XENON1T excess in local $Z_2$ DM models with light dark sector

Recently XENON1T Collaboration announced that they observed some excess in the electron recoil energy around a 2-3 keV. We show that this excess can be interpreted as exothermic scattering of excited dark matter (XDM), $XDM + e_{atomic} \rightarrow DM + e_{free}$ on atomic electron through dark photon exchange. We consider DM models with local dark $U(1)$ gauge symmetry that is spontaneously broken into its $Z_2$ subgroup by Krauss-Wilczek mechanism. In order to explain the XENON1T excess with the correct DM thermal relic density within freeze-out scenario, all the particles in the dark sector should be light enough, namely $\sim O(100)$ MeV for scalar DM and $\sim O(1-10)$ MeV for fermion DM cases. And even lighter dark Higgs $\phi$ plays an important role in the DM relic density calculation: $X X^\dagger \rightarrow Z' \phi$ for scalar DM ($X$) and $\chi \bar{\chi} \rightarrow \phi \phi$for fermion DM ($\chi$) assuming $m_{Z'}>m_\chi$. Both of them are in the $p$-wave annihilation, and one can easily evade stringent bounds from Planck data on CMB on the $s$-wave annihilations, assuming other dangerous $s$-wave annihilations are kinematically forbidden.


I. INTRODUCTION
Recently XENON 1T Collaboration reported that they found electron recoil excess around 2-3 keV with 3.5 σ significance analyzing data for an exposer of 0.65 ton-year [1]. There is an issue on the tritium contamination to be resolved. This energy region is sensitive to solar axion search, but the interpretation of this excess in terms of solar axion is in conflict with astrophysical bounds on the axion coupling to electron. XENON1T Collaboration also interprets the excess in the context of magnetic moment of solar neutrino and absorption of light bosonic dark matter [1]. After the accouncement of XENON1T Collaboration, there appeared a number of papers that address various issues related with this excess .
In this paper, we interpret this electron recoil excess in terms of exothermic DM scattering on atomic electron bound to Xe in the inelastic DM models. We shall consider both complex scalar [51] and Dirac fermion DM models [52] with local U (1) dark gauge symmetry which is spontaneously broken into its Z 2 subgroup by Krauss-Wilczek mechanism [53]. In this framework, the mass difference (δ) between the DM and the excited DM (XDM) is generated by dark Higgs mechanism, and there is no explicit violation of local gauge symmetry related with the presence of dark photon. On the other hand, in a number of literature, the mass difference δ is often introduced by hand in terms of dim-2 (3) operators for scalar (fermion) DM. Then local gauge symmetry is broken explicitly and softly. Introducing dark gauge boson (or dark photon) in such a case would be theoretically inconsistent, since the current dark gauge fields couple is not a conserved current. There will appear some channels where high energy behavior of the scattering amplitudes violate perturbative unitarity, in a similar way with the W L W L → W L W L scattering violates unitarity if W boson mass is put in by hand. One of the present authors pointed out this issue in fermionic DM model (see Appendix A of Ref. [52]).
Local Z 2 scalar [51] and fermion DM models [52]  to consider the DM annihilation should be mainly in p-wave, and not in s-wave, in order to avoid strong constraints from CMB (see [54,55] and references therein).
For this purpose it is crucial to have dark Higgs (φ), since they can play a key roles in the p-wave annihilations of DM at freeze-out epoch: where X and χ are complex scalar and Dirac fermion DM, respectively. At freeze-out epoch, the mass gap is too small (∆m T ) and we can consider DM as complex scalar or Dirac fermion. In the present Universe, we have T ∆m and so we have to work in the two component DM picture for XENON1T electron recoil. It can not be emphasized enough that these channels would not be possible without dark Higgs φ, and it would be difficult to make the DM pair annihilation be dominated by the p-wave annihilation.

II. MODELS FOR (EXCITED) DM
A. Scalar DM model The dark sector has a gauged U (1) X symmetry. There are two scalar particles in the dark sector X and φ with U (1) X charges 1 and 2, respectively. They are neutral under the SM gauge group. After φ gets VEV, whereX µν (B µν ) is the field strength tensors of U (1) X (U (1) Y ) gauge boson in the interaction basis.
We decompose the X as and H and φ as in the unitary gauge.
The dark photon mass is given by where we neglected the corrections from the kinetic mixing, which is second order in parameter. The masses of X R and X I are obtained to be and the mass difference, δ ≡ m R − m I µv φ / √ 2m X . Since the original U (1) X symmetry is restored by taking µ = 0, small µ does not give rise to fine-tuning problem. The mass spectrum of the scalar Higgs sector can be calculated by diagonalising the mass-squared which is obtained in the (h H , h φ ) basis. We denote the mixing angle to be α H and the mass eigenstates to be (H 1 , H 2 ), where H 1 is the SM Higgs-like state and H 2 (≡ φ) is mostly dark Higgs boson. Since we work in the small α H in this paper, the VEV of φ is approximated The mass eigenstates Z µ and Z µ of the neutral gauge bosons can be obtained using the procedure shown in Ref. [56]. In the linear order approximation in we can write the covariant derivative as where Q em (Q X ) is the electric (U (1) X ) charge and A µ is the photon field. We note that Z couples to the electric charge but not to the weak isospin component T 3 . For example, Z does not couple to neutrinos at this order of .
To evade the bound from the DM scattering off the nuclei we are considering sub GeV scale DM. To calculate the relic abundance of the DM we take the X as the physical state with mass m X instead of X I and X R , i.e. m X m R m I , because the mass difference δ is much smaller than the freezeout temperature T f ∼ m X /10 [9]. For this light DM the CMB constraint rules out the s-wave annihilation of the DM pair. So the contribution to the DM annihilation should start from p-wave. We suppress the XX † → Z Z by choosing m Z > m X . We also make XX † → H 2 H 2 subdominant. To achieve this we suppress the direct coupling of the DM to φ and H by taking small λ φX and λ HX . We also take small λ φH to evade the bound from the Higgs invisible decay. However, the coupling λ φH should not be too small to make the DM in thermal contact with the SM plasma. Too small λ φH also makes the H 2 lifetime too long, causing cosmological problems. For example, H 2 with mass ∼ 1 GeV decays dominantly into muon (or strange quark) pairs through mixing λ φH , whose decay with is given by We require that H 2 lives shorter than 1 sec to evade the constraints from Big Bang nucleosynthesis (BBN) and the mixing angle is small. For m H 2 = 1 GeV it is translated into α H > 9.5 × 10 −9 . When the muon channel is kinematically forbidden but m H 2 > 2m e , it decays into electron-positron pair. The decay rate is obtained by replacing m µ by m e in (8).
For example, for m H 2 = 10 MeV, we need α H > 2.0 × 10 −5 . When m H 2 < 2m e , the scalar particle decays into two photons. In this paper we consider m H 2 2m e . The small mixing parameters are also technically natural by extending the Poincaré symmetry [57,58].
The Z can also decay into charged SM particles through mixing . The decay width for Z → e + e − is given by Its lifetime is much shorter than 1 sec in the parameter space of our interest.
The light φ and/or Z may contribute to the effective neutrino number N eff , which is another possible constraint in the model. Since the mediators Z and φ decay before 1 sec, there is no relativistic extra degree freedom which mimics the neutrino at the recombination era (T CMB ≈ 4 eV). Another potential source for ∆N eff is light mediator with mass below 1 MeV. It mainly decays into e ± or γ not into ν, making the difference between the temperatures T γ and T ν larger than the one given by the standard cosmology by imparting its entropy only to γ [59]. This also causes ∆N eff = 0. We evade this problem by taking their masses greater than 1 MeV.
In this restricted region of parameter space the main channel for the relic density is Fig. 1 ) [67]. The leading contribution to cross section is p-wave with the cross section where m φ ≡ 2λ φ v φ m H 2 . The resulting dark matter density is obtained by [60] Ω where a and b are defined by σv = a + bv 2 , and the additional factor 2 comes from the fact that X is a complex scalar instead of real scalar. For example, with m X = 1 GeV, m Z = 1.2 GeV, m φ = 0.2 GeV, λ φ = 4.5 × 10 −5 , g * ≈ 10, and x f ≈ 10, we can explain the current DM relic abundance: Ω X h 2 ≈ 0.12. Other p-wave contributions include XX † → Z * → ff where f is an SM fermion. But these contributions are suppressed by 2 compared to the above annihilation, and we neglect them. The SM Z boson contributions are further suppressed by both small mixing angle 2 and small mass ratio m 4 Z /m 4 Z . To calculate the inelastic down-scattering cross section for the XENON1T anomaly, instead of X and X † we now consider two real scalars X R , X I with mass difference δ. With the kinetic mixing term given in (1) we get the dark-gauge interactions with the DM and the electron [56] where c W is the cosine of the Weinberg angle, Z and Z are mass eigenstates, and we assumed that (∼ 10 −4 ) is small. The cross section for the inelastic scattering X R e → X I e for m X m e and small momentum transfer is given by where α em 1/137 is the fine structure constant and α X ≡ g 2 X /4π. This can be used to predict the differential cross section of the dark matter scattering off the xenon atom for the DM velocity v, which reads where E R is the recoil energy, q is the momentum transfer, K(E R , q) is the atomic excitation factor. From energy conservation we obtain the relation [9], where θ is the angle between the incoming X R and the momentum transfer q = p e − p e .
The integration limits are [9], Then we can obtain the differential event rate for the inelastic scattering of DM with electrons in the xenon atoms given by where n T ≈ 4 × 10 27 /ton is the number density of xenon atoms and n R ≈ 0.15 GeV/m R /cm 3 is the number density of the heavier DM component X R , assuming n R = n I . Integrating over E R , we get the event rate Since X R is a dark matter component in our model with the same abundance with X I , its lifetime should be much longer than the age of the universe. It can decay via X R → X I γγγ as shown in [9]. Its decay into three-body final state, X R → X I νν, is also possible in our model. The relevant interactions are The decay width is given by Although this channel is much more effective than X R → X I γγγ considered in [9], the lifetime of X R is still much longer than the age of the universe.
In the right panel of Fig. 1  on the right allowed region is from the kinematic boundary, m Z +m φ < 2m R . It is nontrivial that we could explain the XENON1T excess with inelastic DM models with spontaneously broken U (1) X → Z 2 gauge symmetry. In particular it is important to include light dark Higgs for this explanation. It would be straightfoward to scan over all the parameters to get the whole allowed region.

B. Fermion DM model
We start from a dark U (1) model, with a Dirac fermion dark matter (DM) χ appointed with a nonzero dark U (1) charge Q χ and dark photon. We also introduce a complex dark Higgs field φ, which takes a nonzero vacuum expectation value, generating nonzero mass for the dark photon. We shall consider a special case where φ breaks the dark U (1) symmetry into a dark Z 2 symmetry with a judicious choice of its dark charge Q φ .
where g X is the dark coupling constant, and Q X denotes the dark charge of φ and χ: Q φ = 2, Q χ = 1, respectively. Then U (1) X dark gauge symmetry is spontaneously broken into its Z 2 subgroup, and the Dirac DM χ is split into two Majorana DM χ R and χ I defined as with We assume y > 0 so that δ ≡ m R − m I = 2yv φ > 0. Then the above Lagrangian is written as where h φ is neutral CP-even component of φ as defined in (3).
When we calculate the DM relic density, we can assume the mass difference is small compared to the DM mass as in the case of the scalar DM, i.e. m χ m R m I . For the fermionic DM the annihilation processes into the scalar pair, To evade the CMB constraint we suppress the s-wave annihilation by assuming 2m χ < 2m Z , m Z + m φ . The calculation of the annihilation process (the top panel of Fig. 2) yields σv = where The current DM relic abundance is obtained by (11).
It turns out that in the limit m χ m e , σ e has exactly the same form with (13) of the scalar DM case.
We require the χ R to be long-lived so that it is also a main component of the dark matter.
It decays mainly via the SM Z-mediating χ R → χ I νν, using the interactions The expression for the decay with, Γ(χ R → χ I νν), also agrees exactly with (20). As shown in (20), the lifetime of χ R is much longer than the age of the universe, which guarantees the χ R is as good a dark matter as χ I .
In the bottom panel of Fig. 2  is kinematically allowed, and the experimental constraint is weaker in the we are interested in, compared with the scalar DM case in Fig. 1 (right). We also show the current experimental bounds by NA64 [66].
Note that the kinetic mixing ∼ 10 −7±1 , which is much smaller than the scalar DM case.
We have checked if the gauge coupling g X and the quartic coupling of dark Higgs (λ φ ) remain in the perturbative regime. The solid (dashed) lines denote the region where g X satisfy (violate) perturbativity condition, depending α X < 1 or not. Within this allowed region, λ φ remain perturbative. Again it is nontrivial that we could explain the XENON1T excess with inelastic fermion DM models with spontaneously broken U (1) X → Z 2 gauge symmetry. In particular it is important to include light dark Higgs for this explanation as in the scalar DM case.

III. CONCLUSION
In this paper, we showed that the electron recoil excess reported by XENON1T Collaboration could be accounted for by exothermic DM scattering on atomic electron in Xe, with sub-GeV light DM: m X ∼ O(100) MeV for the scalar and m χ ∼ O(10) MeV for the fermion DM, and dark Higgs φ neing even lighter that DM particle for both cases. Dark photon should be heavier than DM in order that we can forbid the DM pair annihilation into the Z Z channels. This scenario could be described by DM models with dark U (1) gauge symmetry broken into its Z 2 subgroup by Krauss-Wilczek mechanism. And dark photon Z and dark Higgs φ in such dark gauge models play important roles in DM phenomenology.
In particular in the calculation of thermal relic density, new channels involving a dark Higgs can open XX † → φZ and χχ → φφ, which are p-wave annihilations. Then one could evade the stringent constraints from CMB for such light DM. Other dangerous s-channel annihilations can be kinematically forbidden by suitable choice of parameters. Thus the exothermic scattering in inelastic Z 2 DM models within standard freeze-out scenario can explain the XENON1T excess without modifying early universe cosmology. We emphasize again that the existence of dark Higgs φ is crucial for us to get the desired DM phenomenology to explain the XENON1T excess with the correct thermal relic density in case of both scalar and fermion DM models within the standard freeze-out scenario.

Note Added
While we were preparing this manuscript, there appeared a few papers which explain the XENON1T excess in terms of scalar or fermion exothermic DM [9,11,21,29,30,42]. Our paper is different from these previous works in that we consider dark U (1) gauge symmetry broken to its Z 2 subgroup by dark Higgs mechanism, and include the light dark Higgs in the calculations of thermal relic density for the two component DM in the standard freeze-out scenario: XX † → Z φ for scalar DM and χχ → φφ. Other s-wave channels are kinematically forbidden by suitable choice of mass parameters. This possibility of dark Higgs in the final state in the (co)annihilation channels are not included in other works. By including these new channels, we could achieve the correct thermal relic density and the desired heavier DM fluxes on the Xe targets simultaneously, without conflict with strong constraints on light DM annihilations from CMB. And the models considered in this paper is renormalizable and DM stability is guaranteed by underlying U (1) dark gauge symmetry and its unbroken Z 2 subgroup.