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.

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).