Investigating Earth shadowing effect with DAMA/LIBRAphase1
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Abstract
In the present paper the results obtained in the investigation of possible diurnal effects for lowenergy singlehit scintillation events of DAMA/LIBRAphase1 (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 crosssection 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.
Keywords
Dark Matter Dark Matter Candidate Dark Matter Particle Galactic Halo Sidereal Time1 Introduction
The present DAMA/LIBRA experiment [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], as the former DAMA/NaI [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43] has the main aim to investigate the presence of dark matter (DM) particles in the galactic halo by exploiting the modelindependent 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 lowenergy singlehit scintillation events collected by DAMA/LIBRAphase1 (1.04 ton \(\times \) year exposure) have been released and analysed in terms of a DM second order modelindependent 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 setup. It is worth noting that a similar effect can be pointed out only for candidates with high crosssection 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, 49, 50, 51, 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 crosssection with ordinary matter (i.e. present in the galactic halo with small abundance).
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 crosssection 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
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]
Layer (L)  R\(_{min}\)–R\(_{max}\) (km)  Mass percentage (\(\delta _i\))  Density (\(\rho _L\)) (kg/m\(^3\)) 

Inner Core  0–1221.5  Fe (79 %)  12839 
Ni (21 %)  
External Core  1221.5–3480  Fe (86 %)  10901 
S (12 %)  
Ni (2 %)  
Mantle  3480–6371  O (44.9 %)  4605 
Si (21.6 %)  
Mg (22.8 %)  
Fe (5.8 %)  
Ca (2.3 %)  
Al (2.2 %)  
Na (0.4 %) 
3.2 Interactions of the considered DM candidates and path reconstruction

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 stepbystep 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 \(\sum _k^{n_{hit}} x_k \le d_L\).
4 The expected interaction rate
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/LIBRAphase1 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/LIBRAphase1 [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/LIBRAphase1 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.
5 Data analysis
The results, obtained by analysing in the framework of the Earth Shadow Effect the DAMA/LIBRAphase1 (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 upper limits on \(\xi \) can be compared with the positive results from the DM annual modulation signature achieved by DAMA.^{4} In particular, DAMA/LIBRAphase1 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].
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 lowenergy rate of the singlehit scintillation events in the DAMA/LIBRAphase1 data (exposure: 1.04 ton \(\times \) year) reported in Ref. [12]. For the considered DM candidates having high interaction crosssections 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/LIBRAphase1 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.
Footnotes
 1.
This equation holds in the limit where the form factor can be neglected. For high velocity, high mass DM candidate crossing the Inner Core it is only an approximation. For simplicity we do not consider further this issue here, while the form factor is considered in obtaining the energy loss (see later).
 2.
It is important to remark that the spin independent form factor depends on the target nucleus and there is not an universal formulation for it. Many profiles are available in literature and whatever profile needs some parameters whose value are also affected by some uncertainties. The form factor profiles can differ – in some intervals of the transferred momentum – by orders of magnitude and the chosen profile strongly affects whatever model dependent results [25].
 3.
Here only the first order terms are shown (i.e. the interference terms are omitted).
 4.
We recall that DAMA/LIBRA and the former DAMA/NaI have cumulatively reached a model independent evidence at 9.3\(\sigma \) CL for the presence of DM particles in the galactic halo on the basis of the exploited DM annual modulation signature [4].
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