Semi-annihilating $Z_3$ Dark Matter for XENON1T Excess

The recently reported result from XENON1T experiment indicates that there is an excess with $3.5\sigma$ significance in the electronic recoil events. Interpreted as new physics, new sources are needed to account for the electronic scattering. Here, we show that a dark fermion $\psi$ from semi-annihilation of $Z_3$ dark matter $X$ and subsequent decay may be responsible for the excess. The relevant semi-annihilation process is $X+X\rightarrow \overline{X}+V_\mu (\rightarrow \psi + \overline{\psi})$, in which the final $\psi$ has a box-shape energy spectrum. The fast-moving $\psi$ can scatter with electron through an additional gauge boson that mixes with photon kinetically. The shape of the signals in this model can be well consistent with the observed excess.


I. INTRODUCTION
Recently, XENON1T collaboration has reported an excess with 3.5σ significance in the electronic recoil events with an exposure of 0.65 tonne-year [1] data. The excess is observed around energy range 2 − 3 keV, which could be the relevant range for solar axion search 1 .
However, the interpretation of vanilla solar axion is in strong conflict with other astrophysical bounds on stellar cooling [4][5][6][7]. Alternative explanations are then needed for such an excess.
In this study, we present an explanation of the excess in the framework of semiannihilating dark matter with Z 3 symmetry. The model was originally proposed and investigated in different contexts [44,45] by the present authors. Here, we show that in different parameter space, the semi-annihilation of dark matter can provide an unstable gauge boson that decays into a pair of dark fermion. The resulting boosted dark fermions can then interact with the electrons through dark photon with kinetic mixing and induce the electronic recoil signals. The event spectrum can be well consistent with the XENON1T observation. We also put an upper bound on the semi-annihilation cross section of dark matter and lower bound on the the electron-scattering cross section. This paper is organized as follows. In Sec. II we present the model setup by introducing the particle contents and Lagrangian. Then in Sec. III we discuss the detailed kinematics that would be relevant for later investigations. Later, we give both analytic estimation and numerical illustration how the signal in this model can fit the XENON1T excess in Sec. IV and present constraints on the relevant cross sections in Sec V. Finally, we summarize our paper.

II. THE MODEL
The Lagrangian we are considering is the following one with 2 complex scalars X and Φ, a fermion ψ, and two dark U (1) D × U (1) D gauge groups with two dark gauge bosons, V µ for The charges assigments of these fields are listed in with g and f are the gauge couplings for U (1) D and U (1) D , respectively. Here we have assigned the U (1) D charges of X and Φ to be 1 and 3, respectively. They are neutral under U (1) D however. The fermion ψ may have a different U (1) D change Q ψ , but should also be charged under U (1) D . The U (1) D gauge field A µ has the kinetic mixing with ordinary photon A µ , and will induce ψ-electron scattering for XENON1T excess. The mass of A can originate the usual Higgs mechanism or Stückelberg trick, which does not affect our discussions in this paper. A may be connected to other hidden sector, which would relax its experimental constraints. Different implementations of Z 3 symmetry in other contexts have been investigated in [46][47][48][49][50][51][52][53][54][55][56][57].
After the spontaneous symmetry breaking of U (1) D , the scalar Φ has a non-zero vacuum 2, and the gauge field V µ gets its mass. In the scalar potential we also have the cubic term (X 3 + X †3 ) that preserves the discrete Z 3 symmetry, X → exp(i2nπ/3)X, which makes X a stable dark matter candidate but have the semiannihilating process with emission of V µ or (dark) Higgs (see Refs. [44,45] for more detail).
The masses and interaction terms of scalar fields are included collectively in the potential where H is the Higgs doublet in standard model. Interaction with H can make the dark sector in contact with thermal bath, which would be needed for thermal production of X. For non-thermal production, we do not have to specify the form and strength in V (X, Φ, H).

III. KINEMATICS
The relevant semi-annihilation process is shown in Fig. 1, where the dark matter X with Z 3 symmetry semi-annihilates into its antiparticle and a dark photon V µ which decays into dark ψ pair subsequently. Choosing the mass differences properly, the resulting ψ with velocity v ψ ∼ 0.1c is the boosted dark fermion that interacts with electron by an additional gauge interaction A µ that mixes photon through kinetic term.
In principle, there is another annihilation process X + X → V * µ → ψ + ψ where we have the corresponding energy and velocity, Here and after, m i , i = X, V, ψ are the masses for particles, X, V µ , ψ, respectively. Simple estimation implies that we have v ψ ∼ 0.1c for m ψ 0.995m X , and v ψ ∼ 0.87c for m ψ 0.5m X . However, this process is velocity suppressed at the present time when dark matter in the Milky way is moving with velocity v ∼ 10 −3 c, and the cross section would be ∼ 10 −6 smaller than that in semi-annihilation. Therefore, we shall not consider this channel in our later discussions.
There are also contributions from annihilation X + X → V µ + V µ , whose cross section depends on the gauge coupling ∼ g 4 , in comparison with that for the semi-annihilation ∼ g 2 λ 2 X . For simplicity, we shall focus on the parameter region λ X g such that the semi-annihilation always dominates in our later discussions. Note that the opposite limit λ X g would make X + X → V µ + V µ dominant, which also works with the boosted V µ that decays, although with a different kinematics 2 .
In the semi-annihilation we have the relations for the energies of final states, The The final ψ particles have an energy distribution with box shape, where θ(E − , E + ) = 1 for E − < E < E + and zero otherwise, It can be also translated into velocity distribution f ψ , where v ψ = 1 − m 2 ψ /E 2 ψ . The energy interval of the distribution is E V β V β * ψ , which depends on the three masses, and the relative half-width is δ = β V β * ψ . For small mass differences, m X − m V m V and/or m V /2 − m ψ m ψ , the spectrum will be very narrow around E V /2.

IV. EVENT RATE
The fast-moving ψ can scatter with electron through A µ interaction and induce prompt scintillation events (S1) at XENON1T experiment. The recoil energy of electron is E R 2m e v 2 ψ for m ψ m e . Since the events of excess are centered around E R ∼ 2.5keV, we would need v ψ 0.05c. The differential rate of such events can be estimated as where n T ∼ 4.6 × 10 27 /ton, Φ ψ is the flux of ψ and σ e is the scattering cross section between ψ and electron. To explain the excess, we would need dR/dE ∼ 30/(ton yr keV), which gives Φ ψ σ e 2.4 × 10 −35 /(s keV).
The · denotes that we shall take into account the smearing effect due to energy resolution and efficiency of the experiments [1]. The energy resolution [64] is parametrized by Gaussian distribution with uncertainty σ, For the reconstructed energy at E 2.7keV, the relative resolution is about 19.45%. All of these effects are included in our later numerical illustrations.
The flux Φ ψ from annihilation of dark matter of our galaxy is given by where the ρ X is the energy density of X with an NFW profile, dN /dE is the number distribution of ψ at production and the integration is performed over the light-of-sight s and all-sky solid angle Ω.
The above analytic estimation shows that the ψ-e scattering cross section should be around σ e ∼ 10 −30 cm 2 for GeV-scale dark matter with canonically thermal annihilation cross section. Increasing the mass of dark matter would decrease the flux of ψ and correspondingly requires larger scattering cross section σ e for compensation of the flux loss.
In Fig. (2) we illustrate with a box-shape energy spectrum having relative half-width, excess. The value of δ then can be directly translated into the requirements on the mass differences through the relation δ = β V β * ψ . Together with v ψ 0.05c, we can determine the two mass ratios of m V /m X and m ψ /m V .
To precisely determine the favored parameter region, dedicated statistical analysis would be needed and is beyond the scope of this paper. Instead, we only show several contours for the relative ratios of m V /m X and m ψ /m V in Fig. 3

V. CONSTRAINTS
As we have demonstrated in previous sections, the event rate depends on the product of the dark matter semi-annihilation cross section and ψ-e scattering cross section, σ ann v σ e .
Larger σ ann v implies smaller σ e . There would be some degeneracy in these two values.
However, there is an upper bound on σ ann v from astrophysics even if all the final annihilation products are in dark sector. Because the resulting X from semi-annihilation is also fast moving and may escape from the dark halo in our galaxy, we should require the annihilation rate is smaller than 1 per Hubble time, then we obtain the upper bound, We should note that if σ ann v 3 × 10 −26 cm 3 /s, X can not be produced thermally in the early universe since its relic density would be too small. In such a case, other production mechanism would be needed. For instance, X could be produced from other heavy particles's decay, or was never in thermal equilibrium due to low reheating temperature after inflation.
In order to explain the XENON1T excess, the above upper bound correspondingly gives a lower bound on σ e , σ e 10 −38 cm 2 × m X GeV .
This limit can be easily satisfied for GeV-scale dark matter as we shall show below. Note that the constraints from cosmic-ray scattering with dark matter [65,66] do not apply here because the density of ψ particles in the galactic background is much lower than dark matter X. If the Higgs-portal and V µ kinetic portal couplings are small, then the interaction between X and standard model particles would also be suppressed, therefore the scattering between X and cosmic rays is rare and the constraints can be relaxed, but potentially detectable in future.
The above requirement of σ e can be satisfied with a viable kinetic mixing parameter Here α is the fine-structure constant α = 1/137 and α = f 2 /4π is the constant for A . Such Eq. (13) still holds even if we have ∼ 10 −7 , a value well below the current limit.

VI. SUMMARY
In this paper, we have presented an explanation of the recently observed excess in electronic recoil events at XENON1T experiment with a dark matter model with local Z 3 discrete symmetry. The semi-annihilation of dark matter X can provide an unstable gauge boson V µ which subsequently decays into dark fermion pair ψ + ψ. The energy spectrum of dark fermion has a box shape, with the width depending on the relative mass differences. Because of the semi-annihilation and decay, X + X → X + V µ (→ ψ + ψ), the final dark fermion can be fast-moving with velocity around 0.05c. Then it scatters with an electron through the dark photon that has kinetic mixing with ordinary photon field. We have shown the resulting spectrum shape can be well consistent with the observed data by XENON1T with viable parameter values.