Electrodisintegration of Deuteron into Dark Matter and Proton Close to Threshold

Abstract

We discuss an investigation of the dark matter decay modes of the neutron, proposed by Fornal and Grinstein (2018–2020), Berezhiani (2017, 2018) and Ivanov et al. (2018) for solution of the neutron lifetime anomaly problem, through the analysis of the electrodisintegration of the deuteron d into dark matter fermions $\chi$ and protons p close to threshold. We calculate the triple-differential cross section for the reaction $e^{-}+d\to \chi +p+e^{-}$ and propose to search for such a dark matter channel in coincidence experiments on the electrodisintegration of the deuteron $e^{-}+d\to n+p+e^{-}$ into neutrons n and protons close to threshold with outgoing electrons, protons, and neutrons in coincidence. An absence of neutron signals should testify to a detection of dark matter fermions.

1. Introduction

Recently, Fornal and Grinstein [1,2,3,4,5] proposed a solution to the neutron lifetime anomaly (NLA) problem, related to a discrepancy between experimental values of the neutron lifetime measured in bottle and beam experiments, through a contribution of the neutron dark matter decay mode $n\to \chi +e^{-}+e^{+}$, where $\chi$ is a dark matter fermion and $(e^{-}{e}^{+})$ is the electron-positron pair. However, according to experimental data [6,7,8], the decay mode $n\to \chi +e^{-}+e^{+}$ is suppressed. So at first glimpse it seems that the decay mode $n\to \chi +e^{-}+e^{+}$ cannot explain the NLA problem. In order to overcome such a problem we have assumed [9] that an unobservability of the decay mode $n\to \chi +e^{-}+e^{+}$ may only mean that the production of the electron-positron pair in such a decay is below the reaction threshold, i.e., a mass $m_{\chi }$ of dark matter fermions obeys the constraint $m_{\chi }>m_n-2m_e$, where $m_n$ and $m_e$ are masses of the neutron and electron (positron), respectively. Then, we have proposed that the NLA problem can be explained by the decay mode $n\to \chi +{\nu }_e+{\overline{\nu }}_e$, where $({\nu }_e{\overline{\nu }}_e)$ is a neutrino-antineutrino pair [9]. Since neutrino ${\nu }_e$ and electron $e^{-}$ belong to the same doublet in the Standard Electroweak Model (SEM) [10,11] (see also [12,13]), neutrino-antineutrino $({\nu }_e{\overline{\nu }}_e)$ pairs couple to the neutron-dark matter current with the same strength as electron-positron $(e^{-}{e}^{+})$ pairs [9]. We have extended this effective quantum field theory by a gauge invariant quantum field theory model of the neutron- and lepton-dark matter interactions invariant under the $U_{Y^{\prime }}^{\prime }(1)\times U_{Y^{″}}^{″}(1)$ gauge symmetry. In the physical phase the dark matter sectors with $U_{Y^{\prime }}^{\prime }(1)$ and $U_{Y^{″}}^{″}(1)$ symmetries are responsible for the effective interaction ($n\chi \ell \overline{\ell }$) [9] and interference of the dark matter into dynamics of neutron stars [14,15,16,17], respectively. The dark matter sector with $U_{Y^{″}}^{″}(1)$ symmetry we have constructed in analogue with scenario proposed by Cline and Cornell [17]. This means that dark matter fermions with mass $m_{\chi }<m_n$ couple to a very light dark matter spin-1 boson $Z^{″}$ providing a necessary repulsion between dark matter fermions in order to give a possibility for neutron stars to reach masses of about $2M_{\odot }$ [18], where $M_{\odot }$ is the mass of the Sun [10]. The corrected version of the dark matter sectors, invariant under the $U_{Y^{\prime }}^{\prime }(1)\times U_{Y^{″}}^{″}(1)$ gauge symmetry is expounded in [19].

In connection with different approaches to the explanation of the NLA, we have to mention another mechanism proposed by Berezhiani [20,21], which is not related to the decay of the neutron into a dark matter but based on the $n⟷n^{\prime }$ transitions, where $n^{\prime }$ is a mirror neutron [22,23]. Since, such a mechanism is far from being applicable to the analysis of the electrodisintegration of the deuteron, we will not discuss it in this paper. Another mechanism for the explanation of the NLA proposed by Berezhiani [24], assuming the existence of the neutron decays into dark matter particles, is similar to that by Fornal and Grinstein [1,2,3,4,5]. So our approach to the NLA seems to be a modification of both Berezhiani’s, and Fornal and Grinstein’s mechanism.

We would like to notice that the existence of the reaction $n\to \chi +{\nu }_e+{\overline{\nu }}_e$ for the explanation of the NLA problem entails the existence of the reactions $n+n\to \chi +\chi$, $n+n\to \chi +\chi +{\nu }_e+{\overline{\nu }}_e$ and $\chi +\chi \to n+n$, which together with the reaction $n\to \chi +{\nu }_e+{\overline{\nu }}_e$ can serve as URCA processes for the neutron star cooling [25,26,27].

We would like to emphasize that a possibility to explain the NLA problem by the neutron dark matter decay mode $n\to \chi +{\nu }_e+{\overline{\nu }}_e$ is not innocent and demands to pay the following price. As has been pointed out in [9], the explanation of the neutron lifetime ${\tau }_n=888.0(2.0)\mathrm{s}$ [1] within the Standard Model with the axial coupling constant $g_A=1.2764$ [28,29], reproducing the neutron lifetime ${\tau }_n=879.6(4)\mathrm{s}$ [30], is not possible and demands the account for the contributions of interactions beyond the Standard Model such as the Fierz interference term b [31,32,33,34,35,36,37,38] equal to $b=-1.44\times {10}^{-2}$ [39]. However, as has been shown in [39], the Fierz interference term $b=-1.44\times {10}^{-2}$ does not contradict the existing experimental data on the correlation coefficients and asymmetries of the neutron beta decay. It is obvious that the decay channel is $n\to \chi +{\nu }_e+{\overline{\nu }}_e$, since all decay particles are neutral. Hence, such mechanism of the NLA can be confirmed experimentally by experimental investigations of reactions, where an emission of a dark matter fermion $\chi$ is accompanied by an emission of charged standard model particles.

Having assumed that the results of the experimental data [6,7,8] can be also interpreted as a production of electron-positron pairs below reaction threshold, we may assume that the neutron dark matter decay $n\to \chi +{\nu }_e+{\overline{\nu }}_e$ can be confirmed, for example, in the process of the electrodisintegration of the deuteron into dark matter fermions and protons $e^{-}+d\to \chi +p+e^{-}$ close to threshold, induced by the $(n\chi e^{-}{e}^{+})$ interaction [9].

The paper is organized as follows. In Section 2 we calculate the triple-differential cross section for the electrodisintegration of the deuteron $e^{-}+d\to \chi +p+e^{-}$ into dark matter fermions $\chi$ and protons p. In Section 3 we discuss the obtained results, make an estimate of the triple-differential cross section, calculated in Section 2, and propose an experimental observation of dark matter fermions in coincidence experiments on the electrodisintegration of the deuteron $e^{-}+d\to n+p+e^{-}$ close to threshold by detecting outgoing electrons, protons and neutrons in coincidence. An absence of neutron signals should testify to an observation of dark matter fermions.

2. Triple-Differential Cross Section for Electrodisintegration of Deuteron into Dark Matter Fermions and Protons ${\mathit{e}}^{-}+\mathit{D}\to \mathbf{\chi }+\mathit{P}+{\mathit{e}}^{-}$

For the solution of the neutron lifetime anomaly problem we have proposed to use the following effective interaction [9,19]

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{DMBL}}(x)=-\frac{G_F}{\sqrt{2}}{V}_{ud}[{\overline{\psi }}_{\chi }(x){\gamma }_{\mu }(h_V+{\overline{h}}_A{\gamma }^5){\psi }_n(x)][{\overline{\mathrm{\Psi }}}_e(x){\gamma }^{\mu }(1-{\gamma }^5){\mathrm{\Psi }}_e(x)],\end{array}$$

(1)

where $G_F=1.1664\times {10}^{-11}{\mathrm{MeV}}^{-2}$ is the Fermi weak coupling constant, $V_{ud}=0.97370(14)$ is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element [10], extracted from the $0^{+}\to 0^{+}$ transitions [10]. The phenomenological coupling constants $h_V$ and ${\overline{h}}_A$ define the strength of the neutron-dark matter $n\to \chi$ transitions. Then, ${\psi }_{\chi }(x)$ and ${\psi }_n(x)$ are the field operators of the dark matter fermion and neutron, respectively. For the quantum field theoretic derivation of the interaction Equation (1) we refer to [19]. According to the SEM [10] (see also [12]), the field operator ${\mathrm{\Psi }}_e(x)$ is the doublet with components $({\psi }_{{\nu }_e}(x),{\psi }_e(x))$, where ${\psi }_{{\nu }_e}(x)$ and ${\psi }_e(x)$ are the field operators of the electron-neutrino (electron-antineutrino) and electron (positron), respectively. The leptonic current ${\overline{\mathrm{\Psi }}}_e(x){\gamma }^{\mu }(1-{\gamma }^5){\mathrm{\Psi }}_e(x)$ has the $V-A$ structure, since electron-neutrinos are practically left-handed. The amplitude of the reaction $e^{-}+d\to \chi +p+e^{-}$ is defined by

$$\begin{array}{ccc}\hfill M{(e^{-}d\to \chi p{e}^{-})}_{{\sigma }_e,{\lambda }_d}^{{\sigma }_{\chi },{\sigma }_p,{\sigma }_e^{\prime },}& =& -\frac{G_F}{\sqrt{2}}{V}_{ud}\langle p({\overrightarrow{k}}_p,{\sigma }_p)\chi ({\overrightarrow{k}}_{\chi },{\sigma }_{\chi })|[{\overline{\psi }}_{\chi }(0){\gamma }_{\mu }(h_V+{\overline{h}}_A{\gamma }^5){\psi }_n(0)]|d({\overrightarrow{k}}_d,{\lambda }_d)\rangle \hfill \\ & & \times [{\overline{u}}_e(\overrightarrow{k}{}_e^{\prime },{\sigma }_e^{\prime }){\gamma }^{\mu }(1-{\gamma }^5)u_e({\overrightarrow{k}}_e,{\sigma }_e)],\hfill \end{array}$$

(2)

where ${\lambda }_d=0,\pm 1$ define the polarization states of the deuteron, ${\overline{u}}_e(\overrightarrow{k}{}_e^{\prime },{\sigma }_e^{\prime })$ and $u_e({\overrightarrow{k}}_e,{\sigma }_e)$ are Dirac wave functions of free electrons in the final and initial states of the reaction. In the matrix element of the $d\to \chi +p$ transition $\langle p({\overrightarrow{k}}_p,{\sigma }_p)\chi ({\overrightarrow{k}}_{\chi },{\sigma }_{\chi })|$ and $|d({\overrightarrow{k}}_d,{\lambda }_d)\rangle$ are the wave functions of the dark matter fermion and proton in the final state and the deuteron in the initial one. They are defined by [40]

$$\begin{array}{c}\hfill \langle p({\overrightarrow{k}}_p,{\sigma }_p)\chi ({\overrightarrow{k}}_{\chi },{\sigma }_{\chi })|=\langle 0|a_{\chi }({\overrightarrow{k}}_{\chi },{\sigma }_{\chi })a_p({\overrightarrow{k}}_p,{\sigma }_p)\end{array}$$

(3)

and

$$\begin{array}{ccc}\hfill |d({\overrightarrow{k}}_d,{\lambda }_d=\pm 1)\rangle & =& \frac{1}{{(2\pi )}^3}\int \frac{d^3{q}_p}{\sqrt{2E_p({\overrightarrow{q}}_p)}}\frac{d^3{q}_n}{\sqrt{2E_n({\overrightarrow{q}}_n)}}\sqrt{2E_d({\overrightarrow{q}}_p+{\overrightarrow{q}}_n)}{\delta }^{(3)}({\overrightarrow{k}}_d-{\overrightarrow{q}}_p-{\overrightarrow{q}}_n)\hfill \\ \hfill & & \times {\mathrm{\Phi }}_d\left(\frac{{\overrightarrow{q}}_p-{\overrightarrow{q}}_n}{2}\right)a_p^{†}({\overrightarrow{q}}_p,\pm 1/2)a_n^{†}({\overrightarrow{q}}_n,\pm 1/2)|0\rangle ,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill |d({\overrightarrow{k}}_d,{\lambda }_d=0)\rangle & =& \frac{1}{{(2\pi )}^3}\int \frac{d^3{q}_p}{\sqrt{2E_p({\overrightarrow{q}}_p)}}\frac{d^3{q}_n}{\sqrt{2E_n({\overrightarrow{q}}_n)}}\sqrt{2E_d({\overrightarrow{q}}_p+{\overrightarrow{q}}_n)}{\delta }^{(3)}({\overrightarrow{k}}_d-{\overrightarrow{q}}_p-{\overrightarrow{q}}_n)\hfill \\ \hfill & & \times {\mathrm{\Phi }}_d\left(\frac{{\overrightarrow{q}}_p-{\overrightarrow{q}}_n}{2}\right)\frac{1}{\sqrt{2}}[a_p^{†}({\overrightarrow{q}}_p,+1/2))a_n^{†}({\overrightarrow{q}}_n,-1/2)+a_p^{†}({\overrightarrow{q}}_p,-1/2))a_n^{†}({\overrightarrow{q}}_n,+1/2)]|0\rangle ,\hfill \end{array}$$

(4)

where $|0\rangle$ is the vacuum wave function, $a_j^{†}({\overrightarrow{p}}_j,{\sigma }_j)$ and $a_j({\overrightarrow{p}}_j,{\sigma }_j)$ are operators of creation and annihilation of a fermion $j=n,p,\chi$ with a 3-momentum ${\overrightarrow{p}}_j$ and polarization ${\sigma }_j=\pm 1/2$ obeying standard relativistic covariant anti-commutation relations [40]. Then, ${\mathrm{\Phi }}_d(\overrightarrow{k})$ is the component of the wave function of the bound $np$-pair in the ${}^3{\mathrm{S}}_1$ state defined in the momentum representation. It is normalized to unity [40] (see also [41]):

$$\begin{array}{c}\hfill \int |{\mathrm{\Phi }}_d(\overrightarrow{k}){|}^2{d}^{3}k/{(2\pi )}^3=1.\end{array}$$

(5)

We neglect the contribution of the component of the wave function of the bound $np$-pair in the ${}^3{\mathrm{D}}_1$-state [42,43,44] (see also [41,45]), which is not important for the analysis of the electrodisintegration of the deuteron into dark matter fermions and protons. The wave function of the deuteron Equation (4) is normalized by [40]:

$$\begin{array}{c}\hfill \langle d(\overrightarrow{k}{}_d^{\prime },\lambda {}_d^{\prime }|d({\overrightarrow{k}}_d,{\lambda }_d)\rangle ={(2\pi )}^{3}2E_d({\overrightarrow{k}}_d){\delta }^{(3)}(\overrightarrow{k}{}_d^{\prime }-{\overrightarrow{k}}_d){\delta }_{\lambda {}_d^{\prime }{\lambda }_d}.\end{array}$$

(6)

In the non-relativistic approximation for heavy fermions and in the laboratory frame, where the deuteron is at rest, the amplitudes of the electrodisintegration of the deuteron into dark matter fermions and protons are determined by

$$\begin{array}{ccc}\hfill & & M{(e^{-}d\to \chi p{e}^{-})}_{{\sigma }_e,{\lambda }_d=\pm 1}^{{\sigma }_{\chi },{\sigma }_p,{\sigma }_e^{\prime }}=-\sqrt{4m_d{m}_n{m}_{\chi }}{G}_F{V}_{ud}{\mathrm{\Phi }}_d({\overrightarrow{k}}_p){\delta }_{{\sigma }_p,\pm 1/2}(h_V[{\phi }_{\chi }^{†}({\sigma }_{\chi }){\phi }_n(\pm 1/2)]\hfill \\ & & \times [{\overline{u}}_e^{\prime }({\overrightarrow{k}}_e^{{}^{\prime }},{\sigma }_e^{\prime }){\gamma }^0(1-{\gamma }^5)u_e({\overrightarrow{k}}_e,{\sigma }_e)]-{\overline{h}}_A[{\phi }_{\chi }^{†}({\sigma }_{\chi })\overrightarrow{\sigma }{\phi }_n(\pm 1/2)]·[{\overline{u}}_e^{\prime }({\overrightarrow{k}}_e^{{}^{\prime }},{\sigma }_e^{\prime })\overrightarrow{\gamma }(1-{\gamma }^5)u_e({\overrightarrow{k}}_e,{\sigma }_e)]),\hfill \\ & & M{(e^{-}d\to \chi p{e}^{-})}_{{\sigma }_e,{\lambda }_d=0}^{{\sigma }_{\chi },{\sigma }_p,{\sigma }_e^{\prime }}=-\sqrt{2m_d{m}_n{m}_{\chi }}{G}_F{V}_{ud}{\mathrm{\Phi }}_d({\overrightarrow{k}}_p)\{{\delta }_{{\sigma }_p,+1/2}(h_V[{\phi }_{\chi }^{†}({\sigma }_{\chi }){\phi }_n(-1/2)]\hfill \\ & & \times [{\overline{u}}_e^{\prime }({\overrightarrow{k}}_e^{{}^{\prime }},{\sigma }_e^{\prime }){\gamma }^0(1-{\gamma }^5)u_e({\overrightarrow{k}}_e,{\sigma }_e)]-{\overline{h}}_A[{\phi }_{\chi }^{†}({\sigma }_{\chi })\overrightarrow{\sigma }{\phi }_n(-1/2)]·[{\overline{u}}_e^{\prime }({\overrightarrow{k}}_e^{{}^{\prime }},{\sigma }_e^{\prime })\overrightarrow{\gamma }(1-{\gamma }^5)u_e({\overrightarrow{k}}_e,{\sigma }_e)])\hfill \\ & & +{\delta }_{{\sigma }_p,-1/2}(h_V[{\phi }_{\chi }^{†}({\sigma }_{\chi }){\phi }_n(+1/2)][{\overline{u}}_e^{\prime }({\overrightarrow{k}}_e^{{}^{\prime }},{\sigma }_e^{\prime }){\gamma }^0(1-{\gamma }^5)u_e({\overrightarrow{k}}_e,{\sigma }_e)]-{\overline{h}}_A[{\phi }_{\chi }^{†}({\sigma }_{\chi })\overrightarrow{\sigma }{\phi }_n(+1/2)]\hfill \\ & & ·[{\overline{u}}_e^{\prime }({\overrightarrow{k}}_e^{{}^{\prime }},{\sigma }_e^{\prime })\overrightarrow{\gamma }(1-{\gamma }^5)u_e({\overrightarrow{k}}_e,{\sigma }_e)])\},\hfill \end{array}$$

(7)

where ${\phi }_{\chi }({\sigma }_{\chi })$ and ${\phi }_n({\sigma }_n)$ are the Pauli wave functions of the dark matter fermion and neutron, respectively, $\overrightarrow{\sigma }$ are $2\times 2$ Pauli matrices, $m_d=m_n+m_p+{\epsilon }_d$ is the deuteron mass, $m_n$ and $m_p$ are masses of the neutron and proton, and ${\epsilon }_d=-2.224575(9)\mathrm{MeV}$ is the deuteron binding energy [46], and $m_{\chi }$ is the dark matter fermion mass. The differential cross section for the reaction $e^{-}+d\to \chi +p+e^{-}$, averaged over polarizations of the incoming electron and deuteron and summed over polarizations of the fermions in the final state, is equal to

$$\begin{array}{ccc}\hfill d^9\sigma (E_e^{\prime },E_e,{\overrightarrow{k}}_e^{{}^{\prime }},{\overrightarrow{k}}_e,{\overrightarrow{k}}_{\chi },{\overrightarrow{k}}_p)& =& (1+3g_A^2)\frac{G_F^2{|V_{ud}|}^2}{8{\pi }^5}\frac{{\zeta }^{(\mathrm{dm})}}{{\beta }_e}\left(1+a^{(\mathrm{dm})}\frac{{\overrightarrow{k}}_e^{{}^{\prime }}·{\overrightarrow{k}}_e}{E_e^{\prime }{E}_e}\right){|{\mathrm{\Phi }}_d({\overrightarrow{k}}_p)|}^2\delta (E_e^{\prime }+E_p+E_{\chi }-m_d-E_e)\hfill \\ & & \times {\delta }^{(3)}({\overrightarrow{k}}_{\chi }+{\overrightarrow{k}}_p+{\overrightarrow{k}}_e^{{}^{\prime }}-{\overrightarrow{k}}_e)d^3{k}_{\chi }{d}^3{k}_p{d}^3{k}_e^{\prime },\hfill \end{array}$$

(8)

where ${\beta }_e=k_e/E_e$ is the incoming electron velocity and $g_A=1.27641(56)$ is the axial coupling constant [28,29] introduced in [9] for convenuence. The correlation coefficients ${\zeta }^{(\mathrm{dm})}$ and $a^{(\mathrm{dm})}$ are defined by [9]

$$\begin{array}{c}\hfill {\zeta }^{(\mathrm{dm})}=\frac{1}{1+3g_A^2}(|h_V{|}^2+3|{\overline{h}}_A{|}^2)=\frac{0.018}{{(m_n-m_{\chi })}^5},a^{(\mathrm{dm})}=\frac{|h_V{|}^2-{|{\overline{h}}_A|}^2}{|h_V{|}^2+3{|{\overline{h}}_A|}^2},\end{array}$$

(9)

where $m_n-m_{\chi }$ is measured in MeV. In the non-relativistic approximation for the dark matter fermion and proton and in the center-of-mass frame of the $\chi p$-pair we transcribe Equation (8) into the form

$$\begin{array}{ccc}\hfill d^9\sigma (E_e^{\prime },E_e,{\overrightarrow{k}}_e^{{}^{\prime }},{\overrightarrow{k}}_e,\overrightarrow{p},\overrightarrow{k})& =& (1+3{\lambda }^2)\frac{G_F^2{|V_{ud}|}^2}{8{\pi }^5}\frac{{\zeta }^{(\mathrm{dm})}}{{\beta }_e}\left(1+a^{(\mathrm{dm})}\frac{{\overrightarrow{k}}_e^{{}^{\prime }}·{\overrightarrow{k}}_e}{E_e^{\prime }{E}_e}\right)|{\mathrm{\Phi }}_d\left(\overrightarrow{k}+\frac{m_p}{m_p+m_{\chi }}\overrightarrow{p}\right){|}^2\hfill \\ & & \times \delta \left(\frac{p^2}{2M}+\frac{k^2}{2\mu }-({\mathcal{E}}_0+E_e-E_e^{\prime })\right){\delta }^{(3)}(\overrightarrow{p}+{\overrightarrow{k}}_e^{{}^{\prime }}-{\overrightarrow{k}}_e)d^{3}p{d}^{3}k{d}^3{k}_e^{\prime },\hfill \end{array}$$

(10)

where ${\mathcal{E}}_0=m_n-m_{\chi }+{\epsilon }_d$, $\overrightarrow{p}$ and $\overrightarrow{k}$ are the total and relative 3-momenta of the $\chi p$-pair, related to the 3-momenta of the dark matter fermion ${\overrightarrow{k}}_{\chi }$ and the proton ${\overrightarrow{k}}_p$ as follows

$$\begin{array}{c}\hfill {\overrightarrow{k}}_{\chi }=-\overrightarrow{k}+\frac{m_{\chi }}{m_p+m_{\chi }}\overrightarrow{p},{\overrightarrow{k}}_p=\overrightarrow{k}+\frac{m_p}{m_p+m_{\chi }}\overrightarrow{p}.\end{array}$$

(11)

Then, $M=m_p+m_{\chi }$ and $\mu =m_p{m}_{\chi }/(m_p+m_{\chi })$ are the total and reduced masses of the $\chi p$-pair. Having integrated over $\overrightarrow{p}$ we arrive at the expression

$$\begin{array}{ccc}\hfill d^6\sigma (E_e^{\prime },E_e,{\overrightarrow{k}}_e^{{}^{\prime }},{\overrightarrow{k}}_e,\overrightarrow{k})& =& \frac{\mu }{2{\pi }^2}\frac{{\zeta }^{(\mathrm{dm})}}{{\tau }_n{f}_n{\beta }_e}\left(1+a^{(\mathrm{dm})}\frac{{\overrightarrow{k}}_e^{{}^{\prime }}·{\overrightarrow{k}}_e}{E_e^{\prime }{E}_e}\right)|{\mathrm{\Phi }}_d\left(\overrightarrow{k}+\frac{m_p}{m_p+m_{\chi }}\overrightarrow{q}\right){|}^2\hfill \\ & & \times \delta \left(k^2-2\mu ({\mathcal{E}}_0+E_e-E_e^{\prime })+\frac{\mu }{M}{q}^2\right)d^{3}k{d}^3{k}_e^{\prime },\hfill \end{array}$$

(12)

where $\overrightarrow{q}={\overrightarrow{k}}_e-{\overrightarrow{k}}_e^{{}^{\prime }}$ [47] and $q^2={\overrightarrow{q}}^2$, and we have used the definition of the neutron lifetime $1/{\tau }_n=(1+3g_A^2)G_F^2{|V_{ud}|}^2{f}_n/2{\pi }^3$ [30]. Following Arenhövel [47] and Arenhövel et al. [48] we define the triple-differential cross section in the center-of-mass frame of the $\chi p$-pair and in the laboratory frame of incoming and outgoing electrons

$$\begin{array}{ccc}\hfill & & \frac{d^5\sigma (E_e^{\prime },E_e,{\overrightarrow{k}}_e^{{}^{\prime }},{\overrightarrow{k}}_e,\overrightarrow{n})}{d{E}_e^{\prime }d{\mathrm{\Omega }}_{e}d{\mathrm{\Omega }}_{\overrightarrow{n}}}=\frac{\mu }{4{\pi }^2}\frac{{\zeta }^{(\mathrm{dm})}}{{\tau }_n{f}_n{\beta }_e}{k}_e^{\prime }{E}_e^{\prime }\left(1+a^{(\mathrm{dm})}\frac{{\overrightarrow{k}}_e^{{}^{\prime }}·{\overrightarrow{k}}_e}{E_e^{\prime }{E}_e}\right)\mathrm{\Theta }\left(m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2\right)\hfill \\ \hfill & & \times \sqrt{2\mu ({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{\mu }{M}{q}^2}|{\mathrm{\Phi }}_d\left(\overrightarrow{n}\sqrt{2\mu ({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{\mu }{M}{q}^2}+\frac{m_p}{m_p+m_{\chi }}\overrightarrow{q}\right){|}^2,\hfill \end{array}$$

(13)

where $\mathrm{\Theta }(z)$ is the Heaviside function, $\overrightarrow{n}=\overrightarrow{k}/k$, $d{\mathrm{\Omega }}_e=2\pi sin{\theta }_{e}d{\theta }_e$ and $d{\mathrm{\Omega }}_{\overrightarrow{n}}=sin\theta d\theta d\varphi$ with the standard definition of the kinematics of the electrodisintegration of the deuteron [47,48] (see Figure 1 of Refs. [47,48]), where ${\overrightarrow{k}}_e^{{}^{\prime }}·{\overrightarrow{k}}_e=k_e^{\prime }{k}_{e}cos{\theta }_e$ and $\overrightarrow{n}·\overrightarrow{q}=qcos\theta$. Then, it is convenient to transcribe Equation (13) into the form

$$\begin{array}{ccc}\hfill \frac{d^5\sigma (E_e^{\prime },E_e,{\overrightarrow{k}}_e^{{}^{\prime }},{\overrightarrow{k}}_e,\overrightarrow{n})}{d{E}_e^{\prime }d{\mathrm{\Omega }}_{e}d{\mathrm{\Omega }}_{\overrightarrow{n}}}& =& \frac{m_p}{8{\pi }^2}\frac{{\zeta }^{(\mathrm{dm})}}{{\tau }_n{f}_n{\beta }_e}{k}_e^{\prime }{E}_e^{\prime }\left(1+a^{(\mathrm{dm})}\frac{{\overrightarrow{k}}_e^{{}^{\prime }}·{\overrightarrow{k}}_e}{E_e^{\prime }{E}_e}\right)\mathrm{\Theta }\left(m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2\right)\hfill \\ \hfill & & \times \sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2}|{\mathrm{\Phi }}_d\left(\overrightarrow{n}\sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2}+\frac{1}{2}\overrightarrow{q}\right){|}^2.\hfill \end{array}$$

(14)

For the wave function of the deuteron ${\mathrm{\Phi }}_d(\overrightarrow{\ell })$ we may use the expression

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_d(\overrightarrow{\ell })=\sqrt{\frac{8\pi }{{\displaystyle \sum _{i=1}^n\sum _{j=1}^n\frac{C_i{C}_j}{m_i{m}_j{(m_i+m_j)}^3}}}}\sum _{i=1}^n\frac{C_i}{m_i^2+{\ell }^2},\int \frac{d^3\ell }{{(2\pi )}^3}{|{\mathrm{\Phi }}_d(\overrightarrow{\ell })|}^2=1\end{array}$$

(15)

with parameters $C_i$ and $m_i$ taken from the paper by Machleidt et al. [42]. Also we may follow Gilman and Gross [45] and describe the wave function ${\mathrm{\Phi }}_d(\overrightarrow{\ell })$ by the expression

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_d(\overrightarrow{\ell })=\frac{\sqrt{8\pi \sqrt{-m_N{\epsilon }_d}}}{{\displaystyle (-m_N{\epsilon }_d+{\ell }^2)\left(1+\frac{{\ell }^2}{p_0^2}\right)}}{\left(1-\frac{m_N{\epsilon }_d}{p_0^2}\right)}^{3/2},\int \frac{d^3\ell }{{(2\pi )}^3}{|{\mathrm{\Phi }}_d(\overrightarrow{\ell })|}^2=1,\end{array}$$

(16)

where $-m_N{\epsilon }_d=0.940\times 2.224\times {10}^{-3}{\mathrm{GeV}}^2=2.09\times {10}^{-3}{\mathrm{GeV}}^2$ and $p_0^2=0.15{\mathrm{GeV}}^2$ [45]. The squared 3-momenta ${\ell }^2$ and $q^2$ are defined by

$$\begin{array}{ccc}\hfill {\ell }^2& =& m_p({\mathcal{E}}_0+E_e-E_e^{\prime })+q\sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2}cos\theta ,\hfill \\ \hfill q^2& =& {({\overrightarrow{k}}_e-{\overrightarrow{k}}_e^{{}^{\prime }})}^2={(k_e-k_e^{\prime })}^2+4k_e^{\prime }{k}_e{sin}^2\frac{{\theta }_e}{2}.\hfill \end{array}$$

(17)

Using the definition of the correlation coefficient ${\zeta }^{(\mathrm{dm})}$ (see Equation (9)) we may rewrite the triple-differential cross section Equation (16) as follows

$$\begin{array}{ccc}\hfill & & \frac{d^5\sigma (E_e^{\prime },E_e,{\overrightarrow{k}}_e^{{}^{\prime }},{\overrightarrow{k}}_e,\overrightarrow{n})}{d{E}_e^{\prime }d{\mathrm{\Omega }}_{e}d{\mathrm{\Omega }}_{\overrightarrow{n}}}={\sigma }_0\frac{k_e^{\prime }{E}_e^{\prime }}{16{\pi }^2{\beta }_e}\left(1+a^{(\mathrm{dm})}\frac{{\overrightarrow{k}}_e^{{}^{\prime }}·{\overrightarrow{k}}_e}{E_e^{\prime }{E}_e}\right)\mathrm{\Theta }\left(m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2\right)\hfill \\ & & \times \sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2}|{\mathrm{\Phi }}_d\left(\sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })+q\sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2}cos\theta }\right){|}^2,\hfill \end{array}$$

(18)

where all momenta and energies and the dimension of the wave function of the deuteron are measured in $\mathrm{MeV}$. Then, the scale parameter ${\sigma }_0$ is equal to

$$\begin{array}{c}\hfill {\sigma }_0=2m_p\frac{{\zeta }^{(\mathrm{dm})}}{{\tau }_n{f}_n}=6.4{\left(\frac{0.12}{m_n-m_{\chi }}\right)}^5\frac{\mathrm{fb}}{\mathrm{MeV}}.\end{array}$$

(19)

The triple-differential cross section for the reaction $e^{-}+d\to \chi +p+e^{-}$, given by Equation (18), can be used for the analysis of the experimental data on searches for dark matter fermions in coincidence experiments [49,50,51]. For $(m_n-m_{\chi })\simeq 0.023\mathrm{MeV}$ (see [19] ) the scale parameter ${\sigma }_0$ increases by four orders of magnitude ${\sigma }_0\simeq 24.7\mathrm{pb}/\mathrm{MeV}$.

3. Discussion

We have analyzed the electrodisintegration of the deuteron into dark matter fermions and protons $e^{-}+d\to \chi +p+e^{-}$ close to threshold. Such a disintegration is induced by the electron-neutron inelastic scattering $e^{-}+n\to \chi +e^{-}$ with energies of incoming electrons larger than the deuteron binding energy $|{\epsilon }_d|=2.224575(9)\mathrm{MeV}$. The strength of the reaction $e^{-}+n\to \chi +e^{-}$ is caused by the strength of the neutron dark matter decay mode $n\to \chi +{\nu }_e+{\overline{\nu }}_e$, which has been proposed in [9] for an explanation of the neutron lifetime anomaly in case of an unobservability (see [6,7]) of the dark matter decay mode $n\to \chi +e^{-}+e^{+}$ [1], where the production of the electron-positron pair can be below the reaction threshold [9]. Following such an assumption that the production of the electron-positron pair can be below the reaction threshold for a confirmation of an existence of the dark matter decay mode $n\to \chi +e^{-}+e^{+}$ we have proposed in [9] to analyze the low-energy electron-neutron inelastic scattering $e^{-}+n\to \chi +e^{-}$, which can be in principle distinguished above the background of the low-energy electron-neutron elastic scattering $e^{-}+n\to n+e^{-}$.

The effective interaction Equation (1) is supported by the effective quantum field theory model with gauge $S{U}_L(2)\times U_Y(1)\times U_R^{\prime }(1)\times U_L^{\prime \prime }(1)$ symmetry, where the SM and dark matter sectors are described by the effective low-energy Lagrangian ${\mathcal{L}}_{\mathrm{L}œ\mathrm{M}\&\mathrm{SET}\&{\mathrm{DM}}^{\prime }\&{\mathrm{DM}}^{″}}$ [19]. The SM part of this effective field theory, determined by the effective low-energy Lagrangian ${\mathcal{L}}_{\mathrm{L}œ\mathrm{M}\&\mathrm{SET}}$ invariant under $SU{(2)}_L\times U_Y(1)$ gauge symmetry, is gauge invariant and renormalizable [12,13]. This has been demonstrated in [12,13] by examples of the calculation of the radiative corrections of order $O(\alpha E_e/m_N)$ to the neutron lifetime and correlation coefficients of the neutron beta decay [52]. As has been shown in [9,19] such a quantum field theory model allows (i) to derive the effective interaction Equation (1) in the tree approximation for the dark matter spin-1 boson $Z^{\prime }$ exchanges with $h_V=h_A$ (see [19]), and (ii) following the scenario by Cline and Cornell [17] to show that dynamics of dark matter fermions with mass $m_{\chi }<m_n\sim 1\mathrm{GeV}$ and a light dark matter spin-1 boson $Z^{″}$, responsible for a repulsion between dark matter fermions, does not prevent neutron stars to reach masses of about $2M_{\odot }$. It has also been noticed [9] that the processes $n\to \chi +{\nu }_e+{\overline{\nu }}_e$, $n+n\to \chi +\chi$, $n+n\to \chi +\chi +{\nu }_e+{\overline{\nu }}_e$ and $\chi +\chi \to n+n$, allowed in such a quantum field theory model, can serve as URCA processes for the neutron star cooling [25,26,27]. The effective quantum field theory, described by the Lagrangian ${\mathcal{L}}_{\mathrm{L}œ\mathrm{M}\&\mathrm{SET}\&{\mathrm{DM}}^{\prime }\&{\mathrm{DM}}^{″}}$ [19] is fully a low-energy theory. The application of this effective theory to the analysis of the searches of dark matter in the LHC experiments is not straightforward and demands special consideration, which we are planing to carry out in our forthcoming publication.

However, in order to have more processes with particles of the Standard Model in the initial and final states allowing to search dark matter in terrestrial laboratories we have turned to the analysis of the dark matter decay mode $n\to \chi +e^{-}+e^{+}$ through the electrodisintegration of the deuteron $e^{-}+d\to \chi +p+e^{-}$ induced by the interaction $(n\chi e^{-}{e}^{+})$ [9]. We have calculated the triple-differential cross section for the reaction $e^{-}+d\to \chi +p+e^{-}$ (see Equation (18)), which can be used for the analysis of traces of dark matter in coincidence experiments on the electrodisintegration of the deuteron $e^{-}+d\to n+p+e^{-}$ close to threshold [49,50,51]. An important property of this cross section is its independence of the azimuthal angle $\varphi$ between the scattering and reaction planes (see Figure 1 of Ref. [48]). Using the experimental conditions of Ref. [51]: $E_e=50\mathrm{MeV}$, $E_x=E_e-E_e^{\prime }=8\mathrm{MeV}$, ${\theta }_e=40°$, $\theta =0$ and the wave function of the deuteron Equation (16) we predict the triple-differential cross section for the reaction $e^{-}+d\to \chi +p+e^{-}$ equal to

$$\begin{array}{c}\hfill 16{\pi }^2\frac{d^5\sigma (E_e^{\prime },E_e,{\overrightarrow{k}}_e^{{}^{\prime }},{\overrightarrow{k}}_e,\overrightarrow{n})}{d{E}_e^{\prime }d{\mathrm{\Omega }}_{e}d{\mathrm{\Omega }}_{\overrightarrow{n}}}{|}_{E_e,E_x=8\mathrm{MeV},{\theta }_e=40°,\theta =0}={\left(\frac{0.12}{m_n-m_{\chi }}\right)}^5\left\{\begin{array}{cc}\hfill 9,& E_e=50\mathrm{MeV}\hfill \\ \hfill 21,& E_e=85\mathrm{MeV}\hfill \end{array}\right.\frac{\mathrm{fb}}{\mathrm{MeV}},\end{array}$$

(20)

where we have set $a^{(\mathrm{dm})}=0$ [9]. Following [49] we define the ratio $R(\theta )$ of the triple-differential cross section at fixed $E_e,E_x,{\theta }_e$ and $0\le \theta \le 180°$ to the triple-differential cross section at fixed $E_e,E_x,{\theta }_e$ and $\theta =0°$. We get

$$\begin{array}{c}\hfill R(\theta )=\frac{|{\mathrm{\Phi }}_d\left(\sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })+q\sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2}cos\theta }\right){|}^2}{|{\mathrm{\Phi }}_d\left(\sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })+q\sqrt{m_p({\mathcal{E}}_0+E_e-E_e^{\prime })-\frac{1}{4}{q}^2}}\right){|}^2}.\end{array}$$

(21)

In Figure 1 we plot the ratio $R(\theta )$ for the experimental conditions of [51] and $0\le \theta \le 180°$. For $E_e=50\mathrm{MeV}$ and $E_e=85\mathrm{MeV}$ the 3-momentum transferred q is equal to $q=0.16{\mathrm{fm}}^{-1}$ and $q=0.28{\mathrm{fm}}^{-1}$, respectively.

Of course, the value of the triple-differential cross section Equation (18) is sufficiently small. Its strong dependence on the mass difference $m_n-m_{\chi }$ looks rather promising but cannot guarantee a real enhancement of the cross section. Nevertheless, we believe that an observation of the electrodisintegration of the deuteron into dark matter fermions and protons close to threshold can be performed in coincidence experiments on the electrodisintegration of the deuteron. Indeed, in [49,51] the electrodisintegration of the deuteron $e^{-}+d\to n+p+e^{-}$ has been investigated close to threshold detecting outgoing electrons and protons from the $np$-pairs in coincidence.

We propose to detect dark matter fermions from the electrodisintegration of the deuteron $e^{-}+d\to \chi +p+e^{-}$ above the background $e^{-}+d\to n+p+e^{-}$ by detecting outgoing electrons, protons, and neutrons in coincidence. Since the kinetic energies of the center-of-mass and relative motion of the $np$-pairs (or $\chi p$-pairs) are equal to $(T_{(n/\chi )+p}=q^2/2M=0.28\mathrm{MeV}, T_{(n/\chi )p}=k^2/2\mu =5.61\mathrm{MeV})$ and $(T_{(n/\chi )+p}=q^2/2M=0.83\mathrm{MeV}, T_{(n/\chi )p}=k^2/2\mu =5.06\mathrm{MeV})$ for $E_e=50\mathrm{MeV}$ and $E_e=85\mathrm{MeV}$, respectively, $E_x=8\mathrm{MeV}$ and ${\theta }_e=40°$, neutrons should be detected with kinetic energies $T_{(n/\chi )p}<6\mathrm{MeV}$ in the direction practically opposite to the direction of protons. An absence of neutron signals at simultaneously detected signals of protons and outgoing electrons should testify to an observation of dark matter fermions in the final state of the electrodisintegration of the deuteron close to threshold. The electron-energy and angular distribution of these events with an absence of neutron signals should be compared with the distribution given by Equation (18).

Finally we would like notice that it would be very interesting to understand an influence of the reactions $n\to \chi +{\nu }_e+{\overline{\nu }}_e$, $e^{-}+n\to \chi +e^{-}$, $e^{-}+d\to \chi +p+e^{-}$, ${\nu }_e({\overline{\nu }}_e)+n\to \chi +{\nu }_e({\overline{\nu }}_e)$ and ${\nu }_e({\overline{\nu }}_e)+d\to \chi +p+{\nu }_e({\overline{\nu }}_e)$, which are caused by the same interaction Equation (1), on a formation of dark matter in the Universe during the evolution of the Universe [53,54,55,56].