Search for Dark Sector Physics with NA64

Abstract

The NA64 experiment consists of two detectors which are planned to be located at the electron (NA64e) and muon (NA64μ) beams of the CERN SPS and start operation after the LHC long-stop 2 in 2021. Its main goals include searches for dark sector physics—particularly light dark matter (LDM), visible and invisible decays of dark photons ($A{\kern 1pt} '$), and new light particles that could explain the ⁸Be and ${{g}_{\mu }} - 2$ anomalies. Here we review these physics goals, the current status of NA64 including recent results and perspectives of further searches, as well as other ongoing or planned experiments in this field. The main theoretical results on LDM, the problem of the origin of the $\gamma {\text{ - }}A{\kern 1pt} '$ mixing term and its connection to loop corrections, possible existence of a new light $Z{\kern 1pt} '$ coupled to ${{L}_{\mu }} - {{L}_{\tau }}$ current are also discussed.

1 INTRODUCTION

At present the most striking evidence in favour of new physics beyond the Standard model (SM) is the observation of Dark Matter (DM) [1, 2]. The nature of DM is one of challenging questions in physics. If DM is a thermal relic from the hot early Universe then its existence motivates to look for models with nongravitational interactions between dark and ordinary matter. There is a lot of candidates for the role of dark matter [1, 2]. In particular, there are LDM(light dark matter) models [3–7] with the mass of DM particles $\leqslant {\kern 1pt} {\kern 1pt} 0(1)\,\,{\text{GeV}}$. LDM particles with masses below $0(1)\,\,{\text{GeV}}$ were generally expected to be ruled out because they overclose the Universe [8]. However there are models [3–7] with additional light vector boson and LDM particles that avoid the arguments [8] excluding the LDM. The standard assumption that in the hot early Universe the DM particles are in equilibrium with ordinary matter is often used. During the Universe expansion the temperature decreases and at some point the thermal decoupling of the DM starts to work. Namely, at some freeze-out temperature the annihilation cross-section of DM paricles

$${\text{DM}}\,\,{\text{particles}} \to {\text{SM}}\,\,{\text{particles}}$$

becomes too small to obey the equilibrium of DM particles with the SM particles and the DM decouples. The experimental data are in favour of scenario with cold relic for which the freeze-out temperature is much lower than the mass of the DM particle. In other words the DM particles decouple in non-relativistic regime. The value of the DM annihilation cross-section at the decoupling epoch determines the value of the current DM density in the Universe. Too big annihilation cross-section leads to small DM density and vise versa too small annihilation cross section leads to DM overproduction. The observed value of the DM density fraction $\tfrac{{{{\rho }_{d}}}}{{{{\rho }_{c}}}} \approx 0.23$ [9] allows to estimate the DM annihilation cross-section into the SM particles and hence to estimate the discovery potential of the LDM both in direct underground and accelerator experiments. Namely, the annihilation cross-section leading to the correct DM density is estimated to be ${{\sigma }_{{{\text{an}}}}} \sim 1\,\,{\text{pbn}}$ and the value of the cross-section depends rather weakly on the DM mass [1, 2]. Models with the LDM (${{m}_{\chi }} \leqslant 1\,\,{\text{GeV}}$) can be classified by the spins and masses of the DM particles and mediator. The scalar DM mediator models are severely restricted [10, 11] but not completely excluded by rare $K$- and $B$-meson decays. Models with light vector bosons [4, 12, 13] (vector portal) are rather popular now. In these models light vector boson $A{\kern 1pt} '$ mediates between our world and the dark sector [4]. Another possible hint in favour of new physics is the muon ${{g}_{\mu }} - 2$ anomaly which is the 3.6σ discrepancy between the experimental values [14, 9] and the SM predictions [15–18] for the anomalous magnetic moment of the muon. Among several extensions of the SM explaining the ${{g}_{\mu }} - 2$ anomaly, the models predicting the existence of a weak leptonic force mediated by a sub-GeV gauge boson $Z{\kern 1pt} '$ that couples predominantly to the difference between the muon and tau lepton currents, ${{L}_{\mu }} - {{L}_{\tau }}$, are of general interest. The abelian symmetry ${{L}_{\mu }} - {{L}_{\tau }}$ is an anomaly-free global symmetry within the SM [19–21]. The ${{L}_{\mu }} - {{L}_{\tau }}$ gauge symmetry breaking is crucial for the appearance of a new relatively light, with a mass ${{m}_{{Z{\kern 1pt} '}}} \leqslant 1\,\,{\text{GeV}}$, vector boson ($Z{\kern 1pt} '$) which couples very weakly to muon and tau-lepton with the coupling constant ${{\alpha }_{\mu }} \sim O({{10}^{{ - 8}}})$ [22–25] and explain muon ${{g}_{\mu }} - 2$ anomaly. Recent claim [26] of the discovery of $17\,\,{\text{MeV}}$ vector particle observed as a peak in ${{e}^{ + }}{{e}^{ - }}$ invariant mass distribution in nuclear transitions makes the question of possible light vector boson existence extremely interesting and important and enhance motivation for the experimental searches at low energy intensity frontier.

At present the most popular vector mediator model is the model with additional light vector boson $A{\kern 1pt} '$ (dark photon) [4, 13] which couples to the SM electromagnetic current. However other light vector boson models, in particular, model with ${{L}_{\mu }} - {{L}_{\tau }}$ interaction [27–30], are possible as messenger candidates beetween our world and DM world.

The aim of this paper is review of the search for LDM at the NA64 fixed target experiment [31–35] at CERN and related current and future experiments on the search for LDM. Also we review essential part of the phenomenology related with the LDM models. The paper is organized as follows. In section 2 we describe phenomenology of the dark photon model. In particular, we discuss the bound on low energy effective coupling constant ${{\bar {\alpha }}_{d}}({{m}_{{A{\kern 1pt} '}}}) \equiv {{\alpha }_{d}}$ derived from the requirement of the absence of Landau pole singularity up to some scale ${{\Lambda }_{{{\text{pole}}}}}$. We present the main formulae for the $A{\kern 1pt} '$ electroproduction reaction $eZ \to eZA{\kern 1pt} '$ on nuclei. We review muon ${{g}_{\mu }} - 2$ anomaly and the possibility to explain it due to existence of new light vector boson interacting with muons. Also we discuss the problem of the origin of photon-dark photon mixing term $\tfrac{\epsilon }{2}{{F}^{{\mu \nu }}}F_{{\mu \nu }}^{'}$ and its connection with loop corrections. In sections 3 we review current accelerator and nonaccelerator bounds including experiments on direct LDM detection. In section 4 we describe the NA64 experiment on the search for both invisible and visible $A{\kern 1pt} '$ boson decay. In section 5 we review the last NA64 results and discuss future NA64 perspectives on the search for LDM and, in particular, we discuss the NA64 LDM discovery potential with the use of muon beam. In section 6 we outline some other future experiments related with the search for dark photon and LDM at NA64. Section 7 contains the main conclusions. In Appendix A we collect the main formulae used for the approximate DM density calculations. In Appendix B we discuss the discovery potential of NA64 for the case of visible dark photon $A{\kern 1pt} '$ decays $A{\kern 1pt} ' \to {{\chi }_{1}}{{\chi }_{2}} \to {{e}^{ + }}{{e}^{ - }}{{\chi }_{1}}{{\chi }_{1}}$ with large missing energy.

2 A LITTLE BIT OF THEORY

2.1 Model with Dark Photon

In model with “dark photon” [4, 13] new light vector boson (dark photon) $A{\kern 1pt} '$ interacts with the Standard $S{{U}_{c}}(3) \otimes S{{U}_{L}}(2) \otimes U(1)$ gauge model only due to kinetic mixing with $U{\kern 1pt} '(1)$ gauge field $A_{\mu }^{'}$. Dark photon interacts also with LDM. In renormalizable models DM particles have spin 0 or 1/2. The Lagrangian of the model has the form

$$L = {{L}_{{{\text{SM}}}}} + {{L}_{{{\text{SM}}{\text{,dark}}}}} + {{L}_{{{\text{dark}}}}},$$

(1)

where ${{L}_{{{\text{SM}}}}}$ is the SM Lagrangian,

$${{L}_{{{\text{SM}}{\text{,dark}}}}} = - \frac{\epsilon }{{2cos{{\theta }_{w}}}}{{B}^{{\mu \nu }}}F_{{\mu \nu }}^{'},$$

(2)

${{B}^{{\mu \nu }}} = {{\partial }^{\mu }}{{B}^{\nu }} - {{\partial }^{\nu }}{{B}^{\nu }}$, $F_{{\mu \nu }}^{'} = {{\partial }_{\mu }}A_{\nu }^{'} - {{\partial }_{\nu }}A_{\mu }^{'}$ and the ${{L}_{{{\text{dark}}}}}$ is the DM Lagrangian¹. For Dirac LDM $\chi$ the DM Lagrangian is

$$\begin{gathered} {{L}_{{{\text{dark}}}}} = - \frac{1}{4}F_{{\mu \nu }}^{'}{{F}^{{'\mu \nu }}} + i\bar {\chi }{{\gamma }^{\mu }}{{\partial }_{\mu }}\chi - {{m}_{\chi }}\bar {\chi }\chi \\ + \,\,{{e}_{d}}\bar {\chi }{{\gamma }^{\mu }}\chi A_{\mu }^{'} + \frac{{m_{{A{\kern 1pt} '}}^{2}}}{2}A_{\mu }^{'}{{A}^{{'\mu }}}, \\ \end{gathered} $$

(3)

The abelian gauge symmetry

$$A_{\mu }^{'} \to A_{\mu }^{'} + {{\partial }_{\mu }}\alpha ,$$

(4)

$$\chi \to \exp (i{{e}_{d}}\alpha )\chi $$

(5)

is explicitly broken due to the mass term $\tfrac{{m_{{A{\kern 1pt} '}}^{2}}}{2}A_{\mu }^{'}{{A}^{{'\mu }}}$ in the Lagrangian (3). However we can use the Higgs mechanism for dark photon $A_{\mu }^{'}$ mass creation, namely we can use the Lagrangian

$${{L}_{\phi }} = ({{\partial }_{\mu }}\phi - i{{e}_{d}}A_{\mu }^{'}\phi )({{\partial }^{\mu }}\phi - i{{e}_{d}}{{A}^{{'\mu }}}\phi ){\text{*}} - \lambda {{(\phi {\text{*}}\phi - {{c}^{2}})}^{2}}.$$

(6)

Here ϕ is scalar field. The spontaneous breaking of the gauge symmetry (4), (5) due to $\left\langle \phi \right\rangle \ne 0$ leads to nonzero dark photon mass. As a consequence of the mixing term ${{L}_{{{\text{SM}}{\text{,dark}}}}} = - \tfrac{\epsilon }{{2cos{{\theta }_{w}}}}{{B}^{{\mu \nu }}}F_{{\mu \nu }}^{'}$ the low energy interaction between dark photon $A_{\mu }^{'}$ and the SM fermions is described by the effective Lagrangian

$${{L}_{{A{\kern 1pt} ',{\text{SM}}}}} = \epsilon eA_{\mu }^{'}J_{{em}}^{\mu },$$

(7)

where $J_{{em}}^{\mu }$ is the SM electromagnetic current. The invisible and visible decay rates of $A{\kern 1pt} '$ for fermion DM particles $\chi$ are given by

$$\Gamma (A{\kern 1pt} ' \to \chi \bar {\chi }) = \frac{{{{\alpha }_{{\text{D}}}}}}{3}{{m}_{{A{\kern 1pt} '}}}\left( {1 + \frac{{2m_{\chi }^{2}}}{{m_{{A{\kern 1pt} '}}^{2}}}} \right)\sqrt {1 - \frac{{4m_{\chi }^{2}}}{{m_{{A{\kern 1pt} '}}^{2}}}} ,$$

(8)

$$\Gamma (A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}) = \frac{{{{\epsilon }^{2}}\alpha }}{3}{{m}_{{A{\kern 1pt} '}}}\left( {1 + \frac{{2m_{e}^{2}}}{{m_{{A{\kern 1pt} '}}^{2}}}} \right)\sqrt {1 - \frac{{4m_{e}^{2}}}{{m_{{A{\kern 1pt} '}}^{2}}}} .$$

(9)

Here $\alpha = \tfrac{{{{e}^{2}}}}{{4\pi }} = {1 \mathord{\left/ {\vphantom {1 {137}}} \right. \kern-0em} {137}}$ and ${{\alpha }_{a}} = \tfrac{{e_{d}^{2}}}{{4\pi }}$ is the analog of the electromagnetic fine coupling constant for dark photon. For scalar DM particles $\chi$ the invisible decay width is

$$\Gamma (A{\kern 1pt} ' \to \chi \chi *) = \frac{{{{\alpha }_{{\text{D}}}}}}{{12}}{{m}_{{A{\kern 1pt} '}}}\left( {1 - 4\frac{{m_{\chi }^{2}}}{{m_{{A{\kern 1pt} '}}^{2}}}} \right)\sqrt {1 - \frac{{4m_{\chi }^{2}}}{{m_{{A{\kern 1pt} '}}^{2}}}} .$$

(10)

2.2 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 $p$ up to some scale $n = 0$ [36]. One loop $\beta$-function for $\mathop {\bar {\alpha }}\nolimits_D (\mu )$ is

$$\beta ({{\bar {\alpha }}_{{\text{D}}}}) = \frac{{\bar {\alpha }_{{\text{D}}}^{2}}}{{2\pi }}\left[ {\frac{4}{3}\left( {Q_{F}^{2}{{n}_{F}} + Q_{S}^{2}\frac{{{{n}_{S}}}}{4}} \right)} \right].$$

(11)

Here $\beta ({{\bar {\alpha }}_{{\text{D}}}}) \equiv \mu \tfrac{{d{{{\bar {\alpha }}}_{{\text{D}}}}}}{{d\mu }}$ and ${{n}_{F}}$ (${{n}_{s}}$) is the number of fermions (scalars) with the $U{\kern 1pt} '(1)$ charge ${{Q}_{F}}({{Q}_{S}})$. For the model with pseudo-Dirac fermion [37] we have to introduce an additional scalar with ${{Q}_{S}} = 2$ to realize nonzero splitting between fermion masses, so one loop $\beta$-function is $\beta ({{\bar {\alpha }}_{{\text{D}}}}) = \tfrac{{4\bar {\alpha }_{{\text{D}}}^{2}}}{{3\pi }}$. For the model with Majorana fermion we also have to 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 DM to create nonzero dark photon mass in a gauge invariant way we have to introduce additional scalar field with ${{Q}_{S}} = 1$, so one loop $\beta$-function is $\beta = {{{{\alpha }^{2}}} \mathord{\left/ {\vphantom {{{{\alpha }^{2}}} {3\pi }}} \right. \kern-0em} {3\pi }}$. From the requirement that $\Lambda \geqslant 1$ TeV [36] we find that ${{\alpha }_{{\text{D}}}} \leqslant 0.2$ for pseudo-Dirac and Majorana fermions and ${{\alpha }_{{\text{D}}}} \leqslant 0.8$ for charged scalars ². Here ${{\alpha }_{{\text{D}}}}$ is an effective low energy coupling constant at scale $\mu \sim {{m}_{{A{\kern 1pt} '}}}$, i.e. ${{\alpha }_{{\text{D}}}} = {{\bar {\alpha }}_{{\text{D}}}}({{m}_{{A{\kern 1pt} '}}})$. In our calculations as a reper point we used the value ${{m}_{{A{\kern 1pt} '}}} = 10$ MeV. In the assumption that dark photon model is valid up to Planck scale, i.e. $\Lambda = {{M}_{{{\text{PL}}}}} = 1.2 \times {{10}^{{19}}}$ GeV, we find that for pseudo-Dirac and Majorana fermions ${{\alpha }_{{\text{D}}}} \leqslant 0.05$ while for scalars ${{\alpha }_{{\text{D}}}} \leqslant 0.2$. In the SM the $S{{U}_{c}}(3)$, $S{{U}_{L}}(2)$ and $U(1)$ gauge coupling constants are equal to $\sim {\kern 1pt} ({1 \mathord{\left/ {\vphantom {1 {30}}} \right. \kern-0em} {30}} - {1 \mathord{\left/ {\vphantom {1 {50}}} \right. \kern-0em} {50}})$ at the Planck scale. It is natural to assume that the effective gauge coupling ${{\bar {\alpha }}_{{\text{D}}}}(\mu = {{M}_{{{\text{PL}}}}})$ is of the order of $S{{U}_{c}}(3)$, $S{{U}_{L}}(2)$ and $U(1)$ gauge coupling constants, i.e. ${{\bar {\alpha }}_{{\text{D}}}}(\mu = {{M}_{{{\text{PL}}}}}) \sim ({1 \mathord{\left/ {\vphantom {1 {30}}} \right. \kern-0em} {30}} - {1 \mathord{\left/ {\vphantom {1 {50}}} \right. \kern-0em} {50}})$. As a result of this assumption we find that the values of the low energy coupling ${{\alpha }_{{\text{D}}}}$ in the range ${{\alpha }_{{\text{D}}}} \sim (0.015{\kern 1pt} - {\kern 1pt} 0.02)$ are the most natural.

2.3 Some Comments on the Origin of the Mixing Parameter $\epsilon$

In Holdom paper [13]³ the origin of the mixing $\epsilon$ parameter was assumed to be related with radiative corrections. To clarify this statement consider the simplest model with two free $U(1) \otimes U{\kern 1pt} '(1)$ gauge fields ${{A}_{\mu }}$ and $A_{\mu }^{'}$. The Lagrangian of the model is

$$\begin{gathered} {{L}_{o}} = - \frac{1}{4}{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}} - \frac{1}{4}F_{{\mu \nu }}^{'}{{F}^{{'\mu }}} \\ + \,\,\frac{{m_{{0,A{\kern 1pt} '}}^{2}}}{2}{{A}^{{'\mu }}}A_{\mu }^{'} - \frac{1}{2}{{\epsilon }_{{0{\kern 1pt} l}}}F_{{\mu \nu }}^{'}{{F}^{{\mu \nu }}}, \\ \end{gathered} $$

(12)

where ${{F}_{{\mu \nu }}} = {{\partial }_{\mu }}{{A}_{\nu }} - {{\partial }_{\nu }}{{A}_{\mu }}$ and $F_{{\mu \nu }}^{'} = {{\partial }_{\mu }}A_{\nu }^{'} - {{\partial }_{\nu }}A_{\mu }^{'}$. For ${{\epsilon }_{{0{\kern 1pt} l}}} = 0$ the Lagrangian (12) is invariant under two independent discrete symmetries ${{A}_{\mu }} \to - {{A}_{\mu }}$ and $A_{\mu }^{'} \to - A_{\mu }^{'}$. After diagonalization we find that the spectrum of the model for $\left| {{{\epsilon }_{{0{\kern 1pt} l}}}} \right| \ll 1$ consists of massless vector particle(photon) and massive vector particle(dark photon) with a mass $m_{{A{\kern 1pt} '}}^{2} = m_{{0,A{\kern 1pt} '}}^{2}(1 + \epsilon _{{0l}}^{2})$. Let us add to the model massive fermion field ${{\psi }_{{\text{M}}}}$ with a mass $M$ which interacts both with ${{A}_{\mu }}$ and $A_{\mu }^{'}$ with the interaction Lagrangian

$$\Delta L = e\bar {\psi }{{\gamma }^{\mu }}\psi {{A}_{\mu }} + e{\kern 1pt} '\bar {\psi }{{\gamma }^{\mu }}\psi A_{\mu }^{'}.$$

(13)

At one-loop level the propagator $\int {{{e}^{{ipx}}}} \left\langle {T({{A}_{\mu }}(x){{A}_{\nu }}(0))} \right\rangle {{d}^{4}}x$ depends on virtual momentum ${{p}^{2}}$. It means that one-loop correction ${{\epsilon }_{{1l}}}$ depends on virtual momentum ${{p}^{2}}$, namely

$${{\epsilon }_{{1l}}}({{p}^{2}}) = \frac{{ee{\kern 1pt} '}}{{16{{\pi }^{2}}}}\int\limits_{ - 1}^1 {(1 - {{\eta }^{2}})} \ln \left[ {\frac{{4{{M}^{2}} - {{p}^{2}}(1 - {{\eta }^{2}})}}{{{{\mu }^{2}}}}} \right]d\eta .$$

(14)

Here $\mu$ is some renormalization point, so one-loop contribution to the tree level ${{\epsilon }_{{0l}}}$ parameter depends on the renormalization scheme. To our mind the most natural choice of the renormalization point $\mu$ is to require that radiative corrections to the tree level ${{\epsilon }_{{0l}}}$ parameter vanish at the $A{\kern 1pt} '$ mass shell

$${{\epsilon }_{{1l}}}({{p}^{2}} = m_{{0,A{\kern 1pt} '}}^{2}) = 0.$$

(15)

The renormalization condition (15) guarantees us that radiative corrections don’t modify the tree level formula $m_{{A{\kern 1pt} '}}^{2} = m_{{0,A{\kern 1pt} '}}^{2}(1 + \epsilon _{{0l}}^{2})$ for the pole dark photon mass. The renormalization condition (15) leads to well defined value of the $\epsilon$ parameter at one-loop level

$$\begin{gathered} {{\epsilon }_{{0 + 1l}}}({{p}^{2}}) = {{\epsilon }_{{0l}}} + \frac{{ee{\kern 1pt} '}}{{16{{\pi }^{2}}}} \\ \times \,\,\int\limits_{ - 1}^1 {(1 - {{\eta }^{2}})} \ln \left[ {\frac{{4{{M}^{2}} - {{p}^{2}}(1 - {{\eta }^{2}})}}{{4{{M}^{2}} - m_{{0,A{\kern 1pt} '}}^{2}(1 - {{\eta }^{2}})}}} \right]d\eta {\kern 1pt} . \\ \end{gathered} $$

(16)

For the normalization condition (15) one-loop contribution to the ${{\epsilon }_{{0l}}}$ parameter vanishes as ${{\epsilon }_{{1l}}} \sim \tfrac{1}{{{{M}^{2}}}}$ for large fermion masses $M \gg {{m}_{{0,A{\kern 1pt} '}}}$ that agrees with the decoupling expectations. For the model with two massive fermions ${{\psi }_{1}}$, ${{\psi }_{2}}$ with masses ${{M}_{1}}$, ${{M}_{2}}$, the charges $e,e{\kern 1pt} '$ and $e, - e{\kern 1pt} '$ one-loop correction to the ${{\varepsilon }_{{0l}}}$ parameter is ultraviolet finite and it does not depend on the renormalization point $\mu$

$$\begin{gathered} \epsilon _{{1l}}^{{{\text{naive}}}}({{p}^{2}}) = \frac{{ee{\kern 1pt} '}}{{16{{\pi }^{2}}}} \\ \times \,\,\int\limits_{ - 1}^1 {(1 - {{\eta }^{2}})} \ln \left[ {\frac{{4M_{1}^{2} - {{p}^{2}}(1 - {{\eta }^{2}})}}{{4M_{2}^{2} - {{p}^{2}}(1 - {{\eta }^{2}})}}} \right]d\eta {\kern 1pt} . \\ \end{gathered} $$

(17)

However the $\epsilon _{{1l}}^{{{\text{naive}}}}(0) = \tfrac{{ee{\kern 1pt} '}}{{12{{\pi }^{2}}}}\ln \left[ {\tfrac{{M_{1}^{2}}}{{M_{2}^{2}}}} \right]$ does not vanish for ${{M}_{1}} \to \infty$, ${{M}_{2}} \to \infty$ in contradiction with naive decoupling expectations. To cure this situation we can add one-loop finite counter-term $- \tfrac{{{{\Delta }_{{1l}}}}}{2}F_{{\mu \nu }}^{'}{{F}^{{\mu \nu }}}$ to the Lagrangian (12) with ${{\Delta }_{{1l}}} = - {{\epsilon }_{{1l}}}({{p}^{2}} = m_{{A{\kern 1pt} '}}^{2})$, so one-loop expression for ${{\epsilon }_{{1l}}}({{p}^{2}})$ reads

$${{\epsilon }_{{1l}}}({{p}^{2}}) = \epsilon _{{1l}}^{{{\text{naive}}}}({{p}^{2}}) - {{\epsilon }^{{{\text{naive}}}}}({{p}^{2}} = m_{{0,A{\kern 1pt} '}}^{2}).$$

(18)

One can find that ${{\epsilon }_{{1l}}}(0) \to 0$ for ${{M}_{1}} \to \infty$, ${{M}_{2}} \to \infty$ in accordance with decoupling expectations. Let us formulate our main conclusion—within the abelian $U(1) \otimes U{\kern 1pt} '(1)$ gauge model we can’t predict the value of the mixing parameter $\epsilon$ and to our mind the most natural renormalization scheme is based on the use of the condition that loop corrections to the $\epsilon ({{p}^{2}})$ vanish at the $A{\kern 1pt} '$ mass shell, so ${{\epsilon }_{{0l}}}$ is free arbitrary parameter of the model.

The situation with the $\epsilon$ prediction changes drastically if we assume that one of the $U(1)$ abelian gauge groups arises due to gauge symmetry breaking of nonabelian gauge group. As a simplest example consider the model where dark photon originates from $SU{\kern 1pt} '(2)$ gauge symmetry breaking $SU{\kern 1pt} '(2) \to U{\kern 1pt} '(1)$. The unbroken $U(1) \otimes SU{\kern 1pt} '(2)$ gauge symmetry prohibits the mixing term $- \tfrac{\epsilon }{2}{{F}^{{\mu \nu }}}F_{{\mu \nu }}^{'}$. Suppose $SU{\kern 1pt} '(2)$ gauge symmetry is broken to $U{\kern 1pt} '(1)$ due to the Higgs field ${{\Phi }_{b}}$$(b = 1,2,3)$ in adjoint representation. The $U(1) \otimes U{\kern 1pt} '(1)$ mixing term arises as a result of $SU{\kern 1pt} '(2)$ breaking due to the effective term $\tfrac{{{{\Phi }_{a}}}}{\Lambda }F_{{\mu \nu }}^{{'{\kern 1pt} a}}{{F}^{{\mu \nu }}}$. Suppose we have doublet(under $SU{\kern 1pt} '(2)$) of vector-like fermions ${{\psi }_{a}}$$(a = 1,2)$ with the mass $M$ and the $U(1)$ charge $e$. The Yukawa interaction of vector-like fermions with scalar triplet ${{\Phi }_{b}}$ is ${{L}_{{{\text{Yuk}}}}} = - h{{\Phi }_{a}}\bar {\psi }{{\sigma }_{a}}\psi$. Nonzero vacuum expectation value $\left\langle {{{\Phi }_{3}}} \right\rangle \ne 0$ leads to $SU{\kern 1pt} '(2) \to U{\kern 1pt} '(1)$ gauge symmetry breaking and to the splitting of fermion masses for fermion doublet ${{\psi }_{a}}$, namely ${{M}_{{1,2}}} = M \pm h\left\langle {{{\Phi }_{3}}} \right\rangle$. As a consequence of fermion doublet mass splitting we find nonzero one-loop contribution to the $\epsilon$ parameter, namely

$${{\varepsilon }_{{1l}}} = \frac{{eg}}{{6{{\pi }^{2}}}}{\text{ln}}\left[ {\frac{{M + h\left\langle {{{\Phi }_{3}}} \right\rangle }}{{M - h\left\langle {{{\Phi }_{3}}} \right\rangle }}} \right].$$

(19)

Here $g$ is the $SU(2)$ gauge coupling. The expresssion (19) vanishes for $\left\langle {{{\Phi }_{3}}} \right\rangle = 0$ and for $M \to \infty$. For $M \gg h\left\langle {{{\Phi }_{3}}} \right\rangle$ the $\varepsilon$ parameter is

$${{\epsilon }_{{1l}}} = \frac{{eg}}{{6{{\pi }^{2}}}}\frac{{2h\left\langle {{{\Phi }_{3}}} \right\rangle }}{M}.$$

(20)

So we find that for the model with nonabelian extension of one of the $U(1)$ gauge groups the $\epsilon$ parameter arises as a result of nonabelian gauge symmetry breaking and in principle but not in practise we can predict it as a function of the parameters of the model. To conclude we can say that at present state of art we can’t predict reliably the value of the $\varepsilon$ parameter.

2.4 Dark Photon Production

There are several $A{\kern 1pt} '$ production mechanisms [4]. In proton nucleus collisions the $A{\kern 1pt} '$ are produced mainly in ${{{{\pi }^{0}}} \mathord{\left/ {\vphantom {{{{\pi }^{0}}} \eta }} \right. \kern-0em} \eta }$ decays ${{{{\pi }^{0}}} \mathord{\left/ {\vphantom {{{{\pi }^{0}}} \eta }} \right. \kern-0em} \eta } \to \gamma A{\kern 1pt} '$. The use of visible $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ decay allows to detect dark photon $A{\kern 1pt} '$ as a peak in the ${{e}^{ + }}{{e}^{ - }}$ invariant mass distribution. Also direct $A{\kern 1pt} '$ production in proton nucleus collisions is possible in full analogy with the photoproduction in proton nucleus collisions.

Other perspective way is the $A{\kern 1pt} '$ production in electron nucleus interactions, namely the use of the reaction

$${{e}^{ - }}(p) + Z({{P}_{i}}) \to {{e}^{ - }}(p{\kern 1pt} ') + Z({{P}_{f}}) + A{\kern 1pt} '(k).$$

(21)

Here $p = ({{E}_{0}},\vec {p})$ is the 4-momentum of incoming electron, ${{P}_{i}} = (M,0)$ denotes the $Z$ nucleus 4‑momentum in the initial state, final state $Z$ nucleus momentum is defined by ${{P}_{f}} = (P_{f}^{0},{{\vec {P}}_{f}})$, the $A{\kern 1pt} '$-boson momentum is $k = ({{k}_{0}},\vec {k})$ and $p{\kern 1pt} ' = (e{\kern 1pt} ',\vec {p}{\kern 1pt} ')$ is the momentum of electron recoil. In the improved Weizsacker–Williams (IWW) approximation the differential and total cross-sections for the reaction (21) for ${{m}_{{A{\kern 1pt} '}}} \gg {{m}_{e}}$ can be written⁴ [39] as

$$\begin{gathered} \frac{{d\sigma _{{WW}}^{{A{\kern 1pt} '}}}}{{dx}} = (4{{\alpha }^{3}}{{\epsilon }^{2}}{{\chi }_{{{\text{eff}}}}}) \\ \times \,\,(1 - x + {{{{x}^{2}}} \mathord{\left/ {\vphantom {{{{x}^{2}}} 3}} \right. \kern-0em} 3}){{\left( {m_{{A{\kern 1pt} '}}^{2}\frac{{1 - x}}{x} + m_{e}^{2}x} \right)}^{{ - 1}}}, \\ \end{gathered} $$

(22)

$$\sigma _{{WW}}^{{A{\kern 1pt} '}} = \frac{4}{3}\frac{{{{\epsilon }^{2}}{{\alpha }^{3}}}}{{m_{{A{\kern 1pt} '}}^{2}}}{{\chi }_{{{\text{eff}}}}}{\text{log}}(\delta _{{A{\kern 1pt} '}}^{{ - 1}}),$$

(23)

$${{\delta }_{{A{\kern 1pt} '}}} = {\text{max}}\left[ {\frac{{m_{e}^{2}}}{{m_{{A{\kern 1pt} '}}^{2}}},\frac{{m_{{A{\kern 1pt} '}}^{2}}}{{E_{0}^{2}}}} \right],$$

(24)

where ${{\chi }_{{{\text{eff}}}}}$ is an effective flux of photons

$${{\chi }_{{{\text{eff}}}}} = \int\limits_{{{t}_{{{\text{min}}}}}}^{{{t}_{{{\text{max}}}}}} {dt} \frac{{t - {{t}_{{{\text{min}}}}}}}{{{{t}^{2}}}}[G_{2}^{{{\text{el}}}}(t) + G_{2}^{{{\text{inel}}}}(t)],$$

(25)

and $x = \tfrac{{{{E}_{{A{\kern 1pt} '}}}}}{{{{E}_{o}}}}$. Here ${{t}_{{{\text{min}}}}} = {{m_{{A{\kern 1pt} '}}^{4}} \mathord{\left/ {\vphantom {{m_{{A{\kern 1pt} '}}^{4}} {4E_{0}^{2}}}} \right. \kern-0em} {4E_{0}^{2}}}$, ${{t}_{{{\text{max}}}}} = m_{{A{\kern 1pt} '}}^{2} + m_{e}^{2}$ and $G_{2}^{{{\text{el}}}}(t)$, $G_{2}^{{{\text{inel}}}}(t)$ are elastic and inelastic form-factors respectively. For NA64 energies $E \leqslant 100\,\,{\text{GeV}}$ the elastic form-factor dominates. The elastic form-factor can be represented in the form [39]

$$G_{2}^{{{\text{el}}}} = {{\left( {\frac{{{{a}^{2}}t}}{{1 + {{a}^{2}}t}}} \right)}^{2}}{{\left( {\frac{1}{{1 + {t \mathord{\left/ {\vphantom {t d}} \right. \kern-0em} d}}}} \right)}^{2}}{{Z}^{2}},$$

(26)

where $a = {{111{{Z}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 3}} \right. \kern-0em} 3}}}}} \mathord{\left/ {\vphantom {{111{{Z}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 3}} \right. \kern-0em} 3}}}}} {{{m}_{e}}}}} \right. \kern-0em} {{{m}_{e}}}}$, $d = 0.164\,\,{\text{Ge}}{{{\text{V}}}^{2}}{{A}^{{ - 2/3}}}$ and $A$ is atomic number of nuclei. We consider the quasielastic reaction (21) so the inelastic nuclear formfactor is not taken into account. Numerically, ${{\chi }_{{{\text{eff}}}}} = {{Z}^{2}}\log$, where the function $\log \sim (5{\kern 1pt} - {\kern 1pt} 10)$ and it depends weakly on atomic screening, nuclear size effects and kinematics.

2.5 Muon ${{g}_{\mu }} - 2$ Anomaly and the Light Vector Boson $Z{\kern 1pt} '$

Recent precise measurement of the anomalous magnetic moment of the positive muon ${{a}_{\mu }} = ({{g}_{\mu }} - {{2)} \mathord{\left/ {\vphantom {{2)} 2}} \right. \kern-0em} 2}$ from Brookhaven AGS experiment 821 [14] gives result which is about $3.6\sigma$ higher [42, 43] than the SM prediction

$$a_{\mu }^{{{\text{exp}}}} - a_{\mu }^{{{\text{SM}}}} = 288(80) \times {{10}^{{ - 11}}}.$$

(27)

This result may signal the existence of new physics beyond the SM. New light (with a mass ${{m}_{{Z{\kern 1pt} '}}} \leqslant O(1)\,\,{\text{GeV}}$) vector boson (dark photon) which couples very weakly with muon with ${{\alpha }_{{Z{\kern 1pt} '}}} \sim O({{10}^{{ - 8}}})$ can explain ${{g}_{\mu }} - 2$ anomaly [22–25]. Vector-like interaction of $Z{\kern 1pt} '$ boson with muon

$${{L}_{{Z{\kern 1pt} '}}} = g{\kern 1pt} '\bar {\mu }{{\gamma }^{\nu }}\mu Z_{\nu }^{'}$$

(28)

leads to additional contribution to muon anomalous magnetic moment [43]

$$\Delta a = \frac{{\alpha {\kern 1pt} '}}{{2\pi }}F\left( {\frac{{{{m}_{{Z{\kern 1pt} '}}}}}{{{{m}_{\mu }}}}} \right),$$

(29)

where

$$F(x) = \int\limits_0^1 {dz} \frac{{[2z{{{(1 - z)}}^{2}}]}}{{[{{{(1 - z)}}^{2}} + {{x}^{2}}z]}},$$

(30)

and $\alpha {\kern 1pt} ' = \tfrac{{{{{(g{\kern 1pt} ')}}^{2}}}}{{4\pi }}$. The relations (29, 30) allow to determine the coupling constant $\alpha {\kern 1pt} '$ which explains the value (27) of muon anomaly. For ${{m}_{{Z{\kern 1pt} '}}} \ll {{m}_{\mu }}$ one can find that

$$\alpha {\kern 1pt} ' = (1.8 \pm 0.5) \times {{10}^{{ - 8}}}.$$

(31)

For another limiting case ${{m}_{{Z{\kern 1pt} '}}} \gg {{m}_{\mu }}$ the $\alpha {\kern 1pt} '$ is

$$\alpha {\kern 1pt} ' = (2.7 \pm 0.7) \times {{10}^{{ - 8}}} \times \frac{{m_{{Z{\kern 1pt} '}}^{2}}}{{m_{\mu }^{2}}}.$$

(32)

However the postulation of the interaction (28) is not the end of the story. The main question: what about the interaction of the $Z{\kern 1pt} '$ boson with other quarks and leptons? The renormalizable $Z{\kern 1pt} '$ interaction with the SM fermions ${{\psi }_{k}}$$({{\psi }_{k}} = e,{{\nu }_{e}},u,d,...)$ has the form

$${{L}_{{Z{\kern 1pt} '}}} = g{\kern 1pt} 'Z_{\mu }^{'}J_{{Z{\kern 1pt} '}}^{\mu },$$

(33)

$$J_{{Z{\kern 1pt} '}}^{\mu } = \sum\limits_k {[{{q}_{{Lk}}}{{{\bar {\psi }}}_{{Lk}}}{{\gamma }^{\mu }}{{\psi }_{{Lk}}} + {{q}_{{Rk}}}{{{\bar {\psi }}}_{{Rk}}}{{\gamma }^{\mu }}{{\psi }_{{Rk}}}]} ,$$

(34)

where ${{\psi }_{{Lk,Rk}}} = \tfrac{1}{2}(1 \mp {{\gamma }_{5}}){{\psi }_{k}}$ and ${{q}_{{Lk}}},{{q}_{{Rk}}}$ are the $Z{\kern 1pt} '$ charges of the ${{\psi }_{{Lk}}},{{\psi }_{{Rk}}}$ fermions. The $Z{\kern 1pt} '$ can interact with new hypothetical particles beyond the SM, for instance, with DM fermions $\chi$

$${{L}_{{Z\,'\chi }}} = {{g}_{{\text{D}}}}Z_{\mu }^{'}\bar {\chi }{{\gamma }^{\mu }}\chi .$$

(35)

There are several models of the current $J_{{Z{\kern 1pt} '}}^{\mu }$. In a model with dark photon [13] $Z{\kern 1pt} '$ boson interacts with photon ${{A}_{\mu }}$ due to kinetic mixing term⁵

$${{L}_{{{\text{mix}}}}} = - \frac{\epsilon }{2}{{F}^{{\mu \nu }}}Z_{{\mu \nu }}^{'}.$$

(36)

As a result of the mixing (36) the field $Z{\kern 1pt} '$ interacts with the SM electromagnetic field $J_{{EM}}^{\mu } = \tfrac{2}{3}\bar {u}{{\gamma }^{\mu }}u - \tfrac{1}{3}\bar {d}{{\gamma }^{\mu }}d - \bar {e}{{\gamma }^{\mu }}e + ...$ with the coupling constant $g{\kern 1pt} ' = \epsilon e$ ($\alpha = \tfrac{{{{e}^{2}}}}{{4\pi }} = \tfrac{1}{{137}}$). However experimental data exclude dark photon model as an explanation of muon ${{g}_{\mu }} - 2$ anomaly. Other interesting scenario is the model [6] where $Z{\kern 1pt} '$ (the dark leptonic gauge boson) interacts with the SM leptonic current, namely

$$\begin{gathered} {{L}_{{Z{\kern 1pt} '}}} = g{\kern 1pt} '[\bar {e}{{\gamma }^{\nu }}e + {{{\bar {\nu }}}_{{eL}}}{{\gamma }^{\nu }}{{\nu }_{{eL}}} + \bar {\mu }{{\gamma }^{\nu }}\mu + {{{\bar {\nu }}}_{{\mu L}}}{{\gamma }^{\nu }}{{\nu }_{{\mu L}}} \\ + \,\,\bar {\tau }{{\gamma }^{\nu }}\tau + {{{\bar {\nu }}}_{{\tau L}}}{{\gamma }^{\nu }}{{\nu }_{{\tau L}}}]Z_{\nu }^{'}. \\ \end{gathered} $$

(37)

In refs. [22–24] for an explanation of ${{g}_{\mu }} - 2$ muon anomaly a model where $Z{\kern 1pt} '$ interacts predominantly with the second and third generations through the ${{L}_{\mu }} - {{L}_{\tau }}$ current

$${{L}_{{Z{\kern 1pt} '}}} = g{\kern 1pt} '[\bar {\mu }{{\gamma }^{\nu }}\mu + {{\bar {\nu }}_{{\mu L}}}{{\gamma }^{\nu }}{{\nu }_{{\mu L}}} - \bar {\tau }{{\gamma }^{\nu }}\tau - {{\bar {\nu }}_{{\tau L}}}{{\gamma }^{\nu }}{{\nu }_{{\tau L}}}]Z_{\nu }^{'}$$

(38)

has been proposed. The interaction (38) is ${{\gamma }_{5}}$-anomaly free, it commutes with the SM gauge group and moreover it escapes (see next section) from the most restrictive current experimental bounds because the interaction (38) does not contain quarks and first generation leptons ${{\nu }_{e}}$, $e$. In ref. [44] a model where $Z{\kern 1pt} '$ couples with a right-handed current of the first and second generation SM fermions including the right-handed neutrinos has been suggested. The model is able to explain the muon ${{g}_{\mu }} - 2$ anomaly due to existence of light scalar and it can be tested in future experiments.

The Yukawa interaction of the scalar field with muon

$${{L}_{{{\text{Yuk}},\phi }}} = - {{g}_{{\mu \phi }}}\phi \bar {\mu }\mu .$$

(39)

leads to additional one loop contribution to muon anomalous magnetic moment [43]

$$\Delta {{a}_{\mu }} = \frac{{g_{{\mu \phi }}^{2}}}{{8{{\pi }^{2}}}}\frac{{m_{\mu }^{2}}}{{m_{\phi }^{2}}}\int\limits_0^1 {\frac{{{{x}^{2}}(2 - x)dx}}{{(1 - x)(1 - {{\lambda }^{2}}x) + {{\lambda }^{2}}x}}} ,$$

(40)

where $\lambda = \tfrac{{{{m}_{\mu }}}}{{{{m}_{\phi }}}}$. For heavy scalar ${{m}_{\phi }} \gg {{m}_{\mu }}$

$$\Delta {{a}_{\mu }} = \frac{{g_{{\mu \varphi }}^{2}}}{{4{{\pi }^{2}}}}\frac{{m_{\mu }^{2}}}{{m_{\phi }^{2}}}\left[ {{\text{ln}}\left( {\frac{{{{m}_{\phi }}}}{{{{m}_{\mu }}}}} \right) - \frac{7}{{12}}} \right],$$

(41)

and for light scalar ${{m}_{\mu }} \gg {{m}_{\phi }}$

$$\Delta {{a}_{\mu }} = \frac{{3g_{{\mu \phi }}^{2}}}{{16{{\pi }^{2}}}}.$$

(42)

2.5.1. LDM and $Z{\kern 1pt} '$boson interacting with ${{L}_{\mu }} - {{L}_{\tau }}$current [27–30]. It is interesting that an extension of the ${{L}_{\mu }} - {{L}_{\tau }}$ model is able to explain today DM density in the Universe. Consider as an example the simplest extension with complex scalar LDM $\chi$⁶. The interaction of the DM $\chi$ with the $Z{\kern 1pt} '$ boson is described by the Lagrangian

$$\begin{gathered} {{L}_{{\chi Z{\kern 1pt} '}}} = ({{\partial }^{\mu }}\chi - i{{e}_{d}}{{Z}^{{'\mu }}}\chi ){\text{*}}({{\partial }_{\mu }}\chi - i{{e}_{d}}Z_{\mu }^{'}\chi ) \\ - \,\,m_{\chi }^{2}\chi {\text{*}}\chi - {{\lambda }_{\chi }}{{(\chi {\text{*}}\chi )}^{2}}. \\ \end{gathered} $$

(43)

The nonrelativistic annihilation cross section $\chi \bar {\chi } \to {{\nu }_{\mu }}{{\bar {\nu }}_{\mu }},{{\nu }_{\tau }}{{\bar {\nu }}_{\tau }}$ for $s \approx 4m_{\chi }^{2}$ has the form⁷

$$\sigma {{v}_{{{\text{rel}}}}} = \frac{{8\pi }}{3}\frac{{{{\epsilon }^{2}}\alpha {{\alpha }_{{\text{D}}}}m_{\chi }^{2}v_{{{\text{rel}}}}^{2}}}{{{{{(m_{{Z{\kern 1pt} '}}^{2} - 4m_{\chi }^{2})}}^{2}}}}.$$

(44)

We use standard assumption that in the hot early Universe DM is in equilibrium with ordinary matter. Using the formulae of Appendix A one can find that

$${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}} = k({{m}_{\chi }}) \times {{10}^{{ - 6}}} \times {{\left( {\frac{{{{m}_{\chi }}}}{{{\text{GeV}}}}} \right)}^{2}} \times {{\left[ {\frac{{m_{{Z{\kern 1pt} '}}^{2}}}{{m_{\chi }^{2}}} - 4} \right]}^{2}}.$$

(45)

Here the coefficient $k({{m}_{\chi }})$ depends logarithmically on DM mass ${{m}_{\chi }}$ and ${{k}_{{DM}}} \sim O(1)$ for $1\,\,{\text{MeV}} \leqslant {{m}_{\chi }} \leqslant 300\,\,{\text{MeV}}$.

As a consequence of (45) we find that for ${{m}_{{Z{\kern 1pt} '}}} \ll {{m}_{\mu }}$ the values ${{\epsilon }^{2}} = (2.5 \pm 0.7) \times {{10}^{{ - 6}}}$ and

$${{\alpha }_{D}} \sim 0.4k({{m}_{\chi }}) \times {{\left( {\frac{{{{m}_{\chi }}}}{{{\text{GeV}}}}} \right)}^{2}} \times {{\left[ {\frac{{m_{{A{\kern 1pt} '}}^{2}}}{{m_{\chi }^{2}}} - 4} \right]}^{2}}$$

(46)

explain both the ${{g}_{\mu }} - 2$ muon anomaly and today DM density.

3 CURRENT EXPERIMENTAL BOUNDS

3.1 The Reactions Used for the Search for LDM

Here we briefly describe the most interesting reactions used(or will be used) for the search for both visible and invisible $A{\kern 1pt} '$ decays at accelerators.

3.1.1. Visible $A{\kern 1pt} '$decays searches. There are a lot of dark photon searches based on the use of visible $A{\kern 1pt} '$ decays $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }},{{\mu }^{ + }}{{\mu }^{ - }}$. The production mechanisms are ${{e}^{ + }}{{e}^{ - }} \to \gamma A{\kern 1pt} '$, $eZ \to eZA{\kern 1pt} '$ reactions, neutral meson decays $pZ \to ({{{{\pi }^{0}}} \mathord{\left/ {\vphantom {{{{\pi }^{0}}} {{{\eta }^{0}}}}} \right. \kern-0em} {{{\eta }^{0}}}} \to A{\kern 1pt} '\gamma ) + ...$ in proton nuclei collisions or direct $A{\kern 1pt} '$ production in proton nuclei reactions [4]. The $A{\kern 1pt} '$ boson is reconstructed as a narrow resonance. Also vertex detection for $A{\kern 1pt} ' \to {{l}^{ + }}{{l}^{ - }}$ decay can be used. Really, the $A{\kern 1pt} '$ decay length is proportional to ${{({{\epsilon }^{2}}{{m}_{{A{\kern 1pt} '}}})}^{{ - 1}}}$ implying that searches for displaced vertices probe low values of the $\epsilon$-parameter. Typical example is NA64 experiment.

3.1.2. Invisible $A{\kern 1pt} '$decays. The DM is produced in the reactions like $eZ \to eZ(A{\kern 1pt} ' \to \chi \bar {\chi })$ or ${{e}^{ + }}{{e}^{ - }} \to \gamma (A{\kern 1pt} ' \to \chi \bar {\chi })$ and identified through the missing energy carried away by the escaping DM particles. The hermeticity of the detector is crusial for background rejection. Resonance hunt in missing mass distribution is very effective for the search for $A{\kern 1pt} '$ invisible decays. For instance, BaBar collaboration [45] used the reaction ${{e}^{ + }}{{e}^{ - }} \to \gamma (A{\kern 1pt} ' \to \chi \bar {\chi })$. The ${{e}^{ + }}$, ${{e}^{ - }}$ and $\gamma$ momenta are measured with good accuracy $O({{10}^{{ - 2}}})$ that allows to restore the missing mass ${{m}_{{{\text{mis}}}}} = \sqrt {{{{({{p}_{{{{e}^{ + }}}}} + {{p}_{{{{e}^{ - }}}}} - {{p}_{\gamma }})}}^{2}}}$. The $A{\kern 1pt} '$ is searched for as a peak in distribution of the missing mass ${{m}_{{{\text{mis}}}}}$.

However there are experiments where the exact measurement of the initial and final particle momenta is impossible. For instance, the NA64 experiment [31] uses the reaction $eZ \to eZ(A{\kern 1pt} ' \to \chi \bar {\chi })$ for the search for $A{\kern 1pt} '$ invisible decays and measures only initial and final electron energies. The typical signature for the LDM detection is missing energy in electromagnetic calorimeter without essential activity in hadronic calorimeter. Good hermeticity of the detector allows to suppress the background at the level $O({{10}^{{ - 11}}})$ or even less that is crusial for the $A{\kern 1pt} '$ detection. The number of signal events at NA64 is proportional to ${{\epsilon }^{2}}$.

3.1.3. Electron and proton beam dump experiments. In beam dump experiments DM is produced in decays ${{{{\pi }^{0}}} \mathord{\left/ {\vphantom {{{{\pi }^{0}}} {{{\eta }^{{(')}}}}}} \right. \kern-0em} {{{\eta }^{{(')}}}}} \to \gamma (A{\kern 1pt} ' \to \chi \bar {\chi })$ or in the reactions $pZ \to pZ(A{\kern 1pt} ' \to \chi \bar {\chi })$, $eZ \to eZ(A{\kern 1pt} ' \to \chi \bar {\chi })$ and it is detected via reactions $e\chi \to e\chi$, $N\chi \to N\chi$ in downstream detectors [4]. These experiments probe LDM twice and they are sensitive to LDM coupling constant ${{\alpha }_{{\text{D}}}} = \tfrac{{e_{{\text{D}}}^{2}}}{{4\pi }}$ with dark mediator $A{\kern 1pt} '$. The number of events is proportional to ${{\epsilon }^{4}}{{\alpha }_{{\text{D}}}}$. Therefore a large proton(electron) flux is required.

3.2 Bound From Electron Magnetic Moment

The experimental and theoretical values for electron magnetic moment coincide at the $2.4\sigma$ level [46]

$$\Delta {{a}_{e}} \equiv a_{e}^{{{\text{exp}}}} - a_{e}^{{{\text{SM}}}} = - (0.87 \pm 0.36) \times {{10}^{{ - 12}}}.$$

(47)

The $A{\kern 1pt} '$ boson contributes to the $\Delta {{a}_{e}}$ at one loop level, see formulae (40)–(42). From the value (47) of $\Delta {{a}_{e}}$ it is possible to restrict the couplng constants ${{g}_{{Ve}}}$ and ${{g}_{{Ae}}}$. For the model with equal muon and electron vector couplings ${{g}_{{Ve}}} = {{g}_{{V\mu }}}$ and ${{g}_{{Ae}}} = {{g}_{{A\mu }}} = 0$ the ${{g}_{\mu }} - 2$ muon anomaly explanation is excluded for ${{M}_{{A'}}} \leqslant 20\,\,{\text{MeV}}$ [47].

3.3 Visible $A{\kern 1pt} '$ Decays

3.3.1. Fixed target electron experiments. Fixed target experiments APEX [48] and A1 at MAMI(Mainz Microtron) [49] searched for $A{\kern 1pt} '$ in electron-nucleus scatterings using the $A{\kern 1pt} '$ bremsstrahlung production ${{e}^{ - }}Z \to {{e}^{ - }}ZA'$ and subsequent $A{\kern 1pt} '$ decay into electron-positron pair $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$. The absence of the resonance peak in the invariant ${{e}^{ + }}{{e}^{ - }}$ mass spectrum allows to obtain upper limits on the $A{\kern 1pt} '$ boson coupling constants ${{g}_{{Ve}}}$, ${{g}_{{Ae}}}$ of the $A{\kern 1pt} '$ with electron, see Fig. 1. The A1 collaboration excluded the masses $50\,\,{\text{MeV}} < {{M}_{{A{\kern 1pt} '}}} < 300\,\,{\text{MeV}}$ [49] for ${{g}_{\mu }} - 2$ muon anomaly explanation in the model with equal muon and electon couplings of the $A{\kern 1pt} '$ boson with a sensitivity to the mixing parameter up to ${{\epsilon }^{2}} = 8 \times {{10}^{{ - 7}}}$. APEX collaboration used $\sim {\kern 1pt} 2\,\,{\text{GeV}}$ electron beam at Jefferson Laboratory and excluded masses $175\,\,{\text{MeV}} < {{M}_{{A{\kern 1pt} '}}} < 250\,\,{\text{MeV}}$ for ${{g}_{\mu }} - 2$ muon anomaly explanation in the model with equal muon and electon couplings of the $A{\kern 1pt} '$ boson. Recently NA64 collaboration studied long lived $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ decays and obtained new bounds on mixing parameter $\epsilon$, see sect. 4.

Fig. 1.

Current limits at 90% CL on the mixing parameter ${{\epsilon }^{2}}$ versus the $A{\kern 1pt} '$ mass for visible $A{\kern 1pt} '$ decays, taken from ref. [52].

3.3.2. ${{e}^{ + }}{{e}^{ - }}$experiments. Bar experiment has constrained visible $A{\kern 1pt} '$ decays by using $\Upsilon (1S)$ decays BaBar collaboration [50] looked for visible decays of light $A{\kern 1pt} '$ bosons in the reaction ${{e}^{ + }}{{e}^{ - }} \to \gamma A{\kern 1pt} ',$$A{\kern 1pt} ' \to {{l}^{ + }}{{l}^{ - }}(l = e,\mu )$ as resonances in the ${{l}^{ + }}{{l}^{ - }}$ spectrum. For the model with the $A{\kern 1pt} '$ dark photon the mixing strength values ${{10}^{{ - 2}}}{\kern 1pt} - {\kern 1pt} {{10}^{{ - 3}}}$ are excluded for $0.212\,\,{\text{GeV}} < {{m}_{{A{\kern 1pt} '}}} < 10\,\,{\text{GeV}}$ [50] in the assumption that visible $A{\kern 1pt} '$ decays into the SM particles dominate, see Fig. 1. The KLOE experiment at the DA$\Phi$NE $\Phi$-factory in Fraskati searched for $A{\kern 1pt} '$ in decays $\Phi \to \eta A{\kern 1pt} ' \to \eta {{e}^{ + }}{{e}^{ - }}$ and $\Phi \to \gamma (A{\kern 1pt} ' \to {{\mu }^{ + }}{{\mu }^{ - }})$ [51]. The obtained bounds are weaker than those from NA48/2 [52] and MAMI [49] bounds.

Recently BaBar collaboration used the reaction ${{e}^{ + }}{{e}^{ - }} \to Z{\kern 1pt} '{{\mu }^{ + }}{{\mu }^{ - }},$$Z{\kern 1pt} ' \to {{\mu }^{ + }}{{\mu }^{ - }}$ to search for the $Z{\kern 1pt} '$ boson coupled with muon. The use of this process allows to restrict directly the muon coupling ${{g}_{{V\mu }}}$ of the $Z{\kern 1pt} '$ boson. The obtained results exclude the model with ${{L}_{\mu }} - {{L}_{\tau }}$ interaction as possible explanation of ${{g}_{\mu }} - 2$ muon anomaly for ${{m}_{{Z{\kern 1pt} '}}} > 214\,\,{\text{MeV}}$ [53].

3.3.3. Fixed target proton experiments. The NA‑48/2 experiment used simultaneous ${{K}^{ + }}$ and ${{K}^{ - }}$ secondary beams produced by $400\,\,{\text{GeV}}$ primary CERN SPS protons for the search for light $A{\kern 1pt} '$ boson in ${{\pi }^{0}}$ decays [52]. The decays ${{K}^{ \pm }} \to {{\pi }^{ \pm }}{{\pi }^{0}}$ and ${{K}^{ \pm }} \to {{\pi }^{0}}{{\mu }^{ \pm }}\nu$ have been used to obtain tagged ${{\pi }^{0}}$ mesons. The decays ${{\pi }^{0}} \to \gamma A{\kern 1pt} '$, $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ have been used for the search for $A{\kern 1pt} '$ boson. The $A{\kern 1pt} '$ boson manifests itself as a narrow peak in the distribution of the ${{e}^{ + }}{{e}^{ - }}$ invariant mass spectrum. For the model with dark photon the obtained bounds exclude the ${{g}_{\mu }} - 2$ muon anomaly explanation for $A{\kern 1pt} '$ boson masses $9\,\,{\text{MeV}} < {{m}_{{A{\kern 1pt} '}}} < 70\,\,{\text{MeV}}$ [52], see Fig. 1. It should be noted that the decay width ${{\pi }^{0}} \to \gamma A{\kern 1pt} '$ is proportional to ${{({{g}_{{Vu}}}{{q}_{u}} - {{g}_{{Vd}}}{{q}_{d}})}^{2}}{{ = {{{(2{{g}_{{Vu}}} + {{g}_{{Vd}}})}}^{2}}} \mathord{\left/ {\vphantom {{ = {{{(2{{g}_{{Vu}}} + {{g}_{{Vd}}})}}^{2}}} 9}} \right. \kern-0em} 9}$ and for the models with nonuniversal $A'$-boson couplings⁸, for instance, for the model with ${{L}_{\mu }} - {{L}_{\tau }}$ interaction current the NA‑48/2 bound [52] is not applicable.

3.3.4. ATLAS and CMS bounds on light particles in Higgs boson decays. ATLAS collaboration searched for new light particles ${{\gamma }_{d}}$ in Higgs boson decays $h \to 2{{\gamma }_{d}} + X$, $h \to 4{{\gamma }_{d}} + X$ [55]. In the assumption that new boson ${{\gamma }_{d}}$ decays mainly into muon pair bounds on ${\text{Br}}(h \to 2{{\gamma }_{d}} + X)$ and ${\text{Br}}(h \to 4{{\gamma }_{d}} + X)$ have been otained [55]. It should be stressed that for the model with dark photon the bound on $\epsilon$ parameter is rather weak.

CMS collaboration also searched for new particles [56] in the Higgs boson decay $h \to 2a + X \to 4\mu + X$. Bounds similar to the ATLAS bounds have been obtained.

3.3.5. LHCb bound on $A{\kern 1pt} ' \to {{\mu }^{ + }}{{\mu }^{ - }}$decays. Recently LHCb collaboration performed the search for $A{\kern 1pt} '$ bosons on the base of visible $A{\kern 1pt} ' \to {{\mu }^{ + }}{{\mu }^{ - }}$ decay. In the assumption that the $A{\kern 1pt} '$ production arises as a result of $\gamma A{\kern 1pt} '$ mixing the bound on mixing parameter $\epsilon$ has been derived for wide range of $A{\kern 1pt} '$ masses from $214\,\,{\text{MeV}}$ up to $70\,\,{\text{GeV}}$ for prompt decays and for $214\,\,{\text{MeV}} < {{m}_{{A{\kern 1pt} '}}} < 350\,\,{\text{MeV}}$ for long lived $A{\kern 1pt} '$ [57]. No evidence for signal has been found and upper bound on $\epsilon$ parameter has been derived. The obtained bounds are the most stringent to date for the masses $10.6\,\,{\text{GeV}} < {{m}_{{A{\kern 1pt} '}}} < 70\,\,{\text{GeV}}$.

3.4 Invisible $A{\kern 1pt} '$ Decays

3.4.1. Constraints from $K \to \pi + nothing$decay. Light vector boson $A{\kern 1pt} '$ can be produced in the $K \to \pi A{\kern 1pt} '$ decay in the analogy with the SM decay $K \to \pi \gamma {\text{*}}$ of K-meson into pion and virtual photon. For the model with the dominant $A{\kern 1pt} '$ decay into invisible modes nontrivial bound on the $A{\kern 1pt} '$ boson mass and the coupling constant arises. Namely, the results of BNL E949 and E787 experiments [58] on the measurement of the ${{K}^{ + }} \to {{\pi }^{ + }}\nu \bar {\nu }$ decay width were used to obtain an upper bound on the ${\text{Br}}({{K}^{ + }} \to {{\pi }^{ + }}A{\kern 1pt} ')$ decay as a function of the $A{\kern 1pt} '$ mass in the assumption that $A{\kern 1pt} ' \to {\text{invisible}}$ decay dominates. In the model where the $A{\kern 1pt} '$ is dark photon, the explanation of muon ${{g}_{\mu }} - 2$ anomaly due to the $A{\kern 1pt} '$ existence is excluded for ${{M}_{{A{\kern 1pt} '}}} > 50\,\,{\text{MeV}}$ except the narrow region around ${{m}_{{A'}}} = {{m}_{\pi }}$ [59–61]. Note that in models with non-electromagnetic current interactions of $A{\kern 1pt} '$ with quarks and leptons, for instance, in the model where the $A{\kern 1pt} '$ interacts with the ${{L}_{\mu }} - {{L}_{\tau }}$ current only, the bound from $K \to \pi + {\text{nothing}}$ decay does not work or it is rather weak [60].

3.4.2. The use of the reaction $eZ \to eZA{\kern 1pt} '$, $A{\kern 1pt} ' \to invisible$. The NA64 collaboration [32, 33] used the reaction $eZ \to eZA\,'$, $A{\kern 1pt} ' \to {\text{invisible}}$ for the search for invisible dark photon decays into LDM particles. The obtained bounds exclude the dark photon model as an explanation of muon ${{g}_{\mu }} - 2$, see Fig. 2.

Fig. 2.

Limits at 90% C.L. on the mixing parameter $\epsilon$ versus the $A{\kern 1pt} '$ mass for invisible $A{\kern 1pt} '$ decays, taken from ref. [34].

3.4.3. ${{e}^{ + }}{{e}^{ - }}$experiments. Recently BaBar collaboration [62] used the reaction ${{e}^{ + }}{{e}^{ - }} \to \gamma A{\kern 1pt} '$, $A{\kern 1pt} ' \to {\text{invisible}}$ for the search for invisible decays of $A{\kern 1pt} '$. In the assumption that $A{\kern 1pt} '$ invisible decays dominate the bound $\epsilon \leqslant {{10}^{{ - 3}}}$ has been obtained for ${{m}_{{A\,'}}} \leqslant 9.5\,\,{\text{GeV}}$, see Fig. 2.

3.4.4. Electron beam dump experimemts. In electron beam dump experiments the reaction $eZ \to eZA{\kern 1pt} '$ is used for the $A{\kern 1pt} '$ production. After some shield the $A{\kern 1pt} '$ bosons are manifested as visible decays $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }},{{\mu }^{ + }}{{\mu }^{ - }}$. If $A'$ decays mainly into LDM particles $A{\kern 1pt} ' \to \chi \bar {\chi }$ the use of elastic scattering $\chi e \to \chi e$, $\chi N \to \chi N$ in the far detector allows to detect LDM particles. The results of electron beam dump experiments [63, 64] at SLAC and FNAL have been used [65] to constrain the couplings of light gauge boson $A{\kern 1pt} '$. For the case of dominant $A{\kern 1pt} '$ decays into visible particles electron beam dump experiments exclude ${{10}^{{ - 7}}} \leqslant \epsilon \leqslant {{10}^{{ - 6}}}$ for ${{m}_{{A\,'}}} \leqslant 20\,\,{\text{MeV}}$. For the case where the $A{\kern 1pt} '$ decays dominantly into LDM particles the experiment E137 gives the most stringent bounds and it excludes the parameter $y \equiv {{\epsilon }^{2}}{{\alpha }_{{\text{D}}}}{{\left( {\tfrac{{{{m}_{\chi }}}}{{{{m}_{{A{\kern 1pt} '}}}}}} \right)}^{4}} \geqslant {{10}^{{ - 11}}}({{10}^{{ - 9}}})$ for ${{m}_{{A{\kern 1pt} '}}} \leqslant 1(100)\,\,{\text{MeV}}$.

3.4.5. Proton beam dump experiments. In proton beam dump experiments the main source of the $A{\kern 1pt} '$ arises as a result of ${{\pi }^{0}}(\eta )$ production $pZ \to {{\pi }^{0}}(\eta ) + ...$ with the subsequent ${{\pi }^{0}}(\eta ) \to \gamma A{\kern 1pt} ';$$A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ decays, see e.g [66, 67]. In the case of dominant $A{\kern 1pt} '$ decay into LDM particles $A{\kern 1pt} ' \to \chi \bar {\chi }$ the reactions $\chi e \to \chi e$ and $\chi N \to \chi N$ are used for dark matter identification.

The LSND (Liquid Scintillarion Neutrino Detector) [68] at Los Alamos was constructed to detect neutrino. Neutrino arise mainly from the reaction $pZ \to {{\pi }^{ + }} + ...$ with the subsequent ${{\pi }^{ + }} \to {{\mu }^{ + }}{{\nu }_{\mu }}$ decays. LSND data with $N = {{10}^{{24}}}$ POT also allow to restrict the dark photon couplings. Dark photons $A{\kern 1pt} '$ are produced mainly in the reaction $pZ \to ({{\pi }^{0}} \to \gamma A{\kern 1pt} ') + ...$. The LSND bound on the parameter $y \equiv {{\epsilon }^{2}}{{\alpha }_{{\text{D}}}}{{\left( {\tfrac{{{{m}_{\chi }}}}{{{{m}_{{A{\kern 1pt} '}}}}}} \right)}^{4}}$ is by factor $O(10)$ more strong that the corresponding bound from electron beam dump experiment E137. The MiniBoone experiment at FNAL is also proton beam dump experiment which uses the FNAL $8\,\,{\text{GeV}}$ Booster proton beam. As in LSND dark photons are produced mainly in ${{\pi }^{0}}$ decays and detected in a 800 tonn mineral oil Cherenkov detector situated $\sim {\kern 1pt} 500\,\,m$ downstream of the beam dump. Recently MiniBoone experiment has obtained bound [69] on $y \leqslant {{10}^{{ - 8}}}$ for ${{\alpha }_{{\text{D}}}} = 0.5$ and for DM masses $0.01 < {{m}_{\chi }} < 0.3\,\,{\text{GeV}}$ in a dedicated run with $1.86 \times {{10}^{{20}}}$ protons delivered to a steel beam dump.

3.4.6. COHERENT at ORNL. The primary goal of the COHERENT experiment [70] at Oak Ridge National Laboratory(USA) is to measure coherent elastic neutrino scattering ($CE\nu NS$) process and to check the ${{N}^{2}}$ dependence of the cross section. Recently the COHERENT experiment measured the $CE\nu NS$ process [71] and the results are in agreement with the SM expectations. The COHERENT is beam-dump experiment and LDM can be produced mainly in ${{\pi }^{0}} \to \gamma A{\kern 1pt} ' \to \gamma \chi \bar {\chi }$ decays. DM particles scatter in scintillating cristals and liquid argon detectors at the Apallation Neutron Source at ORNL. The DM particles(if they exist) are produced via ${{{{\pi }^{0}}} \mathord{\left/ {\vphantom {{{{\pi }^{0}}} \eta }} \right. \kern-0em} \eta } \to \gamma A{\kern 1pt} '$ decays and they can be identified through coherent scattering leading to detectable nuclear recoil. In ref. [72] recent COHERENT data [71] have been used for the derivation of the bounds for LDM. For $1 < {{m}_{\chi }} < 90\,\,{\text{MeV}}$ the bound on $\epsilon e_{d}^{{1/2}}$ is between ${{10}^{{ - 5}}}$ and ${{10}^{{ - 4}}}$.

3.5 Bound from the Neutrino Trident Process ${{\nu }_{\mu }}N \to {{\nu }_{\mu }}N{{\mu }^{ + }}{{\mu }^{ - }}$

The neutrino trident ${{\nu }_{\mu }}N \to {{\nu }_{\mu }}N{{\mu }^{ + }}{{\mu }^{ - }}$ events allow to restrict a model where $Z{\kern 1pt} '$ boson interacts with ${{L}_{\mu }} - {{L}_{\tau }}$ current. The data of the CHARM and the CCFR experiments exclude the ${{g}_{\mu }} - 2$ muon anomaly explanation for ${{m}_{{Z{\kern 1pt} '}}} \geqslant 400\,\,{\text{MeV}}$ [73].

3.6 Nonaccelerator Bounds

3.6.1. CMB bound. The residual annihilation of DM particles after equilibrium annihilation and before recombination can still reionize hydrogen and hence modify the CMB (cosmic microwave background) power spectrum. The Planck experiment constraint [74] rules out thermal DM below 10 GeV if the annihilation is s-wave (velocity independent). The p-wave annihilation is allowed since at recombination epoch the temperature is $T \sim {\text{eV}}$ and the p-wave annihilation is suppressed by factor ${T \mathord{\left/ {\vphantom {T {{{m}_{\chi }}}}} \right. \kern-0em} {{{m}_{\chi }}}}$. Also models with pseudo-Dirac LDM [4, 37] escape the CMB bound.

3.6.2. Constraints from stars. Light $A{\kern 1pt} '$ boson can be produced in stars. The energy loss of the stars through the $A{\kern 1pt} '$ places strong limits $\epsilon \leqslant O({{10}^{{ - 14}}})$ on the $A{\kern 1pt} '$ couplings for ${{m}_{{A{\kern 1pt} '}}} \leqslant 0.01\,\,{\text{MeV}}$ [75–83]. The constraints on the $A{\kern 1pt} '$ couplings result from the requirement that the energy loss by the $A{\kern 1pt} '$ emission has to be less than 10 percent of the solar energy in photons [76]. Also for ${{m}_{{A{\kern 1pt} '}}} \leqslant 0.3\,\,{\text{MeV}}$ similar but more weak limit on $\epsilon$ can be derived from horizontal branch stars and red giants where the temperatures are higher than in the Sun [76].

3.6.3. Supernnova 1987A bounds. Bounds from Supernnova 1987A are based on the fact that if dark photons are produced in sufficient quantity, they reduce the amount of energy emitted in the form of neutrinos, in conflict with observations. In ref. [77] bounds on $\epsilon$ parameter were obtained for the model with dark photon. Bounds on $\epsilon$ parameter exist for ${{m}_{{A{\kern 1pt} '}}} \leqslant 120\,\,{\text{MeV}}$ [77]. For the most interesting case ${{m}_{{A'}}} \geqslant 2{{m}_{e}}$ the value $\epsilon \geqslant O({{10}^{{ - 7}}})$ does not contradict to data drom Supernova 1987A [77]. It means that the bounds from Supernova 1987A don’t restrict severely the LDM hypothesis.

3.6.4. Constraints from BBN. Big Bang nucleosynthesis (BBN) can also provide the constraints on $A{\kern 1pt} '$ coupling constants. During the first several minutes after the Big Bang, the temperature of the Universe rapidly decreased as a consequence of the Universe expansion. During the Universe expansion some light elements are produced and the predictions of their abundance from BBN agree with experimental data [78]. The constraints on new interactions are based on the fact that new relativistic particle increases the expansion rate of the Universe through an additional degree of freedom which usually expressed in terms of extra neutrinos $\Delta {{N}_{\nu }}$. The larger Universe expansion rate increases the freeze-out temperature, therefore the ${n \mathord{\left/ {\vphantom {n p}} \right. \kern-0em} p}$ ratio and as a consequence the $^{4}{\text{He}}$ abundance is increased. The observed value of the $^{4}{\text{He}}$ abundance leads to the bound on $\Delta {{N}_{\nu }}$ that is equivalent to the bounds on coupling constants of new relativistic particle. For dark photon model BBN constraints have been obtained in ref. [81]. The $A{\kern 1pt} '$ dark photon model with ${{m}_{{A{\kern 1pt} '}}} \leqslant O(1)\,\,{\text{MeV}}$ is excluded [79] as a mediator explaining current DM abundance. Note that in ref. [82] lower bound ${{m}_{\chi }} \geqslant O(1)\,\,{\text{MeV}}$ on the mass of the LDM particle was obtained from the experimental bound on effective number of neutrinos.

3.7 Direct LDM Detection

The main problem of the LDM detection via elactic LDM scattering at nuclei is the size of the nuclear recoil energy [4]. The velocity of DM is ${{v}_{\chi }} \sim {{10}^{{ - 3}}}\,\,{\text{s}}$ and the maximum possible energy transfer is proportional to the square of the reduced mass ${{\mu }_{{{\text{red}}}}} = \tfrac{{{{m}_{{{\text{nuclei}}}}}{{m}_{\chi }}}}{{{{m}_{{{\text{nuclei}}}}} + {{m}_{\chi }}}}$. The nuclear recoil energy is [4]

$$\begin{gathered} {{E}_{{{\text{NR}}}}} = \frac{{{{q}^{2}}}}{{2{{m}_{{{\text{nuclei}}}}}}} \leqslant \frac{{2{\kern 1pt} \mu _{{{\text{red}}}}^{2}v_{\chi }^{2}}}{{{{m}_{{{\text{nuclei}}}}}}} \leqslant 190\,\,{\text{eV}} \times {{\left( {\frac{{{{m}_{\chi }}}}{{500\,\,{\text{MeV}}}}} \right)}^{2}} \\ \times \,\,\,\left( {\frac{{16\,\,{\text{GeV}}}}{{{{m}_{{{\text{nuclei}}}}}}}} \right) \\ \end{gathered} $$

(48)

that makes the detection of LDM with masses ${{m}_{\chi }} \leqslant O(1)\,\,{\text{GeV}}$ at nuclei extremely difficult. The remaining possibility is the use of electron LDM elastic scattering [4]. For electron LDM scattering the maximum energy transfer to electron is

$${{E}_{e}} \leqslant \frac{1}{2}{{m}_{\chi }}v_{\chi }^{2} \leqslant 3\,\,{\text{eV}}\,\,\left( {\frac{{{{m}_{\chi }}}}{{{\text{MeV}}}}} \right).$$

(49)

Bound electrons with binding energy $\Delta {{E}_{{\text{B}}}}$ can produce measurable signal at [4]

$${{m}_{\chi }} \geqslant 0.3\,\,{\text{MeV}} \times \left( {\frac{{\Delta {{E}_{B}}}}{{1\,\,{\text{eV}}}}} \right).$$

(50)

The elasic nonrelativistic cross-section of scalar or fermion LDM in dark photon model at ${{m}_{\chi }} \gg {{m}_{e}}$ is [4, 84]

$$\sigma (e\chi \to e\chi ) = \frac{{16\pi m_{e}^{2}\alpha {{\epsilon }^{2}}{{\alpha }_{D}}}}{{(m_{{A{\kern 1pt} '}}^{4})}},$$

(51)

while the elastic Majorana cross-section is suppressed by factor ${{k}_{{\text{M}}}} = \tfrac{{2m_{e}^{2}}}{{m_{\chi }^{2}}}v_{\chi }^{2}$

$$\sigma (e{{\chi }_{{{\text{Majorana}}}}} \to e{{\chi }_{{{\text{Majorana}}}}}) = \frac{{16\pi m_{e}^{2}\alpha {{\epsilon }^{2}}{{\alpha }_{D}}}}{{(m_{{A{\kern 1pt} '}}^{4})}} \times {{k}_{{\text{M}}}}$$

(52)

that makes the direct detection of Majorana LDM in dark photon model extremely difficult or even hopeless.

Recently XENON1T collaboration has published new record results [85] on the search for direct electron LDM scattering. New bounds on elasic electron LDM cross sections were obtained for ${{m}_{\chi }} \geqslant 30\,\,{\text{MeV}}$. For the model with dark photon the use of the formula (51) and the results of ref. [85] allows to derive bound on ${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}}$. In Fig. 3 the comparison of 90% C.L. upper limits on the cross-sections of LDM electron scattering transmitted by dark photon mediator $A{\kern 1pt} '$ calculated by using NA64 [34] and BaBar bounds and the XENON1T [85] bounds has been presented for ${{\alpha }_{{\text{D}}}} = 0.1$. For ${{m}_{\chi }} \leqslant 50\,\,{\text{MeV}}$ the NA64 bound is stronger than the XENON1T bound. For pseudo-Dirac fermions with not too small $\delta = \tfrac{{{{m}_{{{{\chi }_{2}}}}} - {{m}_{{{{\chi }_{1}}}}}}}{{{{m}_{{{{\chi }_{1}}}}}}}$ the reaction of ${{\chi }_{2}}$ electroproduction ${{\chi }_{1}}e \to {{\chi }_{2}}e$ for nonrelativistic LDM ${{\chi }_{1}}$ is prohibited due to energy conservation law, while elastic ${{\chi }_{1}}e \to {{\chi }_{1}}e$ scattering is absent at tree level that extremely complicates the direct LDM detection for pseudo-Dirac fermions.

Fig. 3.

Comparison of 90% C.L. upper limits on LDM-electron scattering cross-sections calculated by using NA64 [34] and BaBar constraints on kinetic-mixing from Fig. 2 with results of direct searches by XENON1T [85]. The blue curves are calculated for ${{\alpha }_{{\text{D}}}} = 0.1$, while the dashed blue for ${{\alpha }_{{\text{D}}}} = 0.5$. The Yellow dashed line shows the XENON1T limit obtained without considering signals with <12 produced electrons.

4 NA64 EXPERIMENT

4.1 Invisible Mode

NA64 experiment [31] at the CERN SPS employs the electron beam from the H4 beam line in the North Area (NA). The beam delivers $\approx 5 \times {{10}^{6}}{{e}^{ - }}$ per SPS spill of $44.8s$ produced by the primary $400\,\,{\text{GeV}}$ proton beam with an intensity of a few ${{10}^{{12}}}$ protons on target. The NA64 experiment is a fixed target experiment searching for dark sector particles at the CERN Super Proton Synchrotron(SPS) by using active beam dump technique combined with missing energy approach [31, 86–88]. If new light boson $A{\kern 1pt} '$ exists it could be produced in the reaction of high energy electrons scattering off nuclei. Compared to the traditional beam dump experiment the main advantage of the NA64 experiment is that its sensitivity is proportional to the ${{\epsilon }^{2}}$. While for the classical beam dump experiments the sensitivity is proportional to the ${{\epsilon }^{2}} \cdot {{\epsilon }^{2}}$, where one ${{\epsilon }^{2}}$ comes from new particle production in the dump and another ${{\epsilon }^{2}}$ is from the LDM interaction in far detector. Another advantage of the NA64 experiment is that due to the higher energy of the incident beam, the centre-of-mass system is boosted relative to the laboratory system. This boost leads to enhanced hermeticity of the detector providing a nearly full solid angle coverage.

The NA64 method of the search can be illustrated by considering the search for the dark photon $A{\kern 1pt} '$ production for invisible $A{\kern 1pt} '$ decays $A{\kern 1pt} ' \to \chi \bar {\chi }$ into LDM particles. A fraction $f$ of the primary beam energy ${{E}_{{A{\kern 1pt} '}}} = f{{E}_{0}}$ is carried away by $\chi$ LDM particles, which penetrate the target and detector without interactions resulting in zero energy deposition. The remaining part of beam energy ${{E}_{e}} = (1 - f){{E}_{0}}$ is deposited in the target by the scattered electron. The occurrence of the $A{\kern 1pt} '$ production via the reaction $eZ \to eZA{\kern 1pt} '\,;$$A{\kern 1pt} ' \to \chi \bar {\chi }$ would appear as an excess of events with a signature of a single isolated electromagnetic (e-m) shower in the active dump with energy ${{E}_{e}}$ accompanied by a missing energy ${{E}_{{{\text{miss}}}}} = {{E}_{{A{\kern 1pt} '}}} = {{E}_{0}} - {{E}_{e}}$ above those expected from backgrounds. Here we assume that LDM particles $\chi$ traverse the detector without decaying visibly. Currently, the NA64 employs the $100\,\,{\text{GeV}}$ electron beam from $H4$ beam line at the North Area (NA) of the CERN SPS. The beam was optimized to transport the electrons with the maximal intensity $\geqslant {{10}^{7}}$ per SPS spill with the momentum $100\,\,{{{\text{GeV}}} \mathord{\left/ {\vphantom {{{\text{GeV}}} c}} \right. \kern-0em} c}$. The NA64 detector is schematically shown in Fig. 4. The setup utilized the beam defining scintillator (Sc) counters $S1{\kern 1pt} - {\kern 1pt} S3$ and veto V1, and the spectrometer consisting of two successive dipole magnets with the integral magnetic field of $\approx 7T \times m$ and low-material-budget tracker. The tracker is a set of upstream Micromegas chambers $(T1,T2)$ and downstream Micromegas, GEM and Straw tube stations, measuring the beam ${{e}^{ - }}$ momenta, ${{P}_{e}}$ with the precision ${{\delta {{P}_{e}}} \mathord{\left/ {\vphantom {{\delta {{P}_{e}}} {{{P}_{e}}}}} \right. \kern-0em} {{{P}_{e}}}} \approx {{10}^{{ - 2}}}$ [31]. The magnets also serve as an effective filter rejecting the low energy electrons present in the beam. The key feature of NA64 is the use of synchrotron radiation $(SR)$ from high energy electrons in the magnetic field to significantly enhance electron identification and suppress background from a hadron contamination in the beam. A 16 m long vacuum vessel was installed between the magnets and the ECAL to minimize absorption of the SR photons detected immediately at the downstream end of the vessel with a SRD, which is array of ${\text{PbSc}}$ sandwich counters of a very fine longitudinal segmentation assembled from $80{\kern 1pt} - {\kern 1pt} 100\,\,\mu {\text{m}}$${\text{Pb}}$ and $1\,\,{\text{mm}}\,\,{\text{Sc}}$ plates with wave length shifting (WLS) fiber read-out. This allowed to additionally suppress background from hadrons, that could knock off electrons from the output vacuum window of the vessel producing a fake ${{e}^{ - }}SRD$ tag, by about two orders of magnitude. The detector is also equipped with an active target, which is a hodoscopic electromagnetic calorimeter (ECAL) for the measurement of the electron energy deposition, ${{E}_{{{\text{ECAL}}}}}$, with the accuracy ${{\delta {{E}_{{{\text{ECAL}}}}}} \mathord{\left/ {\vphantom {{\delta {{E}_{{{\text{ECAL}}}}}} {{{E}_{{{\text{ECAL}}}}}}}} \right. \kern-0em} {{{E}_{{{\text{ECAL}}}}}}} \approx {{0.1} \mathord{\left/ {\vphantom {{0.1} {\sqrt {{{E}_{{{\text{ECAL}}}}}\,\,[{\text{GeV}}]} }}} \right. \kern-0em} {\sqrt {{{E}_{{{\text{ECAL}}}}}\,\,[{\text{GeV}}]} }}$ as well as the $X$, $Y$ coordinates of the incoming electrons by using the transverse $e - m$ shower profile. The ECAL is a matrix of $6 \times 6$ Shashlik-type counters assembled with ${\text{Pb}}$ and ${\text{Sc}}$ plates with ${\text{WLS}}$ fiber read-out. Each model is $\approx {\kern 1pt} 40$ radiation lengths $({{X}_{0}})$ and has an initial part $\approx {\kern 1pt} 4{{X}_{0}}$ used as a preshower (PS) detector. By requiring the presence of in-time SR signal in all three SRD counters, and using the information of the longitudinal and lateral shower development in the ECAL, the initial level of the hadron contamination in the beam ${\pi \mathord{\left/ {\vphantom {\pi {{{e}^{ - }}}}} \right. \kern-0em} {{{e}^{ - }}}} \leqslant {{10}^{{ - 2}}}$ was further suppressed by more than 4 orders of magnitude, while the electron ID at the level $\geqslant {\kern 1pt} 95{\text{\% }}$. A high-efficiency veto counter $Veto$, and a massive, hermetic hadronic calorimeter (HCAL) of $\approx 30$ nuclear interaction lengths $({{\lambda }_{{{\text{int}}}}})$ were positioned after the ECAL. The $Veto$ is a plane of scintillation counters used to veto charged secondaries incident on the HCAL detectors from upstream ${{e}^{ - }}$ interactions. The HCAL which was an assembly of four modules ${\text{HCAL}}0{\kern 1pt} - {\kern 1pt} {\text{HCAL}}3$ served as an efficient veto to detect muons of hadronic secondaries produced of in the ${{e}^{ - }}A$ interactions ECAL target. The ${\text{HCAL}}$ energy resolution is ${{\delta {{E}_{{{\text{HCAL}}}}}} \mathord{\left/ {\vphantom {{\delta {{E}_{{{\text{HCAL}}}}}} {{{E}_{{{\text{HCAL}}}}}}}} \right. \kern-0em} {{{E}_{{{\text{HCAL}}}}}}} \approx {{0.6} \mathord{\left/ {\vphantom {{0.6} {\sqrt {{{E}_{{{\text{HCAL}}}}}\,\,[{\text{GeV}}]} }}} \right. \kern-0em} {\sqrt {{{E}_{{{\text{HCAL}}}}}\,\,[{\text{GeV}}]} }}$.

Fig. 4.

Schematic illustration of the setup to search for invisible decays of the bremsstrahlung $A{\kern 1pt} '$s produced in the reaction $eZ \to eZA{\kern 1pt} '$ of 100 GeV e^– incident on the active ECAL target.

4.2 Visible Mode

The NA64 setup designed for the searches for decays $X,A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ of the $X$ bosons, which could explain the ⁸Be anomaly (see below 5.1.2) and the $A{\kern 1pt} '$ is schematically shown in Fig. 5. The NA64 experiment for visible $A{\kern 1pt} ' \to {{e}^{ - }}{{e}^{ + }}$ searches employs the optimized electron beam from the H4 beam line in the North Area (NA) of the CERN SPS. The beam delivers 5 ×10⁶ EOT per SPS spill of 4.8s produced by the primary 400 GeV proton beam with an intensity of a few ${{10}^{{12}}}$ protons on target. Two scintillation counters, $S1$ and $S2$ were used for the beam definition, while the other two, $S3$ and $S4$, were used to detect the ${{e}^{ + }}{{e}^{ - }}$ pairs. The detector is equipped with a magnetic spectrometer consisting of two MPBL magnets and a low material budget tracker. The tracker was a set of four upstream Micromegas (MM) chambers $(T1,T2)$ for the incoming e- angle selection and two sets of downstream MM, GEM stations and scintillator hodoscopes $(T3,T4)$ allowing the measurement of the outgoing tracks [31]. To enhance the electron identification the synchrotron radiation (SR) emitted by electrons was used for their effi- cient tagging and for additional suppression of the initial hadron contamination in the beam ${\pi \mathord{\left/ {\vphantom {\pi e}} \right. \kern-0em} e}$ × ${{10}^{{ - 2}}}$ down to the level ${{10}^{{ - 6}}}$ [87]. The use of SR detectors (SRD) is a key point for the hadron background suppression and improvement of the sensitivity compared to the previous electron beam dump searches [31]. The dump is a compact electromagnetic (e-m) calorimeter WCAL made as short as possible to maximize the sensitivity to short lifetimes while keeping the leakage of particles at a small level. The WCAL was assembled from the tungsten and plastic scintillator plates with wave lengths shifting fiber read-out. The first (last) few layers of the WCAL were read separately to form a signal from a preshower (veto ${{W}_{2}}$) counter. Immediately after the ${{W}_{2}}$ there is also one more veto counter $V2$, and several meters downstream the signal counter $S4$ and tracking detectors. These detectors are followed by another e-m calorimeter (ECAL), which is a matrix of $6$ shashlik-type lead—plastic scintillator sandwich modules [89]. Downstream the ECAL the detector was equipped with a high-efficiency veto counter, and a thick hadron calorimeter (HCAL) [31] used as a hadron veto and muon identificator. For the cuts selection, calculation of various efficiencies and background estimation the package for the detailed full simulation of the experiment based on Geant4 [90] is developed. It contains the subpackage for the simulation of various types of DM particles based on the exact tree-level calculation of cross sections [40, 41]. The method of the search for $A{\kern 1pt} ' \to {{e}^{ - }}{{e}^{ + }}$ decays is described in [31]. The application of all further considerations to the case of the $X \to {{e}^{ + }}{{e}^{ - }}$ decay is straightforward. If the $A{\kern 1pt} '$ exists, it could be produced via the coupling to electrons wherein high energy electrons scatter off a nuclei of the active WCAL dump target, followed by the decay into ${{e}^{ + }}{{e}^{ - }}$ pairs:

$${{e}^{ - }} + Z \to {{e}^{ - }} + Z + A{\kern 1pt} ',\,\,\,\,A{\kern 1pt} ' \to {{e}^{ - }}{{e}^{ + }}.$$

(53)

Fig. 5.

Schematic illustration of the setup to search for visible $A{\kern 1pt} ',X \to {{e}^{ + }}{{e}^{ - }}$ decays decays of the bremsstrahlung $A{\kern 1pt} ',X$ produced in the reaction $eZ \to eZA{\kern 1pt} '$ of 100 GeV e^– incident on the active WCAL target.

The reaction (53) typically occurs within the first few radiation lengths $({{X}_{0}})$ of the WCAL. The downstream part of the WCAL serves as a dump to absorb completely the e-m shower tail. The bremsstrahlung $A{\kern 1pt} '$ would penetrate the rest of the dump and the veto counter $V2$ without interactions and decay in flight into an ${{e}^{ + }}{{e}^{ - }}$ pair in the decay volume downstream the WCAL. A fraction (f) of the primary beam energy ${{E}_{1}} = f{{E}_{0}}$ is deposited in the WCAL by the recoil electron from the reaction (51). The remaining part of the primary electron energy ${{E}_{2}} = (1 - f){{E}_{0}}$ is transmitted through the dump by the $A{\kern 1pt} '$, and deposited in the second downstream calorimeter ECAL via the $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ decay in flight. The occurrence of $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ decays produced in $eZ$ interactions would appear as an excess of events with two e-m-like showers in the detector: one shower in the WCAL and another one in the $ECAL$, with the total energy ${{E}_{{{\text{tot}}}}} = {{E}_{{{\text{WCAL}}}}} + {{E}_{{{\text{ECAL}}}}}$ equal to the beam energy $({{E}_{0}})$, above those expected from the background sources.

5 CURRENT AND FUTURE NA64 RESULTS

In this section we briefly discuss last NA64 results and the perspectives of the NA64e(future NA64 experiment with electron beam) and NA64$\mu$ (future NA64 experiment with muon beam).

5.1 NA64e

5.1.1. Invisible mode. Dark photon bounds. The NA64 collected ${\text{NEOT}} = 2.84 \times {{10}^{{11}}}$ statistics in the 2016–2018 years. Recently NA64 collaboration [34] has been analyzed these data and obtained new bounds on $\epsilon$ parameter⁹ by factor $\sim {\kern 1pt} 2.5$ stronger the previous bound [32], see the upper l.h.s. panel in Fig. 6. After the long shutdown (LS2) at CERN the NA64 experiment plans to accumulate ${\text{NEOT}} \gtrsim 5 \times {{10}^{{12}}}$. The NA64e future expected limits on mixing strength $\epsilon$ after the LS2 period assuming the zero-background case are shown in the upper l.h.s. panel in Fig. 6.

Fig. 6.

The upper l.h.s. panel shows the NA64 90% C.L. current bound (solid) [34], and projected boundes for $5 \times {{10}^{{12}}}$(dashed) and ${{10}^{{13}}}$ (dotted) in the (${{m}_{{A{\kern 1pt} '}}},{{\epsilon }^{2}}$) plane. The upper r.h.s plot and lower plots show the required number of EOT for the 90% C.L. exclusion of the $A{\kern 1pt} '$ with a given mass ${{m}_{{A{\kern 1pt} '}}}$ in the (${{m}_{{A{\kern 1pt} '}}},{{n}_{{{\text{EOT}}}}} \times {{10}^{{ - 12}}}$ ) 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) DM models for $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} = 2.5$ (solid), and $= 3$ (dashed), and ${{\alpha }_{{\text{D}}}} =$ 0.1 (red), 0.05 (blue), and 0.02 (green). Upper(lower) black lines correspond to ${{n}_{{{\text{EOT}}}}} = 5 \times {{10}^{{12}}}(2.84 \times {{10}^{{11}}})$. The curves under lower black line are excluded by last NA64 results [34].

To estimate NA64 LDM discovery potential we have used the formulae of Appendix A to calculate the predicted value of ${{\epsilon }^{2}}$ as a function of ${{\alpha }_{{\text{D}}}}$, ${{m}_{\chi }}$ and ${{m}_{{A{\kern 1pt} '}}}$ in the assumption that in the early Universe LDM was in thermo equilibrium. We used the values ${{\alpha }_{{\text{D}}}} = 0.02,0.05,0.1$ and $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} = 2.5,3$. We have made the calculations for the case of scalar, Majorana and pseudo-Dirac LDM with $(\delta \ll 1)$. Our results [91] are presented in Fig. 6. The upper r.h.s plot and lower plots in Fig. 6 show the required number of ${\text{EOT}}$ for the 90% C.L. exclusion of the $A{\kern 1pt} '$ with a given mass ${{m}_{{A{\kern 1pt} '}}}$ in the (${{m}_{{A{\kern 1pt} '}}},{{n}_{{{\text{EOT}}}}} \times {{10}^{{ - 12}}}$) 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) LDM models for $\tfrac{{{{m}_{{A{\kern 1pt} 'A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} = 2.5$ (solid), and = 3 (dashed), and ${{\alpha }_{{\text{D}}}} =$ 0.1 (red), 0.05 (blue), and 0.02 (green). We see that NA64 experiment has already excluded scalar LDM model with ${{\alpha }_{{\text{D}}}} \leqslant 0.1$, $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} \geqslant 3$ and Majorana LDM with ${{\alpha }_{{\text{D}}}} = 0.02$, $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} \geqslant 2.5$. As one can see from Fig. 6 with ${{n}_{{{\text{EOT}}}}} = 5 \times {{10}^{{12}}}$ NA64e will be able to exclude the most interesting and natural LDM scenarios in the $A{\kern 1pt} '$ mass range $1\,\,{\text{MeV}} \leqslant {{m}_{{A{\kern 1pt} '}}} \leqslant 150$ MeV except the most difficult case of pseudo-Dirac LDM with ${{\alpha }_{{\text{D}}}} = 0.1$, ${{\alpha }_{D}} = 0.05$ and $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} = 2.5$.

5.1.2. The problem with resonance region. The expressions for the annihilation cross-sections are proportional to the factor $K = {{\epsilon }^{2}}{{\alpha }_{{\text{D}}}}{{\left( {\tfrac{{m_{{A{\kern 1pt} '}}^{4}}}{{m_{\chi }^{2}}} - 4} \right)}^{{ - 2}}}$. From the assumption that in the early Universe the LDM was in equilibrium with the SM matter we can predict the dependence of $K$ on DM mass ${{m}_{\chi }}$, see Appendix A. In the resonance region ${{m}_{{A{\kern 1pt} '}}} \approx 2{{m}_{\chi }}$ the ${{\varepsilon }^{2}}$ parameter is proportional to ${{K}^{{ - 1}}}$ that can reduce the predicted ${{\epsilon }^{2}}$ value by (2–4) orders of magnitude [92] in comparison with the often used reference point $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} = 3$. It means that NA64 experiment and probably other future experiments will not be able to test the region ${{m}_{{A{\kern 1pt} '}}} \approx 2{{m}_{\chi }}$ completely. It should be mentioned that the values of ${{m}_{{A{\kern 1pt} '}}}$ and ${{m}_{\chi }}$ are arbitrary, so the case ${{m}_{{A{\kern 1pt} '}}} = 2{{m}_{\chi }}$ could be considered as some fine-tuning. It is natural to require the absence of significant fine-tuning. We require that $\left( {\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{2{{m}_{\chi }}}} - 1} \right) \geqslant 0.25$, i.e. ${{m}_{{A{\kern 1pt} '}}} \geqslant 2.5{{m}_{\chi }}$. In our estimates (see Fig. 6) we used two values $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} = 2.5$ and $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} = 3$. As it follows from the previous subsection the NA64 will be able to test the most interesting LDM models for the case of significant fine-tuning absence.

5.1.3. Visible mode. The ⁸Be anomaly. The ATOMKI experiment of Krasznahorkay et al. [26] has reported the observation of a 6.8σ excess of events in the invariant mass distributions of ${{e}^{ + }}{{e}^{ - }}$ pairs produced in the nuclear transitions of excited $^{8}{\text{Be*}}$ to its ground state via internal pair creation. This anomaly can be interpreted as the emission of a new protophobic gauge $X$ boson with a mass of 16.7 MeV followed by its $X \to {{e}^{ + }}{{e}^{ - }}$ decay assuming that the $X$ has non-universal couplings to quarks, coupling to electrons in the range $2 \times {{10}^{{ - 4}}} \lesssim {{\epsilon }_{e}} \lesssim 1.4 \times {{10}^{{ - 3}}}$ and the lifetime ${{10}^{{ - 14}}} \lesssim {{\tau }_{X}} \lesssim {{10}^{{ - 12}}}$ s [54]. It has motivated worldwide theoretical and experimental efforts towards light and weakly coupled vector bosons, see, e.g. [93–99]. Another strong motivation to the search for a new light boson decaying into ${{e}^{ + }}{{e}^{ - }}$ pair is provided by the Dark Matter puzzle discussed previously.

The NA64e combined 90% C.L. exclusion limits on the mixing $\varepsilon$ as a function of the $A{\kern 1pt} '$ mass are shown in Fig. 7 together with the current constraints from other experiments [100]. The NA64 results exclude the X-boson as an explanation of the $^{8}$Be* anomaly for the $X - {{e}^{ - }}$ coupling ${{\epsilon }_{e}} \lesssim 6.8 \times {{10}^{{ - 4}}}$ and the mass value of 16.7 MeV, leaving the still unexplored region $6.8 \times {{10}^{{ - 4}}} \lesssim {{\epsilon }_{e}} \lesssim 1.4 \times {{10}^{{ - 3}}}$ for further searches. Note that in recent paper [101] the last NA64 data [100] has been analyzed. It was shown that at 90% C.L. models with pure vector or axial vector couplings of electron with $X(16.7)$ boson are excluded but the chiral couplings $V \pm A$ are still possible and moreover it is possible to explain both electron ${{g}_{e}} - 2$ and muon ${{g}_{\mu }} - 2$ anomalies [101].

Fig. 7.

The 90% C.L. exclusion area in the $({{m}_{\chi }};\epsilon )$ from the NA64 experiment (blue area). For the mass of $16.7\,\,{\text{MeV}}$, the $X - e$ coupling region excluded by NA64 is $1.2 \times {{10}^{{ - 4}}} < {{\varepsilon }_{e}} < 6.8 \times {{10}^{{ - 4}}}$.

Very recently the ATOMKI group reported a similar excess of events at approximately the same invariant mass in the nuclear transitions of another nucleus, $^{4}$He [102]. This dramatically increases the importance of confirmation of the observed excess by another nuclear physics experiment, as well as independent searches for the X in a particle physics experiment. Therefore, the NA64 experimental approach based on the using two independent electromagnetic calorimeters, one as an activedump (WCAL) for the $X$ boson production and another one (ECAL) for the $X \to {{e}^{ + }}{{e}^{ - }}$ decay detection is extremely timely. To cover the remaining parameter space for the $X - e$ couplings, which corresponds to a very short-lived X boson case with a lifetime ${{\tau }_{X}} \lesssim {{10}^{{ - 13}}}$ s, is very challenging. A more accurate future measurement after LS2 should include also the ${{e}^{ + }}{{e}^{ - }}$ pair invariant mass reconstruction. This requires the use of a high-precision tracker with an excellent two-track resolution capability combined with a magnetic spectrometer for the accurate decay electron and positron momenta measurements to finally reconstruct the invariant mass of the $X$ with a good precision. For this NA64e will need a substantial upgrade of the current setup with a new high-resolution trackers, e.g. based on micromegas detectors, a new WCAL with a better optimised thickness, and a new synchrotron radiation detector with higher granularity. This makes further searching quite challenging but very exciting and important.

5.1.4. NA64e and the search for Z' boson coupled with L_μ – L_τ current. Light $Z{\kern 1pt} '$ boson which couples with ${{L}_{\mu }} - {{L}_{\tau }}$ current will mix with ordinary photon at one-loop level [28]. Namely, an account of one-loop propagator diagrams with virtual $\mu$- and $\tau$-leptons leads to nonzero $\gamma - Z{\kern 1pt} '$ kinetic mixing $- \tfrac{\epsilon }{2}{{F}^{{\mu \nu }}}Z_{{\mu \nu }}^{'}$ where $\epsilon$ is the finite mixing strength given by [13]

$${{\epsilon }_{{1l}}} = \frac{8}{3}\frac{{e{{e}_{\mu }}}}{{16{{\pi }^{2}}}}{\text{ln}}\left( {\frac{{{{m}_{\tau }}}}{{{{m}_{\mu }}}}} \right) = 1.4 \times {{10}^{{ - 2}}} \times {{e}_{\mu }}.$$

(54)

Here $e$ is the electron charge, ${{e}_{\mu }}$ is electron $Z{\kern 1pt} '$ charge and ${{m}_{\mu }},{{m}_{\tau }}$ are the muon and tau lepton masses respectively. It should be stressed that we assume that possible tree level mixing $- \tfrac{{{{\epsilon }_{{{\text{tree}}}}}}}{2}{{F}^{{{\mu \nu }}}}Z_{{\mu \nu }}^{'}$ is absent or much smaller than one-loop mixing $\tfrac{{{{\epsilon }_{{1L}}}}}{2}{{F}^{{\mu \nu }}}{{Z}^{{\mu \nu }}}$. To be precise, we assume that there is no essential cancellation between tree-level and one-loop mixing terms $\left| {{{\epsilon }_{{{\text{tree}}}}} + {{\epsilon }_{{1{\kern 1pt} l}}}| \geqslant |{{\epsilon }_{{1{\kern 1pt} l}}}} \right|$. For ${{m}_{{Z{\kern 1pt} '}}} \ll {{m}_{\mu }}$ the value ${{e}_{\mu }} = (4.8 \pm 0.8) \times {{10}^{{ - 4}}}$ from Eq.(54) leads to the prediction of the corresponding mixing value

$${{\epsilon }_{{1{\kern 1pt} l}}} = (6.7 \pm 1.1) \times {{10}^{{ - 6}}}.$$

(55)

Thus, one can see that the $Z{\kern 1pt} '$ interaction with the ${{L}_{\mu }} - {{L}_{\tau }}$ current induces at one-loop level the $\gamma - Z{\kern 1pt} '$ mixing of $Z{\kern 1pt} '$ with ordinary photon which allows to probe $Z{\kern 1pt} '$ not only in muon or tau induced reactions but also with intense electron beams. In particular, this loophole opens up the possibility of searching the new weak leptonic force mediated by the $Z{\kern 1pt} '$ in experiments looking for dark photons ($A{\kern 1pt} '$). The fact that the $\gamma - Z{\kern 1pt} '$ mixing of Eq. (55) is at an experimentally interesting level is very exciting. We point out further that a new intriguing possibilities for the complementary searches of the $Z'$ in the currently ongoing experiment NA64 [31, 34] exists. Indeed, the NA64 aimed at the direct search for invisible decay of sub-GeV dark photons in the reaction ${{e}^{ - }} + Z \to {{e}^{ - }} + Z + A{\kern 1pt} ';$$A{\kern 1pt} ' \to {\text{invisible}}$ of high energy electron scattering off heavy nuclei [31]. The experimental signature of the invisible decay of $Z'$ produced in the reaction ${{e}^{ - }} + Z \to {{e}^{ - }} + Z + Z{\kern 1pt} ';$$Z{\kern 1pt} ' \to {\text{invisible}}$ due to mixing of Eq. (54) is the same—it is an event with a large missing energy carried away by the $Z'$. Thus, by using Eq. (55) and bounds on the $\gamma - A{\kern 1pt} '$ mixing the NA64 can also set constraints on coupling ${{e}_{\mu }}$.

The current NA64 bounds on the $\epsilon$ parameter for the dark photon mass region $1 \lesssim {{m}_{{Z{\kern 1pt} '}}} \lesssim 10$ MeV are in the range $0.7 \times {{10}^{{ - 5}}} \lesssim \epsilon \lesssim 3 \times {{10}^{{ - 5}}}$ [34]. Taking into account that the sensitivity of the experiment scales as $\epsilon \sim {1 \mathord{\left/ {\vphantom {1 {\sqrt {{{n}_{{{\text{EOT}}}}}} }}} \right. \kern-0em} {\sqrt {{{n}_{{{\text{EOT}}}}}} }}$, results in required increase of statistics by a factor $\simeq {\kern 1pt} {\kern 1pt} 30$ in order to improve sensitivity up to the mixing value of Eq. (55) for this $Z{\kern 1pt} '$ mass region. This would allow either to discover the $Z{\kern 1pt} '$ or exclude it as an explanation of the ${{g}_{\mu }} - 2$ anomaly for the substantial part of the mass range ${{m}_{{Z{\kern 1pt} '}}} \ll {{m}_{\mu }}$ by using the electron beam. The direct search for the $Z{\kern 1pt} '$ in missing-energy events in the reaction $\mu Z \to \mu ZZ{\kern 1pt} '{\kern 1pt} {\kern 1pt} ;$$Z{\kern 1pt} ' \to {\text{invisible}}$ in the dedicated experiment with the muon beam at CERN would then be an important cross check of results obtained with the electron beam. Let us note that the mixing given by the Eq. (55) would also lead to an extra contribution to the elastic $\nu e \to \nu e$ scattering signal in the solar neutrino measurement at the Borexino experiment [103]. The BOREXINO data on the elastic ${{\nu }_{\mu }}e$ scattering [104] lead to lower bound on ${{m}_{{Z{\kern 1pt} '}}} \geqslant (5{\kern 1pt} - {\kern 1pt} 10)\,\,{\text{MeV}}$ by assuming that muon anomaly is explained due to existence of light $Z{\kern 1pt} '$ boson interacting with ${{L}_{\mu }} - {{L}_{\tau }}$ current and there is no tree level mixing between photon and $Z{\kern 1pt} '$, i.e. ${{\epsilon }_{{{\text{tree}}}}} = 0$. The measurement of $\nu - e$ elastic scattering in the LSND experiment [68] set a similar bound to the ${{e}_{\mu }}$ coupling for ${{m}_{{Z\,'}}} \lesssim 10$ MeV [103]. The expected 90% C.L. NA64 exclusion regions in the (${{m}_{{Z{\kern 1pt} '}}},{{e}_{\mu }}$) plane (dashed curves) from the measurements with the electron beam for $\simeq 4 \times {{10}^{{12}}}$ and $\simeq 4 \times {{10}^{{13}}}$ EOT and muon beams for $\simeq {{10}^{{12}}}$ muons on target (MOT) [28] are shown in Fig. 8. Constraints from the BOREXINO [103], CCFR [105], and BABAR [62] experiments, as well as the BBN excluded area [103, 106] are also shown. The parameter space shown in Fig.8 could also be probed by other electron experiments such as Belle II [107], BDX [108, 109], and LDMX [110], which would provide important complementary results.

Fig. 8.

The NA64 90% C.L. expected exclusion regions in the (${{m}_{{Z'}}},{{e}_{\mu }}$) plane (dashed curves) from the measurements with the electron (NA64$e$, $\simeq 4 \times {{10}^{{12}}}$ EOT) and muon (NA64$\mu$, $\simeq {{10}^{{12}}}$ MOT) beams, taken from ref. [111, 112]. Two triangles indicate reference points corresponding to the mass ${{m}_{{Z{\kern 1pt} '}}} = 9$ and 11 MeV, and coupling ${{e}_{\mu }} = 4 \times {{10}^{{ - 4}}}$ and $5 \times {{10}^{{ - 4}}}$, respectively, which are used to explain the IceCube results, see ref. [103] for details.

5.2 The Experiment NA64$\mu$

Recently, the NA64 collaboration proposed to carry out further searches for dark sector and other rare processes in missing energy events from high energy muon interactions in a hermetic detector at the CERN SPS [111, 112].

A dark sector of particles predominantly weakly-coupled to the second and possibly third generations of the SM is motivated by several theoretically interesting models. Additional to gravity this new very weak interaction between the visible and dark sector could be mediated either by a scalar (${{S}_{\mu }}$) or $U{\kern 1pt} '(1)$ gauge bosons (${{Z}_{\mu }}$) interacting with ordinary muons. In a class of ${{L}_{\mu }} - {{L}_{\tau }}$ models the corresponding ${{Z}_{\mu }}$ could be light and have the coupling strength lying in the experimentally accessible region. If such ${{Z}_{\mu }}$ mediator exists it could also explain the muon ${{g}_{\mu }} - 2$ anomaly—the discrepancy between the predicted and measured values of the muon anomalous magnetic moment [111].

The proposed extension of the NA64 experiment called NA64$\mu$ aiming mainly at searching for invisible decays of the ${{Z}_{\mu }}$ either to neutrinos or LDM particles [112]. The primary goal of the experiment in the 2021 pilot run with the $\simeq 100{\kern 1pt} - {\kern 1pt} 160$ GeV M2 beam is to commission the NA64μ detector and to probe for the first time the still unexplored area of the coupling strengths and masses ${{M}_{{{{Z}_{\mu }}}}} \lesssim 200$ MeV that could explain the muon ${{g}_{\mu }} - 2$ anomaly. Another strong point of NA64$\mu$ is its capability for a sensitive search for dark photon mediator ($A{\kern 1pt} '$) of DM production in invisible decay mode in the mass range ${{m}_{{A{\kern 1pt} '}}} \gtrsim {{m}_{\mu }}$, thus making the experiment extremely complementary to the ongoing NA64e and greatly increases the discovery potential of sub-GeV dark matter. Other searches for ${{S}_{\mu }}$’s decaying invisibly to dark sector particles, and millicharged particles will probe a still unexplored parameter areas [112].

5.2.1. Searching for the μ + Z → μ + Z + Z_μ, ${{Z}_{\mu }} \to \nu \bar {\nu }$. The reaction of the ${{Z}_{\mu }}$ production is a rare event. For the previously mentioned parameter space, it is expected to occur with the rate $\lesssim {{{{\alpha }_{\mu }}} \mathord{\left/ {\vphantom {{{{\alpha }_{\mu }}} \alpha }} \right. \kern-0em} \alpha } \sim {{10}^{{ - 6}}}$ with respect to the ordinary photon production rate. Hence, its observation presents a challenge for the detector design and performance. The experimental setup specifically designed to search for the ${{Z}_{\mu }}$ is schematically shown in Fig. 9.

Fig. 9.

Schematic illustration of the NA64$\mu$ setup to search for invisible ${{Z}_{\mu }}$ decays in the reaction $\mu Z \to \mu Z{{Z}_{\mu }}$ [111].

The experiment could employ the upgraded muon beam at the CERN SPS. The beam was designed to transport high fluxes of muons of the maximum momenta in the range between 100 and 225 GeV/c that could be derived from a primary proton beam of 450 GeV/c with the intensity between 10¹² and 10¹³ protons per SPS spill. The detector shown in Fig. 9 utilizes two, upstream and downstream, magnetic spectrometer sections consisting of dipole magnets and a set of low-material budget straw tubes chambers, ST1–ST4 and ST5–ST6, respectively, allowed reconstruction and precise measurements of incident and scattered in a target muons. It also uses scintillating fiber hodoscopes S1, S2, defining the primary muon beam, and S3, S4, and S5 defining the scattered muons, the active target $T$ surrounded by a high efficiency electromagnetic calorimeter (ECAL) serving as a veto against photons and other secondaries emitted from the target at large angles. Downstream the target the detector is equipped with high efficiency forward veto counters V1 and V2 and a massive, completely hermetic hadronic calorimeter (HCAL) located at the end of the setup to detect energy deposited by secondaries from the ${{\mu }^{ - }}A \to anything$ primary muon interactions with nuclei $A$ in the target. The HCAL has lateral and longitudinal segmentation, and also serves for the final state muon identification. For searches at low energies, Cherenkov counters to enhance the incoming muon tagging efficiency can be used.

The method of the search is the following. The bremsstrahlung ${{Z}_{\mu }}$s are produced in the reaction

$$\mu + Z \to \mu + Z + {{Z}_{\mu }},\,\,\,\,{{Z}_{\mu }} \to \nu \bar {\nu },$$

(56)

from the high energy muon scattering off nuclei in the target. The reaction (56) is typically occurred uniformly over the length of the target. The ${{Z}_{\mu }}$ is either stable or decaying invisibly if its mass ${{M}_{{{{Z}_{\mu }}}}} \leqslant 2{{m}_{\mu }}$, or, as shown, it could subsequently decay into a ${{\mu }^{ + }}{{\mu }^{ - }}$ pair if ${{M}_{{{{Z}_{\mu }}}}} > 2{{m}_{\mu }}$. In the former case, the ${{Z}_{\mu }}$ penetrates the T, veto V1, V2 and the massive HCAL without interaction. In the later case, it could decays in flight into a ${{\mu }^{ + }}{{\mu }^{ - }}$ pair, resulting in the di-muon track signature in the detector. The bremsstrahlung ${{Z}_{\mu }}$ then either penetrates the rest of the detector without interactions, resulting in zero-energy deposition in the V1, V2 and HCAL , or it could decay in flight into a ${{\mu }^{ + }}{{\mu }^{ - }}$ pair if its mass is greater than the mass of two muons. A fraction ($f \lesssim 0.3$) of the primary beam energy ${{E}_{\mu }} = f{{E}_{0}}$ is carried away by the scattered muon which is detected by the second magnetic spectrometer arm. For the radiation length ${{X}_{0}} \lesssim 1$ cm, and the total thickness of the target $\simeq 30$ cm the energy leak from the target into the V1 is negligibly small. The remained part of the primary muon energy ${{E}_{2}} = (1 - f){{E}_{0}}$ is transmitted through the “HCAL wall” by the ${{Z}_{\mu }}$, or deposited partly in the HCAL via the ${{Z}_{\mu }}$ decay in flight ${{Z}_{\mu }} \to {{\mu }^{ + }}{{\mu }^{ - }}$. At ${{Z}_{\mu }}$ energies ${{E}_{{{{Z}_{\mu }}}}} \lesssim 50$ GeV, the opening angle ${{\Theta }_{{{{\mu }^{ + }}{{\mu }^{ - }}}}} \simeq {{{{M}_{{{{Z}_{\mu }}}}}} \mathord{\left/ {\vphantom {{{{M}_{{{{Z}_{\mu }}}}}} {{{E}_{{{{Z}_{\mu }}}}}}}} \right. \kern-0em} {{{E}_{{{{Z}_{\mu }}}}}}}$ of the decay ${{\mu }^{ + }}{{\mu }^{ - }}$ pair is big enough to be resolved in two separated tracks in the M1 and M2 so the pairs are mostly detected as a double track event. The HACL is served as a dump to absorb completely the energy of secondary particles produced in the primary pion or kaon interaction in the target. In order to suppress background due to the detection inefficiency, the detector must be longitudinally completely hermetic. To enhance detector hermeticity, the hadronic calorimeter has the total thickness of $\simeq 28{{\lambda }_{{{\text{int}}}}}$ (nuclear interaction lengths) and placed behind the DV.

The signature of the reaction (56) is

• the presence of incoming muon with energy around 150 GeV,

• the presence of scattered muon with energy $\lesssim 80$ GeV,

• no energy deposition in the HCAL,

• no energy deposition in the HCAL EE.

The occurrence of ${{Z}_{\mu }}$ produced in ${{\mu }^{ - }}Z$ interactions would appear as an excess of events with a single low energy muon accompanied by zero-energy deposition in the detector. The backgrounds for the reaction (56) have been analyzed in ref. [111, 112]. The main backgrounds are due to $\mu$ low-energy tail, HCAL nonhermeticity, $\mu$ induced photonuclear reactions and $\mu$ trident events [111, 112]. These backgrounds were estimated in ref. [111, 112] and they are rather small $\lesssim {{10}^{{ - 12}}}$.

The expected sensitivity of this experiment for ${{\alpha }_{\mu }}$ for different ${{Z}_{\mu }}$ masses and for ${{10}^{{12}}}$ muons on target is shown in Fig. 10. Note that in refs. [113–115] the possibility to use muon beam for the search for light scalar particles has been discussed.

Fig. 10.

Expected constraints on the ${{\alpha }_{\mu }}$ coupling constant as a function of the ${{Z}_{\mu }}$ mass for ${{10}^{{12}}}$$\mu$ at energy ${{E}_{\mu }} =$ 150 GeV [111, 112].

In the $A{\kern 1pt} '$ dark photon model muons and electrons interact with the dark photon with the same coupling constant. Hence, similar to the reaction of Eq. (53), the dark photons will be also produced in the reaction of Eq. (56) with the same experimental signature of the missing energy. For the $A{\kern 1pt} '$ mass region ${{m}_{{A{\kern 1pt} '}}} \gg {{m}_{e}}$, the total cross-section of the dark photon electroproduction $eZ \to eZA{\kern 1pt} '$ scales as $\sigma _{{A{\kern 1pt} '}}^{e} \sim {{\epsilon _{e}^{2}} \mathord{\left/ {\vphantom {{\epsilon _{e}^{2}} {m_{{A{\kern 1pt} '}}^{2}}}} \right. \kern-0em} {m_{{A{\kern 1pt} '}}^{2}}}$. On the other hand, for the dark photon masses, ${{m}_{{A{\kern 1pt} '}}} \lesssim {{m}_{\mu }}$, the similar $\mu Z \to \mu ZA{\kern 1pt} '$ cross-section can be approximated in the bremsstrahlung-like limit as $\sigma _{{A{\kern 1pt} '}}^{\mu } \sim {{\epsilon _{\mu }^{2}} \mathord{\left/ {\vphantom {{\epsilon _{\mu }^{2}} {m_{\mu }^{2}}}} \right. \kern-0em} {m_{\mu }^{2}}}$. Let us now compare expected sensitivities of the $A{\kern 1pt} '$ 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 $A{\kern 1pt} '$ produced by the 100 GeV electron and muon beam can approximated, respectively, as follows

$$\begin{gathered} N_{{A{\kern 1pt} '}}^{e} \approx \frac{{\rho {{N}_{{{\text{av}}}}}}}{A} \times {{n}_{{{\text{EOT}}}}}{{L}^{e}}\sigma _{{A{\kern 1pt} '}}^{e}, \hfill \\ N_{{A{\kern 1pt} '}}^{\mu } \approx \frac{{\rho {{N}_{{{\text{av}}}}}}}{A} \times {{n}_{{{\text{MOT}}}}}{{L}^{\mu }}\sigma _{{A{\kern 1pt} '}}^{\mu }, \hfill \\ \end{gathered} $$

(57)

where ${{L}^{e}} \simeq {{X}_{0}}$ and ${{L}^{\mu }} \simeq 40{{X}_{0}}$ are the typical distances that are passed by an electron and muon, respectively, before producing the $A{\kern 1pt} '$ with the energy ${{E}_{{A{\kern 1pt} '}}} \gtrsim 50$ GeV in the NA64 active Pb target of the total thickness of $\simeq 40$ radiation length (${{X}_{0}}$) [111]. The detailed comparison of the calculated $A'$ sensitivities of NA64e and NA64μ is shown in Fig. 11, 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 NA64μ experiments. The limits were obtained for the background free case by using exact-tree-level (ETL) cross-sections rather than the improved Weizsacker-Williams (IWW) ones calculated for NA64e in ref. [41], and for the NA64$\mu$ case in this work. The later are shown in Fig. 12 as a function of ${{{{E}_{{A{\kern 1pt} '}}}} \mathord{\left/ {\vphantom {{{{E}_{{A{\kern 1pt} '}}}} {{{E}_{\mu }}}}} \right. \kern-0em} {{{E}_{\mu }}}}$ for the Pb target and mixing value $\epsilon = 1$. One can see that in a wide range of masses, $20\,\,{\text{MeV}} \lesssim {{m}_{{A{\kern 1pt} '}}} \lesssim 1\,\,{\text{GeV}}$, the total IWW 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 IWW case. For ${{n}_{{{\text{EOT}}}}} = {{n}_{{{\text{MOT}}}}} = 5 \times {{10}^{{12}}}$ the sensitivity of NA64e is enhanced for the mass range ${{m}_{e}} \ll {{m}_{{A{\kern 1pt} '}}} \simeq 100$ MeV while for the $A{\kern 1pt} '$ masses ${{m}_{{A{\kern 1pt} '}}} \gtrsim 100$ MeV NA64$\mu$ allows to obtain more stringent limits on $\epsilon$ in comparison with NA64e.

Fig. 11.

The NA64e 90% C.L. current [34] and expected exclusion bounds obtained with $2.84 \times {{10}^{{11}}}$ EOT and $5 \times {{10}^{{12}}}$ EOT, respectively, in the (${{m}_{{A'}}},\varepsilon$) plane. The NA64$\mu$ projected bounds calculated for ${{n}_{{{\text{MOT}}}}} = 5 \times {{10}^{{12}}}$ and $5 \times {{10}^{{13}}}$ are also shown.

Fig. 12.

Cross-section of dark photon production by muons as a function of $x = {{{{E}_{{A{\kern 1pt} '}}}} \mathord{\left/ {\vphantom {{{{E}_{{A{\kern 1pt} '}}}} {{{E}_{\mu }}}}} \right. \kern-0em} {{{E}_{\mu }}}}$ for various masses ${{m}_{{A{\kern 1pt} '}}}$ and $\epsilon = 1$. Solid lines represent ETL cross-sections and dashed lines show the cross-sections calculated in IWW approach.

5.3 Combined LDM Sensitivity of NA64e and NA64$\mu$ [91]

The estimated NA64e and NA64$\mu$ limits on the $\gamma - A{\kern 1pt} '$ 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. 13. As discussed in Appendix A, as a result of the $\gamma - A{\kern 1pt} '$ 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 ($y \equiv {{\epsilon }^{2}}{{\alpha }_{D}}{{({{{{m}_{\chi }}} \mathord{\left/ {\vphantom {{{{m}_{\chi }}} {{{m}_{{A{\kern 1pt} '}}}}}} \right. \kern-0em} {{{m}_{{A{\kern 1pt} '}}}}})}^{4}};{{m}_{\chi }}$) plane and restrict the LDM models with the masses ${{m}_{\chi }} \lesssim 1$ GeV.

Fig. 13.

The NA64 90% C.L. current (solid) [34] and expected (dotted light blue) exclusion bounds for $5 \times {{10}^{{12}}}$ EOT in the (${{m}_{\chi }},y$) and (${{m}_{\chi }},{{\alpha }_{{\text{D}}}}$) planes. The combined limits from NA64e and NA64$\mu$ are also shown for ${{10}^{{13}}}$ EOT plus $2 \times {{10}^{{13}}}$ MOT (dashed blue). The limits are calculated for ${{\alpha }_{{\text{D}}}} = 0.1$ and 0.5, and ${{m}_{{A{\kern 1pt} '}}} = 3{{m}_{\chi }}$. The results are also shown in comparison with bounds obtained from the results of the LSND [68], E137 [63], BaBar [62] and MiniBooNE [69] experiments.

The combined limits [34] obtained from the data sample of the 2016, 2017 and 2018 runs and expected from the run after the LS2 are shown in the top panels of Fig. 13 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 ${{\epsilon }^{4}}{{\alpha }_{{\text{D}}}}$ as in beam dump experiments. Therefore, for sufficiently small values of ${{\alpha }_{{\text{D}}}}$ the NA64 limits will be much stronger. This is illustrated in the upper right panel of Fig. 13, where the NA64 limits are shown for ${{\alpha }_{{\text{D}}}} = 0.1$. One can see that for this or smaller values of ${{\alpha }_{{\text{D}}}}$ 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 $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} = 3$ 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 ${{\alpha }_{{\text{D}}}} \leqslant 0.1$ 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 OTHER FUTURE EXPERIMENTS

There are a lot of planned experiments devoted to the search for both visible and invisible $A{\kern 1pt} {\kern 1pt} '$ decay modes. Here we briefly describe the most interesting future experiments.

6.1 SHiP at CERN

The proposed experiment SHiP [116] at CERN is intended to look for visible decays $A{\kern 1pt} {\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }},{{\mu }^{ + }}{{\mu }^{ - }},{{\pi }^{ + }}{{\pi }^{ - }}$ of long lived $A{\kern 1pt} '$ boson. Also SHiP can search for LDM by detection of the LDM scattering in neutrino detector at the 400 GeV SPS beam line at CERN. The detector consists of OPERA-like bricks of lead and emulsions placed in magnetic field. The LDM detection occurs via electon LDM elastic scattering. The dominant backgrounds are expected related with neutrino scattering processes and can be reduced using several cuts. For $N = {{10}^{{20}}}{\text{POT}}$¹⁰ the sensitivity is $y \equiv {{\epsilon }^{2}}{{\alpha }_{{\text{D}}}}{{\left( {\tfrac{{{{m}_{\chi }}}}{{{{m}_{{A{\kern 1pt} '}}}}}} \right)}^{4}} \geqslant {{10}^{{ - 12}}}$ for ${{m}_{\chi }} \leqslant O(1)\,\,{\text{GeV}}$ [4].

6.2 Belle-II at KEK

Belle-II [117] is a multi-purpose detector with sensitivity to invisible $A{\kern 1pt} '$ decays via mono-photon in the range ${{M}_{{A{\kern 1pt} '}}} \leqslant 9.5\,\,{\text{GeV}}$ can look for $A{\kern 1pt} '$ invisible decays using the reaction ${{e}^{ + }}{{e}^{ - }} \to \gamma (A{\kern 1pt} ' \to {\text{invisible}})$. Belle-II also can search for visible $A{\kern 1pt} '$ decays. First data with full luminosity ${{L}_{t}} = 50\,\,{\text{a}}{{{\text{b}}}^{{ - 1}}}$ are expected in 2025. The future sensitivity is ${{\epsilon }^{2}} \geqslant {{10}^{{ - 9}}}$ for ${{m}_{{A{\kern 1pt} '}}} < 9.5\,\,{\text{GeV}}$.

6.3 MAGIX at MESA

Visible dark photon decay searches with dipole spectrometer MAGIX at the $105\,\,{\text{MeV}}$ polarized electron beam are planned at MESA accelerator complex [118]. The electroproduction reaction $eZ \to eZA{\kern 1pt} '$ and visible decay mode $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ will be used to identify the $A{\kern 1pt} '$ as di-electron resonance. The expected sensitivity to the ${{\epsilon }^{2}}$ parameter is up to ${{10}^{{ - 9}}}$ for $10\,\,{\text{MeV}} < {{m}_{{A{\kern 1pt} '}}} < 60\,\,{\text{MeV}}$.

6.4 PADME at LNF

The reaction ${{e}^{ + }}{{e}^{ - }} \to \gamma (A{\kern 1pt} ' \to invisible)$ is used for the search for dark photon. For ${{10}^{{13}}}$ positron on target the expected sensitivity is ${{\epsilon }^{2}} \geqslant {{10}^{{ - 7}}}$ for ${{m}_{{A{\kern 1pt} '}}} < 24\,\,{\text{MeV}}$ [119]. The collection of data started at the end of 2018.

6.5 VEPP3 at BINP

The proposed experiment at BINP [120] is similar to PADME experiment. The expected sensitivity is planned to be ${{\epsilon }^{2}} \geqslant {{10}^{{ - 8}}}$ in the range $5 < {{m}_{{A{\kern 1pt} '}}} < 22\,\,{\text{MeV}}$.

6.6 BDX at JLab

BDX ar JLab is an electron beam-dump experiment [108, 109]. The experiment is sensitive to elastic DM scattering $e\chi \to e\chi$ in the far detector after electron nuclei production in $eZ \to eZ(A{\kern 1pt} ' \to \chi \bar {\chi })$. The expected sensitivity is $y \geqslant {{10}^{{ - 13}}}$ for $1\,\,{\text{MeV}} < {{m}_{\chi }} < 100\,\,{\text{MeV}}$.

6.7 DarkLight at JLab

In this experiment dark photons are produced in the reaction $ep \to epA{\kern 1pt} '$ colliding the $100\,\,{\text{MeV}}$ electron beam on a gaseous hydrogen target [121, 109]. The main peculiarity of this experiment is the possibility to detect the scattered electron and recoil proton, enabling the reconstruction of invisible $A{\kern 1pt} '$ decays. Also the search for visible $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ decays is possible. The expected sensitivity is ${{\epsilon }^{2}} \geqslant {{10}^{{ - 6}}}$ for $10\,\,{\text{MeV}} < {{m}_{{A{\kern 1pt} '}}} < 80\,\,{\text{MeV}}$.

6.8 LDMX

This experiment is similar to NA64 experiment and will use the electroproduction reaction $eZ \to eZ(A{\kern 1pt} ' \to \chi \bar {\chi })$ for the dark photon search [110]. The LDMX(Light Dark Matter Experiment) will measure both missing energy and missing momentum that is extremely important for background suppression. The expected sensitivity for the $\varepsilon$ parameter is up to ${{10}^{{ - 6}}}$ for ${{m}_{{A{\kern 1pt} '}}} = 1\,\,{\text{MeV}}$ [110]. The extended LDMX will be able to increase sensitivity to the $\epsilon$ parameter by factor $10$.

7 CONCLUSIONS

Active beam-dump searches for dark sector physics in missing energy events have been proven by the NA64 experiment to be very powerful and sensitive via both invisible and visible decays of dark vector mediator. The future combined sensitivity of searches with both electron and muon beams has a great potential to probe a large region of the remaining LDM parameter space, especially towards the higher LDM masses. Remarkably, that with ${\text{the}}$ statistics accumulated during years 2016–2018 NA64 already starts probing the sub-GeV DM parameter space. While with $5 \times {{10}^{{12}}}$ EOT NA64 with electron beam is able to test the scalar and Majorana LDM scenarios for $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} \geqslant 2.5$. The combined NA64 results with electron and muon beams and with ${{10}^{{13}}}$ EOT, $2 \times {{10}^{{13}}}$ 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 ${{Z}_{\mu }}$. This makes NA64e and NA64$\mu$ extremely complementary to each other, as well as to the planned LDMX experiment [110], and greatly increases the NA64 discovery potential of sub-GeV DM.

There are several alternatives [7] to the dark photon model based on the use of gauge symmetries like $U{{(1)}_{{B - L}}}$ or $U{{(1)}_{{B - 3e}}}$. As in the dark photon model the observed value of the LDM density allows to estimate the coupling constant $\epsilon$ of new light $Z{\kern 1pt} '$ boson with electron. The value of the $\varepsilon$ parameter for such models coincides with the $\varepsilon$ value for dark photon model up to some factor $k \leqslant 3$ [7], so NA64e can also test such models. For instance, for the model with $(B{\text{--}}L)$ vector interaction NA64e is able to exclude scalar and Majorana LDM scenarios in full analogy with the case of dark photon model.

However it should be stressed that for ${{m}_{{A{\kern 1pt} '}}} \approx 2{{m}_{\chi }}$ the DM annihilation cross-section is proportional to ${{(m_{{A{\kern 1pt} '}}^{2} - 4m_{\chi }^{2})}^{{ - 2}}}$. As a consequence the predicted value of the ${{\epsilon }^{2}}$ parameter is proportional to ${{\left( {\tfrac{{m_{{A{\kern 1pt} '}}^{2}}}{{4m_{\chi }^{2}}} - 1} \right)}^{2}}$ that can reduce the ${{\epsilon }^{2}}$ value by (2–4) orders of magnitude in comparison with the reference point $\tfrac{{{{m}_{{A{\kern 1pt} '}}}}}{{{{m}_{\chi }}}} = 3$ [92]. It means that NA64 experiment as other future experiments like LDMX [110] are not able to test the region ${{m}_{{A{\kern 1pt} '}}} \approx 2{{m}_{\chi }}$ completely¹¹.

Current accelerator experimental data¹² restrict rather strongly the explanation of the ${{g}_{\mu }} - 2$ muon anomaly due to existence of new light gauge boson but not completely eliminate it. The most popular model where dark photon $A{\kern 1pt} '$ interacts with the SM electromagnetic current due to mixing $\tfrac{\epsilon }{2}{{F}_{{\mu \nu }}}{{F}^{{'\mu \nu }}}$ term is excluded. The Borexino data on neutrino electron elastic scattering exclude the models where $Z{\kern 1pt} '$ interacts with both leptonic and $B - L$ currents. The interaction of the $Z{\kern 1pt} '$ boson with ${{L}_{\mu }} - {{L}_{\tau }}$ current is excluded for ${{m}_{{Z{\kern 1pt} '}}} \geqslant 214\,\,{\text{MeV}}$ while still leaving the region of lower masses unconstrained. NA64$\mu$ is able to test the model with ${{L}_{\mu }} - {{L}_{\tau }}$ interaction at ${{m}_{{Z{\kern 1pt} '}}} \leqslant 214\,\,{\text{MeV}}$ as a model explaining muon ${{g}_{\mu }} - 2$ anomaly.

Appendices

APPENDIX A: DM Density Calculations

The observed homogeneity and isotropy of the Universe enable us to describe the overall geometry and evolution of the Universe in terms of two cosmological parameters accounting for the spatial curvature and the overall expansion (or contraction) of the Universe that is realized in the Freedman–Robertson–Walker metric¹³

$$d{{s}^{2}} = d{{t}^{2}} - {{R}^{2}}(t)\left[ {\frac{{d{{r}^{2}}}}{{1 - k{{r}^{2}}}} + {{r}^{2}}(d{{\theta }^{2}} + si{{n}^{2}}\theta d{{\Phi }^{2}})} \right].$$

(58)

The curvature constant $k$ takes three values $k = 1, - 1,0$ that corresponds to closed, open and spationally flat geometries. The cosmological equations are derived from Einstein’s equations

$${{R}_{{\mu \nu }}} - \frac{1}{2}{{g}_{{\mu \nu }}} = 8\pi {{G}_{N}}{{T}_{{\mu \nu }}} + \Lambda {{g}_{{\mu \nu }}}.$$

(59)

We shall use the standard assumption that an effective energy-momentum tensor ${{T}_{{\mu \nu }}}$ is a perfect fluid, for which

$$T\mu \nu = - p{{g}_{{\mu \nu }}} + (p + \rho ){{u}_{\mu }}{{u}_{\nu }},$$

(60)

where $p$ is the pressure, $\rho$ is the energy-density and $u = (1,0,0,0)$ is the velocity vector for the isotropic fluid in co-moving coordinates. For the metric (58) and the energy-momentum tensor (60) the Einstein equations (59) lead to Friedman–Lemaitre equations

$${{H}^{2}} = \frac{{8\pi {{G}_{N}}\rho }}{3} - \frac{k}{{{{R}^{2}}}} + \frac{\Lambda }{3},$$

(61)

$$\frac{1}{{{{a}^{2}}(t)}}\frac{{{{d}^{2}}a}}{{d{{t}^{2}}}} = \frac{\Lambda }{3} - \frac{{4\pi {{G}_{N}}}}{3}(\rho + 3p),$$

(62)

$$H(t) \equiv \frac{1}{{R(t)}}\frac{{dR}}{{dt}},$$

(63)

where $H(t)$ is the Hubble parameter and $\Lambda$ is cosmological constant. Energy conservation $T_{{;\nu }}^{{\mu \nu }} = 0$ leads to the equation

$$\frac{{d\rho }}{{dt}} = - 3H(\rho + p).$$

(64)

The equation (64) allows to determine today critical density ${{\rho }_{c}}$ that corresponds to flat Universe with $k = 0$ and $\Lambda = 0$ in the equations (61), (62), namely

$${{\rho }_{c}} = \frac{{3{{H}^{2}}}}{{8\pi {{G}_{N}}}} = 1.05 \times {{10}^{{ - 5}}}{{h}^{2}}\,\,{\text{GeV}}\,\,{\text{c}}{{{\text{m}}}^{{ - 3}}}$$

(65)

Here the parameter $h$ is defined by

$$H \equiv 100\,\,{\text{h}}\,\,{\text{km}}\,\,{{{\text{s}}}^{{ - 1}}}\,\,{\text{Mp}}{{{\text{c}}}^{{ - 1}}}$$

(66)

and its experimental value is $h = 0.72 \pm 0.03$ [9]. The cosmological density parameter ${{\Omega }_{{{\text{tot}}}}}$ is defined as the energy density relative to the critical density

$${{\Omega }_{{{\text{tot}}}}} = {\rho \mathord{\left/ {\vphantom {\rho {{{\rho }_{c}}}}} \right. \kern-0em} {{{\rho }_{c}}}}.$$

(67)

One can rewrite the equation (61) in the form

$$\frac{k}{{{{R}^{2}}}} = {{H}^{2}}({{\Omega }_{{{\text{tot}}}}} - 1).$$

(68)

As a consequence of the equation (68) we see that for ${{\Omega }_{{{\text{tot}}}}} > 1$ the Universe is closed, for ${{\Omega }_{{{\text{tot}}}}} < 1$ the Universe is open and for ${{\Omega }_{{{\text{tot}}}}} = 1$ the Universe is spatially flat. It is often necessary to distinguish different contributions to the density ${{\Omega }_{{{\text{tot}}}}}$. It is convenient to define present-day density parameters for pressureless matter ${{\Omega }_{m}}$ and relativistic particles ${{\Omega }_{r}}$ plus the vacuum dark energy density ${{\Omega }_{V}}$ and the dark matter density ${{\Omega }_{d}}$. Current data give [9]

$${{\Omega }_{V}} = 0.73 \pm 0.01,$$

(69)

$${{\Omega }_{d}} = 0.23 \pm 0.01.$$

(70)

It is expected that the early Universe can be described by a radiation-dominated equation of state. In addition it is assumed that through much of the radiation-dominated period, thermal equillibrium is established by the rapid rate of particle interactions relative to the expansion rate of the Universe. In equilibrium thermodynamic quantities like energy density, pressure and entropy are calculable quantities in the ideal gas approximation. The density of states for particle $i$ is given by

$$d{{n}_{i}} = \frac{{{{g}_{i}}{{d}^{3}}\vec {p}}}{{{{{(2\pi )}}^{3}}}}{{\left( {exp\left[ {\frac{{{{E}_{i}} - {{\mu }_{i}}}}{{{{T}_{i}}}}} \right] \pm 1} \right)}^{{ - 1}}}.$$

(71)

Here ${{g}_{i}}$ counts the number of degrees of freedom of particle $i$, $E_{i}^{2} = \mathop {\vec {p}}\nolimits^2 + m_{i}^{2}$, $\pm$ corresponds to either Fermi or Bose statistics, ${{\mu }_{i}}$ is the chemical potential¹⁴ and ${{T}_{i}}$ is the temperature. The energy density, the pressure, the number density and the entropy density are given by the formulae

$${{\rho }_{i}} = \int {{{E}_{i}}d{{n}_{i}}} ,$$

(72)

$${{p}_{i}} = \frac{1}{3}\int {\frac{{\vec {p}_{i}^{2}}}{{{{E}_{i}}}}} d{{n}_{i}},$$

(73)

$${{n}_{i}} = \int {d{{n}_{i}}} ,$$

(74)

$${{s}_{i}} = \frac{{{{\rho }_{i}} + {{p}_{i}} - {{\mu }_{i}}{{n}_{i}}}}{{{{T}_{i}}}}.$$

(75)

For instance, for photons with ${{g}_{\gamma }} = 2$ polarization states the energy density, pressure, density of the number of photons and the entropy density are given by the formulae

$${{\rho }_{\gamma }} = \frac{{{{\pi }^{2}}}}{{15}}{{T}^{4}},$$

(76)

$${{p}_{\gamma }} = \frac{1}{3}{{\rho }_{\gamma }},$$

(77)

$${{s}_{\gamma }} = \frac{{4{{\rho }_{\gamma }}}}{{3T}},$$

(78)

$${{n}_{\gamma }} = 0.243{{T}^{3}}.$$

(79)

The number density of nonrelativistic particles is given by the formula

$${{n}_{{{\text{nonrel}}}}} = g\frac{1}{{{{{(2\pi )}}^{{3/2}}}}}{{(mT)}^{{3/2}}}{\text{exp}}\left( { - \frac{m}{T}} \right),$$

(80)

where $g$ is the number of polarizations. As a consequence of the equations (61), (62) and the definition (75) of the entropy density one can find that the total entropy is conserved, namely

$$\frac{{d(s{{R}^{3}})}}{{dt}} = 0.$$

(81)

At the very high temperatures associated with the early Universe, massive particles are pair produced, and are part of the thermal bath. At high temperature $T \gg {{m}_{i}}$ we can neglect masses and approximate the energy density by including those particles with ${{m}_{i}} \ll T$, namely

$$\rho = \left( {\sum\limits_B {{{g}_{B}}} + \frac{7}{8}\sum\limits_F {{{g}_{F}}} } \right)\frac{{{{\pi }^{2}}}}{{30}}{{T}^{4}} \equiv {{g}_{\rho }}{{T}^{4}},$$

(82)

where ${{g}_{{B(F)}}}$ is the number of degrees of freedom of each boson (fermion) and the sum runs over all bosons and fermions with $m \ll T$. The factor ${7 \mathord{\left/ {\vphantom {7 8}} \right. \kern-0em} 8}$ is due to the difference between ${\text{the}}$ Fermi and Bose integrals (71)–(75). The equation (82) defines the effective number of degrees of freedom. For instance, for temperature ${{m}_{e}} < T < {{m}_{\mu }}$ the effective number ${{g}_{\rho }} = {{43} \mathord{\left/ {\vphantom {{43} 4}} \right. \kern-0em} 4}$.

To obtain estimate of dark matter density we have to solve the Boltzmann equation

$$\frac{{d{{n}_{d}}}}{{dt}} + 3H(T){{n}_{d}} = - \left\langle {\sigma {{v}_{{{\text{rel}}}}}} \right\rangle (n_{d}^{2} - n_{{d,{\text{eq}}}}^{2}).$$

(83)

Here

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

(84)

and ${{f}_{d}}(p,T)$ is DM distribution function. The equilibrium nonrelativistic DM density is

$${{n}_{{d,{\text{eq}}}}} = {{g}_{d}}\frac{1}{{{{{(2\pi )}}^{{3/2}}}}}{{({{m}_{\chi }}T)}^{{3/2}}}{\text{exp}}\left( { - \frac{{{{m}_{\chi }}}}{T}} \right),$$

(85)

where ${{m}_{\chi }}$ is the mass of DM particle. The $\left\langle {\sigma v} \right\rangle$ is thermally pair averaged cross section [1, 122]

$$\begin{gathered} \left\langle {\sigma v} \right\rangle = \frac{1}{{8m_{\chi }^{4}T{{K}_{2}}{{{\left( {\tfrac{{{{m}_{\chi }}}}{T}} \right)}}^{2}}}} \\ \times \,\,\int\limits_{4m_{\chi }^{2}}^\infty {ds\sigma } (s)\sqrt s (s - 4m_{\chi }^{2}){{K}_{1}}\left( {\frac{{\sqrt s }}{T}} \right). \\ \end{gathered} $$

(86)

In nonrelativistic approximation $\left\langle {\tfrac{{{{m}_{\chi }}{{{\vec {v}}}^{2}}}}{2}} \right\rangle = \tfrac{{3T}}{2}$.

The DM relative density parameter ${{\Omega }_{d}}$ is represented in the form

$${{\Omega }_{d}} = \frac{{{{m}_{\chi }}{{s}_{o}}{{Y}_{0}}}}{{{{\rho }_{c}}}},$$

(87)

where ${{s}_{0}} \equiv s({{T}_{0}})$ is today dark entropy density and $Y \equiv \tfrac{{{{n}_{d}}}}{s}$ is approximately constant for iso-entropic Universe $(Y({{t}_{d}}) \approx Y({{t}_{0}}))$. The evolution equation for $Y(t)$ reads

$$\frac{{dY}}{{dt}} = - s\left\langle {{{\sigma }_{{{\text{rel}}}}}} \right\rangle ({{Y}^{2}} - Y_{{{\text{eq}}}}^{2}).$$

(88)

The equation (88) can be rewritten in the form

$$\frac{{dY}}{{dx}} = \frac{1}{{3H}}\frac{{ds}}{{dx}}\left\langle {{{\sigma }_{{{\text{rel}}}}}} \right\rangle ({{Y}^{2}} - Y_{{{\text{eq}}}}^{2}).$$

(89)

Here $x = \tfrac{{{{m}_{\chi }}}}{T}$ and $T$ is photon temperature. Note that for the flat Universe the Hubble parameter $H = {{\left( {\tfrac{8}{3}\pi G\rho } \right)}^{{1/2}}}$. The effective degrees of freedom for the energy and entropy densities are defined by

$$\rho = {{g}_{{{\text{eff}}}}}(T)\frac{{{{\pi }^{2}}}}{{30}}{{T}^{4}},$$

(90)

$$s = {{h}_{{{\text{eff}}}}}(T)\frac{{2{{\pi }^{2}}}}{{45}}{{T}^{3}}$$

(91)

respectively, in such a way that the ${{g}_{{{\text{eff}}}}}(T) = {{h}_{{{\text{eff}}}}}(T) = 1$ for a relativistic species with one internal or spin degree of freedom. Taking into account (91) equation (89) takes the form

$$\frac{{dY}}{{dx}} = - {{\left( {\frac{{45}}{\pi }G} \right)}^{{ - 1/2}}}\frac{{g_{ * }^{{1/2}}{{m}_{\chi }}}}{{{{x}^{2}}}}\left\langle {\sigma {{v}_{{{\text{rel}}}}}} \right\rangle ({{Y}^{2}} - Y_{{{\text{eq}}}}^{2}),$$

(92)

where

$$g_{ * }^{{1/2}} = \frac{{{{h}_{{{\text{eff}}}}}}}{{g_{{{\text{eff}}}}^{{1/2}}}}\left( {1 + \frac{1}{3}\frac{T}{{{{h}_{{{\text{eff}}}}}}}\frac{{d{{h}_{{{\text{eff}}}}}}}{{dT}}} \right).$$

(93)

The equilibrium density ${{Y}_{{{\text{eq}}}}}$ is given by

$${{Y}_{{{\text{eq}}}}} = \frac{{45g}}{{4{{\pi }^{2}}}}\frac{{{{x}^{2}}{{K}_{2}}(x)}}{{{{h}_{{{\text{eff}}}}}\left( {\tfrac{{{{m}_{\chi }}}}{x}} \right)}}.$$

(94)

The solution of the equation (92) allows to determine the freeze-out temperature ${{T}_{d}}$. The decoupling temperature ${{T}_{d}}$ is usually defined by the equation

$$\Delta \equiv Y - {{Y}_{{{\text{eq}}}}} = \delta {{Y}_{{{\text{eq}}}}}.$$

(95)

In the approximation $\tfrac{{d\Delta }}{{dx}} = 0$ the equation

$${{\left( {\frac{{45}}{\pi }G} \right)}^{{ - 1/2}}}\frac{{g_{ * }^{{1/2}}{{m}_{\chi }}}}{{{{x}^{2}}}}\left\langle {\sigma {{v}_{{{\text{rel}}}}}} \right\rangle {{Y}_{{{\text{eq}}}}}\delta (\delta + 2) = - \frac{{dln{{Y}_{{{\text{eq}}}}}}}{{dx}}$$

(96)

allows to determine the decoupling temperature ${{T}_{d}}$. The parameter $\delta$ is usually taken to be $\delta = 1.5$. After the decoupling we can neglect ${{Y}_{{{\text{eq}}}}}$ in the equation (83) and the integration from ${{T}_{d}}$ to ${{T}_{0}}$ gives [1, 122]

$$\frac{1}{{{{Y}_{0}}}} = \frac{1}{{{{Y}_{d}}}} + {{\left( {\sqrt {\frac{\pi }{{45}}} {{M}_{{{\text{PL}}}}}} \right)}^{{ - 1}}}\left[ {\int\limits_{{{T}_{0}}}^{{{T}_{d}}} {(g_{ * }^{{1/2}}(\left\langle {\sigma v} \right\rangle )dT} } \right].$$

(97)

Numerically ${{Y}_{d}} \gg {{Y}_{o}}$ and we can neglect it, so we obtain [1, 122]

$${{Y}_{o}} \approx \sqrt {\frac{\pi }{{45}}} {{M}_{{{\text{PL}}}}}{{\left[ {\int\limits_{{{T}_{0}}}^{{{T}_{{{\text{dec}}}}}} {(g_{ * }^{{1/2}}(\left\langle {\sigma v} \right\rangle )dT} } \right]}^{{ - 1}}}.$$

(98)

The DM relic density can be numerically estimated as

$${{\Omega }_{d}}{{h}^{2}} = 8.76 \times {{10}^{{ - 11}}}\,\,{\text{Ge}}{{{\text{V}}}^{{ - 2}}}{{\left[ {\int\limits_{{{T}_{0}}}^{{{T}_{d}}} {(g_{ * }^{{1/2}}\left\langle {\sigma v} \right\rangle )\frac{{dT}}{{{{m}_{\chi }}}}} } \right]}^{{ - 1}}}.$$

(99)

In nonrelativistic approximation with $\left\langle {\sigma {{v}_{{{\text{rel}}}}}} \right\rangle = {{\sigma }_{o}}x_{f}^{{ - n}}$ one can find that the previous formula takes the form [1, 122]¹⁵

$${{\Omega }_{{{\text{DM}}}}}{{h}^{2}} = 0.1\left( {\frac{{(n + 1)x_{f}^{{n + 1}}}}{{({{{{g}_{{ * s}}}} \mathord{\left/ {\vphantom {{{{g}_{{ * s}}}} {g_{ * }^{{1/2}}}}} \right. \kern-0em} {g_{ * }^{{1/2}}}})}}} \right)\frac{{0.876 \times {{{10}}^{{ - 9}}}\,\,{\text{Ge}}{{{\text{V}}}^{{ - 2}}}}}{{{{\sigma }_{0}}}},$$

(100)

where ${{x}_{f}} = \tfrac{{{{m}_{\chi }}}}{{{{T}_{d}}}}$. The following approximate formula [1] takes place for ${{x}_{f}}$:

$${{x}_{f}} = c - \left( {n + \frac{1}{2}} \right){\text{ln}}(c),$$

(101)

$$c = {\text{ln}}(0.038(n + 1)\frac{g}{{\sqrt {g_{{*}}} }}{{M}_{{{\text{Pl}}}}}{{m}_{\chi }}{{\sigma }_{0}}).$$

(102)

Here $g_{{*}}$, ${{g}_{{ * s}}}$ are the effective relativistic energy and entropy degrees of freedom and g is an internal number of freedom degree. If DM particles differ from DM antiparticles ${{\sigma }_{o}} = \tfrac{{{{\sigma }_{{an}}}}}{2}$.

For s-wave annihilation cross-section with $n = 0$

$$\left\langle {\sigma {{v}_{{{\text{rel}}}}}} \right\rangle = 7.3 \times {{10}^{{ - 10}}}\,\,{\text{Ge}}{{{\text{V}}}^{{ - 2}}} \times \frac{1}{{g_{{ * ,{\text{av}}}}^{{1/2}}}}\left( {\frac{{{{m}_{\chi }}}}{{{{T}_{d}}}}} \right).$$

(103)

Here $g_{{ * ,{\text{av}}}}^{{1/2}} = \tfrac{1}{{{{T}_{d}}}}\int_{{{T}_{o}}}^{{{T}_{d}}} {({{{{g}_{{ * s}}}} \mathord{\left/ {\vphantom {{{{g}_{{ * s}}}} {g_{ * }^{{1/2}}}}} \right. \kern-0em} {g_{ * }^{{1/2}}}})dT}$. The calculations show that $1 \leqslant {{c}_{s}} \equiv \tfrac{{{{m}_{\chi }}}}{{10{{T}_{d}}}} \leqslant 1.5$ at 1 MeV ≤ m_χ ≤ $100\,\,{\text{MeV}}$. So we find that

$$\left\langle {\sigma {{v}_{{{\text{rel}}}}}} \right\rangle = 7.3 \times {{10}^{{ - 9}}}\,\,{\text{Ge}}{{{\text{V}}}^{{ - 2}}} \times \frac{1}{{g_{{ * ,{\text{av}}}}^{{1/2}}}}{{c}_{S}}.$$

(104)

For the Dirac fermion DM $\chi$ with dark photon as a messenger between DM and SM sectors the nonrelativistic annihilation cross-section into electron positron pair is¹⁶

$${{\sigma }_{{an}}}(\chi \bar {\chi } \to {{e}^{ - }}{{e}^{ + }}){{v}_{{{\text{rel}}}}} = \frac{{16\pi {{\varepsilon }^{2}}{{\alpha }_{{\text{D}}}}m_{\chi }^{2}}}{{{{{(m_{{A{\kern 1pt} '}}^{2} - 4m_{\chi }^{2})}}^{2}}}}.$$

(105)

$${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}} = 2.0 \times {{10}^{{ - 8}}}\,\,{\text{Ge}}{{{\text{V}}}^{{ - 2}}} \times \frac{{{{{(m_{{A{\kern 1pt} '}}^{2} - 4m_{\chi }^{2})}}^{2}}}}{{m_{\chi }^{2}}}\frac{{2{{c}_{s}}}}{{g_{{ * ,{\text{av}}}}^{{1/2}}}}.$$

(106)

For ${{m}_{{A{\kern 1pt} '}}} = 3{{m}_{\chi }}$ we find

$${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}} = 0.5 \times {{10}^{{ - 12}}} \times {{\left( {\frac{{{{m}_{\chi }}}}{{{\text{MeV}}}}} \right)}^{2}} \times \frac{{2{{c}_{s}}}}{{g_{{ * ,{\text{av}}}}^{{1/2}}}}.$$

(107)

At $20\,\,{\text{MeV}} \leqslant {{m}_{\chi }} \leqslant 200\,\,{\text{MeV}}$ and $T \leqslant 100\,\,{\text{MeV}}$ the effective value $g_{{ * ,{\text{av}}}}^{{1/2}} \approx 3.3$, so we find that

$${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}} \sim 0.4 \times {{10}^{{ - 12}}} \times {{\left( {\frac{{{{m}_{\chi }}}}{{{\text{MeV}}}}} \right)}^{2}}.$$

(108)

Note that for pseudo-Dirac DM the predicted value for ${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}}$ is bigger than the corresponding value for fermion DM. For the p-wave cross-section in nonrelativistic approximation $\left\langle {\sigma {{v}_{{{\text{rel}}}}}} \right\rangle = \left\langle {Bv_{{{\text{rel}}}}^{2}} \right\rangle = 6B\left( {\frac{T}{{{{m}_{\chi }}}}} \right)$. An analog of the formula (103) is

$$6B = 14.6 \times {{10}^{{ - 10}}}\,\,{\text{Ge}}{{{\text{V}}}^{{ - 2}}} \times \frac{1}{{g_{{ * ,{\text{av}}}}^{{1/2}}}}{{\left( {\frac{{{{m}_{\chi }}}}{{{{T}_{d}}}}} \right)}^{2}}.$$

(109)

Here $g_{{ * ,{\text{av}}}}^{{1/2}} = \tfrac{2}{{T_{d}^{2}}}\int_{{{T}_{o}}}^{{{T}_{d}}} {T({{{{g}_{{ * s}}}} \mathord{\left/ {\vphantom {{{{g}_{{ * s}}}} {g_{ * }^{{1/2}}}}} \right. \kern-0em} {g_{ * }^{{1/2}}}})dT}$. For the p-wave annihilations the estimates are similar to the Dirac fermion case, namely for $1\,\,{\text{MeV}} \leqslant {{m}_{\chi }} \leqslant 200\,\,{\text{MeV}}$ we find that $\tfrac{{{{m}_{\chi }}}}{{{{T}_{d}}}} = 10 \times {{c}_{p}}$ with $1 \leqslant {{c}_{p}} \leqslant 2$.

For the charged scalar DM the nonrelativistic annihilation cross-section into electron-positron pair is

$$\sigma {{v}_{{{\text{rel}}}}} = \frac{{8\pi }}{3}\frac{{{{\epsilon }^{2}}\alpha {{\alpha }_{{\text{D}}}}m_{\chi }^{2}v_{{{\text{rel}}}}^{2}}}{{{{{(m_{{A{\kern 1pt} '}}^{2} - 4m_{\chi }^{2})}}^{2}}}},$$

(110)

An analog of the formula (106) is

$${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}} = 4.0 \times {{10}^{{ - 7}}}\,\,{\text{Ge}}{{{\text{V}}}^{{ - 2}}}\frac{{{{{(m_{{A{\kern 1pt} '}}^{2} - 4m_{\chi }^{2})}}^{2}}}}{{m_{\chi }^{2}}}\frac{{2c_{p}^{2}}}{{g_{{ * ,{\text{av}}}}^{{1/2}}}}.$$

(111)

For ${{m}_{{A{\kern 1pt} '}}} = 3{{m}_{\chi }}$ we find

$${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}} = {{10}^{{ - 11}}} \times {{\left( {\frac{{{{m}_{\chi }}}}{{{\text{MeV}}}}} \right)}^{2}}\frac{{2{{c}_{p}}}}{{g_{{ * ,{\text{av}}}}^{{1/2}}}}.$$

(112)

As a reasonable estimate we take

$${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}} \sim {{10}^{{ - 11}}} \times {{\left( {\frac{{{{m}_{\chi }}}}{{{\text{MeV}}}}} \right)}^{2}}.$$

(113)

For Majorana fermions the typical estimate for ${{\epsilon }^{2}}{{\alpha }_{{\text{D}}}}$ has additional factor $\approx 2$.

APPENDIX B: Detection of Long Lived Particles at NA64

In pseudo-Dirac scenario [4] the Majorana particles ${{\chi }_{1}}$ and ${{\chi }_{2}}$ are produced in the reactions

$$eZ \to eZA{\kern 1pt} ',$$

(114)

$$A{\kern 1pt} ' \to {{\chi }_{1}}{{\chi }_{2}}.$$

(115)

Here we assume that ${{m}_{{{{\chi }_{2}}}}} > {{m}_{{{{\chi }_{1}}}}}$. In pseudo-Dirac model the decay

$${{\chi }_{2}} \to {{\chi }_{1}}{{e}^{ + }}{{e}^{ - }}$$

(116)

allows to avoid GMB restrictions [74] on the $s$-wave DM annihilation cross-section. The decay width ${{\chi }_{2}} \to {{\chi }_{1}}{{e}^{ + }}{{e}^{ - }}$ is given by the formula [123]

$$\Gamma ({{\chi }_{2}} \to {{\chi }_{1}}{{e}^{ + }}{{e}^{ - }}) \approx \frac{{4{{\epsilon }^{2}}\alpha {{\alpha }_{{\text{D}}}}{{\Delta }^{5}}}}{{15\pi m_{{A{\kern 1pt} '}}^{4}}},$$

(117)

where $\Delta = {{m}_{{{{\chi }_{2}}}}} - {{m}_{{{{\chi }_{1}}}}}$. For the case of dominant $A{\kern 1pt} ' \to {{e}^{ + }}{{e}^{ - }}$ decay the dark photon decay length is given by the formula [39]

$$\begin{gathered} {{l}_{{A{\kern 1pt} '}}} = {{\gamma }_{{A{\kern 1pt} '}}}c\tau \\ \approx 0.8\,\,{\text{mm}} \times \left( {\frac{{{{\gamma }_{{A{\kern 1pt} '}}}}}{{10}}} \right) \times {{\left( {\frac{{{{{10}}^{{ - 4}}}}}{\epsilon }} \right)}^{2}} \times \frac{{100\,\,{\text{MeV}}}}{{{{m}_{{A{\kern 1pt} '}}}}}, \\ \end{gathered} $$

(118)

where ${{\gamma }_{{A{\kern 1pt} '}}} = \tfrac{{{{E}_{{A{\kern 1pt} '}}}}}{{{{m}_{{A{\kern 1pt} '}}}}}$. The analogous formula for ${{\chi }_{2}} \to {{\chi }_{1}}{{e}^{ + }}{{e}^{ - }}$ decay length is

$$\begin{gathered} {{l}_{{{{\chi }_{2}}}}} = {{\gamma }_{{{{\chi }_{2}}}}}c\tau \\ \approx 0.8\,\,{\text{mm}} \times \left( {\frac{{{{\gamma }_{{{{\chi }_{2}}}}}}}{{10}}} \right) \times {{\left( {\frac{{{{{10}}^{{ - 4}}}}}{\varepsilon }} \right)}^{2}} \times {{\kappa }^{{ - 1}}} \times \frac{{100\,\,{\text{MeV}}}}{{{{m}_{{A{\kern 1pt} '}}}}}. \\ \end{gathered} $$

(119)

Here ${{\gamma }_{{{{\chi }_{2}}}}} = \tfrac{{{{E}_{{{{\chi }_{2}}}}}}}{{{{m}_{{{{\chi }_{2}}}}}}}$ and $\kappa = \tfrac{{4{{\alpha }_{{\text{D}}}}{{\Delta }^{5}}}}{{5\pi m_{A}^{5}}}$. As a numerical example we use the point [123] ${{m}_{{A{\kern 1pt} '}}} = 3{{m}_{{{{\chi }_{1}}}}}$, $\Delta = 0.4{{m}_{{{{\chi }_{1}}}}}$ and ${{\alpha }_{{\text{D}}}} = 0.1$. For this point we find that

$$\begin{gathered} {{l}_{{{{\chi }_{2}}}}} = {{\gamma }_{{{{\chi }_{2}}}}}c\tau \approx 0.8\,\,{\text{mm}} \times \left( {\frac{{{{\gamma }_{{{{\chi }_{2}}}}}}}{{10}}} \right) \times {{\left( {\frac{{{{{10}}^{{ - 4}}}}}{\epsilon }} \right)}^{2}} \\ \times \frac{{100\,\,{\text{MeV}}}}{{{{m}_{{A{\kern 1pt} '}}}}} \times 0.43 \times {{10}^{6}}. \\ \end{gathered} $$

(120)

For NA64 experiment with $100\,\,{\text{GeV}}$ electron beam the $A{\kern 1pt} '$ energy is $\sim 100\,\,{\text{GeV}}$ and approximately ${{E}_{{{{\chi }_{2}}}}} \sim \tfrac{{{{E}_{{A{\kern 1pt} '}}}}}{2} \approx 50\,\,{\text{GeV}}$. As a crude estimate we shall use ${{\gamma }_{{{{\chi }_{2}}}}} = \tfrac{{50\,\,{\text{GeV}}}}{{{{m}_{{{{\chi }_{2}}}}}}}$. As a result we find

$${{l}_{{{{\chi }_{2}}}}} \equiv {{\gamma }_{{{{\chi }_{2}}}}}c\tau \approx 7.6\,\,{\text{cm}}{{\left( {\frac{{{{{10}}^{{ - 1}}}}}{\epsilon }} \right)}^{2}} \times {{\left( {\frac{{100\,\,{\text{MeV}}}}{{{{m}_{{A{\kern 1pt} '}}}}}} \right)}^{2}}.$$

(121)

For instance, for ${{m}_{{A{\kern 1pt} '}}} = 100\,\,{\text{MeV}}$ and $\epsilon = {{10}^{{ - 2}}}$

$${{l}_{{{{\chi }_{2}}}}} \approx 8\,\,{\text{m}}.$$

(122)

So the problem arises—is it possible to derive bounds on ${{\varepsilon }^{2}}$ at finite ${{l}_{{{{\chi }_{2}}}}}$ from NA64 data? The NA64 experiment for the search for invisible $A{\kern 1pt} '$ decays consists of ECAL with the length $60\,\,{\text{cm}}$. Also we have 3 HCAL modules each with ${{l}_{{{\text{HCAL}}}}} = 170\,\,{\text{cm}}$ and the distance between the end of the ECAL and the begining of the HCAL modules is $80\,\,{\text{cm}}$. So the distance between the begining of ECAL and the end of the last HCAL section(the end of NA64 experiment) is ${{l}_{{{\text{exp}}}}} = 6.5\,\,{\text{m}}$. The active zone of ECAL is ${{l}_{{{\text{ECAL}}{\text{,act}}}}} \approx 45\,\,{\text{cm}}$. Suppose the $A{\kern 1pt} '$ is produced in ECAL and immediately decays into ${{\chi }_{2}}{{\chi }_{1}}$( this assumption is correct since ${{\alpha }_{{\text{D}}}} = 0.1$ and $\Gamma (A{\kern 1pt} ' \to {{\chi }_{2}}{{\chi }_{1}})$ is not small) and ${{\chi }_{2}}$ decays into ${{\chi }_{1}}{{e}^{ + }}{{e}^{ - }}$ with the decay length ${{l}_{{{{\chi }_{2}}}}}$. The probability that ${{\chi }_{2}}$ does not decay within NA64, i.e. between the ECAL and the HCAL, is

$${{P}_{{{{\chi }_{2}}}}}({\text{outside}}\,\,{\text{decay}}) = {\text{exp}}\left( { - \frac{{{{l}_{{{\text{exp}}}}}}}{{{{l}_{{{{\chi }_{2}}}}}}}} \right),$$

(123)

where ${{l}_{{{{\chi }_{2}}}}}$ is the ${{\chi }_{2}}$ decay length. We can use the NA64 results on the search for invisible dark photon decays. The bound on mixing parameter is

$${{\epsilon }^{2}} \leqslant \epsilon _{{{\text{NA64}}{\text{,up}}}}^{2} \times {{({{P}_{{{{\chi }_{2}}}}}({\text{outside}}\,\,{\text{decay}}))}^{{ - 1}}},$$

(124)

where $\epsilon _{{{\text{NA64}}{\text{,up}}}}^{2}$ is the NA64 upper bound [34] obtained in the assumption that ${\text{Br}}(A{\kern 1pt} ' \to {\text{invisible}}) =$ 100%. Also the situation with ${{\chi }_{2}}$ decaying withing the ECAL is possible. In this case we have missing energy due to decay chain $A{\kern 1pt} ' \to {{\chi }_{1}}{{\chi }_{2}} \to {{\chi }_{1}}{{\chi }_{1}}{{e}^{ + }}{{e}^{ - }}$ and nonobservation of 2 ${{\chi }_{1}}$ particles. The average missing energy in this decay is ${{E}_{{{\text{miss}}}}} \approx 0.5{{E}_{{A{\kern 1pt} '}}} + \tfrac{1}{3}{{E}_{{A{\kern 1pt} '}}}$ and it is bigger than the used in NA64 missing energy cut ${{E}_{{{\text{miss}}}}} > {{E}_{{{\text{miss}}{\text{,cut}}}}} = 50\,\,{\text{GeV}}$. So we can detect the events related with the ${{\chi }_{2}}$ decay within ECAL by the measurement of missing energy. The probability that ${{\chi }_{2}}$ decays within ECAL active zone is

$${{P}_{{{{\chi }_{2}}}}}({\text{decaysin}}\,\,{\text{ECAL}}) = 1 - {\text{exp}}\left( { - \frac{{{{l}_{{{\text{ECAL}}{\text{,act}}}}}}}{{{{l}_{{{{\chi }_{2}}}}}}}} \right).$$

(125)

So total probability of the ${{\chi }_{2}}$ detection with the use of energy missing cut is

$$\begin{gathered} {{P}_{{{{\chi }_{2}}}}} = {{P}_{{{{\chi }_{2}}}}}({\text{decaysin}}\,\,{\text{ECAL}}) + {{P}_{{{{\chi }_{2}}}}}({\text{outside}}\,\,{\text{decay}}) \\ \times \,\,\left( {1 - {\text{exp}}\left( { - \frac{{{{l}_{{{\text{ECAL}}{\text{,act}}}}}}}{{{{l}_{{{{\chi }_{2}}}}}}}} \right)} \right) + {\text{exp}}\left( { - \frac{{{{l}_{{{\text{exp}}}}}}}{{{{l}_{{{{\chi }_{2}}}}}}}} \right). \\ \end{gathered} $$

(126)

For arbitrary ${{l}_{{{{\chi }_{2}}}}}$ the expression (126) for ${{P}_{{{{\chi }_{2}}}}}$ has minimal value at

$${{l}_{{\exp }}}\exp \left( { - \frac{{{{l}_{{\exp }}}}}{{{{l}_{{{{\chi }_{2}},\min }}}}}} \right) = {{l}_{{{\text{ECAL}}{\text{,act}}}}}\exp \left( { - \frac{{{{l}_{{{\text{ECAL}}{\text{,act}}}}}}}{{{{l}_{{{{\chi }_{2}},\min }}}}}} \right),$$

(127)

or

$${{l}_{{{{\chi }_{2}},\min }}} = \frac{{{{l}_{{\exp }}} - {{l}_{{{\text{ECAL}}{\text{,act}}}}}}}{{\ln \left( {\frac{{{{l}_{{\exp }}}}}{{{{l}_{{{\text{ECAL}}{\text{,act}}}}}}}} \right)}},$$

(128)

and ${{P}_{{{{\chi }_{2}},{\text{min}}}}}$ is

$$\begin{gathered} {{P}_{{{{\chi }_{2}},{\text{min}}}}} = \left( {1 - exp\left( { - \frac{{{{l}_{{{\text{ECAL}}{\text{,act}}}}}}}{{{{l}_{{{\text{exp}}}}} - {{l}_{{{\text{ECAL}}{\text{,act}}}}}}}ln\frac{{{{l}_{{{\text{exp}}}}}}}{{{{l}_{{{\text{ECAL}}{\text{,act}}}}}}}} \right)} \right. \\ \left. { + \,\,exp\left( { - \frac{{{{l}_{{{\text{exp}}}}}}}{{{{l}_{{{\text{exp}}}}} - {{l}_{{{\text{ECAL}}{\text{,act}}}}}}}ln\frac{{{{l}_{{{\text{exp}}}}}}}{{{{l}_{{{\text{ECAL}}{\text{,act}}}}}}}} \right)} \right). \\ \end{gathered} $$

(129)

Numerically for ${{l}_{{{\text{exp}}}}} = 650\,\,{\text{cm}}$ and ${{l}_{{{\text{ECAL}}{\text{,exp}}}}} = 40\,\,{\text{cm}}$ we find

$${{P}_{{{{\chi }_{2}},{\text{min}}}}} \approx 0.22.$$

(130)

The bound on ${{\epsilon }^{2}}$ reads

$${{\epsilon }^{2}} \leqslant \epsilon _{{{\text{NA64}}{\text{,up}}}}^{2} \times {{({{P}_{{{{\chi }_{2}},{\text{min}}}}})}^{{ - 1}}} \approx 4.5 \times \epsilon _{{{\text{NA64}}{\text{,up}}}}^{2}.$$

(131)

Here $\epsilon _{{{\text{NA64}}{\text{,up}}}}^{2}$ is the NA64 bound for the case of invisible $A{\kern 1pt} '$ decay. So we see that NA64 is able to obtain upper bound on ${{\epsilon }^{2}}$ parameter for the case of visible $A{\kern 1pt} '$ decay with large missing energy in a model independent way. The knowledge of ${{l}_{{{{\chi }_{2}}}}}$ allows to improve the bound (131).