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

We discuss prospects for searching for dark photon mediator ($A'$) of light dark matter (LDM) production by using the combined results from the NA64 experiment at the CERN SPS running in high-energy electron (NA64e) and muon (NA64$\mu$) mode. 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 with $ 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 to cover almost fully the parameter space of the most interesting LDM models. This makes NA64e and NA64$\mu$ extremely complementary to each other and greatly increases the discovery potential of sub-GeV DM.


Introduction
At present the most striking evidence in favour of new physics beyond the Standard model is the observation of Dark Matter (DM). Among the numerous DM models, for a review, see e.g. [1] - [5], those that motivate the existence of light DM messengers with a mass m d ≤ O(1) GeV are rather popular now [6,7]. The main idea is that in addition to gravity a new interaction between visible and dark sector exists which is mediated by a new sub-GeV vector or scalar boson, as a review of the current and future efforts towards light Dark Matter and other New Physics, see e.g. Refs. [7]- [11].
Among several renormalizable light DM extensions of the SM, the model with dark photon, where dark sector includes an abelian gauge field A µ (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, − 2 F ' µν F µν .
In renormalizable models the DM particles interacting with the A have spin 0 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 GMB bound [13,14] while for Dirac fermions the annihilation cross section is s-wave that contradicts to the GMB bound [13,14,15]. For the model with pseudo Dirac fermions [16] it is also possible to avoid the GMB bound.
Let us consider, as an example, charged scalar dark matter interacting with dark photons. The charged dark matter field φ d interaction with the A dark photon field is The nonrelativistic DM annihilation cross section φ dφd → e − e + has the form 1 Here is an analog of the fine-structure constant α = 1/137 for the DM particles interacting with DM photon. We shall use 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 T d the thermal decoupling of the DM starts to work. Namely, at some freeze-out temperature T d the cross-section of annihilation DM particles → SM particles becomes too small to obey the equilibrium of DM particles with the SM particles and DM decouples. The experimental data are in favour of scenario with cold relic for which the freeze-out temperature T d is much lower than the mass of the DM particle. In other words DM particles decouple in the non-relativistic regime. The value of the DM annihilation cross section at the decoupling temperature determines the value of the current DM density in the Universe. Too big annihilation cross section leads to small DM density and vice versa too small annihilation cross section leads to DM overproduction. The observed value of the DM density ρ DM ρc ≈ 0.23 ( here ρ DM , ρ c is the dark matter density, and the total energy density of the Universe, respectively) allows to estimate the DM annihilation cross-section into the SM particles and hence to estimate the discovery potential of light dark matter both in direct underground and accelerator experiments. Very crude estimate for the DM annihilation cross section is [2] < σ an v rel >= O(1) pb · c . (3) As a consequence of the formulae (2,3) for fixed values m A and m DM we can estimate the product 2 α D . Note that fixed target NA64 experiment [17,18] uses the reaction of the dark photon electroproduction on nuclei allowing to obtain only upper bounds on 2 vs m A . Therefore, to test the prediction for the 2 α D we have to know either the α D value or at least an upper bound α D ≤ α o on the α D . 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 Λ allow to obtain upper limit on the coupling constant α D [19] in the formula (2) for the annihilation cross section. The bound on α D depends on the scale Λ logarithmically. Moreover, the scale Λ has to be larger than 1 T eV [19]. In this paper we discuss prospects for searching for A dark photon mediator of light dark matter (LDM) production at the NA64 experiment at CERN SPS by using 100 GeV electron ( NA64e) and muon ( NA64µ) beams. The rest of the paper is organized as follows. In Sec. 2 we discuss upper bounds on α D obtained from the requirement of the absence of Landau pole singularity for the effective coupling constantᾱ D (µ) up to some scale Λ. In Sec. 3 we estimate the NA64e discovery potential of LDM and show that with 5 × 10 12 electrons on target (EOT) the experiment is able to probe the scalar and Majorana scenarios of the LDM models. In Sec. 4 we estimate the NA64µ discovery potential of LDM. We show that NA64µ has better sensitivity to the γ −A kinetic mixing for the A masses m A 100 MeV in comparison with NA64e, and that the combined NA64e and NA64µ results obtained with 10 13 EOT and a few 10 13 MOT, respectively, will allow to cover almost fully the most interesting and natural parameter space of the LDM models. This makes the two approaches extremely complementary to each other and greatly increases the discovery potential of NA64. Sec. 5 contains concluding remarks.
In Appendix we collect the main formulae used for approximate DM density calculations.

Upper bound and range of α D
One can obtain upper bound on α D by the requirement of the absence of Landau pole singularity for the effective coupling constantᾱ D (µ) up to some scale Λ [19]. One loop β Here β(ᾱ D ) ≡ µ dᾱ D dµ and n F (n s ) is the number of fermions (scalars) with the U ' (1) charge Q F (Q S ). For the model with pseudo-Dirac fermion we must have additional scalar with Q d = 2 to realize the splitting between fermion masses, so one loop β function is β(ᾱ D ) =  The expression (2) for the annihilation cross section is proportional to factor K = As a consequence in the resonance region m A ≈ 2m DM the annihilation cross section has resonance peak and the dark matter density bound on 2 is proportional to K −1 and for m A ≈ 2m DM the bound on 2 becomes very weak [20] escaping current and future accelerator bounds. It should be mentioned that in general the values of m A and m DM are arbitrary, so the case m A = 2m DM could be considered as some fine tuning.
So it is natural to require the absence of significant fine tuning. Namely, we shall require The NA64e experiment is designed for a sensitive search for the A mediator of sub-GeV dark matter particle (χ) production in the missing energy events from the reaction of 100 GeV electron scattering on heavy nuclei: at the CERN SPS [21,22].

NA64µ projections for the γ − A mixing strength
The NA64µ experiment [23,24] is proposed to search for dark sector particles weakly coupled to the muon, which could explain the muon (g-2) µ anomaly [25,26]. One of the good examples of such a particle, is a new light vector L µ − L τ Z boson [27] - [34], which interacts predominantly with the L µ − L τ current 4 . Interestingly, the Z could also serve as a new leptophilic mediator of dark force between the ordinary and dark matter, which is charged with respect to U (1) Lµ−Lτ , and explain the relic DM abundance [35,36], see also [37]. 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 [38,39,40]. The NA64µ plans to perform a sensitive search for L µ − L τ Z as a mediator of sub-GeV dark matter particle (χ) production in missing energy events from the reaction of 100-160 GeV muon scattering on heavy nuclei: at the CERN SPS [23,24].
In the A models the interaction of dark photon with the leptons and quarks is given Here, J µ SM is the electromagnetic current. So, we see that muons and electrons interact with the dark photon universaly, 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)  Assuming the same signal efficiency the number of A produced by the 100 GeV electron and muon beam can approximated, respectively, as follows where L e X 0 and L µ 40X 0 are the typical distances that are passed by an electron and muon, respectively, before producing the A with the energy E A 50 GeV in the NA64 active Pb target of the total thickness of 40 radiation length (X 0 ) [18,23]. The detail comparison of the calculated A sensitivities of NA64e and NA64µ is shown in Fig.2, where the 90% C.L. limits on the mixing are shown for different number of particles on target for both the NA64e and NA64µ 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. [41], and for the NA64µ case in this work. The later are shown in Fig. 3   NA64e is enhanced for the mass range m e m A 100 MeV . While for the A masses m A 100 MeV NA64µ allows to obtain more stringent limits on 2 compared to NA64e.

Combined LTM sensitivity of NA64e and NA64µ
The reported previously limits on the γ −A mixing strength, allow us to set the combined NA64e and NA64µ constraints on the LDM models, which are shown in the (y; m χ ) plane in Fig.4. As discussed in Sec. I, as a result of the γ − A mixing the cross-section of NA64e + NA64µ Min iBo oN e Min iBo oN e E137 [45], BaBar [46] and MiniBooNE [47] experiments. should be noted that the χ-yield in the NA64 case scales as 2 , not as 4 α D . Therefore, for sufficiently small values of α 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 α D = 0.005. One can see, that for this, or smaller, values of α D , the direct search for LDM in NA64 excludes models of LDM production via vector mediator for the full mass region m χ 0.05 GeV.
While being combined with the NA64µ limit , the NA64 results for the coupling α D = 0.1 exclude the models for the entire mass region m χ 1 GeV.
The upper bounds on also allow to obtain lower bounds on coupling constant α D by using a relation among the parameters derived from the requirement of the thermal freeze-out of DM annihilation into visible matter through γ − A kinetic mixing [19]: GeV the obtained combined NA64e and NA64µ bounds are more stringent than the limits obtained from the results of NA64e allowing to probe almost the full parameter space.
The limits for the Majorana case shown in the lower right panel of Fig. 4 were calculated by setting f = 3, see [18]. Similar to pseudo-Dirac case the the combined NA64e and NA64µ limits exclude the full remained parameter area. Note, that new constraints for the large pseudo-Dirac fermion splitting can also be derived. They will be more stringent than for the case of the small splitting and similar to the one obtained for the Majorana case.

Conclusions
In this paper we considered the NA64 perspectives for discovery of sub-GeV dark matter by running the experiment in electron and muon modes at the CERN SPS. While with 5×10 12 EOT NA64e is able to test the scalar and Majorana LDM scenarios, the combined NA64e and NA64µ results with 10 13 EOT and a few 10 13 MOT, respectively, will allow to cover almost fully the parameter space of the all most interesting LDM models. This makes NA64e and NA64µ extremely complementary to each other, as well as to the planned LDMX experiment [48], and greatly increases the NA64 discovery potential of sub-GeV DM. and f d (p, T ) is the dark matter distribution function.
Here x f = m DM T d and n = 0 for s-wave and n = 1 for p-wave annihilation. If DM particles differ from DM antiparticles σ o = σan 2 .