TeV dark matter and the DAMPE electron excess

The recent high energy electron and positron flux observed by the DAMPE experiment indicates possible excess events near 1.4 TeV. Such an excess may be evidence of dark matter annihilations or decays in a dark matter subhalo that is located close to the solar system. We give here an analysis of this excess from annihilations of Dirac fermion dark matter which is charged under a new $U(1)_X$ gauge symmetry. The interactions between dark matter and the standard model particles are mediated the $U(1)_X$ gauge boson. We show that dark matter annihilations from a local subhalo can explain the excess with the canonical thermal annihilation cross section. We further discuss the constraints from the relic density, from the dark matter direct detection, from the dark matter indirect detection, from the cosmic microwave background, and from the particle colliders.


INTRODUCTION
Recently, evidence for excess electron and positron events near 1.4 TeV has been reported by the DAMPE experiment [1]. Such an excess was not found previously by the AMS-02 [2][3][4] and by the Fermi-LAT [5]. In the recent paper by the CALET experiment [6], the electron and positron events in the two energy bins near 1 TeV also appear higher than expected. Due to the better energy resolution of the DAMPE experiment (∼ 1% for 1 TeV electrons and positrons) than AMS-02, Fermi-LAT, and CALET, the TeV electron and positron flux can be measured with better accuracy by the DAMPE experiment than measured previously by other experiments. In the DAMPE data, the electron and positron excess events occur only in one energy bin near 1.4 TeV. The localized feature in the energy spectrum of the excess events hints a nearby source of the high energy electrons and positrons. Inspired by this excess, studies with dark matter (DM) explanation [7][8][9] and with astrophysical explanation [10,11] have been carried out. In this paper, we study the possibility of attributing the excess of the TeV electrons and positrons to the DM annihilations in the vicinity of the solar system.

THE MODEL
We consider a U (1) X extension of the standard model (SM) with X µ as the new gauge boson, and χ as the Dirac DM particles which is charged under the U (1) X gauge symmetry. The new Lagrangian terms are where X µ is the new gauge boson, X µν is the field strength, M X is the X boson mass. The X boson is assumed to couple to the DM fermions and to the Dirac DM in the vector current form, J µ = g ff γ µ f + g χχ γ µ χ.
In this paper, we consider three scenarios: (1) DM annihilates into e + e − only via an s-channel exchange of the X µ boson (assuming only g e = 0); (2) DM annihilates into all SM final states universally via the s-channel X µ boson; (3) DM annihilates into a pair of on-shell X bosons which subsequently decay into SM fermions.
The annihilation cross section for the χχ → X →f f process is given by where N f = 1(3) for leptons (quarks), Γ X is the total decay width of the X boson, and we have neglected the mass of the final state SM particle. The partial decay with of the X boson is Γ(X →f f ) = N f g 2 f M X /(12π). If the DM annihilation occurs away from the X resonance and the X boson has a narrow decay width, i.e. Γ X M X , one has σv N f c g 2 f g 2 χ /(m 2 χ π(x 2 − 4) 2 ) where x ≡ M X /m χ . For the case in which the X boson only couples to electrons, in order to obtain the desired 1 pb annihilation cross section, one has |g e g χ | 0.37 |x 2 − 4| where x = M X /(1.5 TeV), if the DM annihilation occurs away from the X resonance.

COSMIC RAY PROPAGATION
The propagation of the electrons and positrons can be described by the following diffusion equation GeV/s and D(E) = D 0 (E/GeV) δ . The diffusion coefficient D(E) depends on the height 2L in the z direction of the cylindrical diffusion zone which is usually assumed for the Milky Way galaxy. We adopt the medium case in Ref. [12] such that L = 4 kpc, D 0 = 11 pc 2 /kyr, and δ = 0.7. For the steady-state case, the Green function of arXiv:1711.11579v1 [hep-ph] 30 Nov 2017 the diffusion equation is given by [13][14][15] G(x, E; where λ is the propagation scale which is given by By using the Green function, the general solution to the diffusion equation can be computed via For DM annihilations, the source function of electrons and positrons is given by where ρ χ (x) is the DM mass density, m χ is the DM mass, σv is the velocity-averaged annihilation cross section, where v is the electron velocity; the unit of the flux is 1/ (GeV · cm 2 · s · sr) [12].

ELECTRON FLUX FROM A LOCAL SUBHALO
We consider an ultra-faint DM subhalo which is located 1 kpc away from us. We assume a NFW density profile [16] for the subhalo The distance between the subhalo and us is denoted as d s which is taken to be in the range (0.1-1) kpc. In the following we consider two different sets of parameters (γ, ρ s , r s , d s ) = (1, 1, 1, 1) (denoted as SHA) and (0.5, 100, 0.1, 0.3) (denoted as SHB) where ρ s is in GeV/cm 3 , and r s and d s are in kpc [17][18][19].
If two DM particles with 1.5 TeV mass annihilate in the subhalo via χχ → e + e − only with σv = 3 × 10 −26 cm 3 /s, the electron flux at E = 1.4 TeV is E 3 Ψ χ = 0.2(47) GeV 2 /(m 2 · s · sr) for the SHA (SHB) case. Thus, in order to produce the right amount of electrons and positrons observed by the DAMPE experiment, we will adopt the SHB assumption throughout the paper.

COSMIC RAY BACKGROUND
To understand the cosmic ray (CR) background is essential for astrophysical observations. To know the background predictions precisely in the measurement of the satellite experiments is the major challenge of the current DM indirect search experiments. Usually the CR background can be modelled via the broken power-law forms. We adopt the parameterization formulas as in Ref. [20] where the background electrons and positrons consists of three components: the primary electrons from the CR sources, φ primary , the secondary positrons/electrons originating from interactions between the primary CR and the interstellar medium, φ secondary , and the extra source, for examples pulsars or DM, φ source . For the primary electrons, the flux is parameterized as φ primary = CE −α /(1 + (E/E b ) β ); the secondary positron flux takes the same formula as the primary electron but with different coefficients. The extra sources contains an exponential cut-off scale E c , which takes the form φ source = CE −γ exp(−E/E c ). The total flux for positrons is given by Φ e + = φ secondary + φ source , and the total flux for electrons is given by Φ e − = φ primary + 0.6 φ secondary + φ source [20]. In our analysis, we used the electron plus positron flux measured in DAMPE to fit the various coefficients. The best fit model is given

DAMPE DATA FITTING
We use Ψ B + Ψ χ to fit the DAMPE data, where Ψ B is the CR background, and Ψ χ is the DM contribution from both the Milky Way (MW) halo and the nearby subhalo. We carry out a χ 2 analysis where Φ th i = Ψ B + Ψ χ is the predicted spectrum of electron plus positron, Φ exp i is the experiment data observed by the DAMPE experiment, and δ i is the uncertainties reported by the DAMPE experiment. For the DM signal, we use PPPC4DM ID [21,22] to generate the energy spectrum for the source function. Fig. (1) shows the DMAPE data and the various contributions to the electron flux. Here we analyze the χχ → X → e − e + annihilation channel only, and a delta function energy spectrum dN/dE = 2δ(E − m χ ) is employed for the injection source. The DM annihilation cross section is taken to be σv = 3 × 10 −26 cm 3 /s and mass m χ = 1.5 TeV. As shown in Fig. (1), the DAMPE excess events are well fitted by the hard spectrum from a local DM subhalo if DM pair-annihilates into e − e + . For comparison, we also calculate the electron and positron flux from DM annihilations in the MW halo, which, however, is two order of magnitude smaller and exhibits a rather flat spectrum over a much larger energy range.
In Fig. (2), we overlay signals from different DM models with the 14 high energy bins in the DAMPE data. Instead of using the global fitting background used in Fig. (1), a simple power law (PL) background CE −γ is employed for the TeV electrons. The best fit PL has C = 2.4 × 10 4 GeV 2 m −2 s −1 sr −1 and γ = 0.78 and the minimum χ 2 = 20.4. We consider three different annihilation modes: (A) χχ → X → e + e − , (B) χχ → XX → 2e + 2e − , (C) χχ → X →f f . In the case A and the case B, the X boson couples only electrons; in the case C, the X boson couples to all SM fermions with universal couplings.
For different DM annihilation channels, we employ Φ B + F Φ 0 χ to fit the 14 high energy data points, where Φ B = CE −γ , Φ 0 χ is the flux corresponding to the canonical thermal cross section σv = 3 × 10 −26 cm 3 /s, F is a overall floating parameter. The DM mass is fixed at 1.5 TeV for case A and C, 3 TeV for case B. For case A, we find C = 4.7 × 10 4 GeV 2 m −2 s −1 sr −1 , γ = 0.89 and F = 1.03, with the minimum χ 2 = 9.0 which is improved by ∆χ 2 = 11.4 from the background-only PL fitting. In the case B, χχ → XX → 2e + 2e − , the energy spectrum of electron and positron is a box-shaped distribution [23][24][25][26][27][28] which depends on the mass gap ∆m = m χ − M X which should be small to explain the sharp energy spectrum in the DAMPE data. We take ∆m = 2 GeV and F = 0.4 which has χ 2 = 15.3 (∆χ 2 = 5.1). In the case C, χχ → X →f f , we used PPPC4DMID to compute the energy spectrum. In our model, the branching ratio BR = 1/24(1/8) for each lepton (quark) final state. We find that χ 2 is 9.4 ( ∆χ 2 = 11) for F = 24. The flux in the case C overlaps with case A, indicating that the electron channel is the dominant mode.

COLLIDER CONSTRAINTS
If the X boson couples to both quarks and leptons, one can also search for the resonance in the dilepton signal at the LHC. ATLAS collaboration also used the fourfermion contact interaction Lagrangian to interpret the most recent LHC results on dilepton signals [29] where i, j = L, R for different chiral interactions, and η ij = ± denotes constructive and destructive interferences with the DM processes. In our model, Λ = √ 4πM X / |g q g |. The most stringent 95% C.L. lower limit on Λ analyzed using the recent ATLAS data [29] comes from the final state for the LL chiral interaction in the constructive case, Λ > 40 TeV, which gives rise to a constraint |g q g | < 0.09(M X /TeV).
The LEPII experiment also constrain the new physics models via the contact interaction operators, in which the constraint is expressed in terms of lower bound on the new physics scale Λ. The LEPII group finds that Λ V V > 15.9 TeV in the e − e + → e − e + process at 95 % C.L. [30] [31]. The theoretical value of Λ V V predicted in our model is Λ V V = √ πM X /g e ; thus the constraints on the electron coupling is given g e 0.11(M X /TeV).

DM DIRECT DETECTION CONSTRAINTS
PandaX experiment [32] provides the best constraints to DM-proton spin-independent (SI) cross section for the 1.5 TeV DM, σ SI χp 1.7 × 10 −45 cm 2 . The theoretical prediction of the SI DM-proton cross section in our model is σ χp = µ 2 χp g 2 χ g 2 p /(πM 4 X ) where g p = 2g u +g d . The above SI cross section upper limit provides a constraint on the coupling |g χ g p | 6 × 10 −2 (M X /TeV).
If the X boson only couples to the SM electrons, we need consider the DM direct detection limits due to electron recoils. Ref. [33] analyzed the Xenon10 and Xenon100 results to constrain the interaction cross section between DM and electron; for the 1.5 TeV DM, the upper bound on the cross section between DM and the free electron is σ χe < 3 × 10 −38 cm 2 . The theoretical prediction of the DM-electron cross section in our model is σ χe = µ 2 χe g 2 χ g 2 e /(πM 4 X ) [33]. Thus the upper bound on the couplings is |g χ g e | 1.8 × 10 2 (M X /TeV).
To interpret the direct limit, we only assume the DM contribution from the MW halo. Although the DM density from the subhalo can be significant, its contribution to the DM direct detection signal is offset by the smaller velocity dispersion in the subhalo and the lack of the relative motion with respect to Earth, since the subhalo is also assumed to rotate around the Galactic Center.

DM INDIRECT DETECTION CONSTRAINTS
For the 1.5 TeV DM annihilating into two SM particles, H.E.S.S. data [34] constrain the annihilation cross sections: σv < 2(6)×10 −26 cm 3 /s for the τ + τ − (W + W − ) channel, which is based on 10 years of the inner 300 pc of the Galactic Center region assuming the Einasto profile. This provides a very strong constraint on the case where the X boson couple universally to all SM fermions.
For the pair-annihilation case, χχ → XX, the most stringent constraint comes from the H.E.S.S. data [35] [34], which is stronger than the limits from the Fermi-LAT data in the direction of the dwarf spheroidal galaxies [36] and from the Planck CMB data which is sensitive to energy injection to the CMB from DM annihilations [37] [38]. The H.E.S.S. limit in the χχ → 4e channel is σv < 6(20) × 10 −25 cm 3 /s in the m χ ∼ M X (m χ M X ) case, for the 3 TeV DM. The required DM annihilation cross section for the DMAPE excess in the 4e case is consistent with the above H.E.S.S. constraint.
The IceCube experiment [39] sets an upper bound on the annihilation cross section for various SM channels by analyzing the neutrino signal σv < O(1)×10 −22 cm 3 /s, which is much larger than the annihilation cross section needed for the DAMPE excess.

PARAMETER SPACE
For the electrophilic case in which the X boson only couples to the electrons, to satisfy the LEPII and direct detection constraints, we select a benchmark model point as follows (δm, g e , g χ ) = (100 GeV, 0.1, 0.4) where δm ≡ 2m χ − M X . The DM annihilation for this benchmark model is σv 1.3 pb which can explain the DMAPE excess and relic density.  For the universal case in which the X boson couples to all SM fermions with equal coupling strength, we select a benchmark model point as follows (δm, g f , g χ ) = (10 GeV, 4×10 −3 , 1). The DM annihilation at the halo is σv 30 pb; the DM annihilation at v = 1/4 (the typical temperature for DM thermal freeze-out) is σv 0.3 pb which is smaller than 1 pb. However, since the annihilation cross section depends on the velocity of the DM, one has to take into account the thermal average at the freezeout. Such an enhancement of the halo annihilation was studied previously in the context of PAMELA positron excess [40][41][42]. Fig. (3) shows the relic density line and the ATLAS and PandaX constraints for the universal X case near the resonance region. The benchmark model point is consistent with all constraints as shown in Fig.  (3). The universal case is in tension with the H.E.S.S. constraint for the τ + τ − channel with an Einasto profile. However, for a different DM profile, the tension with the H.E.S.S. constraint could be alleviated.
For the case where DM annihilates into a pair of onshell X bosons, since the coupling between the X boson and the SM particles can be significantly small, the only relevant constraint comes from the indirect detection limits. In our case, the most stringent constraint comes from the H.E.S.S. data, which, however, is one order of magnitude higher than the thermal annihilation cross section. In order to produce a narrow energy spectrum for the injection source function in this case, the mass difference between the DM and the X boson has to be very small, which compress the phase space of the DM annihilation so that σv(χχ → XX) is very small for perturbative g χ coupling. Thus one has to be in the non-perturbative region of the parameter space to generate a large annihilation cross section and a narrow energy spectrum for this scenario.

CONCLUSIONS
We have proposed a simple dark matter model to explain the high energy electron excess events recently observed by the DAMPE experiment. The morphology of the energy spectrum of the excess events hints a local source for the high energy electrons. We investigated the possibility of the DM annihilations in a local subhalo which is 0.3 kpc away from us to generate such an excess.
Three scenarios in the model were investigated. The case where DM only annihilates into e + e − provides a good fitting to the excess while satisfying the various constraints. In the case where the X boson couples universally with all SM fermions, DM has to annihilate near the X boson resonance to generate a much larger annihilation cross section to explain the excess. In the case where DM annihilates into on-shell X bosons, in order to produce the sharp excess in the energy spectrum, the mass gap between the DM and the X boson has to be in the GeV range.