The new interaction suggested by the anomalous $^8$Be transition sets a rigorous constraint on the mass range of dark matter

The WIMPs are considered one of the favorable dark matter (DM) candidates, but as the upper bounds on the interactions between DM and standard model (SM) particles obtained by the upgraded facilities of DM direct detections get lower and lower. Researchers turn their attentions to search for less massive DM candidates, i.e. light dark matter of MeV scale. The recently measured anomalous transition in $^8$Be suggests that there exists a vectorial boson which may mediate the interaction between DM and SM particles. Based on this scenario, we combine the relevant cosmological data to constrain the mass range of DM, and have found that there exists a model parameter space where the requirements are satisfied, a range of $10.4 \lesssim m_{\phi} \lesssim $ 16.7 MeV for scalar DM, and $13.6 \lesssim m_{V} \lesssim $ 16.7 MeV for vectorial DM is demanded. Then a possibility of directly detecting such light DM particles via the DM-electron scattering is briefly studied in this framework.


I. INTRODUCTION
For the time being, we still do not have solid knowledge on dark matter (DM). One of the preferable DM candidates is the weakly interacting massive particles (WIMPs), with WIMP masses of GeV-TeV scale. The recent DM direct detection experiments [1][2][3][4][5] set stringent constraints on the cross section of DM-target nucleus scattering for GeV-TeV scale DM, and the upper bound of the detection cross section will be reduced to the neutrino limit in next decade(s). On one aspect, the existence of DM is convinced by the astronomical observation, while on another aspect, the DM particles have not been detected by all the sophisticated experiments. One may ask if our conjecture on the potential mass range of DM is astray, which results in DM evading the present DM direct detections, namely, can the DM particles are much less massive to be in a sub-GeV range, e.g. in MeV (see Refs. [6][7][8][9][10][11][12][13] for some earlier work). In this scenario, the interactions of the light DM particles just render the nucleus small recoil energies, which are not observable in available experiments for DM direct detections. In this work, we focus on the MeV scale light DM.
The issue concerning DM refers two aspects, one is the identities of DM, i.e what is (are) DM, and another aspect is how DM particles interact among themselves and with SM particles. It is generally believed, the interactions related to the DM sector must be a new type (new types) beyond the standard model (BSM). In this work, to answer the first question, we do not priori assume its identity, but let experimental data determine; to the second question, we look for a new BSM interaction which may offer an interpretation for the present observation. The recent 8 Be experiment has revealed at 6.8σ an anomalous transition between an excited state 8 Be * and the ground state 8 Be [14]. The authors [14,15] argued that this anomaly may be due to the unknown nuclear reactions, but a more preferable possibility is that it is caused by emitting a vectorial boson X during 8 Be * → 8 Be + X, which instantly decays into e + e − pair. The new boson X may be the mediator that we look forward to between DM and SM particle interactions, and this probable is investigated in this paper. A fitted value of X mass is 16.70 ± 0.35(stat) ± 0.5(sys) MeV [14], and in this work we adopt the central mass m X ≃ 16.7 MeV in calculations. The interactions of the vector boson X with quarks and leptons via a scheme of BSM has been argued in the literatures [15][16][17]. In this work, the vector boson X discussed in Ref. [15] is of our concern.
For the scattering between possible scalar, vectorial, fermionic DM and target nucleus, the spin-independent interaction induced by exchanging the vector boson X is dominant (see e.g. Ref. [18]). The vector boson X couples to electron and u,d quarks, and X may also couples to the second and/or the third generation SM charged leptons and up type/down type quarks with equal couplings to the same type fermions (see, e.g. Ref. [16] for more discussions). For the thermally freeze-out DM with such couplings, the DM mass as low as 0.5 GeV has been excluded by the CRESST-II experiment [1]. Thus, the X-mediated sub-GeV DM needs more attention.
Here we focus on MeV scale DM. The energy released by DM annihilation can modify the cosmic microwave background (CMB), and the recent CMB measurement by the Planck satellite [19] sets a stringent bound on the s-wave annihilation of MeV-scale DM [19,20]. For MeV DM with vector form interaction induced by X, the annihilation of fermionic DM pair is s-wave dominant, so is inconsistent with the CMB observation. Thus, the possibility of DM being fermions is disfavored. By contrast, p-wave annihilations of scalar and/or vector DM candidates at freeze out are tolerant by the CMB result. Thus, we concentrate on the case of scalar and vector DM, then the corresponding model parameter space will be derived.
For DM mass in the range of a few MeV/teens MeV, the big bang nucleosynthesis (BBN) and the effective number of relativistic neutrino N ef f at recombination may be altered by the energy release from dark sector annihilations. Thus corresponding observation results will be taken into account to set a lower bound on DM mass.
As recoils of target nucleus are small, the scattering between DM and nucleus is not sensitive for DM in MeV region, thus the direct detection for DM would turn to the DM-electron scattering which might be employed for the light DM hunting, and the issue was investigated in Refs. [21][22][23]. In this work the search for DM via its scattering with electron will be discussed for our concerned model. This work is organized as follows. After this introduction, we present the concrete forms of interactions between SM and DM with new boson X exchanged, and estimate the DM pwave annihilation rate. Next we take into account the constraints by the BBN and CMB to set the mass range of DM, and numerically evaluate the DM-X coupling for the DM mass range of concern. Then we analyze the detection possibility of the MeV DM via the DM-electron scattering. The last section is devoted to a brief conclusion and discussion.

II. INTERACTIONS BETWEEN SM AND DM
Based on the model where the new vector boson X mediates interaction between the SM particles and scalar/vectorial DM, we will analyze the relevant issues. The couplings of X with SM particles has been discussed in Ref. [15]. The effective X-DM coupling can be set in terms of the DM annihilation cross section at DM thermally freeze out.

A. The couplings
We suppose that X mediates a BSM interaction where the new charge in DM −X interaction is e D . The SM fermions are of equipped with also a new charge to couple to X which is parameterized as eε f (in unit of e), and ε f is relevant to the concerned fermion flavor. Let us first formulate the scattering amplitude between scalar DM and SM particles caused by the new interaction where X stands as the mediator. The new effective interaction is in the form where φ is the scalar DM field. J µ DM , J µ SM are the currents of scalar DM, SM fermions, respectively, with To explain the 8 Be anomalous transition, the ε f of the first generation fermion is derived and its value was presented in Ref. [15] as Moreover, if the vector boson X couples to the muon with |ε µ | ≈ |ε e |, the discrepancy between theory and experiment in muon g − 2 can be moderated [15].
For the vectorial DM field V , the V − X vertices are shown in Fig. 1 The couplings of X in SM sector are the same as that of the scalar DM case. For scalar (vectorial) DM, the annihilation φφ * → X → ff (V V * → X → ff ) is a p-wave process. When the scalar (vectorial) DM mass m φ (m V ) is above the X boson mass m X , the annihilation φφ * → XX (V V * → XX) portal is open, as shown in Fig. 2. However the analysis of Refs. [19,20] indicate that the CMB measurement sets a stringent constraint on the MeV scale DM s-wave annihilation. For DM annihilation channels e + e − and 4e, the upper bounds from CMB on the s-wave annihilations of these two channels are as follows: e.g., for DM with the mass of 5 MeV, the cross sections are about below 2.7 × 10 −30 , 4.3 × 10 −30 (cm 3 /s) for e + e − , 4e, respectively; for DM with the mass of 500 MeV, the cross sections are about below 4.2 × 10 −28 , 3.5 × 10 −28 (cm 3 /s) for e + e − , 4e, respectively. For MeV scale DM, these constraints are much below the required thermally freeze-out annihilation cross section, and some tunings are needed if the DM s-wave annihilation exists. Thus for thermally freeze-out DM, to avoid the s-wave annihilation in the process φφ * → XX (V V * → XX), the constraint of m φ (m V ) < m X is mandatory, i.e. the corresponding annihilation is kinematically closed. In addition, as indicated by the 8 Be anomaly transition, the X boson predominantly decays into e + e − , and this implies that it cannot directly decay into DM, otherwise its decay procedure would be dominated by X → φφ * (V V * ). Thus we must demand another constraint m φ (m V ) > m X /2. Therefore, a mass range of DM is m X /2 < m φ (m V ) < m X , and the p-wave annihilation was overwhelming at DM freeze out. Let us first consider the scalar DM. In the mass range m X /2 < m φ < m X , the s-channel annihilation φφ * → X → ff is overwhelming at DM freeze out, as shown in Fig. 3 (a). In one initial DM particle rest frame, the scalar DM annihilation cross section can be written as where v r is the relative velocity of the two DM particles. The factor 1 2 is due to the required φφ * pair in annihilations, and s is the total invariant squared mass. Γ X is the decay width of X, and m f is the mass of the final fermions. The phase space factor β f is Parameterizing Eq. (5) in forms of we can obtain the result With this parameterization, the thermally averaged annihilation cross section at temperature T is [24,25] [26,27] x f ≃ ln 0.038c(c + 2) where c is a parameter of O(1), and we take c = 1/2 for numerical computations. g is the degrees of freedom of DM, and m Pl = 1.22 × 10 19 GeV is the Planck mass. g * is the total effective relativistic degrees of freedom at the temperature T f , and we will adopt the data given by Ref. [28]. The relic density of DM is [26,27] where h is the Hubble parameter (in units of 100 km/(s·Mpc)).

Vectorial DM
Now consider the vectorial DM. In the mass range m X /2 < m V < m X , the annihilation V V * → ff is overwhelming at DM freeze out, as shown in Fig. 3 (b). In one initial particle rest frame, the vectorial DM annihilation cross section is Again parameterizing Eq. (11) in forms of σ ann v r = a+bv 2 The thermally averaged annihilation cross, the relic density of vectorial DM are similar to that we derived for scalar DM, replacing by corresponding input parameters.

III. ANALYSIS ON X-DM COUPLING
The energy released from thermal MeV DM annihilation in the early universe can alter the BBN result and the effective number of relativistic neutrino N ef f . Even though the effects are not violent, it still can be employed to constrain the lower bound of DM mass. After the DM mass range being set, we will calculate the X-DM coupling by means of the DM thermally freeze-out annihilation cross section.

A. DM mass with constraints of N ef f
In the case of m X /2 < m φ (m V ) < m X , the main annihilation product of DM is e + e − . The DM annihilation might heat the electron-photon plasma before freeze out in the early universe. If this happens at the time that the neutrino decoupled from the hot bath, the ratio of the neutrino temperature relative to the photon temperature will be lowered, which causes a reduction of the number of the effective neutrino degrees of freedom [12,29]. The abundances of light elements stemmed from the primordial nucleosynthesis and the CMB power spectra at the recombination epoch would also be affected. For electron neutrinos, a typical decoupling temperature is T d ∼ 2.3 MeV [30]. The value x f of the thermally freeze-out DM is x f ∼ 20. Thus, for the DM of concern, the freeze out of DM is supposed to be after neutrino decoupling, so the effects of DM annihilation need to be taken into account. For the new boson X, the decay width is With the mass m X ≫ T d and X's lifetime much less than 1 second, the contribution from X's entropy to the BBN is negligible.
Here we focus on the constraints from the primordial abundances of light elements 4 He and deuterium, denoted by Y p and y DP , respectively. The abundance values of 4 He and deuterium are related to the baryon density ω b ≡ Ω b h 2 and the effective number of relativistic neutrinos N ef f (or, in the form of the difference of ∆N ef f ≡ N ef f − 3.046, where N ef f = 3.046 is the standard cosmological prediction value [31,32]). The abundances predicted by the BBN are parameterized as Y p (ω b , ∆N ef f ), y DP (ω b , ∆N ef f ), and the corresponding Taylor expansion forms can be obtained with the PArthENoPE code [33]. If the value ω b = 0.02226 +0.00040 −0.00039 is adopted with the bounds of P lanck TT+lowP+BAO [19], the value of N ef f is also determined by the constraints of 4 He and deuterium abundances. The range of N ef f can be derived with the P lanck data, and that is [19] N ef f = 3.14 +0.
Considering Eqs. (14), (15), an lower bound N ef f > ∼ 2.9 is taken in calculations. In the case that DM mainly couples to electron-photon plasma and DM particles freeze out later than the neutrino decoupling, the effective number N ef f can be written as [36,37] N ef f = 3.046 [ where I(T γ ) is given by and Here T γ is the photon temperature, and the integration variable is y = p DM /T γ . The plus/minus sign is for fermionic/bosonic DM particles, respectively. For bosonic DM of concern, the parameter values of the degrees of freedom g B = 2, g B = 6, the mass m DM = m φ , m V are corresponding to the scalar, vectorial DM, respectively. The effective number N ef f as a function of m DM /T d is shown in Fig. 4. Taking the lower bound N ef f > ∼ 2.9, we can obtain that m DM /T d > ∼ 5.2, 6.8 for scalar, vectorial DM, respectively. As the neutrino decoupling is not a sudden process (for more details, see e.g. Refs. [30][31][32]38]), here we take T d > ∼ 2 MeV as a lower bound. Thus, the mass range of DM is derived, B. Numerical result for the X-DM coupling As the DM mass range being set, we turn to investigate the X-DM coupling. The DM relic density is 0.1197 ± 0.0042 [19]. According to the DM thermally averaged annihilation cross section σ ann v r ≈ 6b/x f at T f , the numerical results of b are shown in Fig. 5, with the solid, dashed curves corresponding to the scalar, vectorial DM, respectively. After the values of b defined in Eq. (7) is obtained, and then the X-DM coupling couplings is also determined. The  numerical results of e 2 D ε 2 e are depicted in Fig. 6. Considering the value of ε e given by Eq. (4), we can obtain e 2 D /4π < 1, and thus the X-DM coupling is sufficiently small that the perturbation may apply.

IV. DM-ELECTRON SCATTERING
Now let us turn to investigate the possibility of detecting the light DM of MeV scale by the earth detector.
For the light DM particles, since the recoil of the target nucleus is too small to be substantially observed, one may not detect arrival of DM via the scattering between the MeV DM and target nucleus. Instead, the DM-electron scattering can be employed for the MeV DM hunting. The DM-electron scattering has been investigated in Refs. [21][22][23]. The target atomic electron is in a bound state, and the typical momentum transfer q is of order αm e as a few eV, which may cause excitation/ionization of the electron in inelastic scattering processes. In this work, we study the signals of individual electrons induced by DM-electron scattering. Here, we take the form of the DM-electron scattering cross section as given by Ref. [39], and for scalar DM, that is with µ φe being the φ-electron reduced mass, and F DM (q) ≃ 1 for m X ≫ αm e .
For vectorial DM, the DM-electron scattering cross section is with µ V e being the V -electron reduced mass. As the value of e 2 D ε 2 e is fixed, the DM-electron scattering cross sectionσ e can be obtained. The numerical result ofσ e is shown in Fig. 7, where it is noted that the scattering cross section is independent of the momentum transfer (F DM (q) = 1). The upper solid, upper dashed curves are for the scalar, vectorial DM, respectively, and the dot-dashed curve is the excluded bound set by the XENON10 data [40]. It can be seen that, considering the constraint of XENON10, there exists parameter spaces for scalar, vectorial DM to satisfy the constraints. Now we give a brief discussion about the background in the DM-electron scattering. One irreducible background is from the neutrino-electron scattering, which sets the ultimate limit to the sub-GeV DM direct detections. Fortunately, the DM annual modulation effect from the motion of the earth can be employed to reduce the neutrino background [39,41,42]. The teens MeV DM of concern could be probed via the inelastic processes of DM-electron scatterings, e.g. the individual electron signals by the future noble gas and semiconductor targets. For Ar, Xe [39] and Ge, Si [43] with 1 kg·year exposure, the exclusion reach at 95% confidence level via single electron detections are also shown in Fig. 7. Further explorations of DM-electron scatterings are needed, both in theory and experiment.

V. CONCLUSION AND DISCUSSION
The MeV scalar and vectorial DM has been studied in this work, with the new boson X indicated by the 8 Be anomalous transition being the mediator. Considering the constraints of the DM direct detection and CMB observation, we find that for the case of m X /2 < m φ (m V ) < m X , the p-wave dominant annihilation of DM at freeze out does not conflict with the observed data so far. The primordial abundances of light elements and the effective number of relativistic neutrino N ef f at recombination are sensitive to the DM with the mass of a few MeV to teens MeV, thus the corresponding observed results have been employed to set a lower bound on the DM mass. Taking the combined lower bounds N ef f > ∼ 2.9 and the neutrino decoupling temperature T d > ∼ 2 MeV, we derive a mass range of DM: 10.4 < ∼ m φ < ∼ 16.7 MeV for scalar DM, and 13.6 < ∼ m V < ∼ 16.7 MeV for vectorial DM. For the teens MeV scalar, vectorial DM of concern, the numerical result of the DM-X coupling is derived in terms of the DM thermally averaged annihilation cross section. Once this coupling is set, the strength of the interaction between DM and SM particles is determined.
The DM-electron scattering is employed for the teens MeV DM hunting. We investigate on the signal of the individual electrons in DM-electron scattering, and the scattering cross section σ e is calculated for the DM mass range of concern. We find that, considering the constraint of XENON10, there are still parameter spaces left for the teens MeV scalar, vectorial DM to be observed. Beside the individual electrons, signals of individual photons, individual ions, and heat/phonons can also be employed to explore the MeV DM-electron scattering (see. e.g. Ref. [39,44] for more), even though the ion signal is probably too weak for detection. The teens MeV DM of concern could be probed by the future noble gas and semiconductor targets via the DM-electron scattering. In fact, the wave function of electron in the bound state for a certain target material needs to be considered to guarantee the prediction power. It is noted that the detection possibilities and efficiency of DM are target dependent.
As discussed in Ref. [45], the new boson X may be detectable at the e + e − collider, such as BESIII and BaBar. The new boson X may also give an interpretation about the NuTeV anomaly [46]. For the teens MeV scalar, vectorial DM of concern, further investigation both in theory and experiment aspects are needed. We look forward to the exploration of the X-portal DM in the future.