Combined search for light dark matter with electron and muon beams at NA64

TL;DR

This paper explores how combined electron and muon beam experiments at NA64 can effectively search for light dark matter mediated by a dark photon, covering significant parameter space and enhancing discovery potential.

Contribution

It demonstrates the potential of combined NA64e and NA64μ results to probe light dark matter models more comprehensively than individual experiments.

Findings

NA64e can probe scalar and Majorana LDM scenarios with 5×10^{12} EOT.

Combined NA64e and NA64μ results with ~10^{13} EOT and 10^{13} MOT cover significant LDM parameter space.

NA64e and NA64μ are highly complementary, increasing discovery potential for sub-GeV dark matter.

Abstract

We discuss prospects of searching for a dark photon ($A'$) which serves as mediator between Standard model (SM) particles and light dark matter (LDM) by using the combined results from the NA64 experiment at the CERN SPS running in high-energy electron (NA64e) and muon (NA64$\mu$) modes. We discuss the most natural values and upper bounds on the $A'$ coupling constant to LDM and show they are lying in the range accessible at NA64. While for the projected $ 5\times10^{12}$ electrons on target (EOT) NA64e is able to probe the scalar and Majorana LDM scenarios, the combined NA64e and NA64$\mu$ results with $\simeq 10^{13}$ EOT and a few $10^{13}$ MOT, respectively, will allow covering significant region in the parameter space of the most interesting LDM models. This makes NA64e and NA64$\mu$ extremely complementary to each other and increases significantly the discovery potential of sub-GeV DM.

Figures

Equations

$$L_{\chi A^{\prime }}=({\partial }^{\mu }\chi -i{e}_D{A}^{\prime \mu }\chi {)}^{\ast }({\partial }_{\mu }\chi -i{e}_D{A}_{\mu }^{\prime }\chi )-m_{\chi }^2{\chi }^{\ast }\chi -{\lambda }_{\chi }({\chi }^{\ast }\chi {)}^2.$$

$${\sigma }_{an}{v}_{rel}=\frac{8\pi }{3}\frac{{ϵ}^2\alpha {\alpha }_D{m}_{\chi }^2{v}_{rel}^2}{(m_{A^{\prime }}^2-4m_{\chi }^2{)}^2}.$$

$$<{\sigma }_{an}{v}_{rel}>=O(1)\text{mbox}pb.$$

$$\beta ({\bar{\alpha }}_D)=\frac{{\bar{\alpha }}_D^2}{2\pi }[\frac{4}{3}(Q_F^2{n}_F+Q_S^2\frac{n_S}{4})].$$

$$e^{-}+Z\to e^{-}+Z+A^{\prime };A^{\prime }\to \chi \chi$$

$${\mu }^{-}+Z\to {\mu }^{-}+Z+Z^{\prime };Z^{\prime }\to \nu \nu ,\chi \chi$$

$$N_{A^{\prime }}^e\approx \frac{\rho N_{av}}{A}\cdot n_{EOT}{L}^e{\sigma }_{A^{\prime }}^e,N_{A^{\prime }}^{\mu }\approx \frac{\rho N_{av}}{A}\cdot n_{MOT}{L}^{\mu }{\sigma }_{A^{\prime }}^{\mu },$$

$$\frac{d{n}_d}{dt}+3H(T)n_d=-<\sigma v_{rel}>(n_d^2-n_{d,eq}^2).$$

$$n_d(T)=\int \frac{d^{3}p}{2{\pi }^3}{f}_d(p,T)$$

$$\Omega_{DM}h^{2}=0.1\Bigl{(}\frac{(n+1)x_{f}^{n+1}}{(g_{*s}/g^{1/2}_{*})}\Bigr{)}\frac{0.876\cdot 10^{-9}\mbox{GeV}^{-2}}{\sigma_{0}}\,,$$

$$x_f=c-(n+\frac{1}{2})\ln (c),$$

$$c=\ln (0.038(n+1)\frac{g}{\sqrt{g_{\ast }}}{M}_{Pl}{m}_{\chi }{\sigma }_0).$$

$$\alpha_{D}\simeq 0.02f(m_{A^{{}^{\prime}}},m_{\chi})\cdot\Bigl{(}\frac{10^{-3}}{\epsilon}\Bigr{)}^{2}\Bigl{(}\frac{m_{A^{\prime}}}{100~{}\mbox{MeV}}\Bigr{)}^{4}\Bigl{(}\frac{10~{}\mbox{MeV}}{m_{\chi}}\Bigr{)}^{2}\,$$

Full text

1 Introduction

Nowadays the most promising evidence in favour of a new physics beyond the SM is the observation of Dark Matter (DM). In particular, various DM models, for a review see e.g. [1] - [5], which motivate the existence of light DM messengers with a mass $m_{\chi}\leq O(1)$ GeV are of a great interest [6, 7]. The main idea is that in addition to gravity a new interaction between visible and dark sector can be mediated by a new sub-GeV vector or scalar boson, as a review of the current and projected limits of LDM and other New Physics, see e.g. Refs. [7]-[11].

Among several renormalizable LDM extensions of the SM, the model with dark photon, where dark sector includes an abelian gauge field $A^{\prime}_{\mu}$ (dark photon) is the most popular now. In these dark photon models, dark sector interacts with the SM particles only through nonzero kinetic mixing of the ordinary photon and dark photon, $-\frac{\epsilon}{2}F^{\prime}_{\mu\nu}F^{\mu\nu}$. In renormalizable models the DM particles interacting with the $A^{\prime}$ have spin [math] or $1/2$. Spin $1/2$ DM particles can be Majorana or pseudo-Dirac particles [7, 12]. The annihilation cross-section for scalar or Majorana DM has $p$-wave suppression that allows to escape the CMB bound [13, 14] while for Dirac fermions the annihilation cross-section is $s$-wave that contradicts to the CMB bound [13, 14, 15]. For the model with pseudo-Dirac fermions [16] it is also possible to avoid the CMB bound.

Let us consider, as an example, charged scalar dark matter interacting with dark photons. The charged dark matter field $\chi$ interaction with the $A^{\prime}$ dark photon field is

[TABLE]

The nonrelativistic DM annihilation cross-section $\chi\bar{\chi}\rightarrow e^{-}e^{+}$ has the form111Here we consider the case $m_{A^{\prime}}>2m_{\chi}$, $m_{A^{\prime}}\gg m_{e}$.

[TABLE]

Here $\alpha_{D}=\frac{e^{2}_{D}}{4\pi}$ is an analogue of the fine-structure constant $\alpha=1/137$ for the DM particles interacting with DM photon. We shall use a standard assumption that in the hot early Universe DM is in equilibrium with ordinary matter [5]. During the Universe expansion the temperature decreases and at some temperature the thermal decoupling of the DM occurs. Namely, at freeze-out temperature $T_{d}$ the cross-section of the annihilation $DM~{}particles\rightarrow~{}SM~{}particles$ becomes too small to obey the equilibrium of the DM particles with the SM particles and the DM decouples. The experimental data are in favour of scenario with cold relic at which the freeze-out temperature $T_{d}$ is much lower than the mass of the DM particle. In other words, the DM particles decouple in the non-relativistic regime. The value of the DM annihilation cross-section at the decoupling temperature determines the value of today’s DM density in the Universe. In particular, relatively large annihilation cross-section leads to a low DM density. On the other hand, small annihilation cross-section leads to DM overproduction. The observed value of DM density fraction, $\rho_{DM}/\rho_{c}\approx 0.23$, (here $\rho_{c}$ is a total energy density of the Universe) allows to estimate the DM annihilation cross-section into the SM particles and hence to estimate the discovery potential of light DM for both direct underground and accelerator experiments. One can roughly estimate the typical DM annihilation cross-section as [2]

[TABLE]

As a consequence of the formulae (2,3) we can estimate the product $\epsilon^{2}\alpha_{D}$ for fixed values $m_{A^{\prime}}$ and $m_{\chi}$. Note that fixed target NA64 experiment [18] uses the reaction of the dark photon electroproduction on nuclei that allows obtaining only upper bounds on $\epsilon^{2}$ vs $m_{A^{\prime}}$. Therefore, to test the prediction for the $\epsilon^{2}\alpha_{D}$ we have to know either the $\alpha_{D}$ value or at least its upper bound $\alpha_{D}\leq\alpha_{o}$. The arguments based on the use of the renormalization group and the assumption of the absence of the Landau pole singularity up to some scale $\Lambda$ allow to obtain upper limit on the coupling constant $\alpha_{D}$ [20]. The bound on $\alpha_{D}$ depends on the scale $\Lambda$ logarithmically. Moreover, the scale $\Lambda$ has to be larger than $1$ TeV [20]. So for fixed values of $m_{A^{\prime}}$ and $m_{\chi}$ the knowledge of the upper bound on $\alpha_{D}$ along with the requirement that the dark photon model correctly reproduces the observed DM density allows obtaining lower bound on $\epsilon^{2}$ as a function of $m_{A^{\prime}}$ or $m_{\chi}$.

In this paper we discuss prospects of searching for $A^{\prime}$ dark photon mediator of LDM production at the NA64 experiment at CERN SPS by using $\simeq$ 100 GeV electron ( NA64e) and muon ( NA64$\mu$) beams. The rest of the paper is organized as follows. In Sec. 2 we discuss upper bounds on $\alpha_{D}$ obtained from the requirement of the absence of Landau pole singularity for the effective coupling constant $\bar{\alpha}_{D}(\mu)$ up to some scale $\Lambda$. In Sec. 3 we estimate the NA64e discovery potential of LDM and show that with $\simeq 5\times 10^{12}$ electrons on target (EOT) the experiment is able to probe the most natural parameter space of scalar and Majorana LDM models. In Sec. 4 we estimate the NA64$\mu$ discovery potential of LDM. We show that NA64$\mu$ has better sensitivity to the $\gamma-A^{\prime}$ kinetic mixing for the $A^{\prime}$ masses $m_{A^{\prime}}\gtrsim 100$ MeV in comparison with NA64e, and that the combined NA64e and NA64$\mu$ results obtained with $\simeq 10^{13}$ EOT and a few $10^{13}$ MOT, respectively, will allow covering significant range of natural parameter space of the LDM models including pseudo-Dirac LDM. This makes the two approaches extremely complementary to each other and increases significantly the discovery potential of NA64. Sec. 5 contains concluding remarks. In Appendix we collect the main formulae used for the DM density calculations.

2 Upper bound and range of $\alpha_{D}$

One can obtain upper bound on $\alpha_{D}$ by the requirement of the absence of Landau pole singularity for the effective coupling constant $\bar{\alpha}_{D}(\mu)$ up to some scale $\Lambda$ [20]. One loop $\beta$ function for $\bar{\alpha}_{D}(\mu)$ is

[TABLE]

Here $\beta(\bar{\alpha}_{D})\equiv\mu\frac{d\bar{\alpha}_{D}}{d\mu}$ and $n_{F}$ ($n_{s}$) is the number of fermions (scalars) with the $U^{{}^{\prime}}(1)$ charge $Q_{F}(Q_{S})$. For the model with pseudo-Dirac fermion we introduce an additional scalar with $Q_{S}=2$ to realize the splitting between fermion masses, so one loop $\beta$ function is $\beta(\bar{\alpha}_{D})=\frac{4\bar{\alpha}_{D}^{2}}{3\pi}$. For the model with Majorana fermions, we also introduce an additional scalar field with the charge $Q_{S}=2$ and additional Majorana field to cancel $\gamma_{5}$ anomalies, so the $\beta$ function coincides with the $\beta$ function for the model with pseudo-Dirac fermions. For the model with charged scalar matter, in order to create nonzero dark photon mass, we have to introduce additional scalar field with $Q_{S}=1$, so one loop $\beta$ function is $\beta=\alpha^{2}/3\pi$. From the requirement that $\Lambda\geq 1$ TeV [20], we find that $\alpha_{D}\leq 0.2$ for pseudo-Dirac and Majorana fermions and $\alpha_{D}\leq 0.8$ for charged scalars 222For smaller values of $\Lambda$ we shall have some charged particles with masses $\leq 1$ TeV that contradicts to the LHC bounds.. Here $\alpha_{D}$ is an effective low energy coupling at scale $\mu\sim m_{A^{\prime}}$, i.e. $\alpha_{D}=\bar{\alpha}_{D}(m_{A^{\prime}})$. In our calculations we used the value $m_{A^{\prime}}=10$ MeV. In the assumption that dark photon model is valid up to Planck scale, i.e. $\Lambda=M_{PL}=1.2\times 10^{19}$ GeV, we find that for pseudo-Dirac and Majorana fermions $\alpha_{D}\leq 0.05$ while for scalars $\alpha_{D}\leq 0.2$. In the SM the $SU_{c}(3)$, $SU_{L}(2)$ and $U(1)$ gauge coupling constants are equal to $\sim(1/30-1/50)$ at the Planck scale. One can show that the gauge coupling $\bar{\alpha}_{D}(\mu=M_{PL})$ is of the order of $\sim(1/30-1/50)$. As a result, we find that the values in the range $\sim(0.014-0.02)$ are the most natural for the low energy coupling constant $\alpha_{D}$.

The expression (2) for the annihilation cross-section is proportional to factor $K=(\frac{m^{2}_{A^{\prime}}}{m_{\chi}^{2}}-4)^{-2}$ and in the resonance region $m_{A^{\prime}}\approx 2m_{\chi}$ the DM density bound on $\epsilon^{2}$ is proportional to $K^{-1}$. So for $m_{A^{\prime}}\approx 2m_{\chi}$ the bound on $\epsilon^{2}$ becomes very weak [21]. It should be mentioned that in general the values of $m_{A^{\prime}}$ and $m_{\chi}$ are arbitrary, so the case $m_{A^{\prime}}=2m_{\chi}$ could be considered as some fine-tuning. It is natural to require the absence of significant fine-tuning. Namely, we require that $|\frac{m_{A^{\prime}}}{2m_{\chi}}-1|\geq 0.25$, i.e. $m_{A^{\prime}}\geq 2.5m_{\chi}$. In our estimates we use two values $\frac{m_{A^{\prime}}}{m_{\chi}}=2.5$ and $\frac{m_{A^{\prime}}}{m_{\chi}}=3$. We studied DM models with charged scalar, Majorana fermion and pseudo-Dirac fermion [12]. For the model with pseudo-Dirac DM we considered the most difficult case of small mass splitting $|\delta|\ll 1$333For a pseudo-Dirac fermion $\chi=(\eta,\phi)$ with $\eta$ and $\phi$ Weyl fermions the mass terms have the form [12] $L_{m}=-m_{\chi}\eta\phi-\frac{\Delta}{2}(\eta\eta+\phi\phi)+h.c.$. The mass eigenstates are $\chi_{1}=\frac{i}{\sqrt{2}}(\eta-\phi)$ , $\chi_{2}=\frac{1}{\sqrt{2}}(\eta+\phi)$ with masses $m_{1,2}=m_{\chi}\mp\Delta$. For $\delta\equiv\frac{\Delta}{m_{\chi}}\ll 1$ DM density calculations coincide with the corresponding calculations for Dirac fermion DM.. Our calculations are based on the approximate formulae (10 - 13) presented in the Appendix. The results of our calculations for pseudo-Dirac DM density coincide with the 20 % accuracy with the corresponding calculations of Ref.[12].

3 Projected LDM sensitivity of NA64e

The NA64e experiment is designed for a sensitive search for the $A^{\prime}$ mediator of sub-GeV dark matter particle ($\chi$) production in the missing energy events from the reaction of 100 GeV electron scattering on heavy nuclei:

[TABLE]

at the CERN SPS [22, 23]. After the long shutdown (LS2) stop at CERN the experiment plan to accumulate $\gtrsim 5\times 10^{12}$ EOT. The NA64e limits on mixing strength $\epsilon$ obtained from the 2016-2018 run with $2.84\times 10^{11}$ EOT [19] and expected after the LS2 period assuming the zero-background case [10] are shown in the upper l.h.s. panel in Fig.1.

The rest of the plots show the required number of EOT for the 90% C.L. exclusion of the $A^{\prime}$ with a given mass $m_{A^{\prime}}$ in the ($m_{A^{\prime}},n_{EOT}\times 10^{-12}$ ) plane for pseudo-Dirac with $\delta\ll 1$ (the upper r.h.s. panel), Majorana (the lower l.h.s. panel), and Scalar (the lower r.h.s. panel) dark matter models for $\frac{m_{A^{\prime}}}{m_{\chi}}=2.5$ (solid), and = 3 (dashed), and $\alpha_{D}=$ 0.1 (red), 0.05 (blue), and 0.02 (green). As one can see, NA64e is able to exclude the most interesting and natural LDM scenarios in the $A^{\prime}$ mass range $1~{}MeV\leq m_{A^{\prime}}\leq 150$ MeV except the most difficult case of pseudo-Dirac DM with $\alpha_{D}=0.1$ and $\alpha_{D}=0.05$, $\frac{m_{A^{\prime}}}{m_{\chi}}=2.5$.

4 NA64$\mu$ projections for the $\gamma-A^{\prime}$ mixing strength

The NA64$\mu$ experiment [24, 25] is proposed to search for dark sector particles weakly coupled to the muon, which could explain the muon (g-2)μ anomaly [26, 27]. One of the good examples of such a particle, is a new light vector $Z^{\prime}$ boson [28] - [35], which interacts predominantly with the $L_{\mu}-L_{\tau}$ current444One loop corrections lead to nonzero interactions with electron, and other quarks and leptons [36].. Furthermore, the $Z^{\prime}$ could also serve as a new leptophilic mediator of dark force between SM sector and dark matter, which is charged with respect to $U(1)_{L_{\mu}-L_{\tau}}$. Moreover, this boson can be associated with the mechanism of the DM relic abundance [36, 37, 38]. Another interesting possibility involves muon-specific scalar mediator which could connect the visible and dark sectors and also account for the (g-2)μ anomaly [39, 40, 41].

The NA64$\mu$ plans to perform a sensitive search for $L_{\mu}-L_{\tau}$ $Z^{\prime}$ as a mediator of sub-GeV dark matter particle ($\chi$) production in missing energy events from the reaction of 100-160 GeV muon scattering on heavy nuclei:

[TABLE]

at the CERN SPS [24, 25].

In the $A^{\prime}$ dark photon model the interaction of dark photon with the leptons and quarks is given by $L_{A^{\prime}}=e\epsilon A^{\prime}_{\mu}J^{\mu}_{SM}$. Here, $J^{\mu}_{SM}$ is the electromagnetic current. So, we see that muons and electrons interact with the dark photon universally, with the same coupling constant. Hence, similar to the reaction of Eq.(5), the dark photons will be also produced in the reaction of Eq.(6) with the same experimental signature of the missing energy.

For the $A^{\prime}$ mass region $m_{A^{\prime}}\gg m_{e}$, the total cross-section of the dark photon electroproduction $eZ\rightarrow eZA^{\prime}$ scales as $\sigma^{e}_{A^{\prime}}\sim\epsilon_{e}^{2}/m_{A^{\prime}}^{2}$. On the other hand, for the dark photon masses, $m_{A^{\prime}}\lesssim m_{\mu}$, the similar $\mu Z\rightarrow\mu ZA^{\prime}$ cross-section can be approximated in the bremsstrahlung-like limit as $\sigma^{\mu}_{A^{\prime}}\sim\epsilon_{\mu}^{2}/m_{\mu}^{2}$. Let us now compare expected sensitivities of the $A^{\prime}$ searches with NA64e and NA64$\mu$ experiments for the same number $\simeq 5\times 10^{12}$ particles on target. Assuming the same signal efficiency the number of $A^{\prime}$ produced by the 100 GeV electron and muon beam can approximated, respectively, as follows

[TABLE]

where $L^{e}\simeq X_{0}$ and $L^{\mu}\simeq 40X_{0}$ are the typical distances that are passed by an electron and muon, respectively, before producing the $A^{\prime}$ with the energy $E_{A^{\prime}}\gtrsim 50$ GeV in the NA64 active Pb target of the total thickness of $\simeq 40$ radiation length ($X_{0}$) [24, 25]. The detailed comparison of the calculated $A^{\prime}$ sensitivities of NA64e and NA64$\mu$ is shown in Fig.2, where the 90% C.L. limits on the mixing $\epsilon$ are shown for a different number of particles on target for both the NA64e and NA64$\mu$ experiments. The limits were obtained for the background free case by using exact-tree-level (ETL) cross-sections rather than the Weizsacker-Williams (WW) ones calculated for NA64e in Ref.[42], and for the NA64$\mu$ case in this work. The later are shown in Fig. 3 as a function of $E_{A^{\prime}}/E_{\mu}$ for the Pb target and mixing value $\epsilon=1$. One can see that in a wide range of masses, $20\mbox{ MeV}\lesssim m_{A^{\prime}}\lesssim 1\mbox{ GeV}$, the total WW cross-sections are larger by a factor $\simeq 2$ compared to the ETL ones. As the result, the typical limits on $\epsilon$ for the ETL case are worse by about a factor $\simeq 1.4$ compared to the WW case.

For $n_{EOT}=n_{MOT}=5\cdot 10^{12}$ the sensitivity of NA64e is enhanced for the mass range $m_{e}\ll m_{A^{\prime}}\simeq 100$ MeV while for the $A^{\prime}$ masses $m_{A^{\prime}}\gtrsim 100$ MeV NA64$\mu$ allows obtaining a more stringent limits on $\epsilon^{2}$ compared to NA64e.

5 Combined LDM sensitivity of NA64e and NA64$\mu$

The estimated NA64e and NA64$\mu$ limits on the $\gamma-A^{\prime}$ mixing strength, allow us to set the combined NA64e and NA64$\mu$ constraints on the LDM models, which are shown in the $(y;~{}m_{\chi}$) plane in Fig.4.

As discussed in Sec. I, as a result of the $\gamma-A^{\prime}$ mixing the cross-section of the DM particles annihilation into the SM particles is proportional to $\epsilon^{2}$. Hence using constraints on the DM annihilation cross-section one can derive constraints in the ($y\equiv\epsilon^{2}\alpha_{D}(m_{\chi}/m_{A^{\prime}})^{4};~{}m_{\chi}$) plane and restrict the LDM models with the masses $m_{\chi}\lesssim 1$ GeV.

The combined limits obtained from the data sample of the 2016 [17, 18], and 2017, 2018 runs [19] and expected from the run after the LS2 are shown in the top panels of Fig. 4 together with combined limits from NA64e and NA64$\mu$ for $10^{13}$ EOT and $2\times 10^{13}$ MOT, respectively. The plots show also the comparison of our results with the limits of other experiments. It should be noted that the $\chi$-yield in the NA64 case scales as $\epsilon^{2}$ rather than $\epsilon^{4}\alpha_{D}$ as in beam dump experiments. Therefore, for sufficiently small values of $\alpha_{D}$ the NA64 limits will be much stronger. This is illustrated in the upper right panel of Fig. 4, where the NA64 limits are shown for $\alpha_{D}=0.1$. One can see, that for this, or smaller, values of $\alpha_{D}$, the direct search for LDM at NA64e with $5\times 10^{12}$ EOT excludes the scalar and Majorana models of the LDM production via vector mediator with $\frac{m_{A^{{}^{\prime}}}}{m_{\chi}}=3$ for the full mass region up to $m_{\chi}\lesssim 0.2$ GeV. While being combined with the NA64$\mu$ limit, the NA64 will exclude the models with $\alpha_{D}\leq 0.1$ for the entire mass region up to $m_{\chi}\lesssim 1$ GeV. So we see that for the full mass range $m_{\chi}\lesssim 1$ GeV the obtained combined NA64e and NA64$\mu$ bounds are more stringent than the limits obtained from the results of NA64e that allows probing the full sub-GeV DM parameter space.

6 Conclusions

In this paper we considered the NA64 discovery perspectives of sub-GeV thermal dark matter by running the experiment in electron and muon modes at the CERN SPS. Remarkably, that with the statistics accumulated during years 2016-2018 NA64 already starts probing the sub-GeV DM parameter space for the conventional value of $\alpha_{D}=0.1$ [19]. While with $5\times 10^{12}$ EOT NA64e is able to test the scalar and Majorana LDM scenarios for $\frac{m_{A^{\prime}}}{m_{\chi}}\geq 2.5$, the combined NA64e and NA64$\mu$ results with $\gtrsim 10^{13}$ EOT and $2\times 10^{13}$ MOT, respectively, will allow to fully explore the parameter space of other interesting LDM models like pseudo-Dirac DM model or the model with new light vector boson $Z^{\prime}_{B-L}$. This makes NA64e and NA64$\mu$ extremely complementary to each other, as well as to the planned LDMX experiment [49], and greatly increases the NA64 discovery potential of sub-GeV DM.

There are several different scenarios [12] of the dark photon model which are based on $U(1)_{B-L}$ or $U(1)_{B-3e}$ gauge symmetries. As in the dark photon model, the observed value of the DM density allows estimating the coupling constant $\epsilon$ of new light $Z^{\prime}$ boson with an electron. The value of the $\epsilon$ parameter for such models coincides with the $\epsilon$ value for dark photon model up to some factor $k\leq 3$ [12], so NA64e can also test these scenarios. For instance, for the model with $(B-L)$ vector interaction NA64e is able to exclude scalar and Majorana dark matter scenarios in a way analogous to the case of dark photon.

However it should be stressed that for $m_{A^{\prime}}\approx 2m_{\chi}$ the DM annihilation cross-section (2) is proportional to $(m^{2}_{A^{\prime}}-4m^{2}_{DM})^{-2}$. As a consequence the predicted value of the $\epsilon^{2}$ parameter is proportional to $(\frac{m^{2}_{A^{\prime}}}{4m^{2}_{\chi}}-4)^{2}$ that can reduce the predicted $\epsilon^{2}$ value by 2 - 4 orders of magnitude in comparison with the often used value $\frac{m_{A^{\prime}}}{m_{\chi}}=3$ [21]. It means that NA64 experiment and other future experiments like LDMX [49] are not able to test the region $m_{A^{\prime}}\approx 2m_{\chi}$ completely555The values of $m_{A^{\prime}}$ and $m_{\chi}$ are arbitrary, so the case $m_{A^{\prime}}\approx 2m_{\chi}$ could be considered as some fine-tuning. It is natural to assume the absence of significant fine-tuning. In this paper we required that $m_{A^{\prime}}\geq 2.5m_{\chi}$..

**Acknowledgements

**We are indebted to Prof. V.A. Matveev and our colleagues from the NA64 Collaborations for many for useful discussions. We thank N. Toro for valuable comments. We also would like to thank R. Dusaev for his help in preparing of Fig.4.

Appendix. Basic formulae for DM density

The relic density of DM in the standard scenario is obtained by solving the Boltzmann equation

[TABLE]

Here

[TABLE]

and $f_{d}(p,T)$ is the dark matter distribution function.

The dark matter relic density can be numerically estimated as [5]

[TABLE]

where $<\sigma v_{rel}>=\sigma_{o}x^{-n}_{f}$ and

[TABLE]

[TABLE]

Here $x_{f}=\frac{m_{\chi}}{T_{d}}$, $n=0(1)$ for $s(p)$-wave annihilation and $g_{*}$, $g_{*s}$ are the effective relativistic energy and entropy degrees of freedom. If DM particles differ from DM antiparticles $\sigma_{o}=\frac{\sigma_{an}}{2}$. The requirement that the dark photon model reproduces correct value of the DM density allows to estimate $\alpha_{D}$ as a function of $\epsilon$, $m_{A^{\prime}}$ and $m_{\chi}$, namely [20]:

[TABLE]

For the pseudo-Dirac DM with $\frac{m_{A^{\prime}}}{m_{\chi}}=3$ and $\delta\ll 1$ the estimates based on the use of the formulae (10 - 12) lead to $f=0.25-0.4$ at $1~{}\mbox{MeV}\leq m_{\chi}\leq 100$ MeV while analogous estimate for Majorana DM gives $f=3-5$.