Annual modulations from secular variations: relaxing DAMA?
Dario Buttazzo, Paolo Panci, Nicola Rossi, Alessandro Strumia
AAnnual modulations from secularvariations: relaxing DAMA?
Dario Buttazzo a , Paolo Panci a,b , Nicola Rossi c , Alessandro Strumia b a INFN, Sezione di Pisa b Dipartimento di Fisica, Universit`a di Pisa c INFN, Sezione di Roma and Laboratori Nazionali del Gran Sasso
The DAMA collaboration reported an annually modulated rate with a phasecompatible with a Dark Matter induced signal. We point out that a slowlyvarying rate can bias or even simulate an annual modulation if data are an-alyzed in terms of residuals computed by subtracting approximately yearlyaverages starting from a fixed date, rather than a background continuous intime. In the most extreme case, the amplitude and phase of the annual mod-ulation reported by
DAMA could be alternatively interpreted as a decennialgrowth of the rate. This possibility appears mildly disfavoured by a detailedstudy of the available data, but cannot be safely excluded. In general, a de-creasing or increasing rate could partially reduce or enhance a true annualmodulation, respectively. The issue could be clarified by looking at the fulltime-dependence of the
DAMA total rate, not explicitly published so far.
Contents a r X i v : . [ h e p - e x ] F e b Introduction
Various experiments are searching for interactions of Dark Matter (DM) with ordinarymatter. Most experiments tried to increase the sensitivity to DM collisions by reducingtheir backgrounds and found no evidence for DM. The
DAMA collaboration followeda different strategy, focused on large statistics: DM scatterings would contribute to thetotal event rate with an excess which is annually modulated due to the rotation of theEarth around the Sun. Indeed, the flux of DM particles hitting the Earth would peakaround June, 2 nd , when the Earth’s orbital velocity is more aligned to the Sun’s motionwith respect to the galactic frame.The DAMA collaboration reported the observation of an annual modulation in single-hit scintillation events in NaI crystals, with the phase expected for a DM signal. The up-graded
DAMA/LIBRA detector confirmed the earlier result of
DAMA/
NaI, collectingmore data and reaching a significance of about 13 σ (loosely speaking) for the cumulativeexposure [1–5]. The ANAIS [6–8] and
COSINE [9–11] experiments employ NaI crystalslike
DAMA , and are also looking for an annual modulation in the counting rate. Sofar they have not reported any modulation, but their sensitivity is not yet sufficient toprobe the
DAMA signal [11, 12]. Furthermore, a DM signal compatible with the
DAMA result has not been confirmed by other experiments which use different detectors andtechniques (see e.g. the review of DM in [13]). These experiments reached high sensitivityto both DM nuclear and electron recoils and, therefore, proposing theoretically viable DMinterpretations of the
DAMA detection progressively becomes very challenging [14–28].The
DAMA result remains an open issue. Apart from the DM hypothesis, manyattempts have been put forward to explain the signal by considering speculative oscillatingbackgrounds with a period of roughly one year (see e.g. [29–32]). Without addressing theplausibility of each proposed background (nor of possible backgrounds that might havebeen overlooked), the fact that the phase of the observed modulation is close to June, 2 nd is considered as a significant argument in favour of the DM interpretation of the signal.Furthermore, the amplitude, time dependence, and event distribution in the detector arrayof different oscillating backgrounds cannot satisfy all the peculiar features attributed tothe measured annual modulation signal (for details see [3] and references therein).The DAMA collaboration published the cumulative total rate of the single-hit scintil-lation events as a function of the electron equivalent recoil energy in [1,3,5]. The full timedependence was never explicitly presented. On the other hand, the collaboration pre-sented the modulated residual as a function of time, computed by subtracting — roughlyevery year and roughly starting from the same date — the weighted average of the totalrate in a given cycle of data-taking. This is a dangerous approach, since a time-scalesimilar to the DM periodicity is introduced in the analysis. Indeed, the analysis proce-dure adopted by
DAMA transforms a rate that varies with time into a sawtooth withperiod given by the duration of the analysis cycles. If cycles start around the beginningof September and last roughly one year, a growing background mimics an annual mod-2lation peaked in June. As a result, a slow time-dependence of the total rate, even ifnot oscillating, becomes a possible source of bias. For example, the
DAMA modulatedamplitude could be generated by a growth of the rate of several percent on a decennialtime-scale.This paper is organized as follows: in section 2 we describe the general idea, and weshow how the amplitude and phase of a modulated signal are related to the time vari-ation of the total rate. We apply this to a practical example. In section 3 we brieflypresent the
DAMA analysis. A toy Monte Carlo simulation is performed in 3.1, showingthat a linearly growing rate produces a sawtooth signal that can appear as a cosine, upto statistical errors. In section 4 we consider the
DAMA data. We study the detailedtime-dependence of the
DAMA residuals in section 4.1, finding that both a cosine and asawtooth provide acceptable fits to the data, with the cosine interpretation being some-what favoured. This extreme possibility is consistent with corollary DAMA studies [5],which include a Fourier analysis. In section 4.2 we infer the energy dependence of thesecular variation. We discuss some possible time-dependent backgrounds in section 4.3.Conclusions are given in section 5.
In this section we discuss, from a general point of view, how an apparently periodic signalcan be mimicked by a time-dependent total rate without any modulation. Let us considera total rate R ( t ) that contains an oscillating signal on top of a slowly varying background R ( t ) R ( t ) = R ( t ) + A cos (cid:18) πtT − φ (cid:19) . (1)If the rate could be measured with arbitrary precision, the modulation would be straight-forwardly obtained by directly fitting R ( t ) to the data. The phase of its peaks woulddiffer from φ in the presence of R ( t ).If instead statistical uncertainties are so large that a single annual period is not clearlyvisible, one needs to combine data of multiple periods. Then, a slow time-dependence ofthe background R ( t ) becomes more dangerous. For constant background, R ( t ) ≡ C , onecan consider the average of the rate over any desired time interval ∆, and subtract thisquantity from R ( t ). If ∆ is a multiple of the period T the sinusoidal signal averages tozero, and this procedure therefore isolates the signal from the background, A cos (cid:18) πT t − φ (cid:19) = R ( t ) − C = R ( t ) − (cid:104) R ( t ) (cid:105) ∆ . (2)As discussed in section 3, the DAMA collaboration followed a procedure along these lines.However, when R ( t ) is not constant, this procedure introduces an artificial modulationwith period ∆. As an example, we illustrate our point by considering the simple case3here the rate varies linearly in time, R ( t ) = C + B t . (3)Subtracting the average of the rate in an interval ∆ centred on t yields the residual S ( t ) ≡ R ( t ) − (cid:104) R ( t ) (cid:105) ∆ = B ( t − t ) , with t − ∆2 < t < t + ∆2 , (4)that becomes a sawtooth wave with period ∆ and amplitude B ∆ / S ( t ) = B ∆ ∞ (cid:88) n =1 ( − n +1 nπ sin (cid:18) πn ∆ ( t − t ) (cid:19) (cid:39) B ∆ π sin (cid:18) π ∆ ( t − t ) (cid:19) + .... , (5)which can be matched to the form of eq. (1) as T = ∆ , A = B Tπ , φ = π πt T . (6)The best-fit sinusoidal wave has an extremum a quarter of period after the beginning ofthe sawtooth. For a decreasing background ( B <
0) this extremum at t peak = T / B >
0) it is a minimum, and the maximum ishalf a period later, at t peak = 3 T / A andphase φ will be modified by the fitting procedure. In the simple case where the durationof the cycles is taken equal to the period of the signal one has A = A + B T π +2 AB Tπ sin (cid:16) φ − πt T (cid:17) , tan φ fit = A sin φ + ( B T /π ) cos(2 πt /T ) A cos φ − ( B T /π ) sin(2 πt /T ) . (7) In order to work out a practical example we consider real data from bibliometrics. Weconsider all publications in high-energy physics (excluding astro-ph) since 1995, and wecompute the bi-monthly averages of their number of references N ref ( t ), and of the numberof citations N cit ( t ) they received up to now, as reported in the InSpire database [34]. Thedata are shown in the left panel of fig. 1 as a function of publication time t . The average This functional form of the total rate was dubbed ‘relaxion’ in [33]. ��� ���� ���� ���� ������������������� ���� � �� � � � �� � � � � � � � � ��� �� �� �� �� � � � � � �� �� - �� %- � % � % � % �� % ����� � � �� �� �� � � � � � � �� � �� � � � �� � � � � � Figure 1. Left:
Average number of citations N cit (red) and of references N ref (blue) of pub-lications in high-energy physics in arXiv as function of time. Both show an annual modulationtogether with a trend on a slower, decennial, time-scale: N cit is decreasing and N ref is growing(smooth curves). Right : Monthly variation computed integrating over the years and subtractingthe slow trend in two different ways: subtracting the discontinuous yearly averages (lighter redand blue) or their smooth averages (darker red and blue). number of references per paper is slowly growing with time because of the overall expan-sion of the field. The number of citations is slowly decreasing with time because morerecent papers have not yet fully accumulated citations. These slow variations produce anapparent annual modulation with fractional amplitude A N ≈ ˙ NN π ≈ , (8)when data are analyzed following the procedure described above ( i.e. by subtracting theyearly averages from N ( t )). The artificial modulation peaks approximately 3 months laterthan the start of the analysis period (January, 1 st ) for citations, since N cit ( t ) grows withtime, and approximately 9 month later for references, since N ref ( t ) decreases with time.A different analysis procedure allows to search for a modulation over a slowly vary-ing background without introducing an artificial modulation that biases the signal. Theresiduals are now computed by subtracting a smooth function that follows the averageslow evolution of the background, rather than the discontinuous yearly averages. Suchfunctions can be obtained, for example, as a 1 st or 2 nd order interpolation of the yearly av-erages. In this language the ‘dangerous’ procedure corresponds to subtracting a 0 th orderdiscontinuous interpolation. The two possibilities are plotted in lighter/darker color infig. 1. The right panel of fig. 1 shows the residuals computed following the two procedures.As expected, the two modulation residuals differ by a few %, and their phase is shifted.The correct procedure shows that N cit oscillates by about 5% and that N ref oscillates byabout 1%; both peak around April. The ‘dangerous’ procedure incorrectly determinesthe modulation amplitudes and their phases. This is, by itself, an original new contribution to bibliometrics. A modulation is present becauseduties, conferences, jobs applications follow a periodic calendar. The modulation amplitude becomesmuch larger and visible by eye if unpublished papers and conference proceedings are included. The DAMA analysis: simulation
We now apply the above considerations to the
DAMA analysis. The experiment is lookingfor an annual modulation in the rate of single-hit scintillation events in NaI crystals, asa possible signal of DM interactions with matter. The total rate is dominated by largebackgrounds due to natural radioactivity of the detector and surrounding environment. Asdiscussed above, a background that evolves slowly on a longer time-scale (say decennial)can simulate an annual modulation if the analysis is performed by subtracting yearlyweighted averages. Since this is the procedure followed by the
DAMA collaboration, westudy the possible impact of this observation on the results.The cumulative total rates expressed in cpd/kg/keVee (where ‘ee’ stands for electronequivalent) have been presented as a function of the electron-equivalent recoil energyin three long-run phases in fig. 1 of [1], [3], and [5] for
DAMA/
NaI,
DAMA/LIBRA
Phase 1, and
DAMA/LIBRA
Phase 2, respectively. A change of the total rate betweenthe three different phases is evident, and is due to improvements in the experimental setupcausing an overall background reduction. However, its time dependence within a singledata-taking phase is not provided by the collaboration, and cannot be inferred from thesedata alone.The time dependence is reported for the residuals. In each of the three long-run phases,the
DAMA residuals are derived from the measured rate of the single-hit scintillationevents, after subtracting the time average of the rate over each cycle (of roughly yearlyduration) such that “the weighted mean of the residuals must obviously be zero over onecycle” (see [1,3,4] and references therein). As discussed in the previous section, if the ratehas the form of eq. (1) with a time-independent background R ( t ) ≡ C , this procedurecorrectly subtracts the constant term. If instead R ( t ) is slowly varying, the secular termcontributes to the residuals, getting transformed to a sawtooth with periodicity equal tothe duration of the DAMA cycles (injected because the analysis procedure is repeatedquite regularly every year). The
DAMA cycles start every year roughly around thebeginning of September, so a growing background can appear like a modulated amplitudepeaked at the beginning of June. On the contrary, the peak of the modulation would bearound the beginning of December for a decreasing background.We now show that a linearly growing background R ( t ) without a periodic term canmimic a modulation that can look like the DAMA signal. We do so by first performing aMonte Carlo simulation, and next computing the residuals of the artificial data followingthe same procedure adopted by
DAMA . The Monte Carlo simulation is done by employing a setup similar to the
DAMA/LIBRA detector. In particular, for both Phase 1 and Phase 2 the total detector mass is taken tobe M = 242 . ����������� � � � � [ � �� / �� / � � � �� ] ��������� ����� ����� ������ χ ���������� / ������ = �� / ��� ���� ���� ���� ���� ���� ���� ���� - ��������� ���� � � � � �� � � � Figure 2. Upper : Monte Carlo events generated assuming no modulation, a slow linear growthin time, B = (0 . π/ yr) cpd / kg / keVee , and the same cumulative rate as DAMA/LIBRA
Phase1 and 2.
Lower : Residuals computed with respect to the smooth average are compatible with nomodulation. in the energy window (2– 6) keVee such that ∆ E = 4 keVee. The exposure of the 7data-taking cycles in Phase 1 is 1.03 ton · yr, and their total duration is ∆ T = 2560 days(table 1 of [4]); the exposure and total duration of the 6 data-taking cycles in Phase 2are 1.13 ton · yr and ∆ T = 2153 days (table 1 of [5]). As a consequence, we assume aconstant duty cycle efficiency ζ = exposure / ( M ∆ T ) of 60% and 80% for Phase 1 andPhase 2, respectively.We generate a number of random events for each day following a Poisson distributionwith mean N ( t ) = ζM ∆ E R ( t ) with R ( t ) = C + B t. (9)We assume that R ( t ) grows linearly with time with a coefficient B = A π/ yr, where A = 0 .
01 cpd / kg / keVee is roughly the amplitude of the DAMA modulated signal. Theconstant C is chosen such that the time average of R over the whole data-taking periodis equal to the DAMA cumulative total rate in the (2–6) keVee energy window: forPhase 1, (cid:104) R (cid:105) = 1 .
03 cpd/kg/keVee in an exposure of 0.53 ton · yr [3] ; for Phase 2, (cid:104) R (cid:105) = 0 .
63 cpd/kg/keVee in an exposure of 1.13 ton · yr [5]. The upper panel of fig. 2shows our simulated rate in cpd/kg/keVee as a function of time. The red lines representthe assumed backgrounds R ( t ) in Phase 1 (on the left) and Phase 2 (on the right). Someshort periods without data points are included consistently with the DAMA/LIBRA data acquisition.We now compute the residuals of the simulated events following the procedure adoptedby
DAMA , i.e. subtracting the average event rate in each cycle. As expected we get asawtooth up to statistical errors. A complication with respect to the ideal analysis of In this case the cumulative total rate is only reported in an exposure of 0.53 ton · yr correspondingto 4 cycles of data-taking that last 1406 days (table 1 of [3]). As a consequence the average of R ( t ) isperformed over a time interval shorter than the total duration of Phase 1, ∆ T = 2560 days. ��� ���� ���� ���� ���� ���� ���� - ���� - ���������������� ���� � � � � �� � � � � � � [ � �� / �� / � � � ] � � � � � � � � � � � � � χ � / ������ = ��� / ��� � �� = ������ ± ��������������� ����� ����� ������ Figure 3.
Residuals obtained from the simulated data of fig. 2 (black points) and computedwithin each data-taking cycle (vertical lines) following the
DAMA procedure. The red curveshows the best fit to the simulated residuals with a DM cosine signal with a period of one yearand peaked on June, 2nd. The zero-signal hypothesis is excluded with a significance of . σ despite that no modulation was assumed in the simulated data. section 2 arises because the DAMA data-taking cycles have slightly irregular durations.The start and end dates of the 7 cycles of Phase 1 and the 6 cycles of Phase 2 can befound in table 1 of [4] and [5] respectively. Subtracting the average rate in the k -th cyclethat lasts ∆ T k = t f,k − t i,k the residuals of eq. (9) follow the irregular sawtooth S ( t ) = ζM ∆ E S ( t ) with S ( t ) = B (cid:18) t − t i,k + t f,k (cid:19) for t i,k < t < t f,k , (10)which is no longer a perfectly periodic function. We collect the residuals in 102 timebins of approximately 1.5 months each, adopting the same binning as the one used inthe DAMA analysis. Fig. 3 illustrates the binned residuals expressed in cpd/kg/keVee.The errors are of the same order of the errors of the
DAMA residuals in the (2– 6) keVeeenergy window.This procedure results in something that looks like a cosine annual modulation. Un-like a true sinusoidal modulation, discontinuities are present between the various cycles.However, the binning procedure can partially wash out these discontinuities: if a time binfalls in two different cycles, as sometimes happens for the
DAMA binning, the sawtoothwill approximately average to zero over the bin, and the residual number of events in thebin will be small. Averaging the rate over 25 crystals with different efficiencies and dutycycles could also have an impact in this respect.The consistency of the residual rate with a DM signal can be quantified by fitting theresidual rate with a cosine with a period T = 1 year, and peaked at t peak = 152 . nd . For our simulated Monte Carlo sample, the fit results in acosine amplitude A = (0 . ± . / kg / keVee, consistent with the injected valueof 0.01 cpd/kg/keVee. The goodness of the fit is given by χ = 141 for 101 degrees offreedom. The cosine signal is preferred over the zero-signal hypothesis at 12 . σ , despitethat no cosine modulation was present in the simulated data.If instead the residuals are recomputed by subtracting the average rate as fitted to a8 ���� ����� ����� ����� ����������������� ��� �� �� �� � ������ � ��������������� [ ���� - � ] � � � � � ��������� ℬ = ⨯ π / ����������� = ���� ��� / �� / ����� �� % ���� �������������� ����� ����� ���� ����� ����� ����� ����� ����� ����� ����� ��������������������� �� �� � ������ � ������ � ��������������� [ ���� - � ] � � � � � �������� �� % ���� ������ ℬ = ( ������ ± ������ ) �� - � ���� / ����� ����� � + � Figure 4. Left:
Power spectra computed from Monte Carlo simulations assuming a sawtooth(blue line and band) and a cosine modulation (red band), with A = B · yr /π = 0 .
01 cpd / kg / keVee .The colored bands represent local σ intervals. The red dashed line shows the global 90% C.L.limit for a sinusoidal signal. Right:
Power spectrum of the
DAMA residuals (black) andannual rate (red) for Phase 1 and Phase 2 combined, taken from [5], together with the 90%C.L. allowed region (green) obtained from Monte Carlo simulations assuming a sawtooth with B = 0 . ± . / kg / keVee / yr as in the combined DAMA/LIBRA best fit of Table 1. linear function of time over the full data-taking period, the residuals (shown in the lowerpanel of fig. 2) are consistent with no modulation within one sigma ( χ / d . o . f . = 94 / A = B · yr /π = 0 .
01 cpd/kg/keVee. We computed the power spectrum following theprocedure adopted by the DAMA collaboration in [5]. In the figure, the colored regionscorrespond to local 1 σ intervals, computed by simulating 100 Monte Carlo samples; thered dashed line shows the global 90% C.L. allowed region, where all the peaks are expectedto fall under the assumption of a sinusoidal signal of given amplitude (eq. (4) of [35]). Weobserve that: (cid:5) The power spectra at frequencies around and above 1/yr are obtained from the residuals computed within each cycle, and collected in bins of one day. The numberof data points used to obtain the spectra is 4341 days, to match the value usedin [5]. We see that both the sawtooth and the cosine produce comparable peaks at ν ≈ / yr. The sawtooth contains extra Fourier modes with frequency ν ≈ n/ yr and n integer: the corresponding power is small because their amplitudes scale as 1 /n (see eq. (5)), and the cycles are not exactly annual. (cid:5) Following [5], the power spectra at low frequencies below 1/yr are computed fromaverage event rates in each (roughly annual) cycle. The main low-frequency peaksare due to the change in the total rate between Phase 1 and Phase 2; they do nothave a statistically significant power because they are based on 7+7 time bins only.9 ��� ���� ���� ���� ���� - ���� - ���������������� ���� � � � � �� � � � � � � [ � � � � � / �� / � � � �� ] � �� ��� �� � �� ��� � � � � � � � � � � � � � �� ������ ��� χ � / ������ = ����� / ����������� ��� χ � / ������ = ����� / ��� Figure 5.
The black data points are the
DAMA residuals in the (2 –
6) keVee energy window,taken from [1, 5]. The curves are fits to a cosine annual modulation peaked on June, 2nd (redcurve), as expected for a DM signal, and to the irregular sawtooth obtained from a continu-ously growing background (blue curve). The roughly annual data-taking cycles of
DAMA/
NaI , DAMA/LIBRA
Phase 1, and
DAMA/LIBRA
Phase 2 are shown as vertical lines. (cid:5)
A power spectrum computed from daily total event rates was not considered in [3–5].In the sawtooth simulation, it would exhibit significant power at low frequencies.To summarize, a sawtooth modulation resulting from taking yearly residuals of a linearlygrowing background is statistically compatible with a sinusoidal modulation, given theavailable data. This conclusion is obtained both by fitting the explicit time series, and byperforming a frequency analysis.
We now consider the real data collected by the
DAMA experiment. The total rate as afunction of time has not been explicitly published, and therefore we cannot directly knowwhether a secular evolution might have affected the determination of the modulationamplitude or its phase. We hence consider the published
DAMA residuals, and showthat their detailed time dependence is consistent even with the most extreme possibility:a sawtooth originating from a slowly growing background. A superposition of cosineand sawtooth, with the main contribution coming from the sinusoidal signal, is howeverpreferred by a global fit.
We again assume a linearly growing total rate as in eq. (3). This assumption is motivatedby the amplitude of the observed modulation seeming constant over the Phase 1 andPhase 2 cycles, which requires a slow constant growth of the rate if the modulation isinterpreted as a sawtooth. Furthermore, a constant growth is a good approximation ifthe time-scale of the secular evolution is long compared to the duration of the data-takingcycles. However, the time dependence of the background is unknown, and details of ouranalysis depend of course on it. 10 ����� ����� ����� ����� ����� ����� ����� - ���������������������������� ������ ��������� [ ��� / �� / ����� ] � � � � �� � � � � � ��� �� � ℬ ⨯ � � [ � �� / �� / � � � �� ] ���� / ���������� � ���� / ���������� ����� / ����� % � �� % � ���� % �������� ( � - � ) ����� - ���� ���� ���� ���� ���������������� ������ ��������� [ ��� / �� / ����� ] � � � � �� � � � � � ��� �� � ℬ ⨯ � � [ � �� / �� / � � � �� ] ���� / ����� ����� � ( � - � ) ��������� / ����� ����� � ( � - � ) ����� ���� / ��� ( � - � ) ������� % � �� % � ���� % �������� Figure 6.
Fit of the DAMA residual rates to a sawtooth plus a cosine with period and phaseas predicted by DM.
Left: fit of residuals in the (2 –
6) keVee energy window for
DAMA/
NaI (gray),
DAMA/LIBRA
Phase 1 (red) and Phase 2 (blue).
Right : fit of residuals in the (2 – energy window for DAMA/
NaI (gray) and
DAMA/LIBRA
Phase 1 (red), and in the (1 –
3) keVee energy window for
DAMA/LIBRA
Phase 2.
The experimental residual rates as a function of time in three different energy intervalsare taken from [3], [4] and [5] for
DAMA/
NaI,
DAMA/LIBRA
Phase 1 and Phase 2respectively. The residual rate in the (2–6) keVee energy interval is extracted from [3, 5]and shown in fig. 5 as black points, together with the timespan of the 20 annual cycles ofthe experiment. We fit all these data points with the following two signal functions:1. A cosine modulation A cos[2 π ( t − t peak ) /T ] with T = 1 yr and t peak = June, 2 nd , aspredicted for a DM-induced annual modulation signal;2. The irregular sawtooth generated by a growing background B t , taking into accountthe details of the DAMA cycles as in eq. (10).In the fit we allow for two different values of the sawtooth amplitude: B LIBRA forthe two phases of
DAMA/LIBRA , and B NaI for
DAMA/
NaI, as both the experimentalapparatus and the total mass of the crystals were different. The best-fit curves for thetwo cases are shown in fig. 5. We find that both the cosine and the sawtooth provide goodfits to the data, as they have a χ per degree of freedom of order one. More precisely, thesawtooth fit has χ / d . o . f . = 137 . / χ / d . o . f . = 118 . / The best-fit value for the cosine amplitude is in perfect agreement with the resultreported by the DAMA collaboration, A = (0 . ± . We follow the counting of degrees of freedom adopted in
DAMA papers, although the time binswithin any given cycle can get correlated by the analysis procedure. ���� ����� ����� ����������������������������� ������ ��������� [ ��� / �� / ����� ] � � � � � � �� � � � � � �� � [ � � � � ] ���� �������� ( � - � ) ���������� ��� ������� � = � �������� + �������� �� % � �� % � ���� % �������� ����� ����� ����� ����� ����� ����� ����������������� ������ ��������� [ ��� / �� / ����� ] � � � � � � �� � � � � � �� � [ � � � � ] ���� / ����� ����� � ( � - � ) ������������ � = � ������� ��������� + ���������� % � �� % � ���� % �������� Figure 7. Left:
Fit of the cosine amplitude and its phase to
DAMA residuals, allowing for asawtooth component (colored regions) or setting it to zero (gray regions).
Left: combined fit inthe (2 –
6) keVee energy window.
Right: fit of
DAMA/LIBRA
Phase 2 residuals in the (1 – energy window. sawtooth, the best-fit values of the B coefficients are B NaI = (0 . ± . / kg / keVee / yr , (11a) B LIBRA = (0 . ± . / kg / keVee / yr , (11b)for DAMA/
NaI and
DAMA/LIBRA , respectively, which correspond to a yearly growthof the total rate of a few percent. Allowing for different coefficients for Phase 1 andPhase 2, and even for each individual cycle, gives similar results.The DAMA collaboration also reports residual rates in different energy windows. Weperform a fit in all the energy intervals of
DAMA/
NaI,
DAMA/LIBRA
Phase 1 andPhase 2, and we find that both the cosine and the sawtooth give acceptable fits to thedata. The two theoretical hypotheses result in similar χ / d . o . f . in all cases, except forthe higher energy bin of DAMA/LIBRA
Phase 2 where a cosine is favoured. We alsoprovide combined results with a single B for the three phases in the (2– 6) keVee energyinterval where this is possible. The results of all our fits are summarized in table 1.Keeping in mind that both the cosine and sawtooth provide acceptable fits to the data,we compare the two hypotheses by means of a likelihood ratio test. Indeed, when fittingdata with many degrees of freedom, a likelihood-ratio test is a more powerful statisticalindicator than the χ test. The two tests answer different questions: the χ compares theconsidered model to a generic model, such that all statistical fluctuations contribute. Onthe other hand, the ∆ χ compares two specific models such that only those statisticalfluctuations that discriminate among the two models contribute. We perform a ∆ χ fitof the data to a generic superposition of a sawtooth plus a cosine. We fix the period of12itted Fit to a cosine modulation Fit to a secular variationdata A [cpd/kg/keVee] χ / d . o . f . B [cpd/kg/keVee/yr] χ / d . o . f . DAMA/NaI (2-4) keVee . ± . . /
36 0 . ± . . / (2-5) keVee . ± . . /
36 0 . ± . . / (2-6) keVee . ± . . /
36 0 . ± . . / (2-4) keVee . ± . . /
49 0 . ± . . / (2-5) keVee . ± . . /
49 0 . ± . . / (2-6) keVee . ± . . /
49 0 . ± . . / (1-3) keVee . ± . . /
51 0 . ± . . / (1-6) keVee . ± . . /
51 0 . ± . . / (2-6) keVee . ± . . /
51 0 . ± . . / (2-6) keVee . ± . . /
101 0 . ± . . / (2-6) keVee . ± . . /
138 0 . ± . . / Table 1.
Best-fit values of the DM cosine amplitude A and the sawtooth coefficient B obtainedby fitting the DAMA residuals in different energy intervals, together with the corresponding χ / d.o.f. Our fits of the cosine amplitudes agree with those reported by DAMA/LIBRA . the cosine to one year, and its peak to June 2nd, as predicted for a DM-induced annualmodulation signal, so that the only free parameter is the amplitude A . The sawtooth alsohas one free parameter, the slope B , since its phase and duration are fixed by the DAMA analysis choices. Fig. 6 shows the results of the fit as 68%, 95% and 99.7% allowed contoursof A and B . On the left we consider the (2– 6) keVee energy interval for the three phasesof DAMA , while on the right we fit data in the lower energy window ((2 – 4) keVee forboth
DAMA/
NaI and
DAMA/LIBRA
Phase 1, and (1– 3) keVee for
DAMA/LIBRA
Phase 2). In the (2 – 6) keVee interval the earlier
DAMA/
NaI data resemble more asawtooth, while the more precise later Phase 2 data favour a cosine-dominated fit. Datain the lower energy interval are fitted equally well by both possibilities, a cosine or asecular variation.Finally, we consider the cosine phase as an additional free parameter, still keeping theperiod fixed to one year, and quantify how much a possible sawtooth component relaxesthe determination of the cosine amplitude and phase. In the left panel of fig. 7 we showthe regions of A and t peak preferred at 68%, 95% and 99.7% C.L. by the combined DAMA data-set in the (2– 6) keVee energy interval, both allowing for a free sawtooth component13 � � � �� - �������������������� ������ [ ����� ] = ℬ ⨯ ( � � / π ) [ � �� / �� / � � � �� ] ���� / ����� ������ + ���� / ��� �� �������������� � � � � �� - �������������������� ������ [ ����� ] = ℬ ⨯ ( � � / π ) [ � �� / �� / � � � �� ] ���� / ����� ������ �� �������������� Figure 8.
Energy dependence of the modulated signal observed by
DAMA . The black pointsare the amplitudes of the cosine signal reported by DAMA in energy bins [5]; forcomparison we also show the DM cosine amplitudes in the wider energy intervals discussedabove (red points), and our best-fit values for a linearly growing background (blue points). Theblack curves fit the data assuming an exponential energy dependence.
Left:
Combination of
DAMA/LIBRA
Phase 1 and
DAMA/
NaI.
Right:
DAMA/LIBRA
Phase 2. or setting it to zero. In the right panel we show the same but only for
DAMA/LIBRA
Phase 2 in the (1– 3) keVee energy interval. The best-fit regions assuming zero sawtooth(in gray) reproduce the results reported by
DAMA (see e.g. fig. 15 of [5]). Allowing fora sawtooth component substantially affects the best fit regions. In particular, the phaseof the cosine becomes more uncertain and can more significantly differ from the valuecharacteristic of a DM signal, especially when fitting data at lower energy.The
DAMA collaboration analyzed the data by adopting two additional methodsthat are potentially relevant to indirectly clarify the time-dependence of the background.In [3–5] a frequency analysis of the residuals has been presented. In section 5 of [5] thisresult is complemented at frequencies below about 1/yr with a spectral analysis of theyearly averaged total rate. We have summarized relevant details of the procedure at theend of our section 3.1. We report in the right panel of fig. 4 the power spectrum ofthe combined
DAMA/LIBRA
Phase 1 and Phase 2 data, taken from [5], as black andred curves for the high and low frequencies, respectively. This power spectrum can becompared to what expected from a sawtooth or cosine annual modulation (shown in theleft panel of fig. 4). The green region in the figure shows the global 90% C.L. upper limitobtained from Monte Carlo simulations, assuming a sawtooth with coefficient B as in ourbest fit to DAMA/LIBRA data of eq. (11). One can see that both the sawtooth andcosine hypotheses are compatible with the experimental data, reproducing the peak at afrequency of about 1/yr, without giving significant extra power at different frequencies.In addition, a direct fit of signal plus background to the total event rate has beenperformed (see e.g. section 6 of [5]). In this case, however, the constant part of thebackground — b jk in the notation of [5] — is left free within each cycle [36]. This iseffectively equivalent to fitting residuals, and does not avoid the issues discussed here.14 � � � �������������� ������ [ ����� ] � � � � � � � � � [ � �� / �� / � � � �� ] ���� / ����� ����� � + ���� / ������� / ���������� � � � � � �������������������� ������ [ ����� ] Δ � [ � �� / �� / � � � �� ] � � ( � ) = + ℬ �� � ( � ) = �������� Figure 9. Left:
Rate averaged over all data-taking time for
DAMA/LIBRA
Phase 2 (bluepoints), and for
DAMA/LIBRA
Phase 1 and
DAMA/
NaI combined (red points).
Right:
Difference in the total rate between the end of Phase 1 and the beginning of Phase 2, calculatedassuming both a constant rate (black points), and a background that linearly grows with time(green points). In the latter case the values of B and C are chosen in order to reproduce themodulated signal and the cumulative rates; the errors come from the uncertainty in the residuals. We now discuss the possible energy spectrum of the background rate R ( t ) of eq. (3).The DAMA collaboration provided the energy dependence of the cosine signal ampli-tude (see for example fig. 10 of [5]), showing separately the results for DAMA/LIBRA
Phase 2, and for the combined
DAMA/
NaI and
DAMA/LIBRA
Phase 1. We reportthese data in the two panels of fig. 8 in black. The decrease of the rate with energyis approximately exponential as expected for DM-induced recoils. Above 6 keVee themodulated signal amplitude is compatible with zero.We cannot repeat our fit over the whole energy range to extract the full energy de-pendence of the B coefficient, since the detailed time-dependence of the modulation isavailable only in the few energy bins discussed above. We however notice that in thesethree energy bins the values of B determined by fitting the data with a sawtooth, if com-pared with the cosine fit of DAMA , are close to the na¨ıve expectation B (cid:39) A π/T (seeeq. (7)). This is shown in fig. 8, where the red points represent the amplitude of the mod-ulated rate reported by DAMA , while the blue points correspond to our best-fit values ofthe sawtooth coefficient (rescaled by π/ year). As one can see, the two sets of points areroughly compatible with each other, especially for the earlier experimental phases, and inthe lower energy bins. We therefore use the detailed energy spectrum provided by DAMA(black points in the figure) as an approximate estimate of the energy dependence of B . The constant part of the background C is determined by the energy spectrum of thetotal rate averaged over the whole data-taking period, shown in the left panel of fig. 9.These cumulative total rates differ between DAMA/LIBRA
Phase 1 (red points taken Since data is provided only for the combined
DAMA/
NaI and
DAMA/LIBRA
Phase 1 datasets,we consider a single B for the two setups, as opposed to what was done in table 1. C which is trivially given by the difference of the cumulativerates. If a slowly growing term B t is present, by averaging over the data-taking periodone gets that the variation of the total rate between the end of Phase 1 and the beginningof Phase 2 is given by∆ R ≡ R ( t , end ) − R ( t , start ) = R − R + B T + ∆ T ) , (12)where R , , and ∆ T , are the average rate and total duration of the two phases. Sincethe energy spectra of R , and B are different, we report ∆ R as a function of recoil energyin fig. 9 (right) for the two cases where a secular term is absent (black), or is fittedfrom the modulated signal (green). We do not give a physical explanation on the energydependence of ∆ R .Two comments are in order. First, the change in the averaged rate between the phasescould in principle be entirely due to the secular term, if B <
0. A slowly decreasingbackground, however, would produce an oscillation peaked on December, in counterphasewith the observed modulation. Second, the high-energy tails of the black points in fig. 9(right) are approximately constant; one could therefore wonder whether the full differencecould be explained by the energy dependence of the secular term B plus a constant shift∆ R . This is indeed possible for a positive B with a spectrum peaked at low energies(below ∼ A slow variation of the rate could, in principle, be produced by some signal or by somebackground. We here elaborate on the second possibility. The background in under-ground detectors in the keV energy range has spectral features and time behaviours thatare poorly understood. We first consider possible slowly growing backgrounds, as theirpresence would give an apparent
DAMA modulation peaked around June, as shownabove. Various possible effects may create an increasing background, especially in a fixedenergy window around few keV, as in the Region of Interest (RoI) of the
DAMA experi-ment. We list a few generic examples of possible increasing backgrounds, not necessarilyrelated to the specific case of study: (cid:5)
Increase due to out-of-equilibrium physical effects. A classical example, althoughrelevant at different energies, is the increase of the
Po, from a
Pb initial con-tamination in the transient phase after a broken equilibrium in the contaminationchain Pb → Bi → Po. An increase of the backgrounds in NaI crystals could also arisefrom the diffusion of dirty materials from the surface into the bulk, as well as exter-nal gas cumulative contamination, as e.g.
Rn. Other relevant out-of-equilibrium16hysical effects related to an increase of the backgrounds might be possible andcan in principle be tested by dedicated analyses. A quantitative study of the back-grounds in NaI crystals has been reported in several papers by
DAMA (see [37]and references therein) and more recently by other collaborations ( e.g. [38]). (cid:5)
Increase due to instrumental effects. These are related to the photo-multipliers(PMT) and electronics, and they are more difficult to isolate with respect to theprevious ones. For example, helium gas permeating through PMT glass increasesthe after-pulse rate [39] creating fake low energy signals simulating the real scintil-lation. These are not easily rejectable through simple pulse shape discriminations,as suggested in [40] (although with completely different implications that we do notconsider in this work). Another example is the possible increase of the electronicnoise leaking in the acceptance region after pulse shape discrimination due to ageneric loss of performance of the hardware. (cid:5)
Apparent increase due to a general degradation of the detector resolution. This ispotentially relevant in a fixed energy window close to the software threshold. Herethe limit conditions of the reconstruction performances in terms of efficiency andreliability are usually reached.We next consider the opposite scenario where the background is slowly decreasingwith time. In this case the analysis procedure adopted by
DAMA would transform thebackground into a sawtooth that mimics an annual modulation peaked around December,as discussed in section 2. Then, a decreasing background cannot explain the annualmodulation reported by
DAMA . However, the sawtooth could destructively interferewith additional periodic signals, in particular possibly shifting their phase, and a real DMsignal would be distorted and attenuated.A decrease of the total background rate could arise from different radioactive contam-inants present in NaI crystals at the beginning of the data-taking as short and mediumterm cosmogenic activations or intrinsic out-of-equilibrium contaminants [6, 41, 10, 7, 42].The most critical ones are those with a half-life T / of a few tens of years, comparable tothe total time of data-taking, and an energy spectrum in the RoI of the detector. Exam-ples of possible decreasing backgrounds in NaI crystals in the keV energy range include: • Decay of relatively short-lived isotopes. For example,
Cd with T / = 463 daysand Na with T / = 2 . DAMA because they have fully decayed after few years of data acquisition. • Decay of medium-lived isotopes with a time-scale of order ∼
10 yr. Typical con-taminants that could be present at the beginning of data-taking are the cosmogenic H with T / = 12 . Pb with T / = 22 . β -spectrum peaked around 3 keV and with an endpoint of 18.6 keV.17eing produced by cosmogenic activation, this contaminant is present in NaI crys-tals (grown in above-ground facilities) proportionally to the time exposure betweentheir production and installation underground [43].Let us stress that here we do not want to make any claim about the presence or absenceof a given background, and even less to provide an explanation for the DAMA total rate.Since a careful reconstruction of the rate is strongly hampered by the poor knowledge ofthe background details, it would be extremely useful to directly check whether the
DAMA background underwent a secular evolution. The
COSINE (see e.g. [9]) and
ANAIS (see e.g. [7]) experiments recently started data-taking using NaI crystals, and reported thetotal rates of each crystal as a function of time. The rates of these new experiments show,in their starting phase, a significant time-dependence.
The
DAMA collaboration reported an annual modulation in the single-hit scintillationevents that peaks around June, 2 nd , as predicted by a DM signal. The statistical signifi-cance of the DAMA signal is large, roughly 13 σ , such that the annual modulation couldbe in principle visible even by simply plotting the event rate binned as a function of time.However, the DAMA collaboration reported residuals computed by subtracting theweighted average of its rate over cycles of roughly one year. This procedure can bedangerous, because a possible secular variation of the total rate would be transformedinto a sawtooth with period coinciding with the duration of the
DAMA analysis cycles.As a result, a slowly time-dependent rate, even if not oscillating, becomes a possiblesource of apparent modulation. Within the statistical accuracy, the modulation possiblyproduced by the procedure can resemble a cosine with period of one year. Since thedata-analysis cycles usually start around September, the apparent cosine would peak atthe beginning of June if the background slowly grows with time.In section 3 we considered the most extreme possibility: a slowly and steadily growingbackground that fully reproduces the observed modulation amplitude in
DAMA . Bymeans of a Monte Carlo simulation, we have shown that the DM signal can be mimicked,within statistical errors, by a growth of the total rate on decennial time-scale. A rate thatdecreases with time, on the other hand, would suppress the oscillating signal.The
DAMA collaboration so far has not presented data about the time-dependence ofits total rate, that can confirm or exclude the presence of a slowly changing background.We therefore tried to clarify the issue through the available indirect means in section 4.1.The detailed time dependence of the published residuals can be fitted by an irregularsawtooth with a χ / d . o . f . ≈
1, see fig. 5 and table 1. The available data are consistentwith the modulation being interpreted as an artefact due to a yearly growth of the totalrate of a few percent. A DM cosine provides a somehow better fit to parts of the data-set— specifically the higher energy bins of DAMA/LIBRA Phase 2 — therefore preferring a18M-like signal as a dominant explanation of the observed modulation. A time-dependentbackground rate would have an impact on the signal also in this case, since it would affectthe extraction of the cosine amplitude and phase from the data, as shown in fig. 7. Thiseffect can be important since the position of the sinusoid peak plays an important role inthe interpretation of the modulation as a DM signal. A spectral analysis of the residualsand of the annual averaged rate is found to be consistent with a slow variation of the rate,as well as with a true annual modulation.Finally, in section 4.2 we discussed the energy spectrum of the possible slowly-varyingcomponent of the rate, which must be peaked at low energy, and in section 4.3 we specu-late about possible origins of such a background. We do not attempt at giving any realisticexplanation of a possible time-dependence of the
DAMA total rate; we rather just men-tion a few simple physical and instrumental effects that could produce a background thatgrows with time in the
DAMA region of interest. It is worth mentioning that the
ANAIS and
COSINE experiments recently started data-taking with NaI crystals similar to the
DAMA ones, and presented their rates as function of time, finding (in their initial phase)a sizeable time dependence [7, 9].In conclusion, from the data available to us, we could not exclude the possibility thatthe cosine modulation claimed by DAMA is biased by a slow variation in the rate, possiblydue to some background or signal. The most estreme possibility of no cosine modulationseems disfavoured, but we could not safely exclude it. More in general, a slowly increasingor decreasing rate would partially enhance or mask a true annual modulation, respectively.The potential significance of our observation could, of course, be directly clarified bythe measurement of the
DAMA total rate as function of time. The absence of an apprecia-ble variation of the rate over the time-scale of the experiment would exclude the relevanceof the effects discussed in this paper for the
DAMA analysis. A second possibility to testour observations would be to recompute the residual rate following a different procedure,such as subtracting the average background choosing cycles with different starting pointsor different durations. Even better, the above issues would be avoided by performing adetrend with continuous modelling of the possible time dependence of the background —such as a polynomial fit or interpolation — rather than a discontinuous average. Anotherpossibility is to perform a signal processing in which, along with the original time series,both power spectrum and phase information are shown.The direct detection of Dark Matter interactions in underground laboratories is oneof the main challenges of contemporary particle physics. The observation of a DM sig-nal would be an important discovery, so that any alternative interpretation needs to beexplored with care. We therefore hope that an effort will be put by the experimentalcollaborations to settle the potential issues presented here.
Acknowledgements
This work was supported by ERC grant NEO-NAT and MIUR contracts 2017FMJFMW and 2017L5W2PT(PRIN). We thank Sabine Hossenfelder, Aldo Ianni, Marcello Messina and Tobias Mistele for discussions. eferences [1] DAMA
Collaboration, “Search forWIMP annual modulation signa-ture: Results from DAMA/NaI-3 andDAMA/NaI-4 and the global combinedanalysis” , Phys. Lett. B480 (2000) 23[InSpire:Bernabei:2000qi].[2]
DAMA
Collaboration, “Dark mattersearch” , Riv. Nuovo Cim. 26N1 (2003) 1[arXiv:astro-ph/0307403].
DAMA
Collab-oration, “Dark matter particles in theGalactic halo: Results and implicationsfrom DAMA/NaI” , Int. J. Mod. Phys. D13(2005) 2127 [arXiv:astro-ph/0501412].[3]
DAMA
Collaboration, “First results fromDAMA/LIBRA and the combined re-sults with DAMA/NaI” , Eur. Phys. J. C56(2008) 333 [arXiv:0804.2741].[4]
DAMA
Collaboration, “Final modelindependent result of DAMA/LIBRA-phase1” , Eur. Phys. J. C73 (2013) 2648[arXiv:1308.5109].[5]
DAMA
Collaboration, “FirstModel Independent Results fromDAMA/LIBRAPhase2” , Universe 4(2019) 116 [arXiv:1805.10486].[6]
ANAIS
Collaboration, “Analysis of back-grounds for the ANAIS-112 dark matterexperiment” , Eur. Phys. J. C79 (2019) 412[arXiv:1812.01377].[7]
ANAIS
Collaboration, “First Results onDark Matter Annual Modulation from theANAIS-112 Experiment” , Phys. Rev. Lett.123 (2019) 031301 [arXiv:1903.03973].[8]
ANAIS
Collaboration, “ANAIS-112 sta-tus: two years results on annual modu-lation” [arXiv:1910.13365].[9]
COSINE
Collaboration, “Search for aDark Matter-Induced Annual ModulationSignal in NaI(Tl) with the COSINE-100Experiment” , Phys. Rev. Lett. 123 (2019)031302 [arXiv:1903.10098].[10]
COSINE
Collaboration, “Study of cosmo-genic radionuclides in the COSINE-100NaI(Tl) detectors” , Astropart. Phys. 115(2020-02) 102390 [arXiv:1905.12861]. [11]
COSINE
Collaboration, “Comparison be-tween DAMA/LIBRA and COSINE-100in the light of Quenching Factors” , JCAP1911 (2019) 008 [arXiv:1907.04963].[12] G. Tomar, S. Kang, S. Scopel, J-H.Yoon, “Is a WIMP explanation of theDAMA modulation effect still viable?” [arXiv:1911.12601].[13] Particle Data Group, Phys. Rev. D 98 (2018)030001.[14] P. Ullio, M. Kamionkowski, P. Vogel, “Spindependent WIMPs in DAMA?” , JHEP0107 (2000) 044 [arXiv:hep-ph/0010036].[15] D. Tucker-Smith, N. Weiner, “Inelastic darkmatter” , Phys. Rev. D64 (2001) 043502[arXiv:hep-ph/0101138].[16] M. Fairbairn, T. Schwetz, “Spin-independent elastic WIMP scattering andthe DAMA annual modulation signal” ,JCAP 0901 (2008) 037 [arXiv:0808.0704].[17] J. Kopp, T. Schwetz, J. Zupan, “Globalinterpretation of direct Dark Mattersearches after CDMS-II results” , JCAP1002 (2009) 014 [arXiv:0912.4264].[18] M. Farina, D. Pappadopulo, A. Strumia,T. Volansky, “Can CoGeNT and DAMAModulations Be Due to Dark Matter?” ,JCAP 1111 (2011) 010 [arXiv:1107.0715].[19] N. Fornengo, P. Panci, M. Regis, “Long-Range Forces in Direct Dark MatterSearches” , Phys. Rev. D84 (2011) 115002[arXiv:1108.4661].[20] K. Blum, “DAMA vs. the annually modu-lated muon background” [arXiv:1110.0857].[21] E. Del Nobile, C. Kouvaris, P. Panci, F. San-nino, J. Virkajarvi, “Light Magnetic DarkMatter in Direct Detection Searches” ,JCAP 1208 (2012) 010 [arXiv:1203.6652].[22] P. Panci, “New Directions in Direct DarkMatter Searches” , Adv. High Energy Phys.2014 (2014) 681312 [arXiv:1402.1507].
23] C. Arina, E. Del Nobile, P. Panci, “DarkMatter with Pseudoscalar-Mediated In-teractions Explains the DAMA Signal andthe Galactic Center Excess” , Phys. Rev.Lett. 114 (2015) 011301 [arXiv:1406.5542].[24] R. Catena, A. Ibarra, S. Wild, “DAMA con-fronts null searches in the effective the-ory of dark matter-nucleon interactions” ,JCAP 1605 (2016) 039 [arXiv:1602.04074].[25] P. Gondolo, S. Scopel, “Halo-independentdetermination of the unmodulated WIMPsignal in DAMA: the isotropic case” ,JCAP 1709 (2017) 032 [arXiv:1703.08942].[26] J. Herrero-Garcia, A. Scaffidi, M. White,A.G. Williams, “Time-dependent rate ofmulticomponent dark matter: Repro-ducing the DAMA/LIBRA phase-2 re-sults” , Phys. Rev. D98 (2018) 123007[arXiv:1804.08437].[27] B.M. Roberts, V.V. Flambaum, “Electron-interacting dark matter: Implicationsfrom DAMA/LIBRA-phase2 andprospects for liquid xenon detectorsand NaI detectors” , Phys. Rev. D100(2019) 063017 [arXiv:1904.07127].[28] S. Kang, S. Scopel, G. Tomar, “ADAMA/Libra-phase2 analysis in termsof WIMP-quark and WIMP-gluon effec-tive interactions up to dimension seven” [arXiv:1910.11569].[29] V.A. Kudryavtsev, M. Robinson, N.J.C.Spooner, “The expected background spec-trum in NaI dark matter detectors andthe DAMA result” , Astropart. Phys. 33(2009) 91 [arXiv:0912.2983].[30] J.P. Ralston, “One Model ExplainsDAMA/LIBRA, CoGENT, CDMS, andXENON” [arXiv:1006.5255].[31] R.W. Schnee, “Introduction to dark matterexperiments” [arXiv:1101.5205].[32] D. Nygren, “A testable conventional hy-pothesis for the DAMA-LIBRA annualmodulation” [arXiv:1102.0815]. [33] P.W. Graham, D.E. Kaplan, S. Rajendran, “Cosmological Relaxation of the Elec-troweak Scale” , Phys. Rev. Lett. 115 (2015)221801 [arXiv:1504.07551].[34] High-Energy Physics Literature
InSpire
Database. A. Holtkamp, S. Mele, T. Simko,and T. Smith (
InSpire collaboration), “
In-spire: Realizing the dream of a global digitallibrary in high-energy physics ”, 2010. See also“INSPIRE-HEP Documentation” (2018) bythe
InSpire -HEP Collaboration.[35] G. Ranucci, M. Rovere, “Periodogram andlikelihood periodicity search in the SNOsolar neutrino data” , Phys. Rev. D75(2006) 013010 [arXiv:hep-ph/0605212].[36] Private communication.[37]
DAMA
Collaboration, “Dark Matter in-vestigation by DAMA at Gran Sasso” ,Int. J. Mod. Phys. A28 (2013) 1330022[arXiv:1306.1411].[38] G. Adhikari et al., “UnderstandingNaI(Tl) crystal background for darkmatter searches” , Eur. Phys. J. C77 (2017)437 [arXiv:1703.01982].[39] “Photomultiplier tubes”, Hammatsu Pho-tonics K.K., 2007.[40] D. Ferenc, D. Ferenc ˇSegedin, I. FerencˇSegedin, M. ˇSegedin Ferenc, “Helium Mi-gration through Photomultiplier Tubes –The Probable Cause of the DAMA Sea-sonal Variation Effect” [arXiv:1901.02139].[41]
ANAIS
Collaboration, “Performance ofANAIS-112 experiment after the firstyear of data taking” , Eur. Phys. J. C79(2019) 228 [arXiv:1812.01472].[42] B. Suerfu, M. Wada, W. Peloso, M. Souza,F. Calaprice, J. Tower, G. Ciampi, “Growthof Ultra-high Purity NaI(Tl) Crystal forDark Matter Searches” [arXiv:1910.03782].[43] J. Amare et al., “Cosmogenic produc-tion of tritium in dark matter detec-tors” , Astropart. Phys. 97 (2018-01) 96[arXiv:1706.05818]., Astropart. Phys. 97 (2018-01) 96[arXiv:1706.05818].