented by the neutrino mass eigenstate N1, which is assumed to be odd with respect to a Z2 symmetry to ensure stability. The other two neutrinos are responsible for the generation of active

neutrino masses and mixing through type-I seesaw [44, 45, 83, 84]:

(ν N )

0 mD mTD MR

ν. N

(2.3)

Taking MR mD we get mν = −mTD M1R mD and mN = MR, as long as MR mD,

where mD = y√v2 and MR = yM√vs . Using the Casas-Ibarra parametrization [85] one

22

22

can straightforwardly reproduce current neutrino data (see e.g. [86]). We emphasize

that the right-handed neutrino N1R is decoupled by construction from this seesaw

mechanism as it is odd under a Z2 symmetry.

2.2 Gauge Anomalies

In order for the new U (1)X symmetry to be anomaly free, the following anomaly cancellation conditions must hold:

[SU (3)c]2 U (1)X → QuX + QdX − 2QqX = 0. [SU (2)L]2 U (1)X → QlX = −3QqX .

(2.4) (2.5)

–5–

[U (1)Y ]2 U (1)X → 6QeX + 8QuX + 2QdX − 3QlX − QqX = 0. U (1)Y [U (1)X ]2 → −(QeX )2 + 2(QuX )2 − (QdX )2 + (QlX )2 − (QqX )2 = 0.

[U (1)X ]3 → (QeX )3 + 3(QuX )3 + 3(QdX )3 − 2(QlX )3 − 6(QqX )3 = 0.

(2.6) (2.7) (2.8)

where l, q, e, u, d stand, respectively, for the lepton and quark doublets, right-handed

charged leptons, and the up-type and down-type right-handed quarks. QX2 is, instead, the charge of the Φ2 ﬁeld. In Appendix A, we describe how to ﬁnd the above relations. By using them, it is possible to ﬁnd several U (1)X anomaly free models [47], including the B − L approached here.

Taking QuX and QdX as free parameters, the anomaly conditions are satisﬁed if the SM spectrum is augmented with three right-handed neutrinos with charge QnX = −(QuX + 2QdX). Using the Yukawa Lagrangian in Eq.(2.1), we obtain QdX − QqX + QX2 = 0, QuX − QqX − QX2 = 0, QeX − QlX + QX2 = 0, which, combined with the anomaly conditions, allow us to express all the charges as function of QuX and QdX:

q

QuX + QdX

QX =

, 2

l

3 QuX + QdX

QX = −

2

,

QeX = − 2QuX + QdX ,

QuX − QdX

QX2 =

.

2

(2.9)

In Appendix A, we describe how to ﬁnd the conditions for anomaly freedom. By using them, it is possible to ﬁnd several U (1)X anomaly free models [47], including the B − L approached here.

2.3 Scalar Potential

The scalar potential in the presence of two scalar doublets, transforming diﬀerently under U (1)X, and a scalar singlet reads:

V (Φ1, Φ2) = m211Φ†1Φ1 + m222Φ†2Φ2 + m2sΦ†sΦs + λ21 Φ†1Φ1 + λ3 Φ†1Φ1 Φ†2Φ2 + λ4 Φ†1Φ2 Φ†2Φ1 +

2 + λ2 2

2

Φ†2Φ2 +

(2.10)

+ λ2s Φ†sΦs 2 + µ1Φ†1Φ1Φ†sΦs + µ2Φ†2Φ2Φ†sΦs + µΦ†1Φ2Φs + h.c. .

Notice that, given the fact that the two Higgs doublets have diﬀerent charged under the U (1)X group, only the tri-scalar operator Φ†1Φ2Φs + h.c. is allowed by the symmetries of the system.

The two doublets and the singlet can be decomposed as:

Φ=

φ+i √ ,

i (vi + ρi + iηi) / 2

(2.11)

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Fields U (1)X U (1)B−L

2HDM with right-handed neutrino dark matter

uR dR

QL

LL

eR

NR

QuX QdX (QuX +2 QdX ) −3(QuX2+QdX ) −(2QuX + QdX ) −(QuX + 2QdX )

1/3 1/3 1/3

−1

−1

−1

Fields U (1)X U (1)B−L

Φ2

(QuX −QdX ) 2

0

Φ1

+ 5QuX 2

7QdX 2

2

Table 1. Field context of our 2HDM-U (1)X model where the lightest right-handed neutrino is dark matter, whose stability is protected by a Z2 symmetry.

1 Φs = √ (vs + ρs + iηs) .

2

From the terms of Eqs.(2.2)-(2.10) we can determine the following relations: QXs = QX1 − QX2 and −QlX − QeX + QnX = 0, leading to QX1 = 5Q2uX + 7Q2dX . The values of the U (1)X charges, both in the general case and for X = B − L, the speciﬁc case of study of this work, for the ﬁeld content of our model are summarized in Table 1.

After EW symmetry breaking and assuming that CP is preserved by the scalar potential, the CP-even and CP-odd components of the Higgs ﬁelds will mix, eventually leading to three CP-even mass eigenstates, which we indicate with the standard notation h, H and hs, two charged states H± and one CP-odd state, A. Throughout this study, we will identify h with the 125 GeV SM-like Higgs and hs with a mostly singlet-like Higgs, with negligible mixing with the other CP-even scalars. In such a case the mass of the latter is simply given by:

m2hs λsvs.

(2.12)

The other relevant parameter for DM phenomenology is the mass of the charged Higgs,

given by [47, 81]:

m2H ±

√ 2µvs − λ4v1v2 v2 . 2v1v2

(2.13)

In order to ensure negligible mixing between the Higgs doublets and singlet the µ parameter should be small but still satisfy:

λ4v1v2 µ> √ .

2vs

(2.14)

Unless diﬀerently stated, we will assume, for our analysis, the assignations λ4 = 0.1, λs = 0.1 and µ = 35 GeV.

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2.4 Relevant Interactions

Let’s start with the interactions of the gauge bosons. The new U (1)X and the hypercharge gauge boson are described by the following Lagrangian:

L = − 1 Bˆ Bˆµν +

Xˆ Bˆµν − 1 Xˆ Xˆ µν.

gauge

4 µν

2 cosθW µν

4 µν

(2.15)

As we added a U (1)X gauge group a kinetic mixing between the two Abelian groups is, in general, allowed by the gauge symmetries. Assuming it to be zero at the

tree level, we can nevertheless not avoid its generation at the one-loop level since the

SM fermion are charged under both the two gauge group. In such case the induced kinetic mixing parameter can be estimated as | | ∼ g gX/(16π2) Q2X ln M 2/µ2 [87, 88] where, again, gX is the gauge coupling of the U (1)X and QX is the charge of the SM fermion. This expression roughly gives ∼ 10−2gX; consequently we can assume that kinetic mixing gives a subdominant contribution to DM phenomenology

and, hence, neglect it in numerical computation.

The Lagrangian responsible for the mass of the gauge bosons is:

L = (Dµφ1)†(Dµφ1) + (Dµφ2)†(Dµφ2) + (Dµφs)†(Dµφs) = + 41 g2v2W − µWµ+ + 18 gZ2 v2Z0 µZµ0 − 41 gZ (GX1v12 + GX2v22)Z0 µXµ + 18 (v12G2X1 + v22G2X2 v22 + vs2Q2Xs gX2 )XµXµ

where we have used the following expression for the covariant derivative:

(2.16)

Dµ = ∂µ + igT aWµa + ig Q2Y Bˆµ + igX Q2X Xˆµ,

(2.17)

while:

G = QYi + g Q . Xi cos θW X Xi

(2.18)

From the equation above we straightforwardly recover the SM value for the mass

of the W boson:

m2W = 14 g2v2,

(2.19)

the masses of the remaining gauge bosons are obtained upon diagonalization of the following mass matrix:

M=

gZ2 v2 −gZ (GX1 v12 + GX2 v22)

−gZ (GX1 v12 + GX2 v22) v12G2X1 + v2G2X2 v22 + qX2 gX2 vs2

(2.20)

which leads to the following rotation:

Zµ

cos ξ − sin ξ

Z µ = sin ξ cos ξ

Z0µ Xµ ,

(2.21)

–8–

where the mixing ξ is deﬁned in general as

2gZ (GX1 v12 + GX2 v22)

tan 2ξ =

m2 − m2

.

Z0

X

(2.22)

However, in this work we will mostly consider the regime m2X m2Z02 (we will further comment on this choice in the next section) and very small mixing angle. In

such a case we can write

m2Z m2Z0 = 14 gZ2 v2,

2

2 vs2 2 2 gX2 v2 cos2 β sin2 β

2

mZ mX = 4 gX QXs +

4 (QX1 − QX2) ,

(2.23)

and

sin ξ

GX1 v12 + GX2 v22 m2Z

m2Z = m2

Z

gX Q cos2 β + Q sin2 β + tan θ .

gZ X1

X2

W

(2.24)

Finally, we write the Lagrangian describing the neutral current interactions as follows:

L = − eJµ A − g cos ξJµ Z − sin ξ

NC

em µ 2 cos θW

NC µ

eJ µ +

g Jµ

em Z 2 cos θW NC

Zµ+

1 + 4 gX sin ξ

QRXf + QLXf ψ¯f γµψf + QRXf − QLXf ψ¯f γµγ5ψf Zµ+

1 − 4 gX cos ξ

QRXf + QLXf ψ¯f γµψf − QLXf − QRXf ψ¯f γµγ5ψf Zµ+

− 14 QN1gX cos ξ cos ξN1γµγ5N1Zµ + 41 QN1gX sin ξN1γµγ5N1Zµ,

(2.25)

where we have again adopted a general notation in terms of generic charges for the DM and the SM fermions under the new gauge symmetry (conditions for an anomaly free symmetry are, of course, automatically assumed).

Besides the neutral current there are other interactions that might be relevant for our phenomenology such Z W +W −, Z W +H−, HZ Z , hZ Z , HZZ . The presence of such interactions is an additional feature that distinguishes our work from previous studies in the literature [84, 90, 91]. In particular, the Z W +H− is proportional to ggXv/2 cos ξ, therefore it cannot be neglected. The details expression for the aforementioned couplings can be found e.g. in [81].

2In the opposite regime the same model can be used to interpret the recent XENON1T anomaly [89].

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