Model-independent comparison of annual modulation and total rate with direct detection experiments
Felix Kahlhoefer Florian Reindl Karoline Schäffner Kai Schmidt-Hoberg Sebastian Wild

Optical conductivity of  overdoped cuprate superconductors: application to LSCO
    P. J. Hirschfeld
    December 30, 2023
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§ INTRODUCTION



When lamenting the absence of conclusive evidence for particle dark matter (or in fact any kind of new physics beyond the Standard Model), it is often forgotten that we have been observing for many years a statistically highly significant signal compatible with predictions for the scattering of dark matter (DM) particles from the Galactic halo. The combined data from the DAMA/NaI and the DAMA/LIBRA experiments show evidence for an annual modulation of the event rate at the 9.3σ level <cit.>, which agrees with the DM hypothesis in both period and phase.[One of the few unexpected features in the DAMA data is a slight time-dependence of the modulation amplitude <cit.>.] Even though the NaI detectors employed by the DAMA collaboration do not allow to distinguish nuclear and electron recoils, it has not been possible to identify a plausible background that could account for the observed signal while satisfying all cross-checks performed by the DAMA collaboration (see refs. <cit.> for recent discussions).

The great trouble with the DAMA signal is its incompatibility with exclusion limits obtained from other direct detection experiments. For the simplest assumptions, i.e. spin-independent or spin-dependent interactions between DM and nuclei, a number of experiments exclude the parameter region preferred by DAMA by several orders of magnitude <cit.>. But even more exotic hypotheses, such as leptophilic DM <cit.>, inelastic DM <cit.> or isospin-violating DM <cit.>, are strongly disfavoured by recent searches (see e.g. ref. <cit.>).

While it is very challenging to construct DM models that bring all experimental results into agreement (see refs. <cit.> for some recent attempts), it is even more difficult to generally exclude a DM explanation of the DAMA signal. The reason is that any comparison between different direct detection experiments requires a number of assumptions on both the astrophysical distribution and the particle physics properties of DM and the conclusions may depend sensitively on these assumptions. A number of halo-independent methods have been developed to deal with the former issue, allowing for a statistically meaningful comparison of different experiments without the need to assume a specific DM density or velocity distribution <cit.>.

The best way to reduce the need to make specific assumptions on the particle physics properties of DM would be an independent test of the DAMA result with NaI-based detectors. Indeed, a number of experimental collaborations are currently pursuing this strategy. Most of these, namely SABRE <cit.>, ANAIS <cit.>, PICO-LON <cit.> and COSINE <cit.> (the latter being a joint effort of the DM-Ice <cit.> and KIMS <cit.> collaborations), aim to search for an annual modulation in their event rates in much the same way as the original DAMA experiment. The COSINUS collaboration <cit.>, in contrast, has the more ambitious goal to develop a cryogenic NaI detector that can distinguish nuclear and electron recoils. This approach will allow to significantly suppress background levels, making it possible for the first time to test the DAMA signal using the absolute rate of nuclear recoils in NaI detectors. At the same time, the possibility to measure phonon signals will allow COSINUS to significantly lower the nuclear recoil energy threshold compared to DAMA, further increasing its sensitivity.

Nevertheless, the comparison of a total rate and a modulation amplitude appears to require specific assumptions on the DM velocity distribution, which determines the modulation fraction of the signal. Existing halo-independent methods deal with this issue by making very conservative assumptions on the modulation fraction, typically allowing modulation fractions up to 100%.[A notable exception are refs. <cit.>, which derive halo-independent constraints on the modulation amplitude from constraints on the total rate. While very similar in spirit, the difference between these works and ours is that we explicitly construct the velocity distribution that maximizes the modulation amplitude rather than deriving conservative upper bounds.] In the present work we show that for the case of several experiments employing the same detector material, significantly stronger consistency requirements can be made, leading to potentially much tighter constraints. Moreover, we will show that halo-independent methods can be generalised to consider not only arbitrary DM velocity distributions but also very general classes of interactions between DM and nuclei. This approach is therefore ideally suited for comparing future results from COSINUS with the DAMA signal.



A particular emphasis is placed on the question how to optimise the design of the COSINUS detector. As in all direct detection experiments, the detector development faces two competing goals: lowering the energy threshold and improving the bound on the event rate above threshold (e.g. by increasing the detector mass). We show that for a halo-independent test of the DAMA signal, it is highly desirable to achieve an energy threshold of about 1.8 keV (based on conservative assumptions on the quenching factors in DAMA and the energy resolution in COSINUS), while a further reduction of the threshold is less advantageous than e.g. an increase in exposure. These conclusions are illustrated in figure <ref>, which shows the bound on the total event rate that COSINUS must achieve in order to exclude the DAMA signal in a halo-independent way as a function of the COSINUS threshold. The different lines correspond to different assumptions on the interactions of DM, which will be explained in the following chapters.

This paper is structured as follows: In section <ref> we review the DAMA experiment and the COSINUS proposal, and summarise the assumptions made for our specific implementation of these experiments. In section <ref> we present the simplest – but also most conservative – way to compare the two experiments by comparing the respective efficiencies. Section <ref> discusses a more constraining approach, valid for rather general classes of recoil spectra. We compare the results obtained from this approach with the ones obtained for more specific assumptions on the interactions of DM in section <ref>. Our conclusions are presented in section <ref>, while additional technical details can be found in appendix <ref>.




§ DIRECT DETECTION WITH DAMA AND COSINUS



To introduce the relevant notation and conventions, let us briefly review the formalism for DM direct detection. The differential event rate with respect to recoil energy is given by

    dR^T/dE_R = ξ_Tρ/m_T  m_χ∫_v_min^∞ v f(v + v_E(t)) dσ^T/d E_Rd^3 v  ,

where m_χ and m_T are the mass of the DM particle and the target nucleus T, ξ_T is the mass fraction of T relative to the total detector mass,  ρ and f(𝐯) are the local DM density and velocity distribution in the rest frame of the Sun, v_E(t) is the velocity of the Earth and v=|v|.[It is conventional to define both f(𝐯) and v_E(t) in the Galactic rest frame, where the velocity distribution often takes a very simple form. However, in the context of halo-independent methods it will be more convenient to specify f(𝐯) in the rest frame of the Sun.] In order to transfer an energy E_R in an elastic collision, the incoming DM particle must have a minimum velocity of

    v_min(E_R) = √(m_T E_R/2  μ_Tχ^2) ,

where μ_Tχ = m_T  m_χ / (m_T + m_χ) is the reduced mass of the DM-nucleus system.

For reasons that will become apparent below, we will focus on cut-and-count experiments considering a single signal region in energy, i.e. disregarding the spectral information of the signal. Such an experiment can be fully characterised by the total exposure ℰ and the efficiency functions ϵ^T(E_R) that quantify the probability for a nuclear recoil of the target nucleus T with energy E_R to lead to an event in the signal region. In terms of these quantities, the number of predicted events is given by

    N = ℰ∑_T∫dR^T/dE_R ϵ^T(E_R)  dE_R  .

The total event rate in the signal region is given by

    R = N/ℰ =  ∑_T∫dR^T/dE_R ϵ^T(E_R)  dE_R  .

A null result allows an experiment to place an upper bound on N for a given exposure or, equivalently, an upper bound on R. For example, in the case of zero observed events, it is possible to exclude N > 2.71 at 90% confidence level (CL), leading to the upper bound R < 2.71 / ℰ.

In general, both N and R depend on time because of the time-dependence of 𝐯_E(t). If the DM velocity distribution is isotropic in the Galactic rest frame, one expects an approximately sinusoidal annual modulation that peaks around the 1st of June <cit.>. This prediction is in good agreement with the time-dependence of the DAMA signal. We therefore define the mean rate[Note that for approximately sinusoidal modulation with period of one year, our definition of R̅ agrees with the time-averaged rate ⟨ R ⟩ = 1t∫ R(t')  dt'.]

    R̅ = 1/2( R(t = 1st of June) + R(t = 1st of December) )   ,

and the modulation amplitude

    S = 1/2( R(t = 1st of June)  - R(t = 1st of December) )   .

The results of the DAMA experiment can then be summarised in terms of the modulation amplitude observed in different energy bins. The greatest evidence for an annual modulation is seen in the energy range E^obs_R∈[ 2.5 keVee, 3.5 keVee], where E^obs_R denotes observed recoil energy after quenching (see below). In this energy range the combined data from DAMA/NaI and DAMA/LIBRA imply <cit.>

    S = (2.34 ± 0.28)· 10^-2 kg^-1 days^-1  .

In the following we will focus on this energy range and study whether COSINUS can test the hypothesis that the signal observed there is due to interactions between DM and nuclei. If COSINUS can exclude this hypothesis, it will be impossible to find a DM model that provides a satisfactory explanation of the DAMA signal. We have checked explicitly that our conclusions do not depend sensitively on the energy range that we consider.


In order to predict the modulation amplitude in the DAMA signal region for a given differential event rate, one must first of all convert the physical recoil energy E_R into a quenched energy E'_R = Q   E_R, where Q denotes the quenching factor.[To distinguish between physical recoil energy and reconstructed (quenched) recoil energy, we use the unit keVee (ee = electron equivalent) when referring to the latter.] For the present analysis we adopt the values Q = 0.3 and 0.09 for sodium and iodine scatters, respectively <cit.>. Some recent measurements indicate even smaller quenching factors <cit.>, which would further strengthen the sensitivity of COSINUS with respect to the DAMA data as the COSINUS detection efficiency is a monotonically increasing function of energy. To calculate the efficiency functions ϵ^T(E_R), we follow refs. <cit.> and assume a Gaussian energy resolution with σ(E'_R) = (0.448 keVee) √(E'_R/keVee) + 0.0091 E'_R. We only consider fluctuations of up to four standard deviations, in order to avoid an unphysical extrapolation.



In contrast to DAMA, COSINUS aims to use NaI within a cryogenic detector, simultaneously acquiring a phonon (heat) and a light signal from a particle interaction. The phonon channel yields a precise and unquenched, thus, particle-independent measurement of the energy deposited in the crystal; the scintillation light provides particle discrimination on an event-by-event basis.  This technology has been shown to be extraordinarily sensitive and
allowed the CRESST collaboration to lead the field of direct dark matter detection on the low DM mass frontier <cit.>.
In the past two years COSINUS successfully proved for the first time that cryogenic operation of NaI is possible in ref. <cit.>, leading to the commissioning of the final detector design <cit.>.[In parallel to detector optimisation the COSINUS collaboration is also preparing measurements of the quenching factors for scatterings off sodium and iodine at cryogenic temperatures by irradiating a COSINUS detector to neutrons from AmBe sources and/or a neutron beam <cit.>.] The light channel already exceeds the final performance goals while further improvements in the phonon channel are necessary in order to further decrease the threshold. 

Our implementation of the COSINUS experiment largely follows ref. <cit.>. We assume that physical recoil energy E_R and observed recoil energy E_R^obs are related via a Gaussian energy resolution with σ_E = 0.2  keV, where once again we only take into account fluctuations of up to 4 σ_E. The probability for detecting a nuclear recoil with observed energy E_R^obs is given by 

    ϵ_det(E_R^obs) = Θ(E_R^obs - E_thres) × 0     E_R^obs < 1 keV ,
    
    0.3 ·E_R^obs/keV - 0.1     1 keV≤ E_R^obs≤ 2 keV ,
    
    0.5      E_R^obs > 2  keV ,

where Θ(x) is the Heaviside step function and E_thres denotes the low-energy threshold of the signal region considered by COSINUS. Unless explicitly stated otherwise, we adopt the value E_thres = 1 keV.

A crucial part of the COSINUS strategy to reduce backgrounds is to reject events with large light yield, as these are much more likely to arise from electron recoils than from nuclear recoils. The signal region is therefore restricted to events with a light yield below the mean of the sodium nuclear recoil band (see figure 2 in ref. <cit.>). The final efficiency function is then given by ϵ(E_R^obs) = ϵ_det(E_R^obs) ×ϵ_ly(E_R^obs), where the latter factor quantifies the probability that an event with reconstructed energy E_R^obs has sufficiently low light yield to fall into the signal region. For sodium recoils, one has by construction ϵ_ly = 0.5, whereas a fit of the iodine nuclear recoil band gives

    ϵ_ly(E_R^obs) ≈ 1 - 0.42 exp(-0.89 E_R^obs/keV)

for iodine recoils.

It is important to emphasise that the ability of the COSINUS detector to discriminate between electron recoils and nuclear recoils is reduced at low energies. In other words, a lowering of E_thres may lead to an increase in the number of observed background events and hence to a weaker bound on R. One of the central questions of the subsequent sections will be to determine under which conditions it will be more desirable for COSINUS to lower E_thres or to strive for a more stringent bound on R.






To conclude this section, we compare in figure <ref> the projected sensitivity of the COSINUS experiment with the parameter region favoured by DAMA for standard assumptions on the properties of DM <cit.>. Specifically we assume spin-independent interactions with

    dσ^T/d E_R = A^2   F^2(E_R) m_T σ_p/2  μ_pχ^2   v^2 ,

where A and F(E_R) denote the mass number and form factor of the target nucleus, and σ_p and μ_pχ denote the DM-proton scattering cross section and reduced mass, respectively. We set ρ = 0.3 GeV   cm^-3 and adopt a Maxwell-Boltzmann distribution with 
most probable speed v_0 = 220 km s^-1 and a cut-off at v_esc = 544 km s^-1 for f(v) in the Galactic rest frame. To obtain the corresponding velocity distribution in the rest frame of the Sun, we take the velocity of the Sun in the Galactic rest frame to be v_sun = 230 km s^-1.

We assume that COSINUS can set an upper bound of R < 0.1  kg^-1  days^-1 (corresponding e.g. to a 95% CL upper limit for an exposure of 105 kg   days and 5 observed events) and illustrate the impact of changing the low-energy threshold by considering three different values of E_thres. Our estimated sensitivity for E_thres = 1keV agrees quite well with the one from ref. <cit.>, which makes use of the optimum interval method <cit.> to derive constraints. Since we implement only a single energy range for DAMA, we do not find two separate signal regions (corresponding respectively to scattering dominantly off sodium and dominantly off iodine) but instead obtain a band of allowed parameters. In particular, our implementation of DAMA is conservative, because it leads to a significantly larger allowed parameter region.

The central conclusion from figure <ref> is that for the assumptions of spin-independent interactions and a Maxwell-Boltzmann velocity distribution COSINUS should be able to exclude the DAMA region by about two orders of magnitude in cross section. However, as pointed out in section <ref>, the DAMA hypothesis is already robustly excluded for standard assumptions by a number of experiments. We will therefore now turn to the much more interesting question how the sensitivity of COSINUS compares to DAMA for less restrictive assumptions.



§ HALO- AND MODEL-INDEPENDENT COMPARISON


We begin by discussing the most general – and hence most conservative – way of comparing DAMA and COSINUS. This approach is based on the very simple observation that the modulation amplitude in a given experiment cannot exceed the mean rate <cit.>:

    S ≤R̅  ,

which directly follows from eqs. (<ref>) and (<ref>). We now define S_DAMA^bound≡ 1.78 · 10^-2 kg^-1 days^-1 as the smallest value of the modulation amplitude compatible with the DAMA measurement at the 95% CL, see eq. (<ref>),  and R_COSINUS^bound as the 95% CL upper bound on the mean total event rate in COSINUS. Using eq. (<ref>), the modulation amplitude S_DAMA in DAMA must then satisfy

    S_DAMA^bound≤ S_DAMA≤R̅_DAMA = ∑_T∫dR̅^T/dE_R ϵ_DAMA^T(E_R)  dE_R ,

where we define dR̅^T/dE_R analogously to eq. (<ref>) as the mean differential recoil rate during the year. On the other hand, the upper bound from COSINUS implies

    R_COSINUS^bound≥R̅_COSINUS = ∑_T∫dR̅^T/dE_R ϵ_COSINUS^T(E_R)  dE_R .

These two equations can be combined to give

    ∑_T∫dR̅^T/dE_Rϵ_DAMA^T(E_R)/S_DAMA^bound dE_R≥∑_T∫dR̅^T/dE_Rϵ_COSINUS^T(E_R)/R_COSINUS^bound dE_R  .


Crucially, the mean differential event rate dR̅^T/dE_R is the same for both DAMA and COSINUS, irrespective of the type of interaction, because they employ exactly the same detector material. This means that eq. (<ref>) can only hold if ϵ_DAMA^T (E_R) / S_DAMA^bound > ϵ_COSINUS^T(E_R)/R_COSINUS^bound for at least one target nucleus T and some value of E_R. Consequently, the COSINUS measurement excludes an explanation of the DAMA signal in terms of DM-nucleus scattering for arbitrary differential event rates at more than 95% CL if

    ϵ_COSINUS^T(E_R)R_COSINUS^bound > ϵ_DAMA^T(E_R)S_DAMA^bound   for all  E_R and all T  .




In figure <ref> we show a comparison of the left-hand side and right-hand side of eq. (<ref>) for different assumptions on the value of R_COSINUS^bound that can be achieved. The left panel corresponds to scattering off sodium, the right to scattering off iodine. As expected, the DAMA sensitivity peaks around E_R≈ 10 keV for sodium and around E_R≈ 33  keV for iodine, both of which correspond to E'_R≈ 3 keVee when taking into account the appropriate quenching factors. The sensitivity of COSINUS, on the other hand, covers a much wider range of energies and in particular extends to significantly lower values of E_R. In order to exclude the DAMA signal for sodium scatters, COSINUS thus needs to place a bound on the total event rate of R_COSINUS^bound≈ 10^-2 kg^-1 days^-1. A slightly weaker bound is sufficient for the case of iodine scatters, because the corresponding efficiency function in COSINUS is slightly larger. Since it is however not known what fraction of the DAMA signal is due to iodine scatters, a fully model-independent exclusion is only possible if COSINUS can exclude the DAMA signal for the case of scattering exclusively on sodium.

Eq. (<ref>) constitutes the most general method for comparing DAMA and COSINUS (or more generally any experiment observing an annual modulation with another experiment constraining the total rate for the same detector material). This approach is valid as long as the signal in both experiments is due to nuclear recoils. In particular it applies to all elastic and inelastic DM-nucleus scattering processes and to arbitrary velocity distributions. The sensitivity that COSINUS must achieve is however quite ambitious, about an order of magnitude beyond what we assumed for figure <ref>. We will therefore now explore whether stronger constraints can be obtained for more restrictive assumptions on the differential event rate.



§ HALO-INDEPENDENT COMPARISON FOR CLASSES OF CROSS SECTIONS



Let us begin by considering a fixed DM mass m_χ and a differential cross section with fixed shape but free normalisation, which is controlled by a parameter σ.[For the moment we assume that the differential event rate is dominated by either scattering off sodium or scattering off iodine. Below we will generalise to the case where both elements contribute at comparable level.] For a given DM velocity distribution f(𝐯) in the rest frame of the Sun, we define R^f_COSINUS(σ) and S^f_DAMA(σ) as the predicted rate and modulation amplitude in COSINUS and DAMA, respectively. With these conventions we define

    S_DAMA^max≡max_f [ S_DAMA^f(σ)  with σ given by  R_COSINUS^f (σ)  = R_COSINUS^bound]   .

In other words, S_DAMA^max is the largest modulation amplitude compatible with the COSINUS bound that can be obtained in DAMA from the assumed cross section for any DM velocity distribution. If S_DAMA^max < S_DAMA^bound, we can therefore conclude that COSINUS excludes DAMA for the assumed cross section in a halo-independent way.

At first sight, calculating S_DAMA^max seems rather challenging, given that it requires varying the three-dimensional function f(𝐯). To simplify the problem, we make the important observation that eq. (<ref>) is linear in f(𝐯), meaning that we can write

    R^f_COSINUS(σ)     = ∫d^3   v_0   f(𝐯_0)  ℛ_COSINUS(𝐯_0, σ)   .

Here ℛ_COSINUS(𝐯_0, σ) is the rate calculated from the assumed differential cross section for a velocity distribution of the form f(𝐯) = δ^3(𝐯 - 𝐯_0), corresponding to a DM stream with a single velocity 𝐯_0 <cit.>. In a fully analogous way, S^f_DAMA can be expressed as a superposition of functions 𝒮_DAMA(𝐯_0, σ).

As shown explicitly in appendix <ref>, it turns out that in order to calculate S_DAMA^max, it is sufficient to consider velocity distributions of the form f(𝐯) = δ^3(𝐯 - 𝐯_0). In other words, the optimum velocity distribution that maximises S_DAMA^f always corresponds to a single DM stream:[Note that for very small DM masses the optimal velocity can be significantly larger than the values typically assumed for the Galactic escape velocity v_esc. Restricting velocities to be smaller than v_esc would lead to even stronger bounds.]

    S_DAMA^max≡max_𝐯_0[ 𝒮_DAMA(𝐯_0, σ)  with σ given by ℛ_COSINUS(𝐯_0, σ) = R_COSINUS^bound]   .

At this point it is essential that we consider only a single energy range for both DAMA and COSINUS. The conclusion that a single DM stream is optimal no longer holds for experiments with several signal regions or experiments performing an unbinned analysis <cit.>. In fact, the problem can be simplified further, because ℛ_COSINUS(𝐯_0, σ) depends only on v_0 = |𝐯_0|, while 𝒮_DAMA(𝐯_0, σ) is maximized for streams that are anti-aligned with the velocity of the Earth on the 1st of June, given by 𝐯_E≡ v_E 𝐞_summer with v_E = 30 km s^-1. It is therefore sufficient to consider only streams with 𝐯_0 = v_0  𝐞_summer.[We note that gravitational focusing may modify the phase of the modulation signal <cit.>, such that streams of a slightly different direction are needed to give the best fit to the DAMA signal.]

We conclude that, for a given cross section, COSINUS excludes DAMA in a halo-independent way if

    𝒮_DAMA(v_0, σ)/S_DAMA^bound <  ℛ_COSINUS(v_0, σ)/R_COSINUS^bound

for all v_0. Note that, since the parameter σ leads to the same rescaling on both sides, it is sufficient to check eq. (<ref>) for one specific value of σ. The quantities ℛ_COSINUS and 𝒮_DAMA are easily calculated, allowing to determine very quickly whether DAMA and COSINUS can be compatible for a specific particle physics hypothesis. Alternatively, eq. (<ref>) can be rearranged to

    R_COSINUS^bound < min_v_0[ ℛ_COSINUS(v_0)/𝒮_DAMA(v_0)] S_DAMA^bound  ,

i.e. we can calculate the bound on the total rate that COSINUS must achieve in order to exclude a specific hypothesis in a halo-independent way. We shall refer to this number as the exclusion power of COSINUS.

The aim of this section is however to compare COSINUS and DAMA for whole classes of cross sections at once. The requirement derived above is insufficient for this purpose, as ℛ_COSINUS and 𝒮_DAMA need to be recalculated as soon as the form of the differential cross section changes. However, we can make use of the same trick used to consider arbitrary velocity distributions to also consider broad classes of cross sections. Let us therefore assume that the differential cross sections of interest can be written as a linear combination of a set of basis functions

    dσ^T/dE_R = ∑_i α^T_i dσ_i^T/dE_R  ,

where T = Na,I denotes the two different target elements and we require α^T_i ≥ 0. For each of these basis functions we can then calculate the total rate in COSINUS and the modulation amplitude in DAMA, called ℛ^T_i,COSINUS and 𝒮^T_i,DAMA, respectively. Note in particular that this calculation needs to be done separately for the two target elements. Following the same reasoning as above, one can immediately show that COSINUS excludes DAMA for any cross section of the assumed form and any DM velocity distribution if

    𝒮^T_i,DAMA(v_0)/S_DAMA^bound <  ℛ^T_i,COSINUS(v_0)/R_COSINUS^bound

for all v_0, i and T. Clearly, this approach requires more calculational effort, but potentially allows to make very general statements.




 §.§ Monotonically decreasing cross sections


Let us now turn to a particularly interesting example and consider the class of models for which the differential cross section is a monotonically decreasing function of recoil energy. More specifically, we assume that the differential cross section can be written as

    dσ^T/dE_R = σ_0/m_T v^2 κ^T(E_R)  Θ(E^T_max(v) - E_R)

with an arbitrary reference cross section σ_0 and dimensionless monotonically decreasing functions κ^T(E_R). In this expression the factor 1/v^2 corresponds to the velocity dependence obtained in the usual case that the matrix element for the scattering does not depend on velocity and the final factor (with E_max^T(v)  = 2 μ_T^2 v^2/m_T) enforces energy and momentum conservation for elastic scattering. Examples for cross sections that can be written in this way are spin-independent and spin-dependent scattering (see section <ref>).[Strictly speaking, the relevant form factors only decrease monotonically for sufficiently small momentum transfer, but this covers the energy range relevant for COSINUS and DAMA.] Now we make use of the fact that any monotonically decreasing function can be written as a (potentially infinite) sum of step functions:

    κ^T(E_R) = ∫_0^∞α^T(E_0)  Θ(E_0 - E_R) dE_0  .

We can therefore write any differential cross section of the form given in eq. (<ref>) as

    dσ^T/dE_R = ∫_0^∞α^T(E_0) dσ^T/dE_R(E_0)  dE_0

in terms of the basis functions

    dσ^T/dE_R(E_0) = σ_0/m_T v^2 Θ(E_0 - E_R)  Θ(E^T_max(v) - E_R)   .

Here the continuous parameter α^T(E_0) takes the place of the discrete parameter α^T_i introduced above.

In complete analogy to the previous results, we conclude that COSINUS excludes DAMA for any monotonically decreasing differential cross section and any DM velocity distribution if

    𝒮^T_DAMA(v_0, E_0)/S_DAMA^bound < ℛ^T_COSINUS(v_0, E_0)/R_COSINUS^bound

for all v_0, E_0 and T. The great virtue of this expression is that ℛ^T(v_0, E_0) and 𝒮^T(v_0, E_0) are in fact very easily calculated. Resubstituting the various definitions, we obtain the quite simple expressions

    ℛ^T(v_0, E_0) = ξ_T ρ σ_0/m_T^2   m_χ  v_0∫ϵ^T(E_R)  Θ(E_0 - E_R)  Θ(v_0 - v^T_min(E_R))  dE_R

and

    𝒮^T(v_0, E_0) ≡1/2[ ℛ^T(v_0^+, E_0) - ℛ^T(v_0^-, E_0) ]

with

    v_0^+ = v_0 + v_E  ,      v_0^- = |v_0 - v_E|   .

Note that ℛ^T(v_0, E_0) and 𝒮^T(v_0, E_0) depend on the DM mass via v^T_min(E_R) and hence eq. (<ref>) must be checked separately for each value of m_χ. The local DM density ρ as well as the reference cross section σ_0, on the other hand, cancel out in the condition (<ref>).

Clearly, for many values of v_0 and E_0 the modulation amplitude in DAMA will be zero or extremely small, either because the corresponding recoil energies are too small to be observable or because they are so large that DAMA is predicted to observe essentially the same rate in summer and winter. A large modulation fraction is obtained only if v_0 and E_0 are such that typical nuclear recoils are mostly below threshold in winter but slightly above threshold in summer. The configuration that maximises the modulation amplitude in DAMA relative to the total rate in COSINUS (for scattering off sodium and a DM mass of m_χ = 10GeV) is illustrated in the left panel of figure <ref>. The dashed (dotted) line indicates the differential event rate in summer (winter). Note that for small recoil energies the rate in winter is larger than in summer by a factor v_0^+ / v_0^-, see eq. (<ref>).



The blue shaded region in the left panel of figure <ref> corresponds to the energy window probed by DAMA in the absence of fluctuations. For perfect energy resolution, one would therefore expect a modulation fraction in DAMA of 100%. The fact that the energy resolution in DAMA is however not perfect (and indeed pretty poor) means that some fraction of events will fluctuate into the signal region even in winter, so that the modulation fraction ends up being only about 46%. To achieve larger modulation fractions, one would have to decrease v_0 further below the DAMA threshold. At this point it becomes crucial that COSINUS has a much lower threshold than DAMA, because reducing v_0 further would reduce the modulation amplitude in DAMA more rapidly than the total rate predicted for COSINUS, leading to a less than optimal configuration. 

The fact that the optimum choice of v_0 and E_0 for monotonically decreasing cross sections does not correspond to 100% modulation fraction in DAMA means that we expect COSINUS to have more exclusion power than in the fully model-independent case discussed in section <ref>. In other words, weaker bounds on the total rate in COSINUS will be sufficient to exclude the DAMA signal. The values of R_COSINUS^bound that COSINUS must achieve for this purpose are shown in the right panel of figure <ref> as a function of m_χ. Blue (red) solid lines correspond to scattering off sodium (iodine). The dashed lines show for comparison the values of R_COSINUS^bound required for a fully model-independent exclusion (see figure <ref>).

We observe that the assumption of a monotonically decreasing cross section enhances the exclusion power of COSINUS by at least a factor of about three, meaning that three times weaker bounds on the rate (or three times less exposure) are sufficient to exclude the DAMA signal. For sufficiently small DM masses, the exclusion power becomes even greater owing to a reduced modulation fraction in DAMA. The nominal sensitivity of COSINUS R_COSINUS^bound = 0.1 kg^-1 days^-1 will be sufficient to exclude the DAMA signal for m_χ < 6  GeV for scattering off sodium and for m_χ < 90 GeV for scattering off iodine.

We emphasise that our approach is only suitable for addressing the question whether COSINUS can exclude the DAMA signal. If there is a combination of v_0 and E_0 such that

    𝒮^T_DAMA(v_0, E_0)/S_DAMA^bound > ℛ^T_COSINUS(v_0, E_0)/R_COSINUS^bound

this does not necessarily imply that a good fit to the DAMA data can be obtained. The reason is that we consider only the modulation amplitude from DAMA in one specific bin and do not include the full information of the energy and time dependence of the DAMA signal.



Nevertheless, we make the interesting observation that the step function shown in the left panel of figure <ref> actually gives a surprisingly good fit to the full DAMA data. The reason is once again that the poor energy resolution of the DAMA detector washes out the sharp feature in the differential event rate. This is illustrated in figure <ref>, which shows the differential modulation amplitude (with respect to observed energy E_R^obs) as well as the time dependence of the modulation amplitude integrated over the energy interval E_R^obs∈[2 keVee,6 keVee] in comparison to the actual data points from DAMA. In fact, the quality of the fit is comparable to the one obtained for standard assumptions on the DM interactions and velocity distribution. This leads to the reassuring conclusion that the approach of decomposing the velocity distribution and the differential cross section into sets of basis functions is not exceedingly conservative.



 §.§ Momentum-dependent cross sections


To conclude this section, let us consider possible alternatives to the assumption of monotonically decreasing cross sections. In principle, one could of course consider arbitrary cross sections, i.e. one could decompose the differential cross section into δ-functions rather than step functions. It is easy to see, however, that in this case one can always arrange for 100% modulation fraction in DAMA, so that one would recover the model-independent bounds obtained in section <ref>.

A more interesting assumption is that the differential cross section can grow no faster than some power n of the recoil energy. In other words, we assume that the differential cross section can be written as

    dσ^T/dE_R = (E_R/m_T)^n σ_0/m_T v^2 κ^T(E_R)  Θ(E^T_max(v) - E_R)

with monotonically decreasing functions κ^T(E_R). For n = 0 this corresponds to the case considered above, but for n > 0 one obtains less restrictive assumptions. As we will see in section <ref>, this enables us to consider certain types of DM interactions that are not captured by the assumption of monotonically decreasing cross sections. The corresponding basis functions are given by

    dσ^T_n/dE_R(E_0) = (E_R/m_T)^n σ_0/m_T v^2 Θ(E_0 - E_R)  Θ(E^T_max(v) - E_R)   .




The values of R_COSINUS^bound required to exclude DAMA for different values of n are shown in figure <ref>. As expected, larger values of n, corresponding to less restrictive assumptions on the differential cross section, imply weaker exclusion power from COSINUS, meaning that stronger bounds on the rate (or larger exposure) are needed to exclude the DAMA signal. In fact, there is also a more intuitive way to understand figure <ref>: As discussed above, much of the exclusion power of COSINUS comes from the fact that it has a lower threshold than DAMA. If the differential cross section grows with energy, the benefit of the lower threshold, and hence the exclusion power of COSINUS, is reduced. 




§ HALO-INDEPENDENT COMPARISON FOR SPECIFIC INTERACTIONS



In this section we compare the halo-independent bounds for different classes of recoil spectra derived above to the halo-independent bounds obtained for specific models that fall into these general categories. Specifically, we will consider spin-independent and spin-dependent scattering as examples for models with monotonically falling recoil spectra and CP-odd interactions as examples for models with momentum-dependent scattering. Towards the end of this section we will discuss inelastic DM as an example of a model that does not fall into one of the categories discussed in section <ref>.

The crucial difference between the more general treatment above and the more specific treatment in this section is that by considering a specific type of interaction we impose certain relations between the scattering rates for different elements. Doing so implies in particular that DAMA faces overwhelming constraints from other direct detection experiments. There is consequently less motivation for performing a comparison between DAMA and COSINUS in the context of these models. Such a comparison is nevertheless instructive in order to explore how the general bounds derived above compare to the bounds obtained within specific models.

As we will see below, as long as the event rate in both DAMA and COSINUS is dominated by the scattering off the same target element, the general bounds derived above are only slightly weaker than the ones obtained within a specific model. However, much stronger exclusion limits are obtained in those regions of parameter space where scattering in DAMA is dominated by a single target element, while both sodium and iodine contribute to the event rate in COSINUS. In this case the performance of COSINUS turns out to strongly depend on its low-energy threshold.



 §.§ Spin-independent scattering


We begin by discussing in detail the case of spin-independent scattering, which was also briefly considered in section <ref> (see eq. (<ref>) and figure <ref>). Since the corresponding differential cross section is of the form of eq. (<ref>), we can immediately read off a conservative estimate of the exclusion power of COSINUS from figure <ref>. A stronger bound can however be obtained by explicitly calculating ℛ_COSINUS(v_0) and 𝒮_DAMA(v_0) for stream-like velocity distributions and making use of eq. (<ref>).



Figure <ref> shows the recoil spectrum (in terms of the true nuclear recoil energy E_R) for the optimal value of v_0, i.e. the one that imposes the strongest requirement on R_COSINUS^bound. The two different rows (columns) correspond to different values of m_χ (E_thres). As in the left panel of figure <ref> the dashed (dotted) curves show the rates in summer (winter). The crucial difference to figure <ref> is that for each case there are now two curves, corresponding to sodium (blue) and iodine (red).

We make a number of pertinent observations: First of all, we note that the rate from scattering off iodine is several orders of magnitude larger than the rate from scattering off sodium, owing to the well-known coherent enhancement for heavy target nuclei. Furthermore, as a result of the form factor suppression, the differential event rates are no longer constant below the cut-off. Finally, we note that for m_χ = 100 GeV the differential event rate for iodine extends to larger recoil energies, while for m_χ = 5 GeV the situation is reversed, due to the different ratios of DM particle mass to target nucleus mass.

For m_χ = 100 GeV the situation is clear: Both the total rate and the modulation amplitude are completely dominated by scattering off iodine, and the contribution from sodium can effectively be neglected.[We note that scenarios in which the modulation amplitude in DAMA is dominated by scattering off iodine are already constrained by existing direct detection experiments based on target nuclei also involving iodine, such as COUPP <cit.>, KIMS <cit.> and PICO-60 <cit.>.] The only difference between spin-independent scattering and the general case considered in figure <ref> is the form factor suppression, which substantially suppresses the event rate in the energy range probed by DAMA, but is less relevant for the lower energies probed by COSINUS. We therefore expect COSINUS to have somewhat higher exclusion power (by a factor of a few) for spin-independent interactions than for general monotonically decreasing scattering. Quantitatively, this is confirmed by comparing the orange and cyan curves in the lower left panel of figure <ref>.

For m_χ = 5 GeV the situation becomes more complicated, as the optimum value of v_0 depends on E_thres. The reason is that the value of v_0 that maximises the modulation fraction corresponds to scattering off sodium, such that scatters off iodine are completely below threshold for DAMA. However, this configuration is of advantage for DAMA only if iodine scatters are also below threshold in COSINUS. If the COSINUS threshold is low enough to be able to observe iodine scatters even if the DAMA signal is dominated by sodium scatters, the resulting bound from COSINUS will be overwhelmingly strong due to the coherent enhancement factor. Thus, for E_thres = 1 keV the optimum value of v_0 still corresponds to the case that scattering off iodine dominates both for DAMA and COSINUS. Scattering off sodium becomes advantageous for DAMA only if the threshold of COSINUS is rather high, starting at about E_thres≃ 2 keV.[We note that even for m_χ = 5 GeV and E_thres = 2 keV the differential event rate due to scattering off iodine is only barely below threshold in COSINUS. As a result, the optimum configuration looks quite different from the one in figure <ref>, with only upward fluctuations contributing to the DAMA signal. Only for even smaller DM masses and/or even higher thresholds the optimum configuration from figure <ref> is recovered.]



 §.§ Spin-dependent and momentum-dependent scattering


Having discussed in detail the case of spin-independent scattering, we now consider a number of alternative possibilities. In each case there is a well-defined relation between the scattering rate off iodine and the one off sodium, meaning that we expect COSINUS to have greater exclusion power than for model-independent assumptions. 

The elastic scattering of Galactic DM particles off nuclei can be fully characterised in terms of 18 non-relativistic effective operators, which can depend on the spin of the DM particle and the nucleon, 𝐬_χ and 𝐬_N, the relative velocity 𝐯 and the momentum transfer 𝐪 <cit.>.[In addition, the cross section may be multiplied by an overall factor that depends on the momentum transfer in a target-independent way. For example, a factor of q_ref^4 / q^4 is present for the case of long-range interactions arising from the exchange of a very light mediator <cit.>.] Within this nomenclature, standard spin-independent scattering corresponds to 𝒪_1, while standard spin-dependent scattering corresponds to 𝒪_4. Without reviewing the formalism in detail, we note that several of these operators predict cross sections that grow with increasing momentum transfer, corresponding to recoil spectra that do not decrease monotonically. Particularly well-known examples are the operators generated by the exchange of a spin-0 mediator with general CP phase <cit.>. These are

    𝒪_10^N     = i  𝐬_N ·𝐪/m_N  ,
    𝒪_11^N     = i  𝐬_χ·𝐪/m_N  ,
    𝒪_6^N     = (𝐬_N ·𝐪)(𝐬_χ·𝐪)/m_N^2  ,

with m_N the nucleon mass.
The differential event rate resulting from each of these operators can be written as some power n of the nuclear recoil energy times a monotonically decreasing function (see eq. (<ref>)). Specifically, since E_R∝ q^2, one finds n = 1 for 𝒪_10,11 and n = 2 for 𝒪_6.
In the following we assume that the corresponding spin-0 mediator couples to each Standard Model quark proportionally to its mass, as required by minimal flavour violation. Using the quark spin contents from ref. <cit.> and quark masses from ref. <cit.>, the neutron-to-proton coupling ratios of the operators are given by ≃ 1 for 𝒪_11^N and -0.26 for 𝒪_6^N and 𝒪_10^N.

The COSINUS exclusion power for each of these operators can be calculated in complete analogy to the case of standard spin-independent scattering discussed above. On the one hand, this enables us to determine which particle physics explanations of the DAMA signal can be excluded by COSINUS in a halo-independent way. On the other hand, we can use the results to determine whether the more model-independent approach from section <ref> is overly conservative. Our results are summarized in figure <ref>. The orange curves correspond to the ones already shown in figure <ref> (solid for sodium, dashed for iodine), while the other curves correspond to specific particle physics assumptions (i.e. to specific non-relativistic operators). The first three panels correspond to n=0,1,2, the final panel will be discussed in the following sub-section.



In the first panel, we show in addition to spin-independent scattering also the case of spin-dependent scattering off protons (denoted SD_p). Due to the lack of coherent enhancement for scattering off iodine, the best-fit configuration for small DM masses corresponds to scattering dominantly off sodium. The transition from scattering off iodine to scattering off sodium leads to a visible kink in the exclusion power. We observe that in both regimes the exclusion power of COSINUS is quite close to the respective bounds derived in a model-independent way in section <ref> (orange lines). Nevertheless, we conclude that the design sensitivity of COSINUS will be sufficient to exclude both spin-independent and spin-dependent scattering as an explanation of the DAMA signal in a halo-independent way, independently of the DM mass. We also show the case of spin-independent long-range interactions containing an additional factor of 1/q^4 (denoted SI×  1/q^4). Due to the steeply falling recoil spectrum, the exclusion power of COSINUS is very large, and it should be easy for COSINUS to rule out any interpretation of DAMA in terms of long-range interactions (as proposed e.g. in ref. <cit.>).

In the second and third panel we see that, as expected, an explanation of the DAMA signal in terms of momentum-suppressed scattering is more difficult to exclude with COSINUS, as its lower threshold is less beneficial. Nevertheless, for the design sensitivity R_COSINUS^bound = 0.1 kg^-1 days^-1, COSINUS will be able to rule out also these scenarios in a halo-independent way for DM masses below ≃ 20GeV, ≃ 40GeV and ≃ 75GeV for scattering mediated by the non-relativistic operators 𝒪_6, 𝒪_10 and 𝒪_11, respectively. The qualitative difference between the behaviour of 𝒪_11 and the one of 𝒪_6 and 𝒪_10 is once again due to the absence of coherent enhancement for the latter two. 




 §.§ Inelastic scattering


It is well known that if DM-nucleus scattering is inelastic, i.e. if it requires the transition between two DM states of slightly different mass, the tension between the DAMA signal and other direct detection experiments is reduced <cit.>. For an inelasticity parameter of δ (corresponding to the mass difference of the two DM states), the minimum velocity required for a recoil of energy E_R is given by

    v_min = |δ + m_T  E_R/μ| 1/√(2   E_R  m_T)  .

The requirement that v_min is smaller than the Galactic escape velocity places a lower bound on E_R, meaning that there are no low-energy nuclear recoils and as a result the modulation fraction can be large. 

The scattering rate for inelastic DM cannot be written in the form of eq. (<ref>), providing a prime example for a scenario in which the rather general constraints derived in section <ref> do not apply. Nevertheless, the fully model-independent bound derived in section <ref> does still apply, so we can immediately obtain a conservative estimate of the exclusion power of COSINUS. In the final panel of figure <ref> we study how this conservative estimate compares to the sensitivity that COSINUS must achieve to exclude a specific model of inelastic scattering as an explanation of the DAMA signal in a halo-independent way. For the latter, we choose for illustration the case of spin-independent scattering with f_n/f_p ≃ -0.72 and a mass splitting δ = 50keV. Similar scenarios have been studied e.g. in refs. <cit.> in order to suppress the scattering rate in xenon- and/or germanium-based experiments.[The idea of inelastic DM was recently revisited in ref. <cit.> for a scenario with proto-philic spin-dependent coupling to nucleons. We find that such a scenario can be probed by COSINUS with much less exposure, as in contrast to spin-independent interactions with f_n/f_p ≃ -0.72 there is no suppression for scatterings off iodine recoils.] 

The mass dependence of the rate required in COSINUS for excluding a DM interpretation of DAMA can be understood as follows. For m_χ≲ 5GeV, the suppression of iodine scatters due to the choice f_n/f_p ≃ -0.72 leads to a situation in which both the rate in COSINUS and DAMA are dominated by sodium recoils. For these DM masses, the range of kinematically allowed energies corresponding to the optimal choice of the stream velocity v_0 gets more and more narrow with increasing m_χ. At m_χ≃ 5GeV, the situation is already nearly equivalent to adopting a single recoil energy E_R = E_0 for scattering off sodium, and hence the fully model-independent bound shown as a solid black curve is almost saturated. For 5 GeV≲ m_χ≲ 10 GeV, iodine recoils start to become relevant in COSINUS but not in DAMA, due the larger threshold of the latter experiment; consequently, a less stringent bound on the rate in COSINUS is necessary for excluding DAMA. Finally, for m_χ≳ 10GeV iodine recoils dominate over sodium both in COSINUS and DAMA (see also the lower panel in figure <ref>). The initial decrease of the required rate for these DM masses is again due to the smaller range of kinematically allowed energies, favouring DAMA over COSINUS. For m_χ≳ 50GeV, the situation is again nearly identical to the case of recoils with a single energy, such that in this case the fully model-independent bound for scattering off iodine is nearly saturated.




§ CONCLUSIONS



While the interpretation of the DAMA annual modulation in terms of DM is strongly challenged by null results from other direct detection experiments, it is extremely difficult to refute this interpretation in a model-independent way. In particular, the different target elements as well as techniques employed by the various experiments, while offering important complementarity, also imply larger uncertainties due to the unknown local velocity distribution as well as the particle physics properties of DM. 

In this article we have discussed how the DM interpretation of the DAMA signal in terms of nuclear scatterings can be tested independently of astrophysical and particle physics unknowns with COSINUS, a future experiment which employs the same target material and is thus sensitive to exactly the same recoil spectrum as DAMA (before taking into account detection efficiencies). Compared to other planned experiments also based on a NaI target, a great advantage of the cryogenic detector developed by the COSINUS collaboration lies in its ability to measure the total DM induced scattering rate with a significantly lower threshold than DAMA.

Our main findings are summarised in figure <ref>, where we show the halo-independent exclusion power of COSINUS, i.e. the bound on the total rate that COSINUS must achieve for excluding DAMA in a halo-independent way, as a function of the assumed threshold in COSINUS. The vertical axis on the right of each panel indicates the required exposure assuming zero observed events. The four panels correspond to different values of the DM mass m_χ, the various lines in each panel result from different assumptions on the differential event rate. We make the following observations:

 
  * The black shaded regions in figure <ref> show which combinations of the bound on the total rate and the threshold in COSINUS suffice to exclude a DM interpretation of DAMA, fully independently of the assumed nuclear recoil spectrum. This is only possible if the COSINUS threshold is significantly lower than the one of DAMA, E_thres≲ 1.8keV.  While this is compatible with the design goal E_thres = 1 keV, the required bound on the total rate is about an order of magnitude stronger than the current sensitivity estimate R_COSINUS^bound = 0.1  kg^-1 days^-1 for COSINUS.
 
  * More restrictive assumptions on the differential scattering cross section of DM lead to greater exclusion power, meaning that a weaker bound on the total rate (or equivalently a smaller exposure) is sufficient to exclude the DAMA signal. In particular, the orange curves in figure <ref> show the required bound for the still very general class of differential scattering cross sections that are monotonically decreasing in energy and have the standard 1/v^2 dependence on velocity. For the design sensitivity of COSINUS, such scenarios can be robustly excluded for DM masses m_χ≲ 6 GeV. If the COSINUS sensitivity is further increased by a factor of ≃ 3, this class of scattering cross sections can be excluded for all relevant DM masses. Interestingly, it follows from figure <ref> that this conclusion would still hold if the threshold in COSINUS is raised to significantly larger values; this is a prime example of how our results can be employed in order to optimise the detector layout of COSINUS for testing the DAMA hypothesis.
 
  * Finally, we show in figure <ref> the required bound on the rate in COSINUS when assuming a given particle physics scenario for the scattering process of DM with nuclei (in this case standard spin-independent interactions). In contrast to the other curves, this fixes the relation between the scattering rate off sodium and iodine. In particular, depending on the COSINUS threshold and the DM mass, it can happen that the modulation signal in DAMA is dominated by sodium recoils, while COSINUS can also observe the corresponding signal from iodine, thanks to its lower threshold. This further lowers the required sensitivity of COSINUS compared to e.g. the more general results based on monotonically decreasing cross sections. Similar results for various other interaction types are shown in figure <ref>.

In summary, we have shown that if COSINUS is able to set a bound on the scattering rate of DM of ≃ (0.01 - 0.1) kg^-1 days^-1 with an energy threshold of ≲ 1.8keV, it can exclude very general classes of interactions in which the signal is dominated by DM scattering off sodium or iodine in a halo-independent way. As other explanations of the DAMA data such as DM-electron scattering <cit.>, scattering off thallium <cit.> or scattering off OH impurities <cit.> are already strongly constrained by other experiments <cit.>, this would strongly point towards a so far unknown source of background being responsible for the modulation seen by the DAMA collaboration.

To conclude, let us mention that for the last few years DAMA has been taking data with improved photo-multiplier tubes, which promise a significant lowering of the threshold <cit.>. Measurements of the modulation amplitude in this new energy range will provide important clues to the origin of the DAMA signal <cit.>. To make full use this new information it will be crucial to extend the methods developed here to also include spectral information when calculating the model-independent exclusion power of COSINUS. Clearly, the fact that COSINUS can achieve an even lower threshold and a much better energy resolution than DAMA will then become particularly important.



§ NOTE ADDED




After the completion of this work, DAMA presented new data with a lower threshold of 1 keVee <cit.>. While the measurements above 2 keVee are in agreement with previous results, two new data points at lower energies provide very interesting additional information, because they exhibit neither the steep rise expected for scattering of low-mass DM, nor the turn-over towards anti-modulation expected for high-mass DM <cit.>. We have performed a combination of the two data sets, assuming that the energy resolution has not changed significantly during the detector upgrade. We find that it is no longer possible to obtain a good fit for DM scattering with the standard assumptions discussed in section <ref>. The best-fit spectra for spin-independent scattering on sodium and spin-independent scattering on iodine are shown in figure <ref>. The goodness of fit of the best-fit point for scattering on sodium is  unacceptable (p ∼ 10^-8 when including the first 12 bins), while the hypothesis of scattering on iodine is only marginally compatible with data (p ≈ 0.003). This observation implies that any interpretation of the DAMA signal in terms of DM requires non-standard interactions or non-standard astrophysical distributions (or both), independently of (but already implied by) the exclusion bounds from other experiments. New experiments based on NaI, like COSINUS, and model-independent methods, like the one presented in this work, will therefore be essential to further investigate the nature of the DAMA signal.



We thank Suchita Kulkarni, Josef Pradler and Thomas Schwetz for discussion and Vanessa Zema for helpful comments on the manuscript. This work is supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676 “Particles, Strings and the Early Universe” and the Emmy Noether Grant No. KA 4662/1-1 as well as the ERC Starting Grant `NewAve' (638528). 





§ BEST-FIT VELOCITY DISTRIBUTION



In this appendix we prove that a single stream is enough to determine S_DAMA^max as defined in eq. (<ref>). First of all, we rewrite this equation as

    S_DAMA^max = max_f S^f_DAMA(σ)/R_COSINUS^f(σ) / R_COSINUS^bound  .

The right-hand side of this equation is in fact independent of σ, as the cross section cancels out in the ratio. It is therefore sufficient to calculate S_DAMA^max for an arbitrary reference cross section, so that we will simply drop the dependence on σ for the rest of this appendix.

Let us now define

    S_DAMA^stream≡max_𝐯_0S^𝐯_0_DAMA/R_COSINUS^𝐯_0 / R_COSINUS^bound

in complete analogy to eq. (<ref>), but considering only DM velocity distributions of the form f(𝐯) = δ(𝐯 - 𝐯_0).  We now want to show that

    S_DAMA^max≡ S_DAMA^stream  .


Since it is obviously true that S_DAMA^max≥ S_DAMA^stream, it remains to be shown that S_DAMA^max≤ S_DAMA^stream. To do so, we denote the velocity that maximises the right-hand side of eq. (<ref>) as 𝐰_0, such that

    S_DAMA^stream = S^𝐰_0_DAMA/R_COSINUS^𝐰_0 / R_COSINUS^bound  .

For a given velocity distribution f(𝐯) we can then consider the following expression:

    R_COSINUS^f  -  S_DAMA^f/S_DAMA^𝐰_0 / R_COSINUS^𝐰_0 = ∫d^3 v_0   f(𝐯_0) (  R_COSINUS^𝐯_0  -  S_DAMA^𝐯_0/S_DAMA^𝐰_0 / R_COSINUS^𝐰_0)
      ,

where we have made use of the fact that the velocity distribution enters linearly into the calculation of R_COSINUS and S_DAMA (see eq. (<ref>)).
By definition of 𝐰_0 we have

    S_DAMA^𝐰_0/R_COSINUS^𝐰_0 ≥ S_DAMA^𝐯_0/R_COSINUS^𝐯_0

for all 𝐯_0, and hence the integrand in the right-hand side of eq. (<ref>) is always positive. It therefore follows immediately that

    R_COSINUS^f  -  S_DAMA^f/S_DAMA^𝐰_0 / R_COSINUS^𝐰_0≥ 0   ,

which can be rearranged to 

    S_DAMA^f/R_COSINUS^f / R_COSINUS^bound≤ S_DAMA^𝐰_0/R_COSINUS^𝐰_0 / R_COSINUS^bound = S_DAMA^stream  .

Since this inequality holds for arbitrary velocity distributions f, it must hold in particular for the velocity distribution that maximises the left-hand side, thus completing the proof.


10

Bernabei:2013xsa
R. Bernabei et al.,
  http://dx.doi.org/10.1140/epjc/s10052-013-2648-7Final model
  independent result of DAMA/LIBRA-phase1,  Eur. Phys. J. C73
  (2013) 2648, [http://arxiv.org/abs/1308.51091308.5109].

Kelso:2017zwr
C. Kelso, C. Savage, P. Sandick, K. Freese, and P. Gondolo, Examining the
  time dependence of DAMA's modulation amplitude,
  http://arxiv.org/abs/1710.037701710.03770.

Blum:2011jf
K. Blum, DAMA vs. the annually modulated muon background,
  http://arxiv.org/abs/1110.08571110.0857.

Pradler:2012qt
J. Pradler, B. Singh, and I. Yavin,
  http://dx.doi.org/10.1016/j.physletb.2013.02.033On an
  unverified nuclear decay and its role in the DAMA experiment,  Phys.
  Lett. B720 (2013) 399–404,
  [http://arxiv.org/abs/1210.55011210.5501].

Pradler:2012bf
J. Pradler and I. Yavin,
  http://dx.doi.org/10.1016/j.physletb.2013.04.045Addendum to 'On
  an unverified nuclear decay and its role in the DAMA experiment”,  Phys. Lett. B723 (2013) 168–171,
  [http://arxiv.org/abs/1210.75481210.7548].

Bernabei:2012wp
R. Bernabei et al.,
  http://dx.doi.org/10.1140/epjc/s10052-012-2064-4No role for
  muons in the DAMA annual modulation results,  Eur. Phys. J. C72 (2012) 2064, [http://arxiv.org/abs/1202.41791202.4179].

Davis:2014cja
J. H. Davis, http://dx.doi.org/10.1103/PhysRevLett.113.081302Fitting the annual modulation in DAMA with neutrons from muons and
  neutrinos,  Phys. Rev. Lett. 113 (2014) 081302,
  [http://arxiv.org/abs/1407.10521407.1052].

Bernabei:2014tqa
R. Bernabei et al.,
  http://dx.doi.org/10.1140/epjc/s10052-014-3196-5No role for
  neutrons, muons and solar neutrinos in the DAMA annual modulation results,
   Eur. Phys. J. C74 (2014), no. 12 3196,
  [http://arxiv.org/abs/1409.35161409.3516].

Klinger:2015vga
J. Klinger and V. A. Kudryavtsev,
  http://dx.doi.org/10.1103/PhysRevLett.114.151301Muon-induced
  neutrons do not explain the DAMA data,  Phys. Rev. Lett. 114
  (2015), no. 15 151301, [http://arxiv.org/abs/1503.072251503.07225].

Angloher:2014myn
CRESST-II, G. Angloher et al.,
  http://dx.doi.org/10.1140/epjc/s10052-014-3184-9Results on low
  mass WIMPs using an upgraded CRESST-II detector,  Eur. Phys. J. C74 (2014), no. 12 3184, [http://arxiv.org/abs/1407.31461407.3146].

Akerib:2016vxi
LUX, D. S. Akerib et al.,
  http://dx.doi.org/10.1103/PhysRevLett.118.021303Results from a
  search for dark matter in the complete LUX exposure,  Phys. Rev.
  Lett. 118 (2017), no. 2 021303,
  [http://arxiv.org/abs/1608.076481608.07648].

Hehn:2016nll
EDELWEISS, L. Hehn et al.,
  http://dx.doi.org/10.1140/epjc/s10052-016-4388-yImproved
  EDELWEISS-III sensitivity for low-mass WIMPs using a profile likelihood
  approach,  Eur. Phys. J. C76 (2016), no. 10 548,
  [http://arxiv.org/abs/1607.033671607.03367].

Cui:2017nnn
PandaX-II, X. Cui et al.,
  http://dx.doi.org/10.1103/PhysRevLett.119.181302Dark Matter
  Results From 54-Ton-Day Exposure of PandaX-II Experiment,  Phys. Rev.
  Lett. 119 (2017), no. 18 181302,
  [http://arxiv.org/abs/1708.069171708.06917].

Aprile:2017iyp
XENON, E. Aprile et al.,
  http://dx.doi.org/10.1103/PhysRevLett.119.181301First Dark
  Matter Search Results from the XENON1T Experiment,  Phys. Rev. Lett.
  119 (2017), no. 18 181301, [http://arxiv.org/abs/1705.066551705.06655].

Amole:2017dex
PICO, C. Amole et al.,
  http://dx.doi.org/10.1103/PhysRevLett.118.251301Dark Matter
  Search Results from the PICO-60 C_3F_8 Bubble Chamber,  Phys.
  Rev. Lett. 118 (2017), no. 25 251301,
  [http://arxiv.org/abs/1702.076661702.07666].

Agnese:2017njq
SuperCDMS, R. Agnese et al.,
  http://dx.doi.org/10.1103/PhysRevLett.120.061802Results from
  the Super Cryogenic Dark Matter Search Experiment at Soudan,  Phys.
  Rev. Lett. 120 (2018), no. 6 061802,
  [http://arxiv.org/abs/1708.088691708.08869].

Agnese:2017jvy
SuperCDMS, R. Agnese et al.,
  http://dx.doi.org/10.1103/PhysRevD.97.022002Low-mass dark
  matter search with CDMSlite,  Phys. Rev. D97 (2018), no. 2
  022002, [http://arxiv.org/abs/1707.016321707.01632].

Agnes:2018fwg
DarkSide, P. Agnes et al., DarkSide-50 532-day Dark Matter Search
  with Low-Radioactivity Argon,  http://arxiv.org/abs/1802.071981802.07198.

Agnes:2018ves
DarkSide, P. Agnes et al., Low-mass Dark Matter Search with the
  DarkSide-50 Experiment,  http://arxiv.org/abs/1802.069941802.06994.

Kopp:2009et
J. Kopp, V. Niro, T. Schwetz, and J. Zupan,
  http://dx.doi.org/10.1103/PhysRevD.80.083502DAMA/LIBRA and
  leptonically interacting Dark Matter,  Phys. Rev. D80 (2009)
  083502, [http://arxiv.org/abs/0907.31590907.3159].

TuckerSmith:2001hy
D. Tucker-Smith and N. Weiner,
  http://dx.doi.org/10.1103/PhysRevD.64.043502Inelastic dark
  matter,  Phys. Rev. D64 (2001) 043502,
  [http://arxiv.org/abs/hep-ph/0101138hep-ph/0101138].

Feng:2011vu
J. L. Feng, J. Kumar, D. Marfatia, and D. Sanford,
  http://dx.doi.org/10.1016/j.physletb.2011.07.083Isospin-Violating Dark Matter,  Phys. Lett. B703 (2011)
  124–127, [http://arxiv.org/abs/1102.43311102.4331].

Aprile:2017yea
XENON, E. Aprile et al.,
  http://dx.doi.org/10.1103/PhysRevLett.118.101101Search for
  Electronic Recoil Event Rate Modulation with 4 Years of XENON100 Data, 
  Phys. Rev. Lett. 118 (2017), no. 10 101101,
  [http://arxiv.org/abs/1701.007691701.00769].

Dolan:2014ska
M. J. Dolan, F. Kahlhoefer, C. McCabe, and K. Schmidt-Hoberg,
  http://dx.doi.org/10.1007/JHEP07(2015)103,
  10.1007/JHEP03(2015)171A taste of dark matter: Flavour constraints on
  pseudoscalar mediators,  JHEP 03 (2015) 171,
  [http://arxiv.org/abs/1412.51741412.5174]. [Erratum:
  JHEP07,103(2015)].

DelNobile:2015lxa
E. Del Nobile, G. B. Gelmini, A. Georgescu, and J.-H. Huh,
  http://dx.doi.org/10.1088/1475-7516/2015/08/046Reevaluation of
  spin-dependent WIMP-proton interactions as an explanation of the DAMA data,
   JCAP 1508 (2015), no. 08 046,
  [http://arxiv.org/abs/1502.076821502.07682].

Scopel:2015eoh
S. Scopel and K.-H. Yoon,
  http://dx.doi.org/10.1088/1475-7516/2016/02/050Inelastic dark
  matter with spin-dependent couplings to protons and large modulation
  fractions in DAMA,  JCAP 1602 (2016), no. 02 050,
  [http://arxiv.org/abs/1512.005931512.00593].

Scopel:2015baa
S. Scopel, K.-H. Yoon, and J.-H. Yoon,
  http://dx.doi.org/10.1088/1475-7516/2015/07/041Generalized
  spin-dependent WIMP-nucleus interactions and the DAMA modulation effect, 
  JCAP 1507 (2015), no. 07 041,
  [http://arxiv.org/abs/1505.019261505.01926].

Catena:2016hoj
R. Catena, A. Ibarra, and S. Wild,
  http://dx.doi.org/10.1088/1475-7516/2016/05/039DAMA confronts
  null searches in the effective theory of dark matter-nucleon interactions,
   JCAP 1605 (2016), no. 05 039,
  [http://arxiv.org/abs/1602.040741602.04074].

Kang:2018dlc
S.-H. Kang, S. Scopel, and J.-H. Yoon, Explaining DAMA with proton-philic
  spin-dependent inelastic Dark Matter (pSIDM): a frequentist analysis,
  http://arxiv.org/abs/1801.029051801.02905.

Fox:2010bz
P. J. Fox, J. Liu, and N. Weiner,
  http://dx.doi.org/10.1103/PhysRevD.83.103514Integrating Out
  Astrophysical Uncertainties,  Phys. Rev. D83 (2011) 103514,
  [http://arxiv.org/abs/1011.19151011.1915].

McCabe:2011sr
C. McCabe, http://dx.doi.org/10.1103/PhysRevD.84.043525DAMA and
  CoGeNT without astrophysical uncertainties,  Phys. Rev. D84
  (2011) 043525, [http://arxiv.org/abs/1107.07411107.0741].

Frandsen:2011gi
M. T. Frandsen, F. Kahlhoefer, C. McCabe, S. Sarkar, and K. Schmidt-Hoberg,
  http://dx.doi.org/10.1088/1475-7516/2012/01/024Resolving
  astrophysical uncertainties in dark matter direct detection,  JCAP
  1201 (2012) 024, [http://arxiv.org/abs/1111.02921111.0292].

DelNobile:2013cva
E. Del Nobile, G. Gelmini, P. Gondolo, and J.-H. Huh,
  http://dx.doi.org/10.1088/1475-7516/2013/10/048Generalized Halo
  Independent Comparison of Direct Dark Matter Detection Data,  JCAP
  1310 (2013) 048, [http://arxiv.org/abs/1306.52731306.5273].

Ibarra:2017mzt
A. Ibarra and A. Rappelt,
  http://dx.doi.org/10.1088/1475-7516/2017/08/039Optimized
  velocity distributions for direct dark matter detection,  JCAP 1708 (2017), no. 08 039, [http://arxiv.org/abs/1703.091681703.09168].

Gondolo:2017jro
P. Gondolo and S. Scopel,
  http://dx.doi.org/10.1088/1475-7516/2017/09/032Halo-independent
  determination of the unmodulated WIMP signal in DAMA: the isotropic case, 
  JCAP 1709 (2017), no. 09 032,
  [http://arxiv.org/abs/1703.089421703.08942].

Catena:2018ywo
R. Catena, A. Ibarra, A. Rappelt, and S. Wild, Halo-independent
  comparison of direct detection experiments in the effective theory of dark
  matter-nucleon interactions,  http://arxiv.org/abs/1801.084661801.08466.

Shields:2015wka
E. Shields, J. Xu, and F. Calaprice,
  http://dx.doi.org/10.1016/j.phpro.2014.12.028SABRE: A New
  NaI(T1) Dark Matter Direct Detection Experiment,  Phys. Procedia
  61 (2015) 169–178.

Amare:2014jta
J. Amaré et al., http://dx.doi.org/10.1016/j.phpro.2014.12.026From ANAIS-25 towards ANAIS-250,  Phys. Procedia 61 (2015)
  157–162, [http://arxiv.org/abs/1404.35641404.3564].

Fushimi:2015sew
PICO-LON, K. Fushimi et al.,
  http://dx.doi.org/10.1088/1742-6596/718/4/042022Dark matter
  search project PICO-LON,  J. Phys. Conf. Ser. 718 (2016),
  no. 4 042022, [http://arxiv.org/abs/1512.046451512.04645].

Thompson:2017yvq
COSINE-100, W. G. Thompson,
  https://inspirehep.net/record/1634544/files/arXiv:1711.01488.pdfCurrent status and projected sensitivity of COSINE-100,  in 15th
  International Conference on Topics in Astroparticle and Underground Physics
  (TAUP 2017) Sudbury, Ontario, Canada, July 24-28, 2017, 2017.
http://arxiv.org/abs/1711.014881711.01488.

Cherwinka:2014xta
DM-Ice17, J. Cherwinka et al.,
  http://dx.doi.org/10.1103/PhysRevD.90.092005First data from
  DM-Ice17,  Phys. Rev. D90 (2014), no. 9 092005,
  [http://arxiv.org/abs/1401.48041401.4804].

Kim:2012rza
S. C. Kim et al., http://dx.doi.org/10.1103/PhysRevLett.108.181301New Limits on Interactions between Weakly Interacting Massive Particles and
  Nucleons Obtained with CsI(Tl) Crystal Detectors,  Phys. Rev. Lett.
  108 (2012) 181301, [http://arxiv.org/abs/1204.26461204.2646].

Angloher:2016ooq
G. Angloher et al.,
  http://dx.doi.org/10.1140/epjc/s10052-016-4278-3The COSINUS
  project - perspectives of a NaI scintillating calorimeter for dark matter
  search,  Eur. Phys. J. C76 (2016), no. 8 441,
  [http://arxiv.org/abs/1603.022141603.02214].

HerreroGarcia:2011aa
J. Herrero-Garcia, T. Schwetz, and J. Zupan,
  http://dx.doi.org/10.1088/1475-7516/2012/03/005On the annual
  modulation signal in dark matter direct detection,  JCAP 1203
  (2012) 005, [http://arxiv.org/abs/1112.16271112.1627].

HerreroGarcia:2012fu
J. Herrero-Garcia, T. Schwetz, and J. Zupan,
  http://dx.doi.org/10.1103/PhysRevLett.109.141301Astrophysics
  independent bounds on the annual modulation of dark matter signals,  Phys. Rev. Lett. 109 (2012) 141301,
  [http://arxiv.org/abs/1205.01341205.0134].

Bozorgnia:2013hsa
N. Bozorgnia, J. Herrero-Garcia, T. Schwetz, and J. Zupan,
  http://dx.doi.org/10.1088/1475-7516/2013/07/049Halo-independent
  methods for inelastic dark matter scattering,  JCAP 1307
  (2013) 049, [http://arxiv.org/abs/1305.35751305.3575].

Blennow:2015gta
M. Blennow, J. Herrero-Garcia, T. Schwetz, and S. Vogl,
  http://dx.doi.org/10.1088/1475-7516/2015/08/039Halo-independent
  tests of dark matter direct detection signals: local DM density, LHC, and
  thermal freeze-out,  JCAP 1508 (2015), no. 08 039,
  [http://arxiv.org/abs/1505.057101505.05710].

Herrero-Garcia:2015kga
J. Herrero-Garcia, http://dx.doi.org/10.1088/1475-7516/2015/09/012Halo-independent tests of dark matter annual modulation signals,  JCAP 1509 (2015), no. 09 012,
  [http://arxiv.org/abs/1506.035031506.03503].

Drukier:1986tm
A. K. Drukier, K. Freese, and D. N. Spergel,
  http://dx.doi.org/10.1103/PhysRevD.33.3495Detecting Cold Dark
  Matter Candidates,  Phys. Rev. D33 (1986) 3495–3508.

Freese:2012xd
K. Freese, M. Lisanti, and C. Savage,
  http://dx.doi.org/10.1103/RevModPhys.85.1561Colloquium: Annual
  modulation of dark matter,  Rev. Mod. Phys. 85 (2013)
  1561–1581, [http://arxiv.org/abs/1209.33391209.3339].

Bernabei:1996vj
R. Bernabei et al., http://dx.doi.org/10.1016/S0370-2693(96)01483-9,
  10.1016/S0370-2693(96)80020-7New limits on WIMP search with
  large-mass low-radioactivity NaI(Tl) set-up at Gran Sasso,  Phys.
  Lett. B389 (1996) 757–766.

Xu:2015wha
J. Xu et al., http://dx.doi.org/10.1103/PhysRevC.92.015807Scintillation Efficiency Measurement of Na Recoils in NaI(Tl) Below the
  DAMA/LIBRA Energy Threshold,  Phys. Rev. C92 (2015), no. 1
  015807, [http://arxiv.org/abs/1503.072121503.07212].

Savage:2008er
C. Savage, G. Gelmini, P. Gondolo, and K. Freese,
  http://dx.doi.org/10.1088/1475-7516/2009/04/010Compatibility of
  DAMA/LIBRA dark matter detection with other searches,  JCAP 0904 (2009) 010, [http://arxiv.org/abs/0808.36070808.3607].

Bernabei:2008yh
DAMA, R. Bernabei et al.,
  http://dx.doi.org/10.1016/j.nima.2008.04.082The DAMA/LIBRA
  apparatus,  Nucl. Instrum. Meth. A592 (2008) 297–315,
  [http://arxiv.org/abs/0804.27380804.2738].

Petricca:2017zdp
CRESST, F. Petricca et al.,
  https://inspirehep.net/record/1637341/files/arXiv:1711.07692.pdfFirst results on low-mass dark matter from the CRESST-III experiment,  in
  15th International Conference on Topics in Astroparticle and
  Underground Physics (TAUP 2017) Sudbury, Ontario, Canada, July 24-28, 2017,
  2017.
http://arxiv.org/abs/1711.076921711.07692.

Angloher:2017sft
COSINUS, G. Angloher et al.,
  http://dx.doi.org/10.1088/1748-0221/12/11/P11007Results from
  the first cryogenic NaI detector for the COSINUS project,  JINST
  12 (2017), no. 11 P11007, [http://arxiv.org/abs/1705.110281705.11028].

Reindl:2017bun
COSINUS, F. Reindl et al.,
  http://inspirehep.net/record/1634542/files/arXiv:1711.01482.pdfResults of the first NaI scintillating calorimeter prototypes by COSINUS,
   in 15th International Conference on Topics in Astroparticle and
  Underground Physics (TAUP 2017) Sudbury, Ontario, Canada, July 24-28, 2017,
  2017.
http://arxiv.org/abs/1711.014821711.01482.

Strauss:2014zia
R. Strauss et al., http://dx.doi.org/10.1140/epjc/s10052-014-2957-5Energy-dependent light quenching in CaWO_4 crystals at mK temperatures,
   Eur. Phys. J. C74 (2014), no. 7 2957,
  [http://arxiv.org/abs/1401.33321401.3332].

Kelso:2013gda
C. Kelso, P. Sandick, and C. Savage,
  http://dx.doi.org/10.1088/1475-7516/2013/09/022Lowering the
  Threshold in the DAMA Dark Matter Search,  JCAP 1309 (2013)
  022, [http://arxiv.org/abs/1306.18581306.1858].

Lewin:1995rx
J. D. Lewin and P. F. Smith,
  http://dx.doi.org/10.1016/S0927-6505(96)00047-3Review of
  mathematics, numerical factors, and corrections for dark matter experiments
  based on elastic nuclear recoil,  Astropart. Phys. 6 (1996)
  87–112.

Yellin:2002xd
S. Yellin, http://dx.doi.org/10.1103/PhysRevD.66.032005Finding an
  upper limit in the presence of unknown background,  Phys. Rev. D66 (2002) 032005, [http://arxiv.org/abs/physics/0203002physics/0203002].

Feldstein:2014gza
B. Feldstein and F. Kahlhoefer,
  http://dx.doi.org/10.1088/1475-7516/2014/08/065A new
  halo-independent approach to dark matter direct detection analysis,  JCAP 1408 (2014) 065, [http://arxiv.org/abs/1403.46061403.4606].

Feldstein:2014ufa
B. Feldstein and F. Kahlhoefer,
  http://dx.doi.org/10.1088/1475-7516/2014/12/052Quantifying
  (dis)agreement between direct detection experiments in a halo-independent
  way,  JCAP 1412 (2014), no. 12 052,
  [http://arxiv.org/abs/1409.54461409.5446].

Lee:2013wza
S. K. Lee, M. Lisanti, A. H. G. Peter and B. R. Safdi,
  http://dx.doi.org/10.1103/PhysRevLett.112.011301Effect of Gravitational Focusing on Annual Modulation in Dark-Matter Direct-Detection Experiments,  Phys. Rev. Lett. 112 (2014), no.1,  011301,
  [http://arxiv.org/abs/1308.19531308.1953].
  
Behnke:2012ys
COUPP, E. Behnke et al.,
  http://dx.doi.org/10.1103/PhysRevD.86.052001,
  10.1103/PhysRevD.90.079902First Dark Matter Search Results from a
  4-kg CF_3I Bubble Chamber Operated in a Deep Underground Site,  Phys. Rev. D86 (2012), no. 5 052001,
  [http://arxiv.org/abs/1204.30941204.3094]. [Erratum: Phys.
  Rev.D90,no.7,079902(2014)].

Lee:2014zsa
H. S. Lee et al., http://dx.doi.org/10.1103/PhysRevD.90.052006Search for Low-Mass Dark Matter with CsI(Tl) Crystal Detectors,  Phys. Rev. D90 (2014), no. 5 052006,
  [http://arxiv.org/abs/1404.34431404.3443].

Amole:2015pla
PICO, C. Amole et al.,
  http://dx.doi.org/10.1103/PhysRevD.93.052014Dark matter search
  results from the PICO-60 CF_3I bubble chamber,  Phys. Rev. D93 (2016), no. 5 052014, [http://arxiv.org/abs/1510.077541510.07754].

Fitzpatrick:2012ix
A. L. Fitzpatrick, W. Haxton, E. Katz, N. Lubbers, and Y. Xu,
  http://dx.doi.org/10.1088/1475-7516/2013/02/004The Effective
  Field Theory of Dark Matter Direct Detection,  JCAP 1302
  (2013) 004, [http://arxiv.org/abs/1203.35421203.3542].

Foot:2012rk
R. Foot, http://dx.doi.org/10.1103/PhysRevD.86.023524Mirror dark
  matter interpretations of the DAMA, CoGeNT and CRESST-II data,  Phys.
  Rev. D86 (2012) 023524, [http://arxiv.org/abs/1203.23871203.2387].

Kahlhoefer:2017umn
F. Kahlhoefer, K. Schmidt-Hoberg, and S. Wild,
  http://dx.doi.org/10.1088/1475-7516/2017/08/003Dark matter
  self-interactions from a general spin-0 mediator,  JCAP 1708
  (2017), no. 08 003, [http://arxiv.org/abs/1704.021491704.02149].

Cheng:2012qr
H.-Y. Cheng and C.-W. Chiang,
  http://dx.doi.org/10.1007/JHEP07(2012)009Revisiting Scalar and
  Pseudoscalar Couplings with Nucleons,  JHEP 07 (2012) 009,
  [http://arxiv.org/abs/1202.12921202.1292].

Patrignani:2016xqp
Particle Data Group, C. Patrignani et al.,
  http://dx.doi.org/10.1088/1674-1137/40/10/100001Review of
  Particle Physics,  Chin. Phys. C40 (2016), no. 10 100001.

Frandsen:2011ts
M. T. Frandsen, F. Kahlhoefer, J. March-Russell, C. McCabe, M. McCullough,
  et al., http://dx.doi.org/10.1103/PhysRevD.84.041301On the DAMA
  and CoGeNT Modulations,  Phys. Rev. D84 (2011) 041301,
  [http://arxiv.org/abs/1105.37341105.3734].

Scopel:2014kba
S. Scopel and K. Yoon,
  http://dx.doi.org/10.1088/1475-7516/2014/08/060A systematic
  halo-independent analysis of direct detection data within the framework of
  Inelastic Dark Matter,  JCAP 1408 (2014) 060,
  [http://arxiv.org/abs/1405.03641405.0364].

Foot:2014xwa
R. Foot, http://dx.doi.org/10.1103/PhysRevD.90.121302Can dark
  matter - electron scattering explain the DAMA annual modulation signal?, 
  Phys. Rev. D90 (2014), no. 12 121302,
  [http://arxiv.org/abs/1407.42131407.4213].

Chang:2010pr
S. Chang, R. F. Lang, and N. Weiner,
  http://dx.doi.org/10.1103/PhysRevLett.106.011301Effect of
  Thallium Impurities in the DAMA Experiment on the Allowed Parameter Space for
  Inelastic Dark Matter,  Phys. Rev. Lett. 106 (2011) 011301,
  [http://arxiv.org/abs/1007.26881007.2688].

Profumo:2014mpa
S. Profumo and F. S. Queiroz,
  http://dx.doi.org/10.1088/1475-7516/2014/05/038GeV WIMPs
  scattering off of OH impurities cannot explain the DAMA signal,  JCAP 1405 (2014) 038, [http://arxiv.org/abs/1401.42531401.4253].

Aprile:2015ade
XENON100, E. Aprile et al.,
  http://dx.doi.org/10.1126/science.aab2069Exclusion of
  Leptophilic Dark Matter Models using XENON100 Electronic Recoil Data, 
  Science 349 (2015), no. 6250 851–854,
  [http://arxiv.org/abs/1507.077471507.07747].

Bernabei:2012zzb
R. Bernabei et al., http://dx.doi.org/10.1088/1748-0221/7/03/P03009Performances of the new high quantum efficiency PMTs in DAMA/LIBRA,  JINST 7 (2012) P03009.

DAMA_talk
R. Bernabei, DAMA/LIBRA, 
  LNGS Scientific Committee Meeting, 26–27 March 2018, <https://agenda.infn.it/conferenceDisplay.py?confId=15474>.



