Explorations of pseudo-Dirac dark matter having keV splittings and interacting via transition electric and magnetic dipole moments

The rate of production of particles $\chi_{1},\chi_{2}$ from the annihilation of SM fermions, as shown in fig. 1 (a), is Hall:2009bx ; Bernal:2017kxu ; Dutra:2019gqz

	$\displaystyle R_{2\rightarrow 2}(T)$	$\displaystyle=$	$\displaystyle\sum_{f}N_{c}^{f}\frac{T}{(2\pi)^{6}2^{4}}\int_{s_{min}}^{\infty}ds\sqrt{s-4m_{f}^{2}}\frac{|\vec{p}_{3}^{\,\,0}|}{\sqrt{m_{\chi_{1}}^{2}+|\vec{p}_{3}^{\,\,0}|^{2}}+\sqrt{m_{\chi_{2}}^{2}+|\vec{p}_{3}^{\,\,0}|^{2}}}$		(5)
			$\displaystyle\times\,K_{1}\left(\frac{\sqrt{s}}{T}\right)\int d\Omega_{3}^{\star}\,|\mathcal{M}|^{2},$

where $N_{c}^{f}$ are the color degrees of freedom of SM fermion $f$ with mass $m_{f}$. The total rate is a sum over the SM fermions $f$. $K_{1}$ is the modified Bessel function of the second kind, $s_{min}=\textrm{Max}\left[(m_{\chi_{1}}+m_{\chi_{2}})^{2},4m_{f}^{2}\right]$, and the 3-momentum of $\chi_{1}$ in the center-of-mass (CM) frame is

$$|\vec{p}_{3}^{\,\,0}|=\sqrt{\left(\frac{\sqrt{s}}{2}+\frac{m_{\chi_{1}}^{2}-m_{\chi_{2}}^{2}}{2\sqrt{s}}\right)^{2}-m_{\chi_{1}}^{2}}.$$

The squared-amplitude $|\mathcal{M}|^{2}$ is summed over initial and final spins Masso:2009mu

$$\int d\Omega_{3}^{\star}\sum\big{|}\mathcal{M}\big{|}_{f}^{2}\simeq\begin{cases}32\pi d_{\chi}^{2}e^{2}q_{f}^{2}\frac{(s-4m_{\chi_{1}}^{2})(s+2m_{f}^{2})}{3s}-128\pi\,\delta d_{\chi}^{2}e^{2}q_{f}^{2}\frac{m_{\chi_{1}}(s+2m_{f}^{2})}{3s}+\mathcal{O}(\delta^{2}),&\text{ for EDM,}\\ 32\pi\mu_{\chi}^{2}e^{2}q_{f}^{2}\frac{(s+8m_{\chi_{1}}^{2})(s+2m_{f}^{2})}{3s}+64\pi\,\delta\mu_{\chi}^{2}e^{2}q_{f}^{2}\frac{m_{\chi_{1}}(s+2m_{f}^{2})}{3s}+\mathcal{O}(\delta^{2}),&\text{ for MDM,}\end{cases}$$

(6)

where we give the expressions upto leading order in $\delta\equiv m_{\chi_{2}}-m_{\chi_{1}}$. In the limit of vanishing SM fermion and DM masses, we find that the production rate scales with temperature as $R(T)\propto T^{6}$. From eq. (2), we see that this leads to UV (UltraViolet) FI with maximum yield produced at the largest temperature of the SM bath, equal to the reheating temperature for models with instantaneous reheating Chen:2017kvz ; Elahi:2014fsa ; Chowdhury:2018tzw ; Giudice:2000ex .

Of phenomenological interest is the parameter space with large coupling which can lead to large scattering rate at DD experiments. We note that for $m_{DM}\geq T_{RH}$, there is kinematic suppression in DM production, requiring large couplings to reproduce the observed relic density. For the light DM we consider in this work, $m_{DM}\leq 1\,{\rm GeV}$, we choose low reheating temperatures ($T_{RH}$ = 5 MeV and 10 MeV) from allowed²²2A robust lower bound on reheating temperatures of $T_{RH}>4$ MeV was derived by combining cosmic microwave background, large scale structure and light element abundances data in ref. Hannestad:2004px . values. We verify our calculations using the expressions given above, with those obtained from micrOMEGAs 5.0 Belanger:2018ccd , for fermionic channels of production.

We discuss the results in section V.

The SM bath consists of charged particles that couple to the photon. An electromagnetic wave propagating through this bath behaves qualitatively differently from that in vacuum. Coherent vibrations of the electromagnetic field and the density of charged particles results in a spin-1 particle with 1 longitudinal and 2 transverse polarizations, propagating at a speed less than the speed of light in vacuum. Its dispersion relation depends on the properties of the SM plasma and is therefore known as the ‘plasmon’. We follow the discussion in refs. Braaten:1993jw ; Raffelt:1996wa for the plasmon properties and decay rates.

The significance of plasmon decay leading to production of DM in FI scenarios was noted in ref. Dvorkin:2019zdi (see also Chu:2019rok ; Chang:2019xva ; Hambye:2019dwd ). The massive plasmon can decay to DM in cases where the DM couples to the photon. This is true in the dipolar interaction case we study here. For DM produced via FI mechanism, the final abundance is accumulated over time and summed over the various production channels of SM particles annihilating and decaying to DM. Therefore, the production from plasmon decay can play a significant role in FI produced DM (as opposed to freeze-out production).

To calculate the DM abundance resulting from plasmon decay, we must begin with the modified dispersion relations which depend on the temperature $T$ and the net density of the charged particles in the SM plasma. Since ref. Braaten:1993jw was addressing the energy loss from the plasma in a supernova with typical energies of $\mathcal{O}(10)$ MeV, only electrons and positrons were considered as charged particles of relevance that modify the dispersion relations. But for the case of a UV FI with large enough $T_{RH}$, all charged particles with masses much smaller than the reheating temperature would add to the plasmon effect (see appendix A for further details). The rate of DM production from plasmon decay is

$\displaystyle R_{\gamma^{*}}(T)$

$\displaystyle=$

$\displaystyle\sum_{\textrm{pol}=T,L}g_{\textrm{pol}}\int\frac{d^{3}k}{(2\pi)^{3}}f(\omega_{\textrm{pol}})\Gamma_{\textrm{pol}},$

(7)

where we assume that only the lighter particle $\chi_{1}$ survives. The transverse and longitudinal polarizations are $g_{T}=2$ and $g_{L}=1$, respectively. The distribution for the plasmon is given by $f(\omega_{\rm pol})=1/({\rm exp}\frac{\omega_{\rm pol}}{T}-1)$.

The decay width of a plasmon $\Gamma_{\textrm{pol}}$ with four momentum $k=(\omega,\vec{k})$ in the medium frame, with a definite polarization ‘pol’ is Raffelt:1996wa

$$\Gamma_{\text{pol}=T,L}=\int\frac{d^{3}p_{\chi_{2}}}{(2\pi)^{3}\,(2E_{\chi_{2}})}\frac{d^{3}p_{\chi_{1}}}{(2\pi)^{3}\,(2E_{\chi_{1}})}(2\pi)^{4}\delta^{4}\left(k-p_{\chi_{1}}-p_{\chi_{2}}\right)\frac{1}{2\omega_{\textrm{pol}=T,L}}\sum_{\text{spins}}|\mathcal{M}|_{\gamma^{*}_{\textrm{pol}}\rightarrow\chi_{1},\chi_{2}}^{2},$$

(8)

with $|\mathcal{M}|_{\gamma^{*}_{\text{pol}}\rightarrow\chi_{1}\chi_{2}}^{2}$ being the squared amplitude for plasmon decay to DM, expressions for which are given in eq. (62). For dipolar DM we find that the rate of DM production from plasmon decay is also maximised at the largest temperatures of the SM bath. We call this a UV plasmon production. We show in fig. 2, the scaled DM yield as a function of temperature, from plasmon production for dipolar interaction (UV plasmon) and millicharged interaction Dvorkin:2019zdi ; Dvorkin:2020xga (Infrared, IR, plasmon). Here, we have considered a DM of mass 1 keV for both the cases. The UV plasmon production can be seen to be maximum at the reheating temperature, taken to be 1 GeV in this figure. As $T_{RH}$ is changed, the IR produced yield is unchanged. While the UV plasmon line would shift towards right (higher temperatures) as $T_{RH}$ is increased .

Although we have only shown the yield for EDM interacting DM in fig. 2 for the UV plasmon, the same is also true for MDM interaction. Note that we show the UV plasmon production for $T_{RH}=1\,{\rm GeV}$ to clearly show the difference in IR vs UV production; the maximum production of DM happens at the largest temperatures even for smaller values of $T_{RH}$.

This is in contrast with IR plasmon production where the maximum production takes place at the lowest temperatures so that the plasmon production mechanism begins to dominate for small DM masses, $m_{DM}\lesssim 400$ keV Dvorkin:2019zdi . This can be understood by noting that the $2\rightarrow 2$ annihilation process becomes ineffective when electrons freeze out, while the plasmon production continues to be effective. While for UV plasmon production (with small $T_{RH}$), the plasmon decay production does not take over the annihilation production as we decrease the DM mass, since both the processes maximize at the largest temperatures of the SM bath. For the small reheating temperatures we consider, mindful of the sub-GeV DM masses that are of interest to the DD discussed in the following, the plasmon production is always subdominant and can be safely ignored. For much larger reheating temperatures $T_{RH}\gtrsim 100$ GeV though, the plasmon production can dominate and must be taken into account.

The maximum velocity of DM falling into the Sun is $\sqrt{(v^{\rm{esc}}_{\odot})^{2}+(v_{\rm{DM}}^{\rm{\,halo}})^{2}}\simeq 4\times 10^{-3}$, where $v^{\rm{esc}}_{\odot}$ is the Solar escape velocity. We take the value of the Solar escape velocity to be the number averaged⁴⁴4$\langle v_{\rm esc}\rangle=\int_{0}^{r_{\rm core}}dr\,v_{\rm esc}(r)n(r)/\int_{0}^{r_{\rm core}}dr\,n(r)$ where $n(r)$ is the electron number density Emken:2021lgc and $r_{\rm core}\simeq 0.2R_{\odot}$. escape velocity, over the Solar core Emken:2021lgc (since most of the scatterings are expected to happen inside the core), giving $v^{\rm{esc}}_{\odot}\simeq\,1307\,\text{km/s}$ Emken:2021lgc . DM particle’s most probable Galactic velocity⁵⁵5Using a halo distribution averaged velocity leads to an overall factor of $\simeq 0.89$ in the upscattered flux. is given by $v_{DM}^{\textrm{halo}}\simeq 220\,\text{km/s}$. This is much smaller than the most probable electron velocity in the Solar core

$$v^{e}_{\odot}(\text{core})=\sqrt{\frac{2T_{\odot}}{m_{e}}}\simeq 0.066\text{, for }T_{\odot}=1.1\,\rm{keV},$$

(9)

where $T_{\odot}$ is the temperature at the Sun’s core. Hence, the DM particles can be understood to be at rest, with Solar electrons scattering against them. The steady state DM number density in the Sun is given by

	$\displaystyle n_{\chi_{1},\odot}$	$\displaystyle=n_{\chi_{1}}^{\rm{\,halo}}\left(1+\left(\frac{v^{\text{esc}}_{\odot}}{v_{\text{DM}}^{\text{\,halo}}}\right)^{2}\right)\left(\frac{v_{\text{DM}}^{\text{\,halo}}}{v^{\rm{esc}}_{\odot}}\right)$
		$\displaystyle\simeq n_{\chi_{1}}^{\rm{\,halo}}\left(\frac{v_{\rm{esc}}}{v_{\text{DM}}^{\text{\,halo}}}\right).$		(10)

Here, the gravitational focusing effect, that enhances the area at spatial infinity, is given by the factor $(1+(v^{\text{esc}}_{\odot})^{2}/(v_{\text{DM}}^{\text{\,halo}})^{2})$, and the factor $(v_{\text{DM}}^{\text{\,halo}}/v_{\rm{esc}})$ accounts for the decrease in the number density of DM owing to its larger velocity near the Sun (from conservation of flux). The velocity distribution of electrons in the Sun is taken to be Maxwell Boltzmann (MB)

$$f_{{\rm MB}}(v_{e})=4\pi v_{e}^{2}\left(\frac{m_{e}}{2\pi T_{\odot}}\right)^{3/2}{\rm exp}\left(-\frac{m_{e}v_{e}^{2}}{2T_{\odot}}\right),$$

(11)

where $m_{e}$ and $v_{e}$ are the electron mass and velocity, respectively. The differential flux of particles $\chi_{2}$ generated with recoil energy $K_{\chi_{2}}$ is given as

	$\displaystyle\frac{d\Phi}{dK_{\chi_{2}}}$	$\displaystyle=$	$\displaystyle n_{e}\left\langle\frac{d\sigma_{\chi_{1}\rightarrow\chi_{2}}}{dK_{\chi_{2}}}v_{e}\right\rangle\frac{n_{\chi_{1},\odot}V_{\odot}}{4\pi(1{\rm AU})^{2}}$		(12)
		$\displaystyle=$	$\displaystyle\frac{n_{e}n_{\chi_{1},\odot}V_{\odot}}{4\pi(1{\rm AU})^{2}}\int_{v_{min}(K_{\chi_{2}})}^{\infty}dv_{e}\,f_{MB}\left(v_{e}\right)\frac{d\sigma_{\chi_{1}\rightarrow\chi_{2}}}{dK_{\chi_{2}}}\,v_{e}$		(13)
		$\displaystyle=$	$\displaystyle\begin{cases}\frac{n_{e}n_{\chi_{1},\odot}V_{\odot}}{4\pi(1{\rm AU})^{2}}\frac{\pi\alpha^{2}m_{e}^{2}}{\mu_{\chi e}^{2}}\frac{T_{\odot}}{m_{e}}\left(\frac{m_{e}}{2\pi T_{\odot}}\right)^{3/2}\bar{\sigma}_{e}\frac{1}{K_{\chi_{2}}}\text{exp}\left(-\frac{m_{e}}{4m_{\chi_{1}}T_{\odot}}\frac{(m_{\chi_{1}}K_{\chi_{2}}/\mu_{\chi e}+\delta)^{2}}{K_{\chi_{2}}}\right),\text{ EDM }\\ \frac{n_{e}n_{\chi_{1},\odot}V_{\odot}}{4\pi(1{\rm AU})^{2}}\left(\frac{m_{e}}{2\pi T_{\odot}}\right)^{1/2}\bar{\sigma}_{e}^{F_{DM}(q)=1}\frac{m_{\chi_{1}}}{\mu_{\chi e}^{2}}\,\text{exp}\left(-\frac{m_{e}}{4m_{\chi_{1}}T_{\odot}}\frac{(m_{\chi_{1}}K_{\chi_{2}}/\mu_{\chi e}+\delta)^{2}}{K_{\chi_{2}}}\right)\\ \qquad\times\Big{(}1+\frac{4m_{\chi_{1}}}{m_{\chi_{1}}-2m_{e}}\left(\frac{m_{e}^{2}}{4\mu_{\chi e}^{2}}+\frac{\delta m_{e}^{2}}{2\mu_{\chi e}m_{\chi_{1}}K_{\chi_{2}}}+\frac{m_{e}T_{\odot}}{m_{\chi_{1}}K_{\chi_{2}}}+\frac{m_{e}^{2}\delta^{2}}{4m_{\chi_{1}}^{2}K_{\chi_{2}}^{2}}\right)\Big{)},\text{ MDM,}\end{cases}$		(14)

where $n_{e}=2\times 10^{25}\,\textrm{cm}^{-3}$ and $V_{\odot}=2.2\times 10^{31}\,\textrm{cm}^{3}$ are the Solar mean electron number density and volume, respectively Bahcall:2000nu ; Baryakhtar:2020rwy . The factor of $1/4\pi(1\textrm{AU})^{2}$ is the scaling in the flux on account of traveling from the Sun to the Earth, with the distance travelled being 1 AU (astronomical unit). The reduced mass is given by $\mu_{i,j}\equiv m_{i}m_{j}/(m_{i}+m_{j})$.

Here, we have plugged in the explicit expressions for the temperature averaged differential cross section for up-scattering, given by

$$\left\langle\frac{d\sigma_{\chi_{1}\rightarrow\chi_{2}}}{dK_{\chi_{2}}}v_{e}\right\rangle=\int_{v_{\rm min}}^{\infty}dv_{e}f_{\rm MB}(v_{e})\frac{d\sigma_{\chi_{1}\rightarrow\chi_{2}}}{dK_{\chi_{2}}}v_{e},$$

(15)

where $v_{\rm min}(K_{\chi_{2}})$ is the minimum velocity an electron must have in order to up-scatter a $\chi_{1}$ into a $\chi_{2}$, with recoil energy $K_{\chi_{2}}$, given as

$$v_{\rm min}=\frac{1}{\sqrt{2K_{\chi_{2}}m_{\chi_{1}}}}\left(\frac{m_{\chi_{1}}K_{\chi_{2}}}{\mu_{\chi_{1},e}}+\delta\right),$$

(16)

and

$$\frac{d\sigma_{\chi_{1}\rightarrow\chi_{2}}}{dK_{\chi_{2}}}\simeq\frac{\bar{\sigma}_{e}m_{\chi_{1}}}{2\mu_{\chi_{1},e}^{2}v_{e}^{2}}|F_{DM}(q)|^{2},$$

(17)

in the limit $\delta<<m_{e}$ and $m_{\chi_{1}}$. Here, $\bar{\sigma}_{e}$ is the DM-electron reference cross section Essig:2011nj with the 3-momentum transfer fixed at $q=(\alpha m_{e})$ (appropriate for atomic processes),

	$\displaystyle\bar{\sigma}_{e}$	$\displaystyle\equiv\frac{\mu_{\chi_{1},e}^{2}}{16\pi m_{\chi_{1}}^{2}m_{e}^{2}}\overline{|\mathcal{M}_{\chi_{1}e}(q)|^{2}}\Big{|}_{q^{2}=(\alpha m_{e})^{2}}$		(18)
		$\displaystyle=\begin{cases}\frac{4d_{\chi}2}{\alpha}\frac{m_{\chi_{1}}^{2}}{(m_{\chi_{1}}+m_{e})^{2}},\text{ with }F_{DM}(q)=\frac{\alpha m_{e}}{q},&\text{ for EDM,}\\ \left.\begin{array}[]{l}\alpha\mu_{\chi}^{2}\frac{m_{\chi_{1}}^{2}}{(m_{\chi_{1}}+m_{e})^{2}}\left(1-\frac{2m_{e}}{m_{\chi_{1}}}\right),\text{ with }F_{DM}(q)=1,\\ \frac{4}{\alpha}\mu_{\chi}^{2}v^{2}\frac{m_{\chi_{1}}^{2}}{(m_{\chi_{1}}+m_{e})^{2}},\text{ with }F_{DM}(q)=\frac{\alpha m_{e}}{q}.\end{array}\right\}&\text{ for MDM. }\end{cases}$		(19)

The $q$-dependence of the matrix elements are encoded in the DM form-factor $F_{DM}(q)$ Essig:2011nj . $\overline{|\mathcal{M}_{\chi_{1}e}(q)|^{2}}$ is the squared-amplitude for DM–electron scattering, averaged (summed) over initial (final) spin states. The MDM contribution consists of one part that is independent of relative velocity $v$ and with a form factor $F_{DM}(q)=1$ similar to that for contact interaction, and another part proportional to $v^{2}$ and with a form factor same as that of EDM.

Note that the $\chi_{2}$ particles that reach the Earth are the ones that overcome the gravitational potential well of the Sun,

$$v_{\chi_{2}}^{2}\simeq\frac{2K_{\chi_{2}}}{m_{\chi_{2}}}>(v^{\rm{esc}}_{\odot})^{2}.$$

(20)

Therefore, the flux on the Earth is attenuated from the produced flux as

$$\frac{d\Phi}{dK_{\chi_{2}}}(K_{\chi_{2}})\Bigg{|}_{Earth}=\frac{d\Phi}{dK_{\chi_{2}}}(K_{\chi_{2}}+m_{\chi_{2}}(v^{\rm esc}_{\odot})^{2}/2)\;\Theta\left(K_{\chi_{2}}-\frac{1}{2}m_{\chi_{2}}(v^{\rm{esc}}_{\odot})^{2}\right).$$

(21)

The Heaviside theta function ensures that the condition in eq. (20) is satisfied. We also shift the flux to account for the reduction in $\chi_{2}$ velocity in overcoming the Sun’s gravitational potential. In the following, we denote the flux on the Earth by $d\Phi/dK_{\chi_{2}}$ and drop the explicit notation. There will be a further suppression in the flux of $\chi_{2}$ from its decay in the time taken to travel from the Sun to the Earth. In fig. 4 we show the flux per unit energy from eq. (14) at production (blue-dashed), the flux that escapes the Sun’s gravitational well from eq. (21) (blue-solid) and the attenuated flux at the Earth after taking into account the $\chi_{2}$ decay in travelling from the Sun to the Earth (black-solid). Note that the flux that overcomes the gravitational potential has a lower cut-off, from the Heaviside theta function in eq. (21).

We note that the plasmon does not play any role in the up-scattering or production of DM in the Sun. The plasmon frequency in the non-relativistic limit Braaten:1993jw

$$\omega_{p}^{2}(T)=\frac{4\pi\alpha\,n_{e}}{m_{e}}\left(1-\frac{5}{2}\frac{T}{m_{e}}\right)$$

(22)

gives a plasmon mass order 1 smaller than the Sun’s average temperature for the Solar electron number density. Therefore, the plasmon effect is subdominant in scattering. In addition, there can be no plasmon-sourced production as we consider DM masses much larger than the Solar temperature.

At lepton-fixed target experiments, an electron beam of fixed energy is dumped against an active target, comprised of some heavy nucleus, that is either a part of the detector itself or a separate target. Dark sector particles can be pair produced from the electrons scattering off of nucleons, giving rise to missing-energy final states Gninenko:2013rka ; Andreas:2013lya . These experiments employ stringent selection criteria making it possible to conduct an essentially background free search for such missing-energy signals Bjorken:2009mm ; Tsai:1973py ; Graham:2021ggy ; Fabbrichesi:2020wbt ; Agrawal:2021dbo .

In particular, we use results from the NA64 experiment at CERN SPS NA64:2017vtt to derive constraints on our model via the production of DM particles from an off-shell photon,

$$e^{-}N\rightarrow e^{-}N\gamma_{\rm vir}\rightarrow e^{-}N\chi_{1}\chi_{2},$$

where $\gamma_{\rm vir}$ is the off-shell photon that subsequently produces the dark sector particles. The leading processes are shown in fig. 5. Note that we do not consider the processes where the virtual photon originates from the nucleus since these processes are suppressed by a factor of $(Zm_{e}/m_{N})^{2}$ for coherent photon emission. We follow the discussion in ref. Chu:2018qrm in the following.

The NA64 experiment employs the optimized 100 GeV electron beam from the H4 beamline at the North Area (NA) of the CERN SPS. The beam is incident upon an electromagnetic calorimeter (ECAL) made up of a $6\times 6$ matrix with Pb and Sc plates, each module being $\simeq 40$ radiation lengths ($X_{0}$) long. The radiation length⁸⁸8For incident electrons with large energies, this is essentially a measure of the strength of the Bremsstrahlung process with a larger radiation length implying smaller cross sections for the process. of Sc is about 1 order larger than that of Pb so $e^{-}$ scattering with Sc is subdominant.
Assuming that the fermion pair is produced within the first radiation length gives the target length to be $L_{\textrm{target}}=X_{0}=0.56\,$cm for Pb. The search region in the ECAL is limited by the energy threshold for detection of electron on one side and the requirement for missing energy to be larger than half the beam energy. This gives the selection criteria for the energy of the outgoing electron $E_{4}$ as Banerjee:2019pds ; Krasnikov:2020trl

$$0.3\,\textrm{GeV}<E_{4}<50\,\textrm{GeV}.$$

(34)

The polar angular coverage for the outgoing electron is

$$0.0<\theta_{4}<0.23\,\,{\rm radians}.$$

(35)

The number of signal events with these geometric and angular cuts are Chu:2018qrm

$$N_{\textrm{sig}}=N_{\textrm{EOT}}\frac{\rho_{\textrm{target}}}{m_{N}}L_{\textrm{target}}\int_{E_{4}^{min}}^{E_{4}^{max}}\epsilon_{eff}(E_{4})\int_{\textrm{cos}\,\theta_{4}^{min}}^{\textrm{cos}\,\theta_{4}^{max}}d\textrm{cos}\,\theta_{4}\frac{d\sigma_{\textrm{prod}}}{dE_{4}d\textrm{cos}\,\theta_{4}},$$

(36)

where, $N_{\textrm{EOT}}=2.84\times 10^{11}$ are the number of electrons incident upon the target Banerjee:2019pds . The target material density and nuclear mass are given by $\rho_{\rm target}$ and $m_{N}$, respectively. The detector efficiency of NA64 is known to depend only marginally on the energy and is taken to be constant, $\epsilon_{eff}(E_{4})\simeq 0.5$ (averaging over the total signal efficiencies of the various runs Banerjee:2019pds ). The integration limits of $E_{4}$ and $\theta_{4}$ are from eqs. (34) and (35), respectively. The double differential cross section for production of the processes shown in figs. 5(a) and 5(b) is given by $d\sigma_{prod}/dE_{4}d\cos\theta_{4}$ with the full expression given in eq. (82).

Since the beam energy is much larger than the $\mathcal{O}$(keV) splitting we consider, and the signal being observed is of missing energy in final state, the constraints on our model do not differ from those of the elastic DM cases discussed in ref. Chu:2018qrm . We follow the discussions in Chu:2018qrm and re-derive the constraints from their run as mentioned in ref. Banerjee:2019pds . The details of the calculation are given in Appendix C. We derive constraints by demanding that $N_{sig}<N^{90\%}=2.3$ where the latter corresponds to the $90\%$ C.L. for the number of signals events given zero observed events. The resulting constraints are shown in figs. 7 and 11.

This type of search for missing-energy in the final state gives stronger bounds for a feebly interacting dark particle, as compared to experiments where the dark particle is detected via its scattering off electrons/nuclei in the main detector (for example in the mQ experiment at SLAC Prinz:1998ua ), since processes of the latter kind are suppressed by further powers of the small-valued DM-SM effective coupling. We, therefore, do not study these latter processes.

Constraints are also derived from proton fixed target experiments, and as shown in ref. Chu:2020ysb the strongest such constraint⁹⁹9Constraints coming from other proton-beam experiments such as COHERENT, JSNS², NO$\nu$A and WA66 are expected to be weaker than that from CHARM-II and LEP Chu:2020ysb . comes from CHARM-II experiment, which used a 450 GeV proton beam on a Be target. The constraints are derived from single electron recoil events at recoil energies $E_{R}\in[3,24]$ GeV, so that the small splitting of $\mathcal{O}$(keV) has negligible effect and the constraints for such inelastic DM are the same as those for the elastic case. These are shown in figs. 7 and 11.

DM can be produced from $e^{+}e^{-}$ collisions at lepton colliders and appear as missing energy ($\not{E}$), since they do not scatter within the collider. Along with initial state radiation (ISR) or final state radiation (FSR), this leads to a particularly clean signature of mono-photon plus missing energy ($\gamma+\not{E}$). The FSR processes are suppressed by the small DM-photon couplings, so we only consider the ISR process, as shown in fig. 6, for deriving constraints.

We follow the discussion in ref. Chu:2019rok for constraints on DM dipole moments from pair production, adapting them for the inelastic case. For DM mass splittings much smaller than the CM energy of the $\chi_{1},\chi_{2}$ system, $\delta\sim\mathcal{O}(\text{keV})<<\sqrt{s_{\chi_{1}\chi_{2}}}$, these constraints for inelastic DM are equal to the ones for elastic DM.