Investigating Earth shadowing effect with DAMA/LIBRA-phase1

Abstract

In the present paper the results obtained in the investigation of possible diurnal effects for low-energy single-hit scintillation events of DAMA/LIBRA-phase1 (1.04 ton $\times$ year exposure) have been analysed in terms of an effect expected in case of dark matter (DM) candidates inducing nuclear recoils and having high cross-section with ordinary matter, which implies low DM local density in order to fulfill the DAMA/LIBRA DM annual modulation results. This effect is due to the different Earth depths crossed by those DM candidates during the sidereal day.

1 Introduction

The present DAMA/LIBRA experiment [1–13], as the former DAMA/NaI [14–43] has the main aim to investigate the presence of dark matter (DM) particles in the galactic halo by exploiting the model-independent DM annual modulation signature (originally suggested in Refs. [44, 45]). In particular, they have cumulatively reached a model independent evidence at 9.3$\sigma$ CL for the presence of DM particles in the galactic halo by exploiting the DM annual modulation signature [4]. Recently the results obtained by investigating the presence of possible diurnal variation in the low-energy single-hit scintillation events collected by DAMA/LIBRA-phase1 (1.04 ton $\times$ year exposure) have been released and analysed in terms of a DM second order model-independent effect due to the Earth diurnal rotation around its axis [12]. In particular, the data were analysed using the sidereal time referred to Greenwich, often called GMST. No diurnal variation with sidereal time has been observed at the reached level of sensitivity, which was not yet adequate to point out the effect searched for there. In the present paper those experimental data are analysed in terms of an effect – named “Earth Shadow Effect” – which could be expected for DM candidate particles inducing nuclear recoils; this effect would be induced by the variation – during the day – of the Earth thickness crossed by the DM particle in order to reach the experimental set-up. It is worth noting that a similar effect can be pointed out only for candidates with high cross-section with ordinary matter, which implies low DM local density in order to fulfill the DAMA/LIBRA DM annual modulation results. Such DM candidates could get trapped in substantial quantities in the Earth’s core; in this case they could annihilate and produce secondary particles (e.g. neutrinos) and/or they could carry thermal energy away from the core, giving potentiality to further investigate them.

Preliminary investigations on DM candidates inducing diurnal variation were performed in Refs. [23, 46, 47] and more recently in Refs. [48–52].

2 The Earth Shadow Effect

During a sidereal day the Earth shields a terrestrial detector with a varying thickness, and this induces a variation of the flux of the DM candidates impinging the detector, mainly because of the modification of their velocity distribution, $f(\vec {v})$. It is worth noting that this Earth Shadow Effect is very small and could be detectable only in case of candidates with high cross-section with ordinary matter (i.e. present in the galactic halo with small abundance).

The detector (and the hosting laboratory) velocity in the Galactic frame can be written as:

$$\begin{aligned} \vec {v}_{lab} (t) = \vec {v}_{LSR} + \vec {v}_{\odot } + \vec {v}_{rev}(t) + \vec {v}_{rot}(t), \end{aligned}$$

(1)

where: (i) $\vec {v}_{LSR}$ is the velocity of the local standard of rest (LSR) because of the rotation of the Galaxy; (ii) $\vec {v}_{\odot }$ is the Sun peculiar velocity with respect to LSR; (iii) $\vec {v}_{rev}(t)$ is the velocity of the orbital motion of the Earth around the Sun and (iv) $\vec {v}_{rot}(t)$ is the velocity of the rotation of the Earth around its axis. The two latter terms change as function of the sidereal time, t. Using the galactic coordinate frame (that is x axis towards the galactic center, y axis following the rotation of the Galaxy and the z axis towards the galactic North pole), one gets: $\vec {v}_{LSR}=(0,v_0,0)$, where $v_0= (220\pm 50)$ km/s (uncertainty at 90 % CL) [22, 53–55] is the local velocity, and $\vec {v}_{\odot } = (9, 12, 7)$ km/s [56].

The DM particles in the galactic halo have a velocity distribution $g(\vec {w})$, which depends on the considered galactic halo model. Reference [22] has shown many possible scenarios for the galactic halo; in the following we consider the isothermal halo model just because of its simplicity:

$$\begin{aligned} g(\vec {w}) = A e^{-\frac{w^2}{v_0^2}} \theta (v_{esc} - |\vec {w}|), \end{aligned}$$

(2)

with A normalization constant and ${v}_{esc}$ escape velocity, assumed in the following equal to 650 km/s, as often considered in literature; however, it is also affected by uncertainty. However, no sizeable differences are observed in the outcome when a different value of $v_{esc}=550\, \mathrm km/s$ is considered, more closer to the 90 % CL range of the RAVE Survey results [57]. In the laboratory frame the DM velocity distribution $f(\vec {v})$ is obtained from Eq. 2 straightforward since $\vec {w}=\vec {v}+\vec {v}_{lab}$.

To evaluate the expected daily variation of the DM particles velocity distribution due to the Earth Shadow Effect, it is necessary to estimate the time dependence of the $\theta$ angle, the “zenith distance” of $\vec {v}_{lab}$ (i.e. the distance between $\vec {v}_{lab}$ and the zenith, see Fig. 1). This can be determined by astrophysical considerations studying the Earth’s rotation around its axis.

Fig. 1

Schematic view of the DM particles impinging direction on a detector; the $x', y', z'$ represent the laboratory frame coordinate system. Left schematic representation of the correlation between the thickness, d, crossed by the considered DM candidates to reach a laboratory (hypotetically placed at the geographic North pole) and the DM impinging angle, $\theta _{in}$. Right schematic representation of the experimental condition considered in the text: detector placed at the Gran Sasso National Laboratory (LNGS) with the $z'$ axis in the vertical direction and the $x'$ axis pointing to the vernal equinox

The simplest way to calculate $\theta (t)$ is in the equatorial coordinate system [12] where the $\hat{e}_1^{ecs}$ axis is directed towards the vernal equinox, and $\hat{e}_1^{ecs}$ and $\hat{e}_2^{ecs}$ are on the equatorial plane; the $\hat{e}_3^{ecs}$ axis is towards the North pole. The right-handed convention is used. To work out the galactic coordinates of those versors, one considers: (i) the equatorial coordinates of the galactic North pole: $RA = 192^\circ .859508$ and $DE = 27^\circ .128336$, where RA is the right ascension and DE is the declination; (ii) the equatorial coordinates of the galactic center: $RA = 266^\circ .405100$ and $DE = -28^\circ .936175$, evaluated at the Epoch J2000.0. In the galactic coordinates, those versors can be written as:

$$\begin{aligned} \hat{e}_1^{ecs}= & {} (-0.05487, 0.49411, -0.86767) \nonumber \\ \hat{e}_2^{ecs}= & {} (-0.87344, -0.44483, -0.19808) \\ \hat{e}_3^{ecs}= & {} (-0.48384, 0.74698, 0.45599). \nonumber \end{aligned}$$

(3)

We define $\vec {v}_{s} = \vec {v}_{LSR} + \vec {v}_{\odot }$. In this section, when a numerical calculation is employed, we assume $v_0 = 220$ km/s; hence, $v_{s}=232.28$ km/s, and in the equatorial coordinate system: $\theta _{ecs} = 42^\circ .18$ (the “zenith distance”) and $\varphi _{ecs} = - 46^\circ .14$ (azimuth angle from $\hat{e}_1^{ecs}$). For simplicity in the following of this section in order to offer an estimate of the Earth Shadow Effect diurnal behaviour we consider $\vec {v}_{rev}(t)$ equal to its annual mean value, i.e. zero. To introduce the Earth motion around its axis, firstly we consider the simplest case of a laboratory at North pole and we define the horizontal coordinate system with $z'$ axis directed as shown in Fig. 1, left and $x'$ axis directed towards a given longitude $\lambda _0$. In this coordinate system (N label) the velocity of the Earth can be written as: $v_{lab}^N = v_{s}$, $\theta _N = \theta _{ecs}$, and $\varphi _N = \varphi _{ecs} - \omega _{rot} (t + t_0) = -(\varphi _0 + \omega _{rot} (t+t_0))$, where $\varphi _0 = -\varphi _{ecs}$, $\omega _{rot} = 2 \pi / T_d$ with $T_d = 1$ sidereal day, t sidereal time referred to Greenwich, and $t_0 = 24\lambda _0/2\pi$ sidereal hours.

The general case of $\vec {v}_{lab}$ in a laboratory at latitude $\phi _0$ can be derived by rotating counterclockwise $\vec {v}_{lab}^N$ around $\hat{y}'$ of an angle $\alpha = \pi / 2 - \phi _0$:

$$\begin{aligned} \vec {v}^{\lambda _0 , \phi _0}_{lab}= & {} R(\alpha ) \vec {v}_{lab}^N \nonumber \\= & {} \left( \begin{array}{c@{\quad }c@{\quad }c} \cos \alpha &{} 0 &{} - \sin \alpha \\ 0 &{} 1 &{} 0 \\ \sin \alpha &{} 0 &{} \cos \alpha \\ \end{array} \right) \nonumber \\&\times \left( \begin{array}{c} v_{s} \sin \theta _{ecs} \cos (\varphi _0 + \omega _{rot} (t+t_0)) \\ -v_{s} \sin \theta _{ecs} \sin (\varphi _0 + \omega _{rot} (t+t_0)) \\ v_{s} \cos \theta _{ecs} \end{array} \right) . \end{aligned}$$

(4)

Thus, $\theta (t)$ for a laboratory position, identified by the longitude $\lambda _0$ and latitude $\phi _0$, can be derived from $\cos \theta (t) = \hat{v}_{lab}^{\lambda _0,\phi _0}\cdot \hat{z}'$ (see Fig. 1, right) obtaining:

$$\begin{aligned} \cos \theta (t)= & {} \sin \theta _{ecs} \cos \phi _0 \cos (\omega _{rot} (t+t_0) +\, \varphi _0)\nonumber \\&+ \cos \theta _{ecs} \sin \phi _0 \nonumber \\= & {} \cos \psi \cos \phi _0 \cos (\omega _{rot} (t+t_0)+\, \varphi _0)\nonumber \\&+ \sin \psi \sin \phi _0, \end{aligned}$$

(5)

where $\psi = \pi /2 - \theta _{ecs} = 47^{\circ }.82$ is the altitude. In case of the LNGS, the longitude is $\lambda _0 = 13^\circ 34'$E and the latitude is $\phi _0 = 42^{\circ } 27'$N, thus $t_0 = 0.904444$ sidereal hours. The $\theta (t)$ behaviour at LNGS is shown in Fig. 2. Note that, before detection, the DM particles at LNGS preferentially cross an Earth maximum thickness at about 08:00 h and an Earth minimum thickness at about 20:00 h (GMST in both cases).

Fig. 2

The angle $\theta$ as a function of the sidereal time in the case of the LNGS (latitude $\lambda _0 = 13^\circ 34'$E, longitude $\phi _0 = 42^{\circ } 27'$N). The Earth shielding is maximum about at 8:00 h and minimum around 20:00 h; see text

3 Deformation of the DM velocity distribution due to the Earth Shadow Effect

To study the experimental data in terms of possible Earth Shadow Effect, a Monte Carlo code has been developed to simulate the propagation of the DM candidates elastically scattering off Earth nuclei in their travel in the Earth towards the underground experimental site. For such a purpose useful information has been gathered about the Earth composition and density. The Monte Carlo code numerically estimates the velocity distribution – in the laboratory coordinate system – of the impinging DM particles after having crossed the Earth; such velocity distribution depends on the mass of the DM candidate, on its cross-section on nucleons, on the initial unperturbed velocity distribution, on the sidereal time, and on the latitude and longitude of the laboratory: $f_{lab} (v, t | m_{DM}, \sigma _n)$. Then, this velocity distribution has been used to evaluate – in an assumed framework – the expected counting rate as a function of the sidereal time in order to be compared with the experimental data.

In this section details are given about the assumptions adopted in the simulation, as in particular: the Earth model, the mean free path and path reconstruction of such DM candidate, the adopted interaction model (nuclear form factor, scaling law, etc.) and the $f_{lab} (v, t | m_{DM}, \sigma _n)$ estimation.

3.1 The Earth model

An Earth model has been assumed in order to estimate the signal variation due to the Earth Shadow Effect; in particular, the matter density and composition of the Earth have to be considered. The simulation adopts a simplified Earth model starting by the preliminary reference Earth model (PREM) [58]. Just three main Earth layers with a constant density and homogeneous distribution in each one (these values are averaged over the PREM density distribution behaviour) are considered: (i) the Inner Core; (ii) the External Core; (iii) the Mantle. The densities and mass percentage for each layer are given in Table 1; in this simplified model the rare isotopes (mass percentage lower than 0.1 %) have been neglected.

Table 1 Density values, $\rho _L$, and ith nucleus mass percentage, $\delta _i$, adopted in the present calculations for the layers of the considered Earth model [58]

3.2 Interactions of the considered DM candidates and path reconstruction

We assume that the considered DM candidates loss their energy elastically scattering off nuclei with spin-independent coupling. The mean free path in the Lth Earth’s layer is:

$$\begin{aligned} \lambda _L = \frac{1}{\displaystyle \sum \nolimits _{i=1}^N \sigma _{DM,i} n_i}, \end{aligned}$$

(6)

where: (i) $n_i = \frac{\rho _L }{m_i}\delta _i$ is the number density of the ith nuclei in the Earth’s layer composed by N nuclear species; (ii) $\sigma _{DM,i}$ is the cross-section on the ith nucleus and $m_i$ is the mass of the ith nucleus; (iii) $\rho _L$ is the layer’s density; (iv) $\delta _i$ is the ith nucleus mass percentage in the layer. We considered a coherent scattering and a scaling law¹:

$$\begin{aligned} \sigma _{DM,i} = \sigma _n A_i^2 \frac{\mu _i^2}{\mu _n^2}, \end{aligned}$$

(7)

where: (i) $A_i$ is the mass number of nucleus i; (ii) $\sigma _n$ is the DM candidate-nucleon cross-section; (iii) $\mu _i$ ($\mu _n$) is the DM candidate – nucleus (nucleon) reduced mass. Thus, assuming $m_i \simeq A_i m_n$, one can write:

$$\begin{aligned} \frac{1}{\lambda _L} = \frac{\sigma _n \rho _L}{ m_n} \sum _{i=1}^N A_i^3 \delta _i \left( \frac{m_{DM}+m_n}{m_{DM} + A_i m_n}\right) ^2. \end{aligned}$$

(8)

For simplicity, the deflection of the DM particles crossing the Earth is neglected (i.e. a linear trajectory is considered); in this assumption the Earth thickness crossed by those DM candidates, d, depends only on the impinging angle with respect to the detector, $\theta _{in}$, according to the relation:

$$\begin{aligned} d = 2 R_\oplus \cos \left( \theta _{in} \right) , \end{aligned}$$

(9)

with $R_\oplus$ Earth radius. This d value is the sum of the distances passed through by the DM candidate in each layer: $d=\sum _Ld_L$. Defining the maximum radii of the Earth layers $R_{ic}$, $R_{ec}$ and $R_m = R_\oplus$ for Inner Core, External Core and Mantle respectively (see Table 1), the number of layers crossed by DM particles in the considered schema is: 0 for $\theta _{in} \ge 90^\circ$; 1 for $33^\circ .11 \le \theta _{in} < 90^\circ$; 3 for $11^\circ .05 \le \theta _{in} < 33^\circ .11$; 5 for $0^\circ \le \theta _{in} < 11^\circ .05$ (considering that $\arcsin \left( R_{ec}/R_\oplus \right) =33^\circ .11$ and $\arcsin \left( R_{ic}/R_\oplus \right) =11^\circ .05$).

Thus, in this scenario, the DM particles move in each Earth’s layer with a mean free path $\lambda _L$, that mainly depends on the interaction cross section $\sigma _n$ (see Eq. 8). The number of interactions in each layer, $n_{hit}$, has been estimated as:

• Case 1, high interaction cross sections, $d_L \ge 50 \lambda _L$: $n_{hit}$ is relatively high and follows a gaussian distribution with mean value and variance equal to $d_L / \lambda _L$;

• Case 2, small interaction cross sections, $d_L < 50 \lambda _L$: a step-by-step approach has been adopted in the simulation. The path between two consecutive interactions, $x_k$, follows the distribution $\lambda _L^{-1} e^{-(x_k/\lambda _L)}$; it can be used to propagate the particle within the layer as long as $\sum _k^{n_{hit}} x_k \le d_L$.

In the considered scenario the DM candidate particles interact via SI elastic scattering on nuclei; thus, their energy-loss for each interaction is given by the induced nuclear recoil energy:

$$\begin{aligned} E_R = E_{in} r \left( \frac{1-\cos \theta ^*}{2} \right) , \end{aligned}$$

(10)

where $E_{in} = m_{DM} v^2 / 2$ and v are the DM energy and the velocity before the interaction, $\theta ^*$ is the angle of diffusion in the center of mass and r is a kinematic factor:

$$\begin{aligned} r=\frac{4 m_{DM} m_i}{(m_{DM}+m_i)^2}. \end{aligned}$$

(11)

The interaction cross-section is given by [18]:

$$\begin{aligned} \frac{d\sigma _{DM,i}}{dE_R} (v,E_R) = \frac{d\sigma _{DM,i}}{dE_R} (v,0)F^2_i(E_R), \end{aligned}$$

(12)

with $F^2_i (E_R)$ nuclear form factor. In addition, for a given velocity v:

$$\begin{aligned} \frac{d\sigma _{DM,i}}{dE_R} (v,0) = \frac{d\sigma _{DM,i}}{d\Omega }\times \frac{d\Omega }{dE_R}, \end{aligned}$$

(13)

with (also see Eq. 10):

$$\begin{aligned} \frac{d\sigma _{DM,i}}{d\Omega } = \frac{\sigma _{DM,i}}{4\pi }, \; \; \; \frac{dE_R}{d\Omega } = \frac{rE_{in}}{4\pi }. \end{aligned}$$

(14)

Hence, assuming the cross section scaling law given in Eq. 7, the interaction cross-section can be written as:

$$\begin{aligned} \frac{d\sigma _{DM,i}}{dE_R} (v,E_R) = \frac{\sigma _{DM,i}}{r E_{in}}F^2_i(E_R) = \sigma _n A_i^2 \frac{m_i}{2 \mu _n^2 v^2}F^2_i(E_R).\nonumber \\ \end{aligned}$$

(15)

Thus, after one interaction, such a DM particle loses an energy in the range $0 \le E_R \le r E_{in}$ with distribution given in Eq. 15. In the following we assume for the nuclear form factor the Helm formula² [59, 60]:

$$\begin{aligned} F_i(qr_0) = \frac{3\left[ \sin (qr_0) + qr_0 \cos (qr_0) \right] }{(qr_0)^3}e^{-\frac{1}{2} q^2s^2}, \end{aligned}$$

(16)

where $r_0 = \sqrt{r_i^2 - 5 s^2}$, $r_i = 1.2 A_i^{1/3}$ fm, $s\simeq 1$ fm and $q^2 = 2m_i E_R$.

In the simulation, once $n_{hit}$ has been evaluated, the energy loss of the considered DM particle, $E_R$, is estimated for each interaction and the output velocity is:

$$\begin{aligned} v_f = \sqrt{ v^2 - \frac{2E_R}{m_{DM}}}. \end{aligned}$$

(17)

Therefore, the net effect is a modification of the velocity distribution, $f_{lab} (v, t | m_{DM}, \sigma _n)$; Fig. 3 shows some examples of the obtained velocity distribution for a detector located at LNGS.

Fig. 3

Some examples of simulated $f_{lab} (v, t | m_{DM}, \sigma _n)$ in arbitrary units for some $m_{DM}$ and $\sigma _n$ values when taking into account the Earth Shadow Effect. In these plots: (i) the considered Galactic Halo model is an isothermal sphere with $v_0 = 220$ km/s; (ii) the velocity distribution at GMST hour 8:00 (continuous black line) and 20:00 (dashed–red on-line–line) – corresponding to the maximum and the minimum of the Earth Shadow Effect in case of a target-detector placed at LNGS

4 The expected interaction rate

In the SI coupling scenario, considered here, the DM candidates scatter off the nuclei in the detector. Their expected interaction rate as a function of the nuclear recoil energy, $E_R$, for a mono-atomic (i nucleus) detector is:

$$\begin{aligned} \frac{dN_i}{dE_R}= & {} N_T \int _{v_{min}(E_R)}^\infty \frac{\rho v}{m_{DM}} f_{lab}(v) \nonumber \\&\times \frac{d\sigma _{DM,i}}{dE_R} (v,0) F^2_i(E_R) dv, \end{aligned}$$

(18)

where $N_T$ is the number density of the target nuclei in the detector and $\rho = \xi \rho _0$ is the DM particles density in the galactic halo ($\xi$ is the relative abundance and $\rho _0$ the overall DM density in the galactic halo). The integral is calculated over all the possible DM particle velocities in the laboratory frame considering the distribution $f_{lab}(v) = f_{lab} (v, t | m_{DM}, \sigma _n)$. The minimal velocity providing $E_R$ recoil energy is $v_{min}(E_R) = \sqrt{\frac{E_Rm_i}{2\mu _i^2}}$. The galactic escape velocity is included in the f(v) definition. Using Eq. 15, the expected rate can be rewritten as:

$$\begin{aligned} \frac{dN_i}{dE_R}= & {} \xi \sigma _n \frac{N_T \rho _0}{m_{DM}} A_i^2 \frac{m_i}{2\mu _n^2} F^2_i(E_R) \nonumber \\&\times \int _{v_{min}(E_R)}^\infty \frac{f_{lab} (v, t | m_{DM}, \sigma _n) }{v} dv \nonumber \\= & {} \xi \sigma _n \frac{dN_i'}{dE_R} (E_R , t | m_{DM}, \sigma _n) . \end{aligned}$$

(19)

Generalizing to detectors with more than one kind of target nuclei [as e.g. in the case of the NaI(Tl) considered here], the expected experimental rate is:

$$\begin{aligned} \frac{dR}{dE_{det}} = \xi \sigma _n \frac{dR'}{dE_{det}} (E_{det}, t | m_{DM}, \sigma _n) , \end{aligned}$$

(20)

where:

$$\begin{aligned}&\frac{dR'}{dE_{det}}(E_{det}, t | m_{DM}, \sigma _n) \nonumber \\&\quad =\int G(E_{det},E') \left[ \sum _i \int K_i(E'|E_R) \right. \nonumber \\&\quad \quad \times \left. \frac{dN_i'}{dE_R}(E_R , t | m_{DM}, \sigma _n) dE_R \right] dE'; \end{aligned}$$

(21)

the $G(E_{det},E')$ kernel takes into account the detector’s energy resolution (generally through a gaussian convolution) and the $K_i(E'|E_R)$ kernel takes into account the energy transformation of the nuclear recoil energy in keV electron equivalent. For example the latter kernel can be written in the simplest case of a constant quenching factor, $q_i$, as: $K_i(E'|E_R) = \delta (E' - q_i E_R)$. For a discussion about the quenching factors see Refs. [5, 8].

The expected differential rate – as well as $f_{lab} (v, t | m_{DM}, \sigma _n)$ – depends on the time through three different effects: (i) the time dependence of the Earth’s orbital motion velocity, $\vec {v}_{rev} (t)$ (see Eq. 1); (ii) the time dependence of the Earth’s rotation velocity around its axis, $\vec {v}_{rot} (t)$ (see Eq. 1); (iii) the possible Earth Shadow Effect which depends on $\sigma _n$. Details about the first two effects, responsible of the model-independent annual and diurnal modulation of the DM signal rate, respectively, are reported in Ref. [12]. Following the same approach the expected rate in an energy interval $\Delta E_k$ can be written as³:

$$\begin{aligned} S_k (t)= & {} \int _{\Delta E_k} \xi \sigma _n \left( \frac{dR'}{dE_{det}} \right) (E_{det}, t | m_{DM}, \sigma _n) dE_{det} \nonumber \\= & {} \xi \sigma _n \left[ S'_{0,k} (m_{DM}, \sigma _n) \right. \nonumber \\&+\left. S'_{m,k}(m_{DM},\sigma _n) \cos (\omega (t-t_0)) \right. \nonumber \\&+ \left. S'_{d,k}(m_{DM},\sigma _n) \cos ( \omega _{rot} (t-t_d))\right. \nonumber \\&+\left. S'_{d,sh,k}(m_{DM},\sigma _n,t) \right] , \end{aligned}$$

(22)

where: (i) $S'_{0,k} (m_{DM}, \sigma _n)$ is the time independent component of the expected signal; (ii) $S'_{m,k} (m_{DM},\sigma _n)$ is the annual modulation amplitude, $\omega = 2\pi /T$ with $T=1$ yr, $t_0 \simeq$ June $2{\mathrm{nd}}$; (iii) $S'_{d,k} (m_{DM},\sigma _n)$ is the diurnal modulation amplitude, $\omega _{rot} = 2\pi / T_d$ with $T_d = 1$ sidereal day, $t_d$ for the case of a detector at the Gran Sasso longitude ranges from 13.94 to 14.07 h depending on the $v_0$ value (see Ref. [12]); (iv) $S'_{d,sh,k} (m_{DM},\sigma _n,t)$ (whose average value over $T_d$ is null) takes into account the signal variation as a function of the sidereal time due to a possible Earth Shadow Effect.

The ratio $R_{dy} = S'_{d,k}(m_{DM},\sigma _n) / S'_{m,k}(m_{DM},\sigma _n)$ is model independent and it is $R_{dy} \simeq 0.016$ at LNGS latitude; thus, considering the DAMA/LIBRA-phase1 experimental result on the DM annual modulation, the expected $\xi \sigma _n S'_{d,k}(m_{DM},\sigma _n)$ is order of $10^{-4}$ counts per sidereal day per kg per keV (cpd$_{sid}$/kg/keV, hereafter) [12]. The reached experimental sensitivity of DAMA/LIBRA-phase1 [12] is not yet enough to observe such a diurnal modulation amplitude; in fact, in the (2–4) keV energy interval considered here, the experimental diurnal modulation amplitude from DAMA/LIBRA-phase1 data is $(2.0 \pm 2.1) \times 10^{-3}$ cpd$_{sid}$/kg/keV [12] ($<$ $5.5 \times 10^{-3}$ cpd$_{sid}$/kg/keV, 90 % CL). Thus, in the following we do not further approach it.

Few examples of the $S'_{d,sh,k} (t)$ behavior for $m_{DM} = 30$ GeV and for different values of the cross-section in the given framework are shown in Fig. 4. For clarity, in the explicative case of $\sigma _n = 10$ pb and $\xi = 1.1 \times 10^{-8}$ so that $\xi \sigma _n = 1.1 \times 10^{-7}$ pb is compatible with the DAMA/LIBRA-phase1 DM annual modulation result, the obtained amplitude of $\xi \sigma _n S'_{d,sh,k}(t)$ is of order of $3 \times 10^{-2}$ cpd$_{sid}$/kg/keV. Such value can be studied in DAMA/LIBRA-phase1 (see later).

Fig. 4

Examples of expected $S'_{d,sh,k}(m_{DM},\sigma _n,t)$ obtained following the approach described in the text. The Q$_I$ quenching factor and the set A of the parameters’ values in presence of the channeling effect have been considered (see text); moreover, $v_0 = 220$ km/s. The energy interval, considered here, is (2–4) keV and the $m_{DM} =$ 30 GeV. The obtained $S'_{d,sh,k}(m_{DM},\sigma _n,t)$ are shown for the cases: (i) $\sigma _n$ $=$ 10 pb (continuous black line); (ii) $\sigma _n$ $=$ 1 pb (dashed–red on-line–line); (iii) $\sigma _n$ $=$ 0.1 pb (dotted–blue on-line–line); iv) $\sigma _n$ $=$ 0.01 pb (dot-dashed–green on-line–line)

5 Data analysis

The results, obtained by analysing in the framework of the Earth Shadow Effect the DAMA/LIBRA-phase1 (total exposure 1.04 ton $\times$ year) data, essentially depend on the most sensitive (2–4) keV interval; thus, this is the energy region considered here.

In the present analysis, as in Refs. [5, 32], three possibilities for the Na and I quenching factors have been considered: (Q$_I$) the quenching factors of Na and I “constants” with respect to the recoil energy $E_{R}$: $q_{Na}\simeq 0.3$ and $q_{I}\simeq 0.09$ as measured by DAMA with neutron source integrated over the 6.5–97 and the 22–330 keV recoil energy range, respectively [15]; (Q$_{II}$) the quenching factors evaluated as in Ref. [61] varying as a function of $E_R$; (Q$_{III}$) the quenching factors with the same behaviour of Ref. [61], but normalized in order to have their mean values consistent with Q$_{I}$ in the energy range considered there.

Another important effect is the channeling of low energy ions along axes and planes of the NaI(Tl) DAMA crystals. This effect can lead to an important deviation, in addition to the other uncertainties discussed above. In fact, the channeling effect in crystals implies that a fraction of nuclear recoils are channeled and experience much larger quenching factors than those derived from neutron calibration (see [5, 30] for a discussion of these aspects). Since the channeling effect cannot be generally pointed out with neutron measurements as already discussed in details in Ref. [30], only modeling has been produced up to now. In particular, the modeling of the channeling effect described by DAMA in Ref. [30] is able to reproduce the recoil spectrum measured at neutron beam by some other groups (see Ref. [30] for details). For completeness, we mention an alternative channeling model, as that of Ref. [62], where larger probabilities of the planar channeling are expected. Moreover, we mention the analytic calculation claiming that the channeling effect holds for recoils coming from outside a crystal and not from recoils produced inside it, due to the blocking effect [63]. Nevertheless, although some amount of blocking effect could be present, the precise description of the crystal lattice with dopant and trace contaminants is quite difficult and analytical calculations require some simplifications which can affect the result. Because of the difficulties of experimental measurements and of theoretical estimate of this channeling effect, in the following it will be either included or not in order to give idea on the related uncertainty.

Thus, the data analysis has been repeated in some discrete cases which allow us to account for the uncertainties on the quenching factors and on the parameters used in the nuclear form factors. The first case (set A) is obtained considering the mean values of the parameters of the used nuclear form factors (see above and Ref. [25]) and of the quenching factors. The set B adopts the same procedure as in Refs. [20, 21], by varying (i) the mean values of the measured $^{23}$Na and $^{127}$I quenching factors up to $+$2 times the errors; (ii) the nuclear radius, $r_i$, and the nuclear surface thickness parameter, s, in the form factor from their central values down to $-$20 %. In the last case (set C) the Iodine nucleus parameters are fixed at the values of case B, while for the Sodium nucleus one considers: (i) $^{23}$Na quenching factor at the lowest value measured in literature; (ii) the nuclear radius, $r_i$, and the nuclear surface thickness parameter, s, in the SI form factor from their central values up to $+$20 %. Finally, three values of $v_0$ have been considered: (i) the mean value: 220 km/s, and (ii) two extreme cases: 170 and 270 km/s.

Because of the large number of the needed simulations, the mass of the DM candidate and of the cross section on nucleon have been discretized as in the following: six $m_{DM}$ (5, 10, 30, 60, 100 and 150 GeV) and eight $\sigma _n$ (10, 5, 1, 0.5, 0.1, 0.05, 0.01 and 0.005 pb).

The expectations are compared with the experimental model-independent diurnal residual rate of the single-hit scintillation events, measured by DAMA/LIBRA-phase1 in the (2–4) keV energy interval, as function of the sidereal time (see in Fig. 2 of Ref. [12]). Two examples of expected signals are reported in Fig. 5. Here, the used sidereal time bin is 1 h (24 time bins in the sidereal day) and the experimental residuals are: $S_{d}^{exp}(t_i) \pm \sigma _{d,i}$. We compute the $\chi ^{2}$ quantity:

$$\begin{aligned} \chi ^2= & {} \displaystyle \sum _{i=1}^{24} \frac{\left( S_{d}^{exp}(t_i) - \xi \sigma _n S'_{d,sh,k}(m_{DM},\sigma _n,t_i) \right) ^2}{ \sigma _{d,i}^2} = \nonumber \\= & {} \chi _0^2 - 2 B \xi + A \xi ^2, \end{aligned}$$

(23)

where

$$\begin{aligned}&\chi _0^2 = \displaystyle \sum _{i=1}^{24} \frac{\left( S_{d}^{exp}(t_i) \right) ^2}{ \sigma _{d,i}^2} \\&B(m_{DM},\sigma _n) = \sigma _n \displaystyle \sum _{i=1}^{24} \frac{S_{d}^{exp}(t_i) \times S'_{d,sh,k}(m_{DM},\sigma _n,t_i)}{\sigma _{d,i}^2} \\&A(m_{DM},\sigma _n) = \sigma _n ^2 \displaystyle \sum _{i=1}^{24} \frac{ \left( S'_{d,sh,k}(m_{DM},\sigma _n,t_i) \right) ^2}{\sigma _{d,i}^2}. \end{aligned}$$

The $S'_{d,sh,k}(m_{DM},\sigma _n,t_i)$ have been evaluated for each set of parameters described above.

Fig. 5

Experimental model-independent diurnal residual rate of the single-hit scintillation events measured by DAMA/LIBRA-phase1 in the (2–4) keV energy interval (crosses) as function of the sidereal time [12], superimposed to two examples of expectations obtained by MC simulation. The template curves are obtained in the same scenario as in Fig. 4, considering $\xi \sigma _n = 1.1 \times 10^{-7}$ pb (compatible with the DAMA/LIBRA-phase1 DM annual modulation result): (i) the dashed (red on-line) line is obtained for $\sigma _n = 10$ pb (and, thus, $\xi =1.1 \times 10^{-8}$); (ii) the dotted (blue on-line) line is obtained for $\sigma _n = 0.1$ pb (and, thus, $\xi =1.1 \times 10^{-6}$). The latter is compatible with the absence of diurnal rate variation in DAMA/LIBRA-phase1, while the former is not

The $\chi ^2$ of Eq. 23 is function of only one parameter, $\xi$. Since the data do not show the presence of significant diurnal variation in the counting rate as already described in Ref. [12], only upper limits for $\xi$ are allowed, once given $m_{DM}$ and $\sigma _n$. We can define:

$$\begin{aligned} \Delta \chi ^{2}\{\xi \}=\chi ^{2}\{\xi \}-\chi ^{2}_0. \end{aligned}$$

The $\Delta \chi ^{2}$ is a $\chi ^{2}$ with one degree of freedom and is used to determine the upper limit of $\xi$ parameter at $2\sigma$ CL.

Two examples to describe the followed procedure are reported in Fig. 6, where the excluded regions (above the dotted lines) in the $\xi$ vs $\sigma _n$ plane for the cases of $m_{DM} =$ 10 and 60 GeV are shown as obtained on the basis of the Earth Shadow Effect in the given model framework.

Fig. 6

Examples of comparisons at $2\sigma$ CL between allowed regions from the DM annual modulation results of DAMA/LIBRA-phase1 (black continuous lines) and exclusion limits from the Earth Shadow Effect (dotted–red on-line–line). In both cases the quenching factors Q$_I$, including the channeling effect, and the set B of the parameters have been considered (see text) for $v_0 = 220$ km/s and for two DM masses. Left $m_{DM} = 10$ GeV, the upper limits on $\xi$ do not constrain the results of annual modulation. Right $m_{DM} = 60$ GeV, the upper limits on $\xi$ do exclude the band with $\sigma _n > 0.05$ pb and $\xi > 10^{-3}$ for the considered model framework. The combined allowed regions are reported as shaded–green on-line–area

The upper limits on $\xi$ can be compared with the positive results from the DM annual modulation signature achieved by DAMA.⁴ In particular, DAMA/LIBRA-phase1 reports an annual modulation amplitude in the (2–4) keV energy interval: $S_m^{exp} = (0.0167 \pm 0.0022)$ cpd/kg/keV, corresponding to 7.6$\sigma$ CL [4].

Here for each set of parameters described above, one can evaluate (see e.g. Eq. 22) the $\xi \sigma _n$ allowed values as:

$$\begin{aligned} \xi \sigma _n = \frac{S_m^{exp}}{S'_{m,(2-4) keV}(m_{DM},\sigma _n) }. \end{aligned}$$

(24)

This corresponds, once including the experimental uncertainties on $S_m^{exp}$, to a band in the $\xi$ vs $\sigma _n$ plane (within the continuous solid line). In Fig. 6 such bands at $2\sigma$ CL are reported. One can see that for the scenario considered there and for $m_{DM}=$ 10 GeV the upper limits on $\xi$ do not constrain the results of the DM annual modulation. On the contrary, for $m_{DM}=$ 60 GeV the upper limits on $\xi$ do exclude the band with $\sigma _n > 0.05$ pb and $\xi > 10^{-3}$. The shaded bands in Fig. 6 corresponds to the allowed regions in the $\xi$ vs $\sigma _n$ plane, for the given $m_{DM}$, from the combined analyses of the DM annual modulation result and of the Earth Shadow Effect in the considered framework.

Finally, for each considered set of parameters the three-dimensional allowed region – calculated as described above – in the parameter’s space: $\xi$, $\sigma _n$, $m_{DM}$, is depicted as a surface in Figs. 7, 8 and 9, for $v_0$ equal to 170, 220, 270 km/s, respectively. We note that the “thickness” of the allowed regions around the shown surfaces is $\le$ $\pm 30 \,\%$; therefore, for simplicity it is not represented in these figures.

Fig. 7

Examples of the mean values of the allowed region of $\xi$ as function of $\sigma _n$ and $m_{DM}$, represented as an allowed surface (see text). The plots have been obtained for $v_0 = 170$ km/s in the following scenarios: a the quenching factors Q$_I$, without channeling and marginalizing over the parameters sets A, B, C; b as in case a including the channeling effect; c the quenching factors Q$_{II}$; d the quenching factors Q$_{III}$. We note that the “thickness” of the allowed regions around the surfaces is $\le$ $\pm 30 \,\%$; therefore, for simplicity it is not represented in these figures. Finally, we recall that other uncertainties not considered in the present paper are present and can extend the result

Fig. 8

Examples of the mean values of the allowed region of $\xi$ as function of $\sigma _n$ and $m_{DM}$, represented as an allowed surface (see text). The plots have been obtained for $v_0 = 220$ km/s in the same sets of parameters as in Fig. 7. See text

Fig. 9

Examples of the mean values of the allowed region of $\xi$ as function of $\sigma _n$ and $m_{DM}$, represented as an allowed surface (see text). The plots have been obtained for $v_0 = 270$ km/s in the same sets of parameters as in Fig. 7. See text

Finally, we recall that other uncertainties not considered in the present paper are present. For example, including other possible halo models sizeable differences are expected in the results as shown e.g. in Refs. [5, 22].

6 Conclusions

The Earth Shadow Effect has been investigated in a given framework considering the model independent results on possible diurnal variation of the low-energy rate of the single-hit scintillation events in the DAMA/LIBRA-phase1 data (exposure: 1.04 ton $\times$ year) reported in Ref. [12]. For the considered DM candidates having high interaction cross-sections and very small halo fraction the obtained results constrain at 2$\sigma$ CL, in the considered scenario, the $\xi$, $\sigma _n$ and $m_{DM}$ parameters (see Figs. 7, 8, 9) when including the positive results from the DM annual modulation analysis of the DAMA/LIBRA-phase1 data [4]. For example, in the considered scenario for quenching factors Q$_I$ with channeling effect, B parameters set, $v_0 = 220$ km/s and $m_{DM} = 60$ GeV, the obtained upper limits on $\xi$ do exclude $\sigma _n > 0.05$ pb and $\xi > 10^{-3}$. When also including other uncertainties as other halo models etc. the results would be extended.