Searching for light WIMPS via their interaction with electrons

TL;DR

This paper explores methods to detect light WIMP dark matter particles through their interactions with electrons, evaluating expected event rates for various detection techniques and energy ranges.

Contribution

It introduces theoretical models to estimate event rates for light WIMPs interacting with electrons across different energy regimes and detection setups.

Findings

Calorimetric detectors can detect meV WIMPs via free electron interactions.

Recoil electrons can be detected for WIMPs heavier than electrons.

Atomic excitations induced by WIMPs can be observed in magnetic fields.

Abstract

In the present work we examine the possibility of detecting light dark matter particles (WIMP) utilizing their possible interactions with the electrons. Employing reasonable theoretical models we evaluate the expected event rates in the following cases: i) For WIMPs in the meV region treating electrons as free and utilizing calorimetric detectors at low temperatures. ii) Detecting recoiling electrons with energy in the eV region ejected out of an atom in the case of WIMPs with a mass more than an order of magnitude heavier than the electron. iii) By observing atomic excitations in the range of 1 meV to 10 eV excitation energy induced by the electron spin interaction in a magnetic field.

Tables

Table 1. Table 1: The ℓ = 0 ℓ 0 \ell=0 bound electron wave functions in momentum space. a = α ​ Z n r + ℓ + 1 ​ m e ​ c 2 ℏ ​ c 𝑎 𝛼 𝑍 subscript 𝑛 𝑟 ℓ 1 subscript 𝑚 𝑒 superscript 𝑐 2 Planck-constant-over-2-pi 𝑐 a=\frac{\alpha Z}{n_{r}+\ell+1}\frac{m_{e}c^{2}}{\hbar c} , α ≈ 1 137 𝛼 1 137 \alpha\approx\frac{1}{137} .

Table 2. Table 2: Listed are the atoms and the indicated binding energy of the corresponding s-electrons. Only electrons with binding energies less than 10 eV are listed.

49In: 0.1 eV	11Na: 0.7 eV	23Al: 0.7 eV	50Sn: 0.9 eV	31Ga: 1.5 eV	12Mg: 2.1 eV	65Cd: 2.2 eV	82Pb: 3.1 eV
31Ge: 5.0 eV	3Li: 5.3 eV	51Sb: 6.7 eV	14Si: 7.6 eV	83Bi: 8.0 eV	33As: 8.5 eV	84Po: 9.0 eV

Table 3. Table 3: the coefficients ( C j 1 , m 1 , j 2 , m 2 , ℓ ) 2 superscript subscript 𝐶 subscript 𝑗 1 subscript 𝑚 1 subscript 𝑗 2 subscript 𝑚 2 ℓ 2 \left(C_{j_{1},m_{1},j_{2},m_{2},\ell}\right)^{2} connecting via the spin operator a given initial state | i ⟩ = | n ​ ℓ , j 1 , m 1 ⟩ ket 𝑖 ket 𝑛 ℓ subscript 𝑗 1 subscript 𝑚 1 |i\rangle=|n\ell,j_{1},m_{1}\rangle with all possible states | f ⟩ = | n ​ ℓ , j 2 , m 2 ⟩ ket 𝑓 ket 𝑛 ℓ subscript 𝑗 2 subscript 𝑚 2 |f\rangle=|n\ell,j_{2},m_{2}\rangle , for ℓ = 0 ,  1 ℓ 0 1 \ell=0,\,1 . Note s-states are favored.

Table 4. Table 4: The same as in table 3 , the coefficients ( C j 1 , m 1 , j 2 , m 2 , ℓ ) 2 superscript subscript 𝐶 subscript 𝑗 1 subscript 𝑚 1 subscript 𝑗 2 subscript 𝑚 2 ℓ 2 \left(C_{j_{1},m_{1},j_{2},m_{2},\ell}\right)^{2} for ℓ = 2 ℓ 2 \ell=2

Equations

$${\Omega }_b=0.0456\pm 0.0015,{\Omega }_{\text{mbox}CDM}=0.228\pm 0.013,{\Omega }_{\Lambda }=0.726\pm 0.015$$

$$\varphi +\varphi \to \chi +\chi$$

$$\sigma ={\lambda }^2\frac{1}{(2m_{\chi }{)}^2}\frac{m_e^2}{m_H^4}\frac{1}{2\pi }2{\mu }_r^2={\sigma }_{0H}\frac{1}{(1+x{)}^2}.$$

$${\sigma }_{0H}=\frac{1}{4\pi }{\lambda }^2\frac{m_e^2}{m_H^4}=8.4\times 10^{-45}\text{mbox}{cm}^2=8.4\times 10^{-9}\text{mbox}pb$$

$$\mathcal{L}=\frac{1}{2\sqrt{2}}{G}_F{J}_{\mu }^Z(\chi )J^{z\mu }(f)=\frac{1}{2\sqrt{2}}{G}_F{K}_{\mu }(\bar{f}{\gamma }^{\mu }(g_V-g_A{\gamma }_5)f$$

$${\mathcal{M}}^2=\frac{1}{8}{G}_F^2{g}_A^2$$

$$d\sigma =\frac{1}{\upsilon }\frac{1}{8}{G}_F^2{q}^{2}dqd\xi \delta (q\upsilon \xi -\frac{q^2}{2{\mu }_r}),{\mu }_r=\text{mbox}reducedmassoftheWIMPelectronsystem,$$

$${\sigma }_e=\frac{1}{8}{G}_F^2\frac{1}{\pi }{\mu }_r^2=\frac{1}{8}{G}_F^2\frac{1}{\pi }{m}_e^2\frac{x^2}{(1+x{)}^2}={\sigma }_{0Z}\frac{x^2}{(1+x{)}^2}$$

$${\sigma }_{0Z}=\frac{1}{8}{G}_F^2\frac{1}{\pi }{m}_e^2\approx 1.0\times 10^{-9}$$

$$\frac{{\sigma }_e}{{\sigma }_N}=\frac{9}{4}\frac{m_e^2{m}_{\chi }^2}{m_p^2(m_e+m_{\chi }{)}^2}\Rightarrow {\sigma }_e={\sigma }_{0Z}\frac{x^2}{(1+x{)}^2},x=\frac{m_e}{m_{\chi }}$$

$${\sigma }_{0Z}=\frac{9}{4}\frac{m_e^2}{m_p^2}{\sigma }_N=3.2\times 10^{-45}\text{mbox}{cm}^2\approx 3.2\times 10^{-9}\text{mbox}pb$$

$${\sigma }_{0H}\approx {\sigma }_{0Z}={\sigma }_0=4.0\times 10^{-45}\text{mbox}{cm}^2=4.0\times 10^{-9}\text{mbox}pb$$

$$d\sigma =\frac{1}{\upsilon }{\lambda }^2{\frac{1}{(2m_{\chi })}}^2\frac{m_e^2}{m_H^4}\frac{1}{(2\pi {)}^2}{d}^3{\mathbf{p}}_{\chi }^{\prime }{d}^3\mathbf{q}\delta ({\mathbf{p}}_{\chi }-{\mathbf{p}}_{\chi }^{\prime }-\mathbf{q})\delta \left(\frac{{\mathbf{p}}_{\chi }^2}{2m_{\chi }}-\frac{{\mathbf{p}}_{\chi }^{\prime 2}}{2m_{\chi }}-\frac{{\mathbf{q}}^2}{2m}\right)$$

$$d\sigma =\frac{1}{2}{\sigma }_{0H}\frac{1}{m_{\chi }^2}\frac{1}{\upsilon }{q}^{2}dqd\xi \delta (q\upsilon \xi -\frac{q^2}{2{\mu }_r}),{\mu }_r=\frac{m_e{m}_{\chi }}{m_e+m_{\chi }}=\text{mbox}reducedmass,$$

$$q=2m_r\upsilon ,\upsilon =\text{mbox}WIMPvelocity,\xi =\hat{\upsilon }.\hat{q}\ge 0$$

$$d\sigma ={\sigma }_0\frac{1}{{\upsilon }^2}\frac{1}{2m_{\chi }^2}{m}_{e}dT={\sigma }_0\frac{1}{2{\upsilon }^2}\frac{1}{x^2}\frac{dT}{m_e},x=\frac{m_{\chi }}{m_e},{\sigma }_0={\sigma }_{0H}$$

$$T=\frac{q^2}{2m_e}=2\frac{1}{m_e}{\mu }_r^2{\upsilon }^2{\xi }^2=2m_e\frac{m_{\chi }^2}{m_e^2+m_{\chi }^2}{\upsilon }^2{\xi }^2=2m_e{\upsilon }^2{\xi }^2\frac{x^2}{(1+x{)}^2}$$

$$d\sigma ={\sigma }_{0Z}\frac{1}{2{\upsilon }^2}\frac{dT}{m_e}$$

$$\frac{T}{T_{\chi }}=4\frac{x}{(1+x{)}^2},x=\frac{m_{\chi }}{m_e}$$

$$\upsilon =\sqrt{\frac{m_{e}T}{2{\mu }_r^2{\xi }^2}}=\left(1+\frac{1}{x}\right)\sqrt{\frac{1}{2}\frac{T}{m_e}}\to \upsilon \ge \left(1+\frac{1}{x}\right)\sqrt{\frac{1}{2}\frac{T}{m_e}}\to {\upsilon }_{min}=\left(1+\frac{1}{x}\right)\sqrt{\frac{1}{2}\frac{T}{m_e}}.$$

$$dR=\sigma_{0}\frac{\rho_{\chi}}{m_{\chi}}\frac{1}{2}\upsilon\nu(x)\frac{dT}{m_{e}}f({\mbox{\boldmath$\upsilon$}})d\upsilon d\Omega,\,\nu(x)=\left\{\begin{array}[]{cc}\frac{1}{x^{2}},&\mbox{{\tiny SW}}\\ 1,&\mbox{{\tiny FW}}\\ \end{array}\right.$$

$$dR=\frac{\rho_{\chi}}{m_{\chi}}\sigma_{0}\frac{dT}{m_{e}}\frac{1}{2\upsilon_{0}}\eta(\upsilon_{\mbox{\tiny{min}}})\times\left\{\begin{array}[]{cc}{x^{2}},&\mbox{{\tiny SW}}\\ 1,&\mbox{{\tiny FW}}\\ \end{array}\right.,\,\eta(\upsilon_{\mbox{\tiny{min}}})=\int_{\upsilon_{\mbox{\tiny{min}}}}^{\upsilon_{esc}}f({\mbox{\boldmath$\upsilon$}})\upsilon d\upsilon d\Omega$$

$$\frac{1}{\pi \sqrt{\pi }}{e}^{-y^{{}^{\prime }2}},y^{{}^{\prime }}=\frac{{\upsilon }^{{}^{\prime }}}{{\upsilon }_0},{\upsilon }_0=220\text{mbox}km/s$$

$${\mathbf{y}}^{{}^{\prime }}\to \mathbf{y}+{\hat{\upsilon }}_s+\delta \left(\sin \alpha \hat{x}-\cos \alpha \cos \gamma \hat{y}+\cos \alpha \sin \gamma {\hat{\upsilon }}_s\right),\delta =\frac{{\upsilon }_E}{{\upsilon }_0}$$

$$dR=\left(\frac{\rho_{\chi}}{m_{\chi}}{\upsilon_{0}}\right)N_{e}\frac{1}{2\upsilon_{0}^{2}}\frac{dT}{m_{e}}\left(\Psi_{0}(y_{min})+\Psi_{1}(y_{min})\cos{\alpha}\right)\times\left\{\begin{array}[]{cc}{\frac{1}{x^{2}}},&\mbox{{\tiny SW}}\\ 1,&\mbox{{\tiny FW}}\\ \end{array}\right.,\,x=\frac{m_{\chi}}{m_{e}},$$

$${\Psi }_0(x)=\frac{1}{2}H\left(y_{esc}-x\right)\left[\text{mbox}erf(1-x)+\text{mbox}erf(x+1)+\text{mbox}erfc(1-y_{\text{mbox}esc})+\text{mbox}erfc(y_{\text{mbox}esc}+1)-2\right],x=y_{\text{mbox}min}$$

$${\Psi }_1(x)$$

$$y_{min}=\frac{{\upsilon }_{min}}{{\upsilon }_0}=\frac{1}{{\upsilon }_0}\left(1+\frac{1}{x}\right)\sqrt{\frac{1}{2}\frac{T}{m_e}},y_{esc}=\frac{{\upsilon }_{esc}}{{\upsilon }_0}$$

$$\frac{dR}{d(T/1\text{mbox}eV)}=\rho \Lambda \left({\Sigma }_0\left(\frac{m_{\chi }}{m_e},\frac{T}{\left(1\text{mbox}eV\right)}\right)+{\Sigma }_1\left(\frac{m_{\chi }}{m_e},\frac{T}{\left(1\text{mbox}eV\right)}\right)\cos \alpha \right),\rho =\frac{1\text{mbox}eV}{2m_e{\upsilon }_0^2}\approx 2$$

$$\Sigma_{i}(x,s)=\frac{1}{x}\Psi_{i}\left(1.23\left(1+\frac{1}{x}\right)\sqrt{\rho s}\right)\times\left\{\begin{array}[]{cc}{\frac{1}{x^{2}}},&\mbox{{\tiny SW}}\\ 1,&\mbox{{\tiny FW}}\\ \end{array}\right.,\,i=0,1,\,s=\frac{T}{1\mbox{eV}}$$

Full text

Dark matter, light WIMP, direct detection, big bounce universe, WIMP-electron scattering, event rates, modulation

I Introduction

The combined earlier results MAXIMA-1 Hanary et al. (2000),Wu et al. (2001),Santos et al. (2002), BOOMERANG Mauskopf et al. (2002),Mosi et al. (2002) DASI Halverson et al. (2002) and COBE/DMR Cosmic Microwave Background (CMB) observations Smoot et al. (1992), Spergel et al. (2003) imply that the Universe is flat Jaffe et al. (2001) and that most of the matter in the universe is Dark Spergel et al. (2003). These results have been confirmed and improved by the recent WMAP Spergel et al. (2007) and Planck data Pla . Combining these data one finds:

[TABLE]

Since, on the other hand, any non exotic component cannot exceed $40\%$ of the above $\Omega_{\mbox{{\tiny CDM}}}$  Bennett et al. (1995), exotic (non baryonic) matter is required.

On the smaller scales there exists firm indirect evidence from the observed rotational curves, see e.g. the review Ullio and Kamioknowski (2001), for a halo of dark matter in galaxies and dwarf galaxies.

Anyway in spite of the above indirect evidence for the existence of dark matter at all scales, it is essential to directly detect such matter in order to unravel the nature of its constituents.

It clear that the direct detection of dark matter depends on the nature of the dark matter constituents and their interactions.

These, called WIMP’s (Weekly interacting particles), are expected to have a velocity distribution with an average velocity, close to the rotational velocity $\upsilon_{0}=220$ km/s of the sun around the galaxy, i.e. they are completely non relativistic. In fact a Maxwell-Boltzmann distribution with a maximum cut off of about 2.84$\upsilon_{0}$ leads to a maximum energy transfer close to the average WIMP kinetic energy $\prec T\succ\approx 0.4\times 10^{-6}mc^{2}$. Thus for GeV WIMPS this average is in the KeV regime, not high enough to excite the nucleus, but sufficient to measure the nuclear recoil energy. For light dark matter particles in the MeV region, which we will also call WIMPs, the average energy that can be transferred is in the eV region.

In the present work we will focus on light WIMPs with a mass less 10 times the electron mass. So they can be detected by measuring the electron recoil, following the WIMP-electron interaction in some targets that posses weakly bound electrons . Much lighter WIMPs can only be detected by special materials involving very weakly bound electrons, like superconductors by measuring the total deposited energy.

The event rate for such a process can be computed from the following ingredients Lewin and Smith (1996): i) The elementary WIMP-electron cross section. ii) The WIMP density in our vicinity obtained from the rotation curves. Due to the assumed smallness of the WIMP mass, this is expected to be about six orders of magnitude lager than that involved in the usual WIMPs considered in nuclear recoils. iii) The WIMP velocity distribution. In the present work we will consider a Maxwell-Boltzmann (MB) distribution in the galactic frame, with the WIMP velocity appropriately transformed in the local frame.

In all recoil experiments, like the nuclear measurements first proposed more than 30 years ago Goodman and Witten (1985), in order to overcome the formidable background problems one can exploit the modulation effect, a periodic signal due to the motion of the earth around the sun. Unfortunately this effect, also proposed a long time ago Drukier et al. (1986) and subsequently studied by many authors Primack et al. (1988); Gabutti and Schmiemann (1993); Bernabei (1995); Lewin and Smith (1996); Abriola et al. (1999); Hasenbalg (1998); Vergados (2003); Green (2003); Savage et al. (2006); FKL , in the case of nuclear recoils.

In spite of these problems many experimental undertook the task of detecting nuclear recoils in WIMP-nucleus scattering, see e.g. Abe et al. (2009); Bruch et al. (2009); Armengaud et al. (2011); Kim et al. (2012); Felizardo et al. (2012); Archambault et al. (2012); Bernabei et al. (2013); CRE ; Aprile et al. (2017); Akerib et al. (2014). None has been detected but very stringent limits on the nucleon cross section have been set which can be found in a recent reviewKST . Furtherore projected sensitivities of Dark Matter direct detection experiments to effective WIMP-nucleus couplings have also appearedLUX .

The above results combined with theoretical motivations stimulated interest in lower mass WIMPs, see e.g. the recent work Essig et al. (2012a). In fact the first direct detection limits on sub-GeV dark matter from XENON10 have recently been obtained Essig et al. (2012b). It is, however, clear that Light WIMPs are quite different in energy, mass. One, thus, needs suitable detectors, which maybe completely different from current WIMP detectors employed for heavy WIMP searches. It is encouraging that light WIMPs in the keV region can be detected employing Superfluiid Helium Schutz and Zurec (2016).

For WIMPs in the mass range of the electron mass, since the available energy is in the eV region, the detection of electron recoils is possible only for electrons with very low binding energies. Furthermore the detector should be able to measure recoil energy in few eV region.

Regarding the elementary WIMP-electron cross section we will consider two models:

i) Scalar WIMPs, which are viable cold dark matter candidates. Their mass, as far as we know, has not been constrained by any experiment. This scalar WIMP couples with ordinary Higgs with a quartic coupling, which has been inferred by the LHC experiments. Thus the WIMP interacts with electrons via Higgs exchange with an amplitude proportional to the electron mass $m_{e}$.

ii)For comparison we will consider a model with a fermion WIMP interacting via a Z-exchange with the electron, with a coupling determined phenomenologically. This model, due to the axial coupling, leads to a spin interaction of the electron

In the present paper we will address the implications of light scalar WIMPs on the expected event rates scattered off electrons. The scalar WIMPs have the characteristic feature that the elementary cross section in their scattering off ordinary quarks or electrons is increasing as the WIMPs get lighter, which leads to an interesting experimental feature, provided, of course, that the low energy electrons can be detected. For comparison we will also consider light Fermion WIMPs interacting with the electrons via Z-exchange.

The paper is organized as follows: In section II we discuss the particle model employed. In section III we study the detection of essentially free electrons in special low temperature detectors, e.g. superconducting materials, which act as caloremeters. We eill exploit the enhancent of the obtained rates due to the scalar nature of the WIMPs. In section IV we discuss the effect of the electron binding on the expected rates in the case of experiments measuring electron recoils111We will not concern ourselves here with two-dimensional targets like those considered recently, see e.g. Hochberg et al. (2017),Derenzo et al. (2017). Such detectors will be considered separately elsewhere Kop . in the case of WIMPs with a mass a bit higher than that of the electron. In section V we discuss the possibility of detecting light WIMPs via atomic excitations. This can occur via the spin induced atomic transitions with excitation energy much smaller than the electron binding energy.

II The particle model.

We will consider two such models:

II.1 Scalar WIMPs interacting with the Higgs particle in a quartic coupling.

Scalar WIMP’s can occur in particle models. Examples are i) In Kaluza-Klein theories for models involving universal extra dimensions (for applications to direct dark matter detection see, e.g., Oikonomou et al. (2007)). In such models the scalar WIMPs are characterized by ordinary couplings, but they are expected to be quite massive. ii) extremely light particles  Boehm and Fayet (2004), which are not relevant to the ongoing WIMP searches ii) Scalar WIMPs such as those considered previously in various extensions of the standard model Ma (2006), which can be quite light and long lived protected by a discrete symmetry.

Here we will consider as WIMP a scalar particle $\chi$ interacting with another scalar $\phi$, e.g. the Higgs scalar, via a quartic coupling Silveira and Zee (1985); Holz and Zee (201); Bento et al. (2001, 2000), and more recently Cheung and Vergados (2015). The interest in such a WIMP has recently been revived due to a new scenario of dark matter production in bounce cosmology Li et al. (2014); Cheung et al. (2014) in which the authors point out the possibility of using dark matter as a probe of a big bounce at the early stage of cosmic evolution. A model independent study of dark matter production in the contraction and expansion phases of the Big Bounce reveals a new venue for achieving the observed relic abundance in which dark matter was produced completely out of chemical equilibriumCheung and Vergados (2015) . In this way, this alternative route of dark matter production in bounce cosmology can be used to test the bounce cosmos hypothesis Cheung and Vergados (2015).

In fact the quartic coupling

[TABLE]

involving the scalar WIMP $\chi$ and the Higgs scalar $\phi=h$ discovered at LHC, leads to the Feynman diagram shown in Fig. 1.

In the case of the proton the cross section has previously been discussed Cheung and Vergados (2015). In the case of the electron the elementary cross section is

[TABLE]

or

[TABLE]

In deriving this scale we have assumed that the quantity $\lambda$ is the same with the quartic coupling appearing in the Higgs potential. This is determined by the LHC data, $\lambda=1/2$. In the context of dark matter interactions this is a rather large cross section. It is the result of the fact that, in the small Yukawa coupling $f=\frac{m_{e}}{v}$, the vacuum expectation value $v$ is canceled by that appearing n the quartic coupling. We thus emphasize that the cross section does not suffer from the suppression expected in the decay $h\rightarrow e^{-}e^{+}$ in which $f$ appears and, thus, it cannot be constrained by the LHC data. To the best of our knowledge it is not constrained by any other data.

II.2 Fermion WIMPs interacting via Z-exchange.

Such a mechanism has been considered in the case of the lightest supersymmetric particle (LSP) for the spin induced hadron cross section and more recently in the WIMP electron scattering Vergados et al. (2018). The resulting cross section depends on the coupling of the dark neutral fermions to the Z-boson, i.e. it depends on the nature of the standard model (SM) fermion and the nature of the dark matter:

[TABLE]

We are interested in the axial current component, since the Fermi-like coupling of the electron vanishes. We will assume that axial current coupling of the WIMP is also unity. $K=g_{chi}=1$. Then the invariant amplitude squared takes the form:

[TABLE]

Proceeding as in the previous subsection we find

[TABLE]

which leads to the total cross section:

[TABLE]

with

[TABLE]

One may try to infer the electron cross section from information on the the corresponding the nucleon cross section. In fact this cross section has been constrained by the WIMP-nucleus scattering for a WIMP mass, e.g. of 2 GeV, i.e. $\mu_{r}=\frac{2}{3}m_{p}$, by the CRESST-TUM40 experiment Angloher et al. (2014) to be $5\times 10^{-3}$ pb.

Using the above constrain we obtain:

[TABLE]

with

[TABLE]

This value is a factor of 3 larger than the elementary cross section obtained above. Both of them are a bit smaller compared to the value $\sigma_{0}=5.0\times 10^{-9}$pb determined phenomenologically Vergados et al. (2018). All of them are smaller than that associated with the scalar WIMP obtained above.

In this work we will assume for simplicity common elementary cross section $\sigma_{0}$,

[TABLE]

III The WIMP-electron rate for free electrons

The evaluation of the rate proceeds as in the case of the standard WIMP-nucleon scattering, but we will give the essential ingredients here to establish notation. We will begin by examining the case of a free electron. i) The case of the scalar WIMPs (SW):

The differential cross when all particles involved are non relativistic and the initial electron is at rest can be cast in the form:

[TABLE]

where the factor $1/(2m_{\chi})^{2}$ the usual normalization for the scalar particles and $m_{H}\approx 126$ GeV the mass of the exchanged Higgs particle. Integrating over the momenta we find:

[TABLE]

From the energy conserving $\delta$ function one finds tat the momentum ${\bf q}$ transferred to the electron is given by

[TABLE]

Integrating over $\xi$ with the use of the delta function one finds :

[TABLE]

Where $T$ is the kinetic energy of the outgoing electron given by:

[TABLE]

ii) The case of the fermion WIMP (FW).

Proceeding as above we find

[TABLE]

From Eq. (13) , after integrating over the angles, we find that the fraction of the energy of the WIMP transferred to the electron is

[TABLE]

We thus see that this ratio becomes unity, i.e. maximum, when $x=1$.

The maximum energy transfer depends on the escape velocity, which is assumed to be $\upsilon_{esc}\approx 2.84\upsilon_{0}$ with $\upsilon_{0}=0.710^{-3}c$ the sun’s velocity round the center of the galaxy. Integrating the energy transfer over the velocity distribution we obtain the average energy transfer. The maximum and the average energy transfer are exhibited in fig. 2.

Thus for MeV WIMP the average energy transfer is in the eV region, which is reminiscent of the standard WIMPs where GeV mass leads to an energy transfer in the keV region. The same the average energy is obtained by the convolution the energy transfer with the differential rate, which will be given below (for more details see Vergados et al. (2018)).

Furthermore for a given energy transfer $T$ we find:

[TABLE]

In other words the minimum velocity consistent with the energy transfer $T$ and the WIMP mass is constrained as above. The maximum velocity allowed is determined by the velocity distribution and it will be indicated by $\upsilon_{esc}$. From this we can obtain the differential rate per electron in a given velocity volume $\upsilon^{2}d\upsilon d\Omega$ as follows:

[TABLE]

where f({\mbox{\boldmath\upsilon}}) is the velocity distribution of WIMPs in the laboratory frame. Integrating over the allowed velocity distributions we obtain:

[TABLE]

The parameter $\eta(\upsilon_{\mbox{\tiny{min}}})$ is a crucial parameter.

Before proceeding further we find it convenient to express the velocities in units of the Sun’s velocity. We should also take note of the fact the velocity distribution is given with respect to the center of the galaxy. For a M-B distribution this takes the form:

[TABLE]

We must transform it to the local coordinate system :

[TABLE]

with $\gamma\approx\pi/6$, ${\hat{\upsilon}}_{s}$ a unit vector in the Sun’s direction of motion, $\hat{x}$ a unit vector radially out of the galaxy in our position and $\hat{y}={\hat{\upsilon}}_{s}\times\hat{x}$. The last term, in parenthesis, in Eq. (20) corresponds to the motion of the Earth around the Sun with $\upsilon_{E}\approx 28$ km/s being the modulus of the Earth’s velocity around the Sun and $\alpha$ the phase of the Earth ($\alpha=0$ around June 3nd). The above formula assumes that the motion of both the Sun around the Galaxy and of the Earth around the Sun are uniformly circular. Since $\delta$ is small we can expand the distribution in powers of $\delta$ keeping terms up to linear in $\delta$.

[TABLE]

where in the above equation the first term in parenthesis represents the average flux of WIMPs, the second term gives the number $N_{e}$ of electrons available for the scattering 222In standard targets $N_{e}=\frac{m_{t}Z_{eff}}{Am_{p}}$, in a target of mass $m_{t}$ containing atoms with mass number $A$, $Z_{eff}$ represents the number of available electrons. The meaning of $Z_{eff}$ becomes clear if one takes into account that the electrons are not free but bound in the atom see section IV. Thus they are not all available for scattering, i.e. $Z_{eff}<<Z$.: . Furthermore for a M-B distribution one finds Vergados et al. (2018):

[TABLE]

and

[TABLE]

with

[TABLE]

In the above expression the Heaviside function $H$ guarantees that the required kinematical condition is satisfied. After this we are going to proceed in evaluating the expected spectrum of the recoiling electrons.

The expression given by Eq. (21 ) can be cast in the form:

[TABLE]

where

[TABLE]

and

[TABLE]

Where $N_{e}$ the number of electrons in the target.

The total event rates are given by:

[TABLE]

The time average rate $R_{0}$ is exhibited in Fig. 3a.

For the time dependence we prefer to present:

[TABLE]

Where $R_{r}$ is essentially independent of $x$ and is exhibited in Fig. 3b.

It is thus obvious for light WIMPs it is necessary to consider special materials in which the electrons are loosely bound, like electron pairs in a superconductor, provided, of course, that the number of these electrons is not very small.

We will now estimate the rate for free electrons, i.e. estimate $\Lambda$ considering the following input:.

• •

the elementary cross section $\sigma_{0}=4\times 10^{-9}pb=4\times 10^{-45}\mbox{cm}^{2}$ both for the Z and Higgs exchage.

• •

The particle density of WIMPs in our vicinity:

[TABLE]

(we use the electron mass in this estimate, since the correct mass dependence has been included through the extra factor of $x$ in Eq. (25)). This value leads to a flux:

[TABLE]

• •

The number of electrons in the target, estimated to be

[TABLE]

We thus using Eq. (26) we obtain

[TABLE]

From Fig. 3a we find:

• •

$x=1\Rightarrow$

[TABLE]

both for Fermion and scalar WIMPs. Maximum for Fermion WIMPs

• •

For scalar WIMPs

[TABLE]

[TABLE]

We should mention, however, that the WIMP detection in calorimetric experiments is still difficult, since, in spite of the large rate in the case of scalar WIMPs, the total amount energy deposited in the detector for such a light WIMP is very small.

Anyway it is encouraging that it seems possible, as it has recently been suggested HPZ , to detect even very light WIMPS, much lighter than the electron, utilizing Fermi-degenerate materials like superconductors at low temperatures. In this case the energy required is essentially the gap energy of about $1.5kT_{c}$ which is in the meV region, i.e the electrons are essentially free. These authors claim that in spite of the small energy in the range of few meV deposited to the system, the detection of very light WIMPs becomes feasible.

IV The WIMP-electron rate for bound electrons

In the presence of bound electrons the WIMP mass must be around the mass of the electron, $x=\frac{m_{\chi}}{m_{e}}\geq 1$. In this case it is advantageous to consider the $Z$-exchange. Thus the differential cross section for bound electrons 333 Since as we have seen in section II

$\sigma_{e}=\sigma_{0Z}\frac{x^{2}}{(1+x)^{2}}$

$\frac{\sigma_{0Z}}{m_{e}^{2}}\leftrightarrow\frac{\sigma_{0Z}}{m_{e}^{2}}\left(1+\frac{1}{x}\right)^{2}\leftrightarrow\frac{\sigma_{e}}{\mu^{2}_{r}}$

(29) The latter form are preferred of the WIMP-electron cross section is determined phenomenologically

takes the form:

[TABLE]

where ${\bf p}_{\chi}$ and ${\bf p}^{\prime}_{\chi}$ are the momenta of the oncoming and outgoing WIMPs with mass $m_{\chi}$ and $\upsilon$ is the velocity of the oncoming WIMP. Further more

[TABLE]

with $\psi_{n_{r},\ell,m}$ the bound electron wave function coordinate space. ${\cal M}({\bf q})$ essentially represents the overlap between the electron bound wave function and the plane wave of the outgoing electron with momentum ${\bf q}$. It can be written as $(2\pi)^{3/2}\Phi_{n_{r},\ell,m}(\bf{a,q})$, with $\Phi_{n_{r},\ell,m}(\bf{a,q})$ the bound electron wave function in momentum space. For $\ell=0$ (s-states), which are of interest in the present work, they appear in table 1.

Note that the energy of the atom is negligible and does not appear in the energy conserving $\delta$ function.

Thus integrating over ${\bf p}_{A}$ with the help of the momentum conserving $\delta$ function we obtain

[TABLE]

Then

[TABLE]

where $T$ is the recoiling energy of the electron $T=q^{2}/(2m_{e})$. Similarly the integration over ${\bf q}$ for s-wave functions yields $\Phi^{2}_{n_{r},\ell}(a,\sqrt{2m_{e}T})4\pi\sqrt{2m_{e}T}m_{e}dT$. Furthermore by writing $\sqrt{2m_{e}T}=ua$ we get

[TABLE]

Thus the cross section becomes

[TABLE]

where having in mind to eventually use the Maxwell-Boltzmann (M-B) velocity distribution we have expressed the velocities in units of $\upsilon_{0}=220$km/s. Measuring now the $b$ and $T$ in eV, which is the expected scale we obtain

[TABLE]

where

[TABLE]

The behavior of the function $\psi^{2}_{n_{r},\ell}(u(T))$ for $\alpha Z\approx\frac{1}{2}$ for various values of $n_{r}$ is exhibited in Fig. 4. One can see that the higher $n_{r}$ are favored. For a given $n_{r}$ it is essentially independent of $T$ for recoiling energies of interest to us.

Returning now to Eq. (32) we find some very useful limits.

i) in folding with the velocity distribution we must integrate between $y_{min}=\sqrt{\frac{2\rho^{\prime}(b+T)}{x}}$ and $y_{esc}=2.84$

ii) for a given $x$ and $b$ the maximum electron energy is

[TABLE]

Thus for a value of $x=5$ and a binding energy 2.5 eV the maximum electron energy is expected to be 3 eV.

iii) For a given binding energy $x$ must be at least $x_{min}=0.90b$

Folding the cross section with the velocity distribution (see Eq. (44) below) including the extra factor of $y$ coming from the flux we obtain:

[TABLE]

The total rate can now be cast in the form

[TABLE]

[TABLE]

where

[TABLE]

with $\rho_{\chi}$ the WIMP density in our vicinity. Note that that $m_{e}$ rather $m_{\chi}$ has been employed in determining the number density of WIMPs with a compensating factor $1/x$ already incorporated into Eq. 36.

There exist few atoms which possess s-state electrons with small binding energies. From atomic data tables Larkins (1977); Sev ; F.T and Freedman (1978) we found and list those with $b\leq 10$ eV in table 2. There exist of course states with binding energies smaller than those of the s-states, but, as we have mentioned for light WIMPs they are not going to contribute significantly to the total rate.

It thus apperars that i) NaI (b=0.7 eV in Na) as scintillator and ii) CdTe (b=2.2 eV in Cd), Ge(Li) (b=5 eV in Ge and Li) and Si (b=7.6 eV) can be used as solid state detectors.

Many of the elements listed in table 2, involving s-electrons with low binding energies can serve as good targets, provided, of course, that recoiling electrons with energies in the few eV can be detected. Once a special target is selected, one must make an orbit by orbit calculation, based on the data of table 2, and sum the cross section over all orbits multiplied with the number of electrons involved.

At this point we will make a simple calculation using $N_{e}=N_{A}=10^{25}$, which corresponds to the number of atoms of a Kg of an $A=60$ target and is an order of magnitude larger than that used in the case of free electrons discussed in the previous section. We thus obtain the results shown in Fig. 5 using $Z_{eff}$ much smaller then $Z$ for a typical atom. In spite of the larger $N_{e}$, for low $x$ the obtained results are smaller than those obtained in the previous section. We can trace this suppression to the atomic parameter $a$, which is of the order of $m_{e}$, much larger than the electron recoiling energies, which, for $x<10$, tend to be in the few eV region.

The results, of course, tend to further increase approximately linearly with $x$ and eventually, for $x>50$, electron recoils become easily detectable. For such values of $x$, of course, all electrons can participate, i.e. $Z_{eff}=Z$

V Atomic excitations

We have seen that detecting low mass WIMPs by observing recoiling electrons is pretty hard, since few electrons can be ejected, due to their binding in the atom. This problem does not persist, if the electrons are not ejected, but promoted to a higher level and the de-excitation photons are observed. In this case an energy difference even much smaller than eV is possible, if the target is placed in a magnetic field at low temperature.

As a matter of fact the axial current present in the Z-mediated WIMP-electron interaction through the electron spin can cause atomic transitions between atomic levels within states, which have the same radial quantum numbers and angular quantum numbers $j_{1},m_{1}$ and $j_{2},m_{2}$. If the atom is placed in a magnetic field the transition matrix element is expressed in terms of the Glebsch-Gordan coefficient and the nine- j symbol:

[TABLE]

When $j_{1}=j_{2}$ the two states are those arising from the splitting of the degeneracy due to the Zeeman effect with an energy difference $\delta E=E_{f}-E_{i}=\mbox{ a few}\mu$eV. If $j_{1}\neq j_{2}$ the two levels correspond the spin orbit partners with energy differences in the eV region. For the readers convenience these matrix elements are tabulated for some cases of practical interest will be given below.

The differential cross section now takes the form:

[TABLE]

where $M$ is the mass of the atom and $q$ the momentum transfer to the atom and $\delta E$ the excitation energy. The recoil energy of the atom is negligible. Integrating over the momentum ${\bf q}$ we find:

[TABLE]

Performing the remaining integration we get

[TABLE]

We must now fold it with the velocity distribution in the local frame, ignoring the motion of the Earth around the sun, i.e.

[TABLE]

The integral over $\xi$ is done analytically to yield:

[TABLE]

or

[TABLE]

The last integral can only be done numerically.

The event rate, omitting the orbit dependent angular momentum coefficient $(C_{\ell,j_{1},m_{1},j_{2},m_{2}})^{2}$takes the form:

[TABLE]

where $\Lambda$ is defined as

[TABLE]

One can easily find that the constraint among the parameters is

[TABLE]

The extra factor of $1/x$ in Eq. (47) comes from the fact that the value of $\Lambda$ employed has been evaluated with WIMP number density associated with a mass $m_{e}$, rather than $m_{\chi}$. It has, of course, been assumed one electron per atom $N_{e}=N_{a}=1.0\times 10^{24}$. We exhibit the obtained rates in Fig. 6b.

It is worth comparing the results obtained above with those of in the of WIMP-electron scattering, see Fig. 3. We see that for a given excitation energy the atomic rates increase with the WIMP mass. Thus, e.g., for $m_{\chi}=m_{e}$ the electron scattering yields $0.36\times 1.7\times 10^{-3}=6.0\times 10^{-4}$ events per year. We will compare this with that associated with $\delta$E=0.1 eV excitation. We get $3.5\times 10^{-5}$, 0.0015 and 0.014 for $x=1,10,100$ respectively. In other words the ratio of atomic to recoil events per year for free electrons becomes $0.06$,25 and $2.3\times^{3}$ in the above order for $x$. Clearly the atomic excitations are much favored for $x>5$. They are also much favored compared to detecting the recoi of bound electrons for light WIMPs. An additional advantage of the atomic experiments is the fact that targets with a number of electrons $N_{e}>10^{24}$ are feasible.

The detection involves measuring the de-excitation of the populated level. It is also possible, following Sikivie’s ideas Sikivie (2014) for axion detection, to concentrate Ver on the population of a preferred atomic level at low excitation provided that it is not otherwise occupied by electrons. Then shine a tunable laser to further excite the electrons to a preferred level and then obseve the de-excitation of the chosen level. This may require to cool system at very low temperatures and use a target, perhaps enriched with an impurity if necessary, so that the system maintains an atomic structure at the necessary low temperature.

The obtained rates in Fig. 6b are in principle detectable, but it should be noted that the angular momentum factors $(C_{\ell,j_{1},m_{1},j_{2},m_{2}})^{2}$ have not been included. These are tabulated in 3-4.

VI Discussion

In the present paper we examined the possibility of detecting light WIMPs by exploiting their possible interactions with electrons. We found that, for WIMPs in the mass range of the electron mass, the electron recoiling energies are in the eV region. It is therefore very difficult for electrons to be ejected by overcoming their binding. Furthermore, for WIMP masses less than 50 times the electron mass, the expected rate is too small to be observed. Scattered electrons may be observed, if they are essentially free, with the use of electron detectors may be a good way to directly detect light WIMPs in the sub-MeV region. The WIMP density in our vicinity becomes quite high due to their small mass and the WIMP-electron cross section section may be quite enhanced for scalar WIMPs.

Such detectors utilizing Fermi-degenerate materials like superconductorsHPZ have recently been suggested. In this case the energy required is essentially the gap energy of about $1.5kT_{c}$ which is in the meV region, i.e the electrons are essentially free. We have seen that event rates can be quite high for very light WIMPs, but the amount of energy deposited in the detector is quite small.

We have also seen that it may be possible to detect light WIMPs using a detector in a magnetic field via atomic excitations due to the well known electron spin interactions

Acknowledgments

J.D.V is happy to acknowledge support of this work by the National Experts Council of China via a ”Foreign Master” grant.