Connecting Low scale Seesaw for Neutrino Mass to Inelastic sub-GeV Dark Matter with Abelian Gauge Symmetry

Motivated by the recently reported excess of electron recoil events by the XENON1T experiment, we propose low scale seesaw scenarios for light neutrino masses within $U(1)_X$ gauge extension of the standard model that also predicts stable as well as long lived dark sector particles. The new fields necessary for seesaw realisation as well as dark matter are charged under the $U(1)_X$ gauge symmetry in an anomaly free way. A singlet scalar field which effectively gives rise to lepton number violation and hence Majorana light neutrino masses either at tree or radiative level, also splits the dark matter field into two quasi-degenerate states. While sub-eV neutrino mass and non-zero dark matter mass splitting are related in this way, the phenomenology of sub-GeV scale inelastic dark matter can be very rich if the mass splitting is of keV scale. We show that for suitable parameter space, both the components with keV splitting can contribute to total dark matter density of the present universe, while opening up the possibility of the heavier dark matter candidate to undergo down-scattering with electrons. We check the parameter space of the model for both fermion and scalar inelastic dark matter candidates which can give rise to the XENON1T excess while being consistent with other phenomenological bounds. We also discuss the general scenario where mass splitting~$\Delta m$ between the two dark matter components can be larger, effectively giving rise to a single component dark matter scenario.


I. INTRODUCTION
Recently, the XENON1T collaboration has reported an excess of electron recoil events over the background in the recoil energy E r in a range 1-7 keV, peaked around 2.4 keV [1].
Though this excess is consistent with the solar axion model at 3.5σ significance and with neutrino magnetic moment signal at 3.2σ significance, both these interpretations face stringent stellar cooling bounds. Since XENON1T collaboration has neither confirmed nor ruled out the possible origin of this excess arising due to beta decay from a small amount of tritium present in the detector, it has created a great interest in the particle physics community to search for possible new physics interpretations for this excess of electron recoil events.
In this work, we try to connect low scale seesaw origin of non-zero masses of light neutrinos with sub-GeV inelastic DM within an abelian gauge extension of the SM. Recently, several low scale seesaw models with additional U (1) X gauge symmetries have been proposed in different contexts; for example, see [14][15][16][17][18] and references therein. Here we extend the SM with a gauged U (1) X symmetry which primarily describes the dark sector of our model comprising of inelastic DM candidates and fields responsible for seesaw realisation. The breaking of U (1) X to a remnant Z 2 symmetry gives rise sub-eV masses of light neutrinos by generating a dimension five effective operator O 1 LLHH/Λ [19] at low energy, where L and H are lepton and Higgs doublets respectively, O 1 is the Wilson coefficient that depends upon different couplings of the UV complete theory and Λ is the scale of U (1) X symmetry breaking. Note that this operator breaks lepton number by two units and hence the corresponding neutrino mass is of Majorana type. After the electroweak phase transition the neutrino mass is given by: m ν ∝ H 2 /Λ. The breaking of U (1) X gauge symmetry to a discrete Z 2 also lead to a small mass splitting between the components of either a Dirac fermion DM or a complex scalar DM, leading to an inelastic DM scenario. We ensure the stability of DM via the remnant Z 2 symmetry. If we assume that the DM mass is of sub-GeV scale with mass splitting of keV or smaller, then both the components can be present today and can give rise interesting phenomenology. In particular, if the mass splitting is a few keV, then we can have a tantalising scenario where the heavier component scatters off the electron and gets converted to the lighter component effectively explaining the XENON1T anomaly [1]. We first discuss viable low scale seesaw models within an U (1) X gauge symmetric framework which also predicts inelastic DM. Since all such seesaw models have similar DM phenomenology, we study the latter for a sub-GeV inelastic DM whose interactions with the SM relies primarily on kinetic mixing of U (1) X gauge symmetry with U (1) Y of the SM. We calculate the relic abundance of DM and constrain the parameter space from all available bounds and the requirement of fitting XENON1T excess. While inelastic DM and its connection to origin of light neutrino masses have been studied in earlier works too (see [20] for example and references therein), most of these works focused on heavy DM regime with masses in the range of electroweak scale to a few TeV. Such DM scenarios are probed typically at nuclear recoil experiments. In this work, we focus on sub-GeV mass regime of inelastic DM in the context of electron recoil experiments while showing possible connections to the origin of light neutrino mass in different neutrino mass models.
The paper is organised as follows. In section II, we discuss different low scale seesaw models for light neutrino mass within a framework U (1) X gauge symmetry which also gives rise to inelastic DM. In section III we discuss the constraints on such low scale U (1) X gauge models followed by detailed discussion of inelastic DM with sub-GeV mass and its implications for XENON1T excess in section IV. We finally conclude in section V.

II. LOW SCALE DARK SEESAW WITH U (1) X GAUGE SYMMETRY
In this section, we describe different seesaw realisations for sub-eV masses of light neutrinos by considering the presence of an Abelian gauge symmetry U (1) X which plays a crucial role in both neutrino and dark matter sectors. In particular, the dark matter phenomenology and light neutrino masses are correlated in the following ways.
• Degenerate two component dark matter implies vanishing light neutrino masses and hence disallowed from experimental data.
• Tiny mass splitting of keV order or below leads to a two component dark matter scenario with the heavier dark matter candidate being long lived on cosmological scales. The mass splitting of keV scale is of particular interest from XENON1T excess point of view.
• Larger mass splitting between the DM candidates make heavier DM unstable leading to a single component DM. This, although can not explain XENON1T excess, remains consistent with other DM and neutrino requirements.
We will discuss the second and third scenarios mentioned above one by one. We adopt a minimalistic approach and consider only the newly introduced fermions and scalars to be charged under the U (1) X gauge symmetry leaving the SM particles to be charge-less under this new symmetry. Additional discrete symmetries are also incorporated to obtain the desired couplings in the Lagrangian for seesaw realisations. With non-minimal particle content one can also consider scenarios with similar seesaw realisation and DM phenomenology without any additional discrete symmetries as have been studied in several works, for example, see [21][22][23][24][25][26][27][28][29][30] and references therein.

A. Inverse seesaw model with inelastic DM
Here we consider an inverse seesaw [31], which is a typical low scale model in contrast to the high scale canonical seesaw scenarios like type I, type II and type III [32][33][34][35][36][37][38][39][40]. The inverse seesaw is realised in a gauged U (1) X extension of a two Higgs doublet model. The gauge group of the theory is thus given by: SU (3) c ⊗SU (2) L ⊗U (1) Y ⊗U (1) X . An additional discrete Z 4 symmetry is also imposed to have the correct mass matrix structure of neutral fermions. As shown in table I (particle content of inelastic fermion DM with inverse seesaw) and table II (particle content of inelastic scalar DM with inverse seesaw), the new degrees of freedoms apart from a second Higgs doublet H 2 are all singlets under the SM gauge group The U (1) X gauge charges of these newly introduced particles are chosen in such a way that give rise to the desired neutrino and DM phenomenology.
While N R , S R are singlet fermions taking part in inverse seesaw, the fields Ψ L,R and η are introduced as viable fermion and scalar DM candidates respectively. When the singlet scalar Φ 2 acquires a vacuum expectation value (vev), the gauged U (1) X symmetry breaks down to a remnant Z 2 symmetry under which the vector-like fermion singlet Ψ(= Ψ L + Ψ R ) is odd. As a result, Ψ behaves as a candidate of fermion DM 1 . We show that the spontaneous breaking of U (1) X gauge symmetry not only generates the lepton number violating mass term for 1 For scalar DM we need another stabilising symmetry as we discuss below  inverse seesaw, but also splits the DM (both fermion and scalar) into two quasi-degenerate components. Note that, we consider only single component DM, either fermion or scalar, not both in the same model. We discuss fermion and scalar DM separately to show their inelastic nature arising from a scalar field taking part in generating light neutrino masses.
It should be noted that the models we considered here are anomaly free by the virtue of the assigned gauge charges of the newly introduced fermions.
The Lagrangian involving the new degrees of freedom consistent with the extended symmetry is given by where L DM describes the Lagrangian for inelastic DM and is discussed below separately for fermion (Ψ) and scalar (η) cases.
The electroweak symmetry is broken when the Higgs doublets H 1 and H 2 acquire nonzero vevs, while the vevs of Φ 1 and Φ 2 break Z 4 × U (1) X symmetry of the hidden sector.
The scalar fields which acquire non-zero vevs can be represented as As can be seen from the Lagrangian in equation 1, the vev of singlet scalar Φ 2 generates the Majorana mass term µ for S R field which consequently appears as 33-term (entry for third row and third column) of neutral lepton mass matrix given in equation (3) and hence is responsible for the light neutrino mass generation through inverse seesaw mechanism. Later we shall show that the vev of Φ 2 also creates a mass splitting between the DM components (both for fermion and scalar DM models).

Light Neutrino Masses:
In the effective theory, the neutral lepton mass matrix can be written in the basis n = where M ν has the structure Assuming that µ << m D < M , the light neutrino mass matrix at leading order can be given as: For a typical choice: m D ∼ 10 GeV, M ∼ 1 TeV and µ ∼ 1 keV, we get sub eV neutrino mass.

Inelastic fermion dark matter:
We now show how inelastic fermion DM arises in this inverse seesaw model. The particle content for inelastic fermion DM realisation is already given in table I. The relevant Lagrangian satisfying U (1) X × Z 4 symmetry can be written as: where D µ = ∂ µ + ig Z µ and B αβ , Y αβ are the field strength tensors of U (1) X , U (1) Y respectively and is the kinetic mixing parameter. We note that the kinetic mixing plays a crucial role in giving rise the DM phenomenology.
The scalar singlet Φ 2 acquires a non-zero vev u 2 and breaks U (1) X spontaneously down to a remnant Z 2 symmetry under which Ψ L,R are odd while all other fields are even. As a result, Ψ L and Ψ R combine to give a stable DM candidate in the low energy effective theory. The vev of Φ 2 also generates Majorana masses for fermion DM: m L = y L u 2 / √ 2 and m R = y R u 2 / √ 2 for Ψ L and Ψ R respectively. We assume m L , m R << M . As a result, the Dirac fermion Ψ = Ψ L + Ψ R splits into two pseudo-Dirac states ψ 1 and ψ 2 with masses as defined earlier, the Lagrangian in terms of these physical mass eigenstates can be written as where sin θ ≈ m − /M . The mass splitting between the two mass eigenstates is given by In order to address the XENON1T anomaly in section IV, we take ∆m ∼ 2.5 keV.

Inelastic scalar dark matter:
We now turn to show the inelastic nature of scalar DM that arises naturally in this inverse seesaw model. The particle content for inelastic scalar DM realisation is already given in table II. Unlike the case of fermion DM, here U (1) X × Z 4 symmetry alone is not enough to stabilise the scalar DM η. This is due to the presence of a term H † 1 H 2 Φ 1 η in the Lagrangian, allowed by U (1) X × Z 4 symmetry. Therefore, we impose an additional Z 2 symmetry under which η is odd while rest of the particles are even as mentioned in table II. The relevant Lagrangian involving η and U (1) X gauge boson can be written as: where D µ = ∂ µ + ig Z µ and B αβ , Y αβ are the field strength tensors of U (1) X , U (1) Y respectively. Here Z is the U (1) X gauge boson and g is the corresponding gauge coupling and is the kinetic mixing parameter defined earlier.
The scalar Φ 2 acquires a vev and breaks U (1) X gauge symmetry spontaneously. We parametrise the scalar singlet DM field η as: Note that the vev of Φ 2 not only gives mass to Z gauge boson: M 2 Z = g 2 (4v 2 2 ), but also creates a mass splitting between η 1 and η 2 which is evident from the following effective Lagrangian obtained by putting the above field parametrisation in equation (7).
Thus the mass splitting between the two states η 1 and η 2 is given by We assume ∆m << m η 1,2 . As a result, the two components of η, i.e. η 1 and η 2 give rise viable inelastic DM candidates. Because of the kinetic mixing between the U (1) X gauge boson Z and the SM Z boson, these DM particles can interact with the SM particles which is evident from the following effective Lagrangian: B. Type II seesaw with inelastic DM Here we consider a variant of type II seesaw along with inelastic DM in a gauged U (1) X × the breaking of U (1) X × Z 4 symmetry which also lead to a low scale type II seesaw.

Light Neutrino Masses:
The Yukawa Lagrangian relevant for the discussion is  where L DM is the Lagrangian for inelastic scalar or fermion DM as discussed below.
The relevant part of the scalar potential is given by: We assume that µ 2 ∆ > 0 and λ ∆ > 0. As a result ∆ L does not acquire any direct vev. However, the vev of Φ 1 and H can induce a small vev for ∆ L as: As a result the light neutrino mass matrix is given by m ν = Y ν v L . The origin of light neutrino masses is shown in figure 1. Note that in conventional type II seesaw, the induced vev is decided by trilinear term µH T ∆ L H and hence for µ ∼ µ ∆ it corresponds to a high scale seesaw like type I. However, here, the trilinear term is dynamically generated as µ = λ 1 Φ 1 via vev of the scalar Φ 1 . Note that Φ 1 vev can be even lower than the electroweak scale.
Thus for λ 1 Φ 1 µ ∆ (which can be achieved by suitable tuning of λ 1 and Φ 1 vev), one can bring down the scale of type II seesaw µ ∆ to a much lower scale.
Now we turn to comment on the viability of inelastic fermion and scalar DMs in this model.

Inelastic fermion dark matter:
The Lagrangian relevant for fermion DM is given by: From Eq. (13), we see that the dimension five terms generate small Majorana masses for

Inelastic scalar dark matter:
Similar to fermion DM realisation with type II seesaw, it is possible to have a scalar DM scenario as well. The corresponding particle content is shown in table IV. While the origin of neutrino mass remains the same as before, the relevant terms in the scalar DM Lagrangian can be written as follows.
From Eq. (14), we see that the vevs of Φ 1,2 generate a mass splitting between the real and imaginary parts of singlet scalar DM η = (η 1 + iη 2 )/ √ 2. The corresponding mass splitting is The mass splitting can be brought down to keV scale, necessary for explaining the anomalous XENON1T excess in section IV, by tuning λ 2 and Φ 1 vev which does not play a role in U (1) X symmetry breaking. Further details of inelastic scalar dark matter remain same as discussed in the previous subsection.

C. Radiative seesaw with inelastic DM
Radiative seesaw has been one of the earliest proposals for low scale seesaw, see [41] for a recent review. Due to additional loop suppressions and free parameters, natural realisation of low scale seesaw becomes possible in such frameworks. While there are many possible radiative seesaw, here we outline just one possibility that suits our desired phenomenology.
In [42], a radiative seesaw model was introduced with the addition of a fermion doublet, a fermion singlet and a scalar singlet. Here we consider an alternate possibility with additional fermion singlet, scalar doublet and scalar singlet. Note that there are simpler realisation of one loop seesaw with dark matter, see for example [43]. However, the requirement of a sub-GeV inelastic scalar DM forces us to consider a complex scalar singlet into account because a scalar doublet with GeV scale components will be ruled out by precision data from Large Electron Positron (LEP) collider experiment [44].
The new particle content of the model is shown in table V. The relevant part of the Lagrangian consistent with U (1) X gauge symmetry is given by The relevant part of the scalar potential is The U (1) X symmetry is broken by a nonzero vev (v φ ) of Φ to a remnant Z 2 symmetry under which Ψ L,R , η, χ are odd while all other fields are even. As a result the lightest among Ψ L,R , η = η 1 + iη 2 and χ = χ 1 + iχ 2 can gives rise to a viable DM candidate. Since χ is a doublet it's mass can not be less than 45 GeV in order to avoid invisible Z-decay width.
On the other hand, the mass of singlet fermion Ψ and singlet scalar η can be much smaller than the mass of χ. In the effective theory, the vev of Φ generate small Majorana masses m L = Y L v φ and m R = Y R v φ for Ψ L and Ψ R respectively. As a result the Dirac fermion Ψ splits up into two pseudo-Dirac states ψ 1 and ψ 2 with masses M 1 and M 2 respectively.
Similarly the vev of Φ creates a mass splitting between the real and imaginary parts of η through the term µ 2 η 2 Φ + h.c. as given in the scalar potential (16). The corresponding mass splitting is given by We will see that this mass splitting is related to the non-zero masses of light neutrinos in this model.
The light neutrino mass arises at one loop level via the diagram shown in figure 2. The contribution to light neutrino mass can be estimated, in the mass insertion approximation [45], to be where M k is the mass of pseudo-Dirac states ψ k with k = 1, 2 and v, v φ are vevs of neutral component of the SM Higgs doublet H and the scalar singlet Φ. The loop function I ν is given by: where the parameters r i are defined as Similar to the seesaw models discussed earlier, here also light neutrino mass is proportional to the term in scalar potential which splits the scalar singlet (η) mass namely µ 2 ηηΦ † . Non-zero µ 2 implies non-zero mass splitting between scalar and pseudoscalar components of η. That is, ∆m 2 = 2µ 2 v φ . Thus, we can rewrite the light neutrino mass formula as  to the new gauge coupling to be M Z /g ≥ 7 TeV [49]. However, since we are interested in the low mass of the gauge boson, bounds from hadron colliders like ATLAS and CMS will not be very relevant. Similarly, LEP bound is also not applicable in such low mass regime.
One can constrain Z − Z mixing and Z mass from electroweak precision measurements as well. However, for GeV scale Z mass with tiny kinetic mixing with SM Z boson, such bounds do not apply [50]. For a detail of the direct search bounds on such a light gauge boson, one may refer to [51]. Recently, a low scale U (1) X model was also studied in the context of flavour anomalies, dark matter and neutrino mass [52].
Since all the scenarios discussed here contain light DM and Z gauge boson at or below GeV scale, SM Higgs can decay into them due to mixing of SM Higgs with singlet scalars and Z − Z mixing respectively. Since our analysis relies upon Z − Z mixing only, we consider Higgs decay to Z Z and ZZ only. The corresponding decay widths are given by Here m h denotes SM Higgs mass. Using these, we constrain the parameter space in − M Z plane using the bound mentioned above. The resulting parameter space ruled out from this bound is shown in figure 3. One can also constrain the model from the LHC measurements of SM Higgs decaying into light gauge bosons with four lepton final states [56]. As will be clear from our final parameter space to be discussed later, such bounds are trivially satisfied for the region of parameter space we focus on.
In some specific seesaw scenarios discussed above, electroweak multiplets are also introduced which couple directly to SM leptons. For example, the type II seesaw scenario considers a SU (2) L triplet which gives rise to a doubly charged physical scalar. Search for same sign dileptons at the LHC puts strict bound on such scalars, depending upon its branching ratio into specific leptonic final states [57]. Roughly, doubly charged scalar with masses below 800 GeV are currently disfavoured from such searches. Precision electroweak data can also rule out certain mass ranges of physical scalars belonging to this triplet [58].
The radiative seesaw model discussed above also contains an additional scalar doublet χ components is much larger than the nuclear recoil energy then such processes are forbidden.
In particular, the SM Z-boson mediated interactions in direct search experiments can be forbidden if the DM is inelastic [60][61][62][63][64]. The main difference between the two scenarios is that in the former case the life time of heavier component is required to be longer than the age of the universe, while in the latter it is not required. We will discuss the corresponding results for the latter case as well where DM in the present universe is effectively in terms of the lighter component only as the heavier component can decay in early epochs due to large mass splitting.
The relic abundance of two component DM can be found by numerically solving the corresponding Boltzmann equations. Let n 2 and n 1 are the total number densities of two dark matter candidates χ 2 and χ 1 respectively. The two coupled Boltzmann equations in terms of n 2 and n 1 are given below, where, n eq i is the equilibrium number density of dark matter species i and H denotes the Hubble expansion parameter. The thermally averaged annihilation and coannihilation processes (χ i χ j → XX) are denoted by σv , where X denotes all particles to which DM can annihilate into. Since we consider GeV scale DM, the only annihilations into light SM fermions can occur, such as e − , µ − , ν e , ν µ , ν τ , u, d, s. The only available channel for annihilation of χ 1,2 to light SM fermions is through Z − Z mixing. Additionally small mass splitting between the two DM components lead to efficient coannihilations while keeping their conversions into each other sub-dominant. We have solved these two coupled Boltzmann equations using micrOMEGAs [65]. Due to tiny mass splitting, we find almost identical relic abundance of two DM candidates. Thus each of them constitutes approximately half of total DM relic abundance in the universe, i.e. n 2 ≈ n 1 ≈ n DM /2. We then constrain the model parameters by comparing with Planck 2018 limit on total DM abundance Ω DM h 2 = 0.120 ± 0.001 [66].
Here Ω DM is the density parameter of DM and h = Hubble Parameter/(100 km s −1 Mpc −1 ) is a dimensionless parameter of order one.
As discussed above, we assume χ 2 is heavier than χ 1 with a small mass splitting ∆m = M 2 − M 1 between the two components. Moreover, we assume ∆m of keV scale in order to explain the XENON1T anomaly. For a fixed incoming velocity v of χ 2 , the differential scattering cross section for χ 2 e → χ 1 e can be given as where m e is the electron mass, σ e is the free electron cross section at fixed momentum transfer q = 1/a 0 , where a 0 = 1 αme is the Bohr radius with α = e 2 4π = 1 137 being the fine structure constant, E r is the recoil energy of electron and K(E r , q) is the atomic excitation factor. We assume the DM form factor to be unity. In this paper the atomic excitation factor is adopted from [67]. For E r = (1 − 5) keV, the scattering happens dominantly with electrons in the 3s shell. The Atomic excitation factor K(E r , q) is independent of E before it reaches the threshold of the next quantum energy level. Since most of the signal events have a recoil energy in a range 2 − 3 keV [1], so one can use K(E r , q) K(∆m, q) K(2 keV, q) [67] for the calculation. In the above equation (24), f (v) is the local DM velocity distribution function which can always be normalised to unity i.e. f (v)dv = 1. f (v) can be taken as a pseudo-Maxwellian distribution given by where A is the normalisation constant, v m is the average velocity which we consider to be v m = 1×10 −3 and σ v is the DM velocity dispersion. The free electron scattering cross-section in this case is given by: where α Z = g 2 4π , α = g 2 4π and is the kinetic mixing parameter between Z and Z mentioned earlier which we take to be ≤ 10 −3 . It should be noted that σ e is independent of DM mass as the reduced mass of DM-electron is almost equal to electron mass for GeV scale DM mass we are considering.
Unlike the elastic case, the limits of integration in Eq. (24) are determined depending on the relative values of recoil energy (E r ) and the mass splitting between the two DM components.
And for E r ≤ ∆m The dependency of atomic excitation factor on the momentum transferred q is shown in figure 4. Here the dominant contribution comes from the bound states with principal quantum number n = 3 as their binding energy is around a few keVs. In the right panel of figure 4, we have shown the plot for the integration of momentum transferred times the atomic excitation factor i .e.K int (E r , q) = q+ q− qdqK(E r , q) as a function of the recoil energy E r for M 1 = 0.3GeV and ∆m = 2keV. The figure shows a peak around E r ∆m since The atomic excitation factor, after the q integration, is plotted as a function of the transferred recoil energy E r .
the q − approaches to zero and the momentum transfer maximising this factor is available.
It is worth mentioning that such kind of enhancement is a characteristic feature of inelastic scattering.
The differential event rate for the inelastic DM scattering with electrons in Xenon atom, i.e χ 2 e → χ 1 e, can be given as: where n T = 4 × 10 27 Ton −1 is the number density of Xenon atoms and n DM is the number density of the dark matter particle.
The detected recoil energy spectrum can be obtained by convolving the above equation (29) with the energy resolution of the detector. Incorporating the detector efficiency, the energy resolution of the detector is given by a Gaussian distribution with an energy dependent width, where γ(E) is the detector efficiency which is reported in figure 2 of [1] and the width σ det is given by where a = 0.3171 and b = 0.0037. Thus the final detected recoil energy spectrum is given by be both up or down type depending upon the mass splitting. While DM-electron scattering cross section is given by equation (26), the spin-independent DM-nucleus scattering cross section is given by where χ denotes the DM particle, A and Z are the mass number and atomic number of the target nucleus respectively and µ N χ = m N mχ m N +mχ is the reduced mass. f p and f N are the interaction strengths for proton and neutron respectively. The occurrence of this process solely depends on the mass splitting between the two states. In fact, the minimum velocity of the DM needed to register a recoil inside the detector is given If the mass splitting is above a few hundred keV, then the inelastic scattering will be forbidden. On the other hand, the elastic DM-nucleon scattering is much smaller due to velocity suppression.
Variation of relic abundance of fermion DM as a function of its mass is shown in figure   8 for a set of fixed benchmark parameters. Clearly, due to tiny mass splitting (2. Since the mass splitting between ψ 1 and ψ 2 is kept at keV scale ∆m = O(keV ), there can be decay modes like ψ 2 → ψ 1 νν mediated by Z −Z mixing. If both the DM components are to be there in the present universe, this lifetime has to be more than the age of the universe that is τ ψ 2 > τ Univ. . The decay width of this process is Γ(ψ 2 → ψ 1 νν) = g 2 g 2 2 (∆m) 5 will lead to an exclusion line in the range g ∼ 10 −4 − 10 −3 which remains weaker than the lifetime bound and hence not shown.
We also incorporate the Planck bound on DM annihilation into charged leptons [66], specially e − e + , µ − µ + pairs which are dominant in the region of our interest. The Cosmic microwave background (CMB) anisotropies, very precisely measured by the Planck experiment, are sensitive to energy injection in the intergalactic medium (IGM) from DM annihilations.
The effective parameter constrained by CMB anisotropies is where f eff (z) is the efficiency factor characterising the fraction of energy transferred by the DM annihilation processes into the IGM. While the efficiency factor is redshift dependent, CMB anisotropies are most sensitive to redshift z ∼ 600. For earlier works on imprint of DM annihilations on CMB please see [69][70][71][72]. Considering the efficiency factor to be close to unity for DM mass ranges of our interest, the Planck 2018 bound P ann < 3 × 10 −11 GeV −3 at 95% C.L. can be used to constrain the model parameters. We consider the dominant s-wave coannihilations of DM into charged leptons and assume 3M DM = M Z to derive the CMB exclusion line in figure 9. Since DM relic is satisfied mostly around the Z resonance, this relation will not change much for other allowed points as well. As can be seen from figure 9, the CMB bound on DM annihilation can be even stronger than the lifetime bound for DM mass beyond 1 GeV. It however leaves plenty of parameter space consistent with DM relic as well as XENON1T favoured scattering cross section of DM with electrons.
The dominance of Z resonance in DM annihilation is visible from the scattered points in figure 9. It can be seen that DM with a particular mass around M Z /2 can satisfy relic independent of portal coupling g . This is due to the resonance feature around M DM ≈ M Z /2 which was also noticed in figure 6.

B. Scalar DM
We now turn to find a viable inelastic scalar DM in a scenario where the SM is augmented

C. DM conversion at late epochs
Although relative abundance of the two DM candidates χ 1 and χ 2 are expected to be approximately the half of total DM relic abundance from the above analysis based on chemical decoupling of DM from the SM bath, there can be internal conversion happening between the two DM candidates via processes like χ 2 χ 2 → χ 1 χ 1 , χ 2 e → χ 1 e until later epochs. While such processes keep the total DM density conserved, they can certainly change the relative proportion of two DM densities. It was pointed out by [9,73] as well as several earlier works including [74,75]. In these works, DM is part of a hidden sector comprising a gauged U (1) X which couples to the SM particles only via kinetic mixing of U (1) X and U (1) Y , denoted by , same as our scenario. Thus, although the DM-SM interaction is suppressed by 2 leading to departure from chemical equilibrium at early epochs, the internal DM conversions like χ 2 χ 2 → χ 1 χ 1 can happen purely via U (1) X interactions and can be operative even at temperatures lower than chemical freeze-out temperature, specially when U (1) X gauge coupling is much larger compared to the kinetic mixing parameter. However, in our work we choose U (1) X gauge coupling to be small enough so that such late conversion between two DM components does not happen. For our choices of couplings and region of interest from XENON1T excess point of view, both DM-SM interactions as well χ 2 χ 2 → χ 1 χ 1 freeze out at same epochs. Similarly, the interaction χ 2 e → χ 1 e also freezes out around the same epoch as other processes.
For a quantitative comparison, we estimate the cross sections of different processes relevant for both fermion and scalar DM masses below 100 MeV. For fermion DM, they are given by For scalar DM they are given by Left (right) panel shows rates of different processes for fermion (scalar) DM.

D. Constraints from indirect detection
In addition to the relevant constraints on such sub-  [76,77], one can constrain the DM parameter space from GeV to tens of TeV mass regime. As these studies give constraints on DM masses from 1 GeV to tens of TeV only, we check for a benchmark DM mass of 1 GeV, the constraints imposed by these limits on our model parameter space. Note that the indirect detection limits on DM annihilation rates to specific final states are imposed assuming the final state to be 100%. In figure   15,   [79]. However, it is also worth noting that the existing bounds for diffuse X-ray photons at energies below 4 keV are rather weak, keeping our scenario safe [79]. While in most of the models discussed here, the dominant electromagnetic mass diagram in figure 2. Using the analysis of [80], one can make a simple estimate of the corresponding decay width as where M χ + is the mass of charged component of the additional scalar doublet introduced in the radiative neutrino mass model. While this two body decay width goes as third power of mass splitting, compared to the three body decay width which depends upon fifth power of mass splitting as discussed before and hence can be dominant, we can choose the Yukawa coupling Y ν , M χ + appropriately to satisfy the lifetime bounds. Even though the experimental bounds on such DM decay into photons with energy 2.5 keV are rather weak, we can show that it is possible to have lifetime several order of magnitudes larger compared to the age of the universe [79], for this two body decay involving a photon in final state. The corresponding parameter space consistent with age of universe bound as well as a conservative lower bound on lifetime τ > 10 26 s is shown in figure 16. While the Yukawa coupling Y ν also appears in neutrino mass generation, we can satisfy the limits from  Other cosmology bounds on such scenario may arise due to late decay of Z into SM leptons. For example, if Z decays after neutrino decoupling temperature T ν dec ∼ O(MeV), it will increase the effective relativistic degrees of freedom which is tightly constrained by Planck 2018 data as N eff = 2.99 +0. 34 −0.33 [66]. As pointed out by the authors of [81], such constraints can be satisfied if M Z O(10 MeV), already satisfied by our scenario. Similar bound also exists for thermal DM masses in this regime which can annihilate into leptons till late epochs. Such constraints from the BBN as well as the CMB measurements can be satisfied if M DM O(1 MeV) [82] which is also satisfied in our models. only plays a crucial role in generating light neutrino masses but also splits the DM field into two quasi-degenerate components. In the limit of vanishing mass splitting between the two DM components, light neutrino mass also tends to zero thereby making the inelastic nature of DM a primary requirement from neutrino mass constraints. If the mass splitting is sufficiently small, say of the order of keV or below, both the DM components can be present in the universe as the lifetime of heavier DM can exceed the age of the universe for suitable choice of parameters. Interestingly, such keV scale mass splitting of sub-GeV DM can give rise to the electron recoil excess recently reported by the XENON1T collaboration.
From minimality point of view, we choose the Abelian gauge symmetry to be dark so that none of the SM particles are charged under it. DM particles can annihilate into SM particles through kinetic mixing of U (1) X and U (1) Y . We constrain the parameter space of the model from the requirement of correct relic abundance as well as reproducing the XENON1T excess.
We also impose bounds on direct detection cross section for such sub-GeV DM. We find that for sub-GeV DM with keV mass splitting the constraints coming from lifetime criteria on