Dirac dark matter, dark radiation, and the type-II seesaw mechanism in alternative $U(1)_X$ standard model

We propose an extra $U(1)_X$ model with an alternative charge assignment for right-handed right-handed neutrinos. The type-II seesaw mechanism by a triplet Higgs field is promising for neutrino mass generation because of the alternative charge assignment. The small vacuum expectation value (VEV) of an additional Higgs doublet naturally leads to a very small VEV of the triplet Higgs field, and as a result, the smallness of neutrino mass can be understood. With the minimal Higgs field for $U(1)_X$ with the charge $1$, right-handed neutrinos are candidates for Dirac dark matter (DM) and dark radiation (DR). We have derived and imposed the LHC bound, the DR constraint and the bound from DM direct searches in the wide range of parameter space. Among various $U(1)_X$ choices, the DM direct search bound is found to be weakest for $U(1)_R$ where the constraints from thermal DM and non-negligible DR can be compatible. Such a number of the effective neutrino species would be interesting from the viewpoint of the so-called Hubble tension.


I. INTRODUCTION
The standard model (SM) of particle physics has been constructed on the basis of the gauge principle and the spontaneous symmetry breaking by the Higgs mechanism at the vacuum. The introduction of an extra U(1) gauge interaction is one of the promising and well-defined extensions of the SM. The B −L (baryon number minus lepton number) appears to be an accidental global symmetry in the SM, indicating that this might be a gauge symmetry in a UV completion of the theory [1][2][3][4]. At the same time, the cancellation of an anomaly for a chiral gauge theory is critical. When the gauge symmetry is extended from the SM gauge group SU(3) C × SU(2) L × U(1) Y to SU(3) C × SU(2) L × U(1) Y × U(1) B−L , the number of right-handed neutrinos and those charge are limited to two choices by the anomaly cancellation conditions. One is three right-handed neutrinos with each U(1) B−L charge −1, which has been usually considered -and might be called -the standard assignment. Under this assignment, it is straightforward to explain observed neutrino masses by the type-I seesaw mechanism [5][6][7][8] with right-handed neutrinos, by introducing the scalar field with the B − L charge 2 that generates the Majorana masses of right-handed neutrinos. In the other charge assignment, two out of three right-handed neutrinos have the U(1) B−L charge of −4 and the one has the charge of +5, which is sometimes called the alternative assignment [9].
A similar extra U(1) extended model can be constructed based on the U(1) R gauge symmetry where only right-handed fermions are charged while left-handed ones are not charged [10]. The U(1) R charged particles are same as in the U(1) B−L model up to chirality, neutrino masses can be generated by the type-I seesaw mechanism at tree level [11] or loop level [12]. The implication due to the chirality dependence [10,13,14], the axial-vector coupling [15], U(1) R charged Higgs scalars [16,17], and U(1) R interacting dark matter (DM) [18][19][20][21], can be found in literature. The anomaly-free most general gauged U(1) extension of the SM is defined as a linear combination of the SM U(1) Y and the U(1) B−L gauge groups. With this convenient parametrization of U(1) X [22][23][24], we can study a wide class of extra U(1) models including the representative model U(1) B−L and U(1) R . In this paper, we consider U(1) X models with the alternative U(1) B−L assignment.
In this paper we propose the triplet Higgs models in the extra U(1) X model with the alternative charge for right-handed neutrinos. The triplet Higgs fields have to be charged under the extra U(1) X to have Yukawa interactions with lepton doublets which are also charged under the U(1) X . Simultaneously, the triplet Higgs field requires the introduction of another doublet Higgs to develop the vacuum expectation value (VEV) by a scalar trilinear term. This second Higgs doublet cannot couple with the SM fermions due to the charge mismatch. In this construction, the charge assignment requires that the Higgs sector must be the combination of so-called type-I two Higgs doublet model (THDM) [41] and triplet Higgs model. The smallness of the generated neutrino mass via the type-II seesaw mechanism is a consequence of the smallness of the second Higgs VEV. In a philosophical sense, this is a neutrinophilic Higgs model [42] in the bosonic sector. Since the small triplet Higgs VEV is naturally realized in the wide range of parameter space, we may expect it easier to observe the Majorana nature through the same sign dilepton signal from the doubly-charged Higgs boson in collider experiments. The simplest way to realize the U(1) X breaking is to introduce an SM singlet scalar with U(1) X charge 1 whose VEV generates not only the mass of pseudoscalar but also the mass of right-handed neutrino DM. In the minimal extension, the singlet scalar plays three roles. This model with only one U(1) X charged scalar predicts a Dirac righthanded neutrino and one massless right-handed neutrino at the renormalizable level. 1 This could be interesting from the viewpoint of cosmology, because those states are candidates of DM and dark radiation (DR), respectively. The DR has been constrained by cosmological observations [43], and the DR constraint on the U(1) B−L model has been studied [44,45].
On the other hand, DR is interesting, because it could relax the so-called Hubble tension 2 which is the discrepancy between the current Hubble parameter H 0 inferred from the cosmic microwave background by Planck [43] and that measured by low-z observations such as the SH0ES collaboration [47]. The preferred value of the number of effective neutrino species N eff is evaluated as 3.2 N eff 3.5 [43] and 3.2 N eff 3.4 [48,49] for different data sets. This paper is organized as follows. In Sec. II, we describe the extra U(1) X model with the alternative charge assignment for right-handed neutrinos and the generation of neutrino masses by the type-II seesaw mechanism. The mass spectrum of the particles, especially the various Higgs particles and right-handed neutrinos will be derived. We also summarize the present experimental constraints on the model. In Sec. III we provide the relevant formula for discussion of DM and DR. In Sec. IV we present the interesting parameter region in terms of the current experimental and cosmological constraints for the different parameter sets of the U(1) X model. Section V is devoted to our summary. ν i R (i = 1, 2, 3), one Higgs doublet Φ 1 , one triplet Higgs ∆ 3 , and one U (1) X Higgs field Φ X are introduced. x H is a real free parameter in the U (1) X charge unfixed by the anomaly-free conditions.
Our model is based on the gauge group SU(3) C × SU(2) L × U(1) Y × U(1) X . The particle content is listed in Table I. Under these gauge groups, three generations of right-handed neutrinos (ν i R with i running 1, 2, 3) have to be introduced for the anomaly cancellation. We consider the alternative U(1) B−L charge assignment for right-handed neutrinos. The U(1) X symmetry is defiend as the linear combination of U(1) Y and U(1) B−L , and the U(1) X charge is parametrized by the relative U(1) Y charge x H normalized by the U(1) B−L charge. For this charge assignment, right-handed neutrinos can not have Yukawa interaction with lefthanded leptons l i L and the SM Higgs field Φ 2 due to the mismatch of the charge. Instead of introducing two doublet Higgs fields and two singlet Higgs fields to generate Majorana masses of right-handed neutrinos and to form Dirac neutrino masses for the type-I seesaw [30], we introduce, as an economical way, one triplet Higgs field ∆ 3 and one doublet field Φ 1 , which can generate neutrino masses by type-II seesaw, and one singlet field Φ X to break the U(1) X symmetry. Moreover, the SM singlet scalar with U(1) X charge 1 generates not only the mass of pseudoscalar but also the mass of right-handed neutrino DM. Thus, this extension of the scalar sector is the minimal and unique. The additional Φ 1 does not have any Yukawa coupling to the SM fermion as in the type-I THDM but is necessary to have a trilinear term of ∆ 3 .
The Yukawa couplings with those additional Higgs fields are given by where the superscript C denotes the charge conjugation, the dot denotes the antisymmetric product of SU(2), Y ∆ and Y ν i R are Yukawa couplings between left-handed lepton doublets and right-handed neutrinos, respectively. After the triplet Higgs field develops a VEV v ∆ , left-handed neutrino masses are generated as After Φ X develops a VEV v X , the Yukawa interactions between ν i R with Φ X give the Dirac mass of right-handed neutrinos as where we define We find that one linear combination is a massless state. Other two components are summarized as by composing a Dirac spinor as There is one massless state ν R and one Dirac fermion χ. Thus, ν R behaves as DR and χ is a candidate for DM. It is worth noting that χ has no direct coupling with the SM particles thanks to its U(1) X charge assignment which guarantees the stability of the DM candidate. This is in a remarkable contrast with right-handed neutrino DM in the minimal U(1) X model with the standard charge assignment, where the extra Z 2 parity has to be introduced by hand to stabilize DM [50][51][52]. 3 The gauge interactions of ν R and χ can be read from those of ν i while similarly, the U(1) X gauge interaction for an SM chiral fermion (f L/R ) can be read from the usual covariant derivative, where Table I.
The scalar potential is given by At a high scale ≫ v ≃ 246 GeV, U(1) X symmetry is spontaneously broken by the VEV of At the U(1) X broken vacuum, the U(1) X gauge boson acquires the mass, as ν R does in Eq. (3). Then, the scalar φ X from Φ X also has the mass m 2 φ X = λ X v 2 X . Thus, the effective scalar potential at a low-energy scale below the scale of v X is given by where the third term is generated from the third term in Eq. (15) by replacing Φ X with its VEV as −µ 2 12 = λ 12 v X / √ 2. µ 2 i =μ i 2 + λ iX v 2 X /2, λ i , and Λ i(j) are coupling constants. Unlike the usual THDM, the (Φ † 1 Φ 2 ) 2 term is absent due to the U(1) X gauge symmetry. The µ 2 12 terms generated by the VEV of v X is essential to give the mass of a CP odd Higgs boson and remove a dangerous Nambu-Goldstone boson from the spectrum.
The stationary conditions are expressed as where v 1 and v 2 are the VEVs of Φ 1 and Φ 2 , respectively. Our notation satisfies v = v 2 1 + v 2 2 ≃ 246 GeV. The condition (20) can be recast as where, in the last approximation, we have assumed the condition for the type-II seesaw mechanism, µ 2 3 ≫ v 2 . For a given Λ 6 and µ 2 3 , our smaller v 1 results a smaller v ∆ , which naturally fits the ρ parameter constraint and enhances the decay rate of the doubly-charged Higgs boson to the same sign dilepton H ±± → ℓ ± i ℓ ± j as we will show. In the small v ∆ limit, the masses of scalars are given by m 2 B. Summary of experimental constraints
A flavor-violating radiative decay ℓ i → ℓ j γ is also induced and its branching ratio for µ → eγ, which gives the most stringent bound, is given by [61] Br The MEG experiment has reported the latest result of Br(µ → eγ) < 4.2 × 10 −13 [64].

Any of those lepton flavor-violating decay gives the lower bound on the VEV of the triplet
Higgs field as v ∆ > O(1) eV [65] for m H ±± 1 TeV.

X boson search at the LHC
In this section we evaluate the production cross section of this process at the LHC for a choice of parameters consistent with the LHC constraints from dilepton channel pp → X → with the auxiliary function [52] F (x H ) = 13 + 16x H + 10x 2 H .
g X has been constrained to be small in previous studies. As in Ref. [33], since the total X boson decay width is very narrow, we use the narrow width approximation to evaluate the X boson production cross section where f q and fq are the parton distribution function (PDF) for a quark and antiquark, s = xys is the invariant mass squared of colliding quarks for the center of mass energy s.
The factor 2 in Eq. (36) counts two ways of q coming from which proton out of two colliding protons. Since the most severe bound is from the dilepton channel (ℓ = e, µ), we calculate and compare it with the ATLAS results [67]. We employ PDFs of CTEQ6L [70] with a factorization scale Q = m X for simplicity. Following the manner to obtain a suitable kfactor presented in Ref. [51], we scale our result by a k-factor of k = 0.947 to match the recent ATLAS analysis in our calculation. The results will be presented after we discuss cosmological constraints.

C. Decay of doubly-charged Higgs boson
The decay rates for principal decay modes of H ±± are [71,72] with Those of other minor modes, which are not relevant for our later discussion, can be found in Ref. [71].
In the left panel of Fig. 1

III. COSMOLOGY
Next, we consider cosmological constraints and implication of our model.

A. Dark radiation
With one U(1) X charged scalar Φ X , out of three right-handed neutrinos, there is one massless state ν R and one Dirac fermion χ. We at first consider the DR constraint due to thermal production of ν R .
The thermal averaged cross section of ν R for the temperature T ≪ m X is expressed as with and the color factor N c = 3 for quarks. The invariant squared amplitude before taking thermal averaging are listed in Appendix A. The decoupling temperature T dec of ν R from the thermal bath is evaluated by where H =ȧ/a is the cosmic expansion rate in the radiation dominated universe described by with a, ρ r , and g * are the scale factor, the energy density of radiation and the number of relativistic degrees of freedom, respectively. M P ≃ 2.4 × 10 18 is the reduced Planck mass.
After ν R decoupled from thermal bath at the decoupling temperature T dec , the energy density of ν R , ρ ν R , decreases as ρ ν R ∝ a −4 . By parametrizing the energy density of ν R with ∆N eff as the total radiation energy density except photons is expressed as where T ν is the temperature of left-handed SM neutrinos and N ν eff is the effective number of neutrinos in the SM. For recent calculations of N ν eff , see e.g., Refs. [73][74][75][76][77].
B. Dark matter

Abundance
We estimate the thermal relic abundance of our Dirac DM χ by solving the Boltzmann where n is the number density of χ, n EQ is its number density at thermal equilibrium, σv is the thermal averaged products of the annihilation cross section and the relative velocity.
The amplitude squared integrated over the scattering angle θ is given by where s is the center-of-mass energy and the total decay width is given by with each partial decay width (32), (33) and (34).
The resultant DM relic abundance is given by where x d = m χ /T d with the decoupling temperature T d [78].

Direct DM detection bound
The DM χ with the mass m χ can scatter off nucleons through the X boson exchange. The spin-independent (SI) weakly interacting massive particle (WIMP)-nucleon cross section for the elastic scattering through vector-vector couplings is given by [79] with Here, Z and A are the atomic number and the mass number of a target. m N is the mass of nucleus.

IV. BENCHMARK POINTS
We examine the constraints on our model for several parameters by collider experiments and cosmology.
The first case is x H = 0, which corresponds to U(1) B−L . On the upper-right panel in model are allowed. The LHC bound becomes most less stringent for x H = −1.2 [52], which is shown in the lower-left panel. In this case, the N eff constraint is comparable or more stringent than the LHC bound for all mass range of m X in the plot.
Finally, we consider the case of x H = −2, which corresponds to U(1) R . This is a case that the constraint from direct DM search becomes weaker by the destructive interference between proton and neutron in the cross section, which can be easily seen by rewriting the SI WIMP-nucleon cross section (56) as For atomic nucleus with A ≃ 2Z, the cancellation occurs around x H ≃ −2. Thus, a larger g X than that in the U(1) B−L model is allowed. As can be seen in the lower-right panel of Fig. 2, m X is not necessarily so close to 2m χ , and more interestingly, there exist parameter points that thermal DM with the mass of 2 TeV and ∆N eff = O(0.1) can be simultaneously realized. We note that another model for the simultaneous realization of thermal DM and DR was proposed in Ref. [81], where DM is flavored U(1) interacting scalar and the gauge boson is light.

V. SUMMARY
We have proposed a simple extension of the SM with an extra U(1) X gauge symmetry with an alternative charge assignment for right-handed neutrinos. The type-II seesaw mechanism is an economical way to generate appropriate neutrino masses in this framework and an additional Higgs doublet to have a trilinear interaction with the triplet Higgs field. Since the tadpole term is proportional to the squared VEV of the additional doublet Higgs field, the smallness of neutrino mass could be understood as the consequence of the smallness of the additional Higgs VEV. Thus, without introducing a very small dimensionful parameter in the Higgs potential, we can observe the dilepton decay of the doubly-charged Higgs boson, which is evidence of nonconservation of lepton number.
We have also derived the constraints for thermal DM and DR. Since we have introduced only one SM singlet U(1) X breaking scalar, it predicts the existence of one massless state and Dirac fermion in the right-handed neutrino sector. The LHC sets a stringent bound on the model parameters for m X a few TeV, while the DR constraint is more significant for Another important but unaddressed subject is baryogenesis, which is beyond the scope of this paper. We, here, note that thermal leptogenesis by heavy-triplet Higgs particles [82][83][84][85] or Affelck-Dine baryogenesis by scalar condensations [86] appears to be promising.