On an unverified nuclear decay and its role in the DAMA experiment

The rate of the direct decay of 40K to the ground state of 40Ar through electron capture has not been experimentally reported. Aside from its inherent importance for the theory of electron capture as the only such decay known of its type (unique third-forbidden), this decay presents an irreducible background in the DAMA experiment. We find that the presence of this background, as well as others, poses a challenge to any interpretation of the DAMA results in terms of a Dark Matter model with a small modulation fraction. A 10ppb contamination of natural potassium requires a 20% modulation fraction or more. A 20ppb contamination, which is reported as an upper limit by DAMA, disfavors any Dark Matter origin of the signal. This conclusion is based on the efficiency of detecting 40K decays as inferred from simulation. We propose measures to help clarify the situation.


I. INTRODUCTION
A 40 K concentration of 0.0117(1)% [1] in naturally occurring potassium, nat K, is an ubiquitous source of radioactivity. Its decays in solidified rocks are responsible for the air's argon content and in a human body it is the element with one of the largest activities. The chemical similarity of sodium and potassium is not only responsible for a per mille fraction of nat K in seawater but may ultimately be the reason why trace amounts of potassium can also be found in ultra-radiopure scintillating crystals grown from NaI powders. These crystals are employed in rare event searches, such as in Dark Matter (DM) direct detection experiments where radioactive backgrounds pose one of biggest challenges limiting sensitivity.
The decay scheme of 40 K is shown in Fig. 1. All kinematically available energy levels in 40 Ar and 40 Ca are populated in the 40 K decay: 1. The dominant mode is the β − decay to the calcium ground state with a half-life T 1/2 (β − ) = 1.407(7) × 10 9 yr and an endpoint energy of Q β − = 1311.07 (11) keV.
3. Positron emission to the ground state of 40 Ar is possible. It has been measured [2] and found to have a small branching BR β + = 0.00100(12)%. 4. Last but not least is EC directly into the ground state of 40 Ar. Little is known empirically about this decay and the current branching quoted in the * jpradler@pha.jhu.edu † ndgroup@mcmaster.ca ‡ iyavin@perimeterinstitute.ca  [1]. The dashed line depicts the direct decay to the ground state of 40 Ar for which no dedicated measurement has been reported. The branching ratio quoted is extracted from the measured β + decay rate and the extrapolated theoretical ratio EC/β + . literature, BR EC = 0.2(1)% [1], is based on the measurement of BR β + and a theoretical extrapolation from the EC to positron ratio for 1 st and 2 nd forbidden-unique transitions.
Experimentally, the most accessible modes are (1) and (2) because the respective electron or gamma-ray are readily detectable and numerous measurements exist. The quoted numbers are world-averages obtained in [3]. The observation of positron emission (3) is challenging because of its low intensity and pair-production background, but it was first reported in [2]. In contrast to the above modes, the rare EC to the ground state of 40 Ar in (4) has never been reported from a dedicated measurement. Since it is a transition from J π ( 40 K) = 4 − to J π ( 40 Ar) = 0 + it is classified as unique third-forbidden (3U) and it is the only such EC decay known [4,5]. As such, its empirical verification would bring an important arXiv:1210.5501v2 [hep-ph] 26 Mar 2013 closure to the theory of unique forbidden decays.
The energy release involved in this special decay is carried almost entirely by the emitted neutrino. Only about 3 keV of energy is observable from the Auger electron emission and the X-ray yield upon K-shell capture. As such, this decay forms an important irreducible background to the DAMA [6] experiment which looks for DM scattering in the laboratory. Since the experiment does not employ any discrimination between electron/gamma and nuclear-recoil activity, the release of only 3 keV associated with the rare EC decay of 40 K shows up directly in the region of interest for dark matter.
The purpose of this letter is to point out that-despite the omnipresent nature of 40 K-a dedicated measurement of one of its decay modes, namely the EC directly to the ground state of 40 Ar, remains outstanding. We begin by discussing what is currently known about this decay, carefully separating what is empirically established from what is based on a theoretical extraction. We then discuss the importance of this background in the DAMA experiment and how it affects the interpretation of the claimed results. We close with a brief discussion of how this special decay may be measured directly.

II. 40 K RARE EC DECAY
Given the absence of a reported measurement for the EC mode, its branching ratio, BR EC , can be inferred from the theoretical ratio BR EC /BR β + . The branching ratio of β + decay, BR β + , has been measured directly. The rational behind this is that to leading order the atomic and nuclear pieces factorize and the branchings BR EC and BR β + are dominated by the same nuclear matrix element, which then cancels in the ratio. Hence, the calculation of the ratio BR EC /BR β + becomes essentially a question of accurately modeling the atomic wave functions of the electron/positron and the neutrino involved.
Explicitly, the rate for positron emission is given by where g is the Fermi constant times the cosine of the Cabibbo angle; p (E) is the positron momentum (energy) with endpoint p 0 (E 0 ); the neutrino momentum is p ν = E 0 −E. The shape factor C 3U contains the nuclear matrix element M 3U and the Coulomb correction in the form of the Fermi function F (Z, E). The neutrino and positron are predominantly emitted in the triplet state allowing for one unit less in lepton orbital angular momentum when mediating a nuclear spin change of J = 4. Denoting by j ν and j e their respective angular momenta, where the sum is subject to the condition j ν + j e = J; g A is the axial vector coupling and R denotes the nuclear radius. We follow [7] in the evaluation of the atomic part.
Turning to EC, this process mainly occurs through Kshell capture and the rate is to a good approximation given by 1 where f K contains the 1S amplitude for the bound state radial electron wave function and Here q K is the momentum of the emitted neutrino. Equations (1) to (4) show that the nuclear matrix element and the strong sensitivity to the nuclear radius cancel in the ratio λ EC /λ β + = BR EC /BR β + . Using Eqs. (1) to (4) directly we find a value BR EC /BR β + = 190 when using the atomic data collected in [8]. When using the approximations to the shape factor given in [9] we find a slightly smaller value of BR EC /BR β + = 150. Early evaluations [2] used λ EC /λ β + = 155 without further reference to literature. The most recent one [1] uses This is in good agreement with what we find using the direct calculation above, however it is important to note that this latter ratio is an extrapolated theoretical expectation and not computed from the above equations.
The LOGFT computer program [10] used in nuclear data evaluations cannot compute the EC/β + for the 3U ratio in (5). Instead the ratios to unique first-(1U) and second-forbidden (2U) transitions are calculated, BR EC /BR 1U = 8.51 (9) and BR EC /BR 1U = 45.20(47), respectively. Assuming a constant increase by the same factor, BR EC /BR 3U = BR EC /BR β + = 240 is obtained from which the value in (5) has been adopted.
There is no doubt of the success of leading order theoretical predictions based on the V-A theory of weak interactions in explaining weak decay strengths and ratios observed throughout the periodic table. Yet, the 40 K decays to the ground states of 40 Ar and 40 Ca are the only known 3U transitions realized in nature [4,5]. Hence, for the EC in question it is not obvious how to truly ensure the validity of (5), e.g. by gauging it against measurements of other 3U transitions. Theoretical attempts to match the calculated β + and β − strengths to actual measurements yield discrepancies by a factor ∼ 3 for 40 K(β − ) to 40 Ca and a factor ∼ 6 for 40 K(β + ) to 40 Ar when simple nuclear shell models are employed [11]. These discrepancies are very likely the result of inaccuracies in the evaluation of the corresponding nuclear matrix elements (which cancel when forming the ratio), but they make it difficult to feel confident about the theoretical estimation in Eq. (5). At the very least, they remind us of the importance of an empirical verification.
It is also possible to obtain a theory-independent estimate of BR EC from measurements of the total half-life of 40 K. Indeed, this is facilitated by a recent high-precision measurement [12], T 1/2 = 1.248(9) × 10 9 yr. 2 Neglecting positron emission we can write, and using the measured values for the β − and EC * branches one obtains, Given the large uncertainties it is clear that the available lifetime measurements are currently not sensitive enough to pin down the EC strength with reasonable accuracy. This special branch thus deserves its own dedicated measurement. Aside from its inherent significance, the precise activity of this branch also forms an important background in the DAMA experiment as we discuss in the following section.

III. POTASSIUM BACKGROUND IN DAMA
The DAMA experiment is situated in the Gran Sasso underground laboratory and searches for DM scattering off nuclei with a target made of 25 NaI(Tl) crystals amounting to almost 250 kg in mass [13]. The detector registers energy depositions between 2 keV to tens of MeV of almost any source: electrons, gamma-and Xrays, muons, alpha particles, and nuclear recoils. Events of the last class with recoil energies below a few keV ee 3 are expected from the scattering of DM particles in the galactic halo against the target nuclei if the DM mass is 10 GeV. Importantly, no discrimination procedure to veto against the other types of recoils is employed. Instead, the most compelling feature of the reported result is the annually modulating "single-hit" 4 event rate between 2 − 6 keV ee with a phase that is compatible with the phase expected from DM (June 2). The collaboration reports the residual recoil rate from which the average count rate of a cycle was subtracted. 5 The residual rate exhibits an annually modulating pattern that can be nicely fitted to the model S m cos ω(t − t 0 ) with ω = 2π/(1 yr), a modulation amplitude of S m 0.04 cpd/kg in the 2 − 4 keV bin and a phase of t 0 ≈ 140 days. This is compatible with the expectation from a Maxwellian DM halo velocity distribution (see for example the recent review [14]).
However, knowledge of the modulated rate, S m , without a detailed insight into the unmodulated rate, R 0 , makes the interpretation of the signal in terms of DM difficult. What is reported is that R 0 ∼ 1 cpd/kg/keV for the spectrum below 10 keV. Essentially no information is available as to what it is comprised of. Radioactive backgrounds in rare event searches arise predominantly from naturally occurring radioactive isotopes (such as 40 K) in the detector material and its surroundings, from cosmogenically activated elements (e.g. 129 I), and from elements in the natural uranium ( 238 U) and thorium ( 232 Th) decay chains.
DAMA addresses all those background sources in [13] where they quote the concentrations of the different radio-isotopes. The main shortcoming of this study is that the influence on the low energy single-hit spectrum is not addressed quantitatively. The observed rate below 10 keV is essentially flat, with the exception of a bump around 3 keV. Flat spectra are typical for beta decays due to the Coulomb corrections at small electron emission energies as discussed below. Bumps in the keV energy region point towards EC. It has been speculated for a long time that the bump seen at 3 keV may very well be associated with K-shell electron capture of 40 K. The nuclear recoil in this decay is negligible and the scintillator detects the entire K-shell electron binding energy of 40 Ar (3.2 keV) released in the form of X-rays and Auger electrons. It is important to note that the position of the peak coincides with the maximum of S m . Although the DAMA collaboration insists that these two bumps are of different origin, the presence of a poorly understood background in the signal region is at least unsettling.
Based on the reported levels of radioactivity, the authors of [16] were the first to try and evaluate the resulting rate in the DM signal region. Their main conclusion was that the unmodulated rate seems to require higher than reported concentrations of some of the isotopes. Moreover, in-situ contaminations of the crystals were found to dominate the low-energy spectrum as external radio-impurities are limited by the prominent backscatter peak they produce for 40 keV. Fig. 2 shows the unmodulated "single-hit" spectrum: the three (red) line segments are reported event rates by DAMA [13]. The gray line is taken from [16] and shows the simulated DAMA "single-hit" spectrum and simulated backgrounds. The three disjont pieces (in red) are DAMA reported rates from [13] and [15]. The statistical uncertainty in the signal region is negligible. The gray line is the spectrum from 129 I, 238 U, and 232 Th obtained in [16] with respective concentrations 0.2 ppt, 5 ppt, and 1.7 ppt. The blue lines show the various 40 K contributions as detailed in the text with a nat K contamination level of cK = 10 ppb. The highest energy α-peak in the gray line is vetoed by the DAMA DAQ system [13]. spectrum from 238 U, 232 Th, and 129 I in-situ decays. The broad peak between 40 − 100 keV is from 129 I with little contribution elsewhere. Hence, the low-energy count rate is dominated by 238 U and 232 Th resulting in an essentially flat spectrum below 10 keV. The (quenched) α-decays in those chains result in peaks above 2 MeV, which are used by DAMA to derive the reported concentrations, but those are found to be insufficient to explain the observed rate below 10 keV. Though the contributed amount from 238 U and 232 Th at low energies is limited by the observation of the spectrum at high energies ( few MeV), it is important to recognize that the contribution from 40 K decay is not similarly constrained.
DAMA reports an upper limit on the contamination level of nat K as c K ≤ 20 ppb [13]. They do so by using the decay of 40 K to the excited state of 40 Ar to look for double-coincidence events where the 3 keV is registered in one crystal and the 1460 keV gamma-ray is registered entirely in another crystal. The contamination level is obtained by dividing the observed double-coincidence rate by the probability for such events to occur. The latter was inferred from Monte Carlo (MC) simulation but no detailed description of this procedure has been provided. Given the indirect nature of the nat K determination and the lack of knowledge of potential systematic uncertainties, it may well be possible that the potassium contamination is larger than what is reported.
40 K contributes to the low energy spectrum in three distinct ways: i) The direct EC decay to the ground state of 40 Ar, which is the principal subject of this paper, contributes solely to the bump at 3.2 keV.
ii) The electron emission associated with the 40 K to 40 Ca decay contributes a flat background that extends all the way up to the end-point energy of 1311 keV.
iii) The EC decay to the excited state of 40 Ar results in the same low energy contribution to the 3 keV bump, but it is followed by the emission of 1460 keV gamma-ray. This decay contributes to the singlehit rate at low energy only when the 1460 keV gamma-ray escapes undetected.
The direct decay (i) is the easiest to calculate since it only depends on the contamination level of nat K in the NaI crystals and the branching ratio of this decay. If we assume a contamination level of c K then the activity from 40 K decays to the ground state is given by, = 0.11 kg NaI day Here N A is Avogadro's number, M u = 1 g/mol is the molar mass constant, T 1/2 is the total lifetime of 40 K, c 40 = 0.0117(1)% is the 40 K fraction in nat K [1], A K = 39.0983(1) is the atomic weight of potassium [17]; c K = 20 ppb corresponds to the upper limit as reported by DAMA [13]. The decay will be perceived as a mono-energetic event at the 40 Ar K-shell binding energy of E K = 3.2 keV. The signal shape is hence dominated by the energy resolution of the detector σ(keV) = 0.448 √ E + 0.0091E [13] where E is in units of keV. We find good agreement with the observed shape of the background and the total rate in the energy bin [E min , E max ] then reads The β − decay to 40 Ca , contribution (ii), is also easy to account for since the emitted electron is entirely contained in the crystal where the decay happened. Thus, the spectrum is found from the 3U shape factor for 40 Ca. 6 At low energy this is a fairly flat background as shown in Fig 2. The shoulder present near the kinematical endpoint of 1311 keV might offer a straightforward way to estimate the level of 40 K contamination level if it can be clearly observed above the other backgrounds. Such a determination would obviate the need to rely on MC for the purpose of determining the concentration of nat K. Our results indicate that given the background levels from other sources (mainly 238 U and 232 Th), this shoulder could be observed or a useful upper bound can be obtained. Unfortunately, the DAMA collaboration has not released the spectrum at this energy range. Assuming that the bump at 3 keV is entirely due to background we predict that a shoulder with a height of about 0.04 cpd/kg/keV should be seen at around 700 keV, associated with the β − decay of 40 K .
Finally, a MC simulation is necessary to estimate the rate associated with (iii) where the 1460 keV gammaray escapes entirely undetected. This was done independently in Ref. [16] where the total contribution from (i), (ii) and (iii) was taken into account with BR EC = 0.2 %. In order to investigate the effect of different values of BR EC we subtracted the contribution of (i) and (ii) from the total 40 K spectrum quoted in Ref. [16] to obtain the spectrum associated with the EC decay to the excited state, contribution (iii). From Fig. 2 we find that (iii) is indeed the dominant source of 40 K background at E R ∼ 3 keV by a factor of about 5 × (0.2/BR EC ). Given the importance of this background it would be interesting to compare this number with DAMA-internal MC simulations.
We close this section by noting that the flat spectrum at low energies associated with the 3U transition to 40 Ca is a universal feature of β − decay. In the non-relativistic limit, the product of Fermi function and phase space volume in β − decay reads, where η = Zα/v is the Sommerfeld parameter and v is the asymptotic electron velocity; for β − decay of 40 K, η > 1 for E 200 keV and the last equality is approximately constant in this energy regime. We suggest that the essentially flat low-energy event rate seen in DAMA (more than 20 data-points reported up to E R ≤ 10 keV and between 20 − 30 keV with negligible error-bar) is strongly suggestive of the presence of such a constant background component (e.g. through an unaccounted βemitter or from low-energy Compton background) at a level of 7 B flat 0.85 cpd/kg/keV. (9) In what follows, we will assume that this is indeed the case and work out the implications for a successful DM interpretation given that potassium is present as well. Hence, B 0 = B flat + B 40 where B 40 is the low-energy background from 40 K. 7 An extended discussion regarding this hypothesis can be found in an addendum to this paper [18].

IV. DM SIGNAL INTERPRETATION IN THE PRESENCE OF BACKGROUNDS
We now proceed to investigate the influence of backgrounds on the DAMA DM signal claim. To better quantify the issue let us write R 0 = B 0 + S 0 where B 0 is background and S 0 is the unmodulated contribution of any tentative DM signal. These quantities depend on E R but are assumed to be stationary in time. The total event rate in the presence of a DM signal is then given by 8 The observed modulating fraction of the event rate has its maximum at a recoil energy of approximately 3 keV, and in the (2 − 4) keV energy bin, Written in this form, it becomes clear that an understanding of the background B 0 is essential for a successful interpretation of the DAMA results. A viable DM model must reproduce the observed modulation amplitude, and in particular have sufficiently large modulation amplitude near 3 keV ee recoil energy. Given a certain background level B 0 we have where s max m ≡ S m /S 0 | 2−4 keV is the maximum modulation fraction. For very low background levels (B 0 /S 0 1) the requirement on the signal's modulation amplitude s max m ≥ 2% can be easily satisfied by most models of DM that aspire to explain the DAMA result. However, considering the bump around 3 keV in the unmodulated rate, it is almost certainly the case that the background levels are not that low.
Using the background contributions just discussed, including the flat contribution in Eq. (9), we compute the required modulation fraction in (2 − 4) keV from Eq. (12) as a function of both the branching ratio BR EC and the contamination level of nat K; The unmodulated signal rate, S 0 , is required to explain the observed single hit rate in that bin, R 0 2.2 cpd/kg. Contours of the required fraction are shown in Fig. 3. Since the contribution from EC into the excited state of 40 Ar (case (iii) in the previous section) is obtained from MC and its uncertainty is unknown we consider two scenarios: fractions including this contribution are shown as solid contours whereas fractions neglecting this contribution are shown as dotted contours. In the respective shaded regions s max m would be larger than 100% and these regions are therefore excluded. As is clear from the figure, any model with small modulation fraction of a few % is already strongly disfavored by the data. Light DM models, which usually predict ∼ 10% modulation fraction (see e.g. [20]), are also in tension with the data. The viability of these models can be better assessed with a measurement of BR EC and a more thorough investigation of the contamination levels from nat K. Finally, inelastic DM models [21] such as MiDM [22] can yield a large modulation fraction ( 30 %) due to the heightened sensitivity to the galactic escape velocity. Such models are unlikely to be ruled out through our simple considerations. It is however remarkable that if the DAMA quoted bound on nat K is indeed saturated (c K = 20 ppb) and the strength of background (iii) is adequately caught by the MC simulation, then the required modulation fraction cannot be attained by any DM model. The necessity of a more comprehensive discussions of backgrounds in the signal region by the DAMA collaboration is evident.

V. COMMENTS ON REALIZING A MEASUREMENT OF THE RARE DECAY OF 40 K
Given the above, a direct measurement of the branching ratio of the rare electron capture decay of 40 K directly to the ground state of 40 Ar is clearly desirable. The very low energy (∼ 3keV) released make such a measurement challenging and higher concentrations of 40 K compared with the natural abundance is likely needed. It is possible to obtain enriched potassium samples with a 40 K concentration of 14% and higher [23]. Such a high concentration would lead to an activity at the level of so that statistics is unlikely to be the limiting factor of any effort to measure BR EC . The main difficulty in measuring the rare decay directly to the ground state is the small branching ratio compared with the more common EC decay 40 K → 40 Ar * (1460). As discussed before, if the 1460 keV γ-ray escapes the detector, only the deposition of 3 keV of energy is registered, which mimics the direct decay. However, with an additional surrounding anti-coincidence veto, this background becomes reducible. Given that the ratio of branching ratios of the two decays is ∼ 0.2%/10% = 2% the veto efficiency only needs to be somewhat better than 1%. Whether this is a realistically attainable efficiency is beyond our expertise to determine. The low energy release of ∼ 3 keV is not an easy energy range for detection, but NaI(Tl) crystals such as the ones used by the DAMA collaboration have been demonstrably sensitive in this range. Germanium based detectors such as the ones employed by the CoGeNT collaboration [24] have an even lower threshold and better energy resolution. A precise measurement of the electron capture decay of 40 K seems possible. Finally, we note that scintillating crystals can also be grown from KI(Tl) powders and have in fact been used in measurements of the 40 K decay scheme [25]. Because of the relatively smaller scintillation light output in comparison with NaI(Tl), it remains to be proven if sensitivity at 3 keV can be attained.
Finally, there has recently been some renewed interest in the exotic possibility of temporal variations in nuclear decay rates [26]. The data used is rather old and suspect, and there has since been several refutations of these claims [27][28][29][30]. However, we point out that no conclusive exclusion of such an effect has been reported in the case of EC decay rates. 9 Thus, it might be of some interest 9 Ref. [28] searched for correlation between the Sun-Earth distance to the ratio of 22 Na/ 44 Ti. 44 Ti decays via EC, whereas 22 Na does so only ≈ 10% so in principle this ratio can be sensitive to variations associated with EC. A power-spectrum analysis of the data (which [28] did not perform) seems to reveal significant power at a period of one year and therefore such variations cannot be robustly excluded.
to search for such modulations in the electron capture decay of 40 K directly to the ground state of 40 Ar. This is especially interesting given the unique nature of this rare decay and its direct relevance to the annual modulations claimed by the DAMA collaboration.

VI. CONCLUSIONS
In this letter we have identified a branch of 40 K decay which is yet experimentally unverified and whose strength is only estimated from theory. Aside from its intrinsic importance as the only such known decay of its kind, this branch has important ramifications for DM direct detection. The in-situ presence of 40 K is well established in the DAMA detector although the precise contamination level is not clear. Interestingly, it presents a background in the very signal region from where the collaboration derives its claim for DM detection. Before closing we would like to highlight some of our findings: • The 40 K EC decay to the ground state of 40 Ar-a third forbidden unique transition and the only one known of its kind-lacks a dedicated measurement to-date.
• Nuclear data evaluations use extrapolations to infer the associated branching. Although we find good agreement when using leading-order theory, a direct experimental verification is nevertheless much desired and called for.
• Depending on the actual concentration of 40 K, a DM explanation of the DAMA signal based on elastic scattering and a Mawellian halo velocity profile may already be excluded.
• With a potassium concentration of 10 ppb (which is below the DAMA inferred upper limit), its β − decay may very well dominate the spectrum at 1 MeV. The 3U shape with its kinematical shoulder could then be used as an independent measurement of the 40 K concentration.
Based on the above findings we propose the following steps to help improve the situation: • A dedicated measurement of the 40 K EC decay into the ground state-potentially over an extended period of time to exclude the possibility of tempo-ral variations-is itself a missing piece in the experimental verification of leading order weak decay calculations and will help to settle the role of this decay in the DAMA experiment.
• We propose that the concentration levels of nat K can be inferred from the activity level associated with the β − shoulder of 40 K at around 1 MeV. This method has been verified impressively in [31] for a NaI prototype crystal by the ANAIS collaboration. A count rate in DAMA much greater than 0.04 cpd/kg/keV will severely undermine a DM interpretation of the signal. This assumes that the other reported concentrations are reliable and that a flat background at a level of 0.85 cpd/kg/keV is present.
• The DAMA spectrum between 10-20 keV should be released as it 1) helps to clarify the hypothesis of a flat β − backgrounds and 2) potentially allows to identify further EC elements if other K-shell capture "bumps" were present.
• We urge the collaboration to provide more details concerning the probability of coincidence events when the 1460 keV gamma-ray from 40 K decay escapes one crystal but is detected elsewhere. A comparison with the independent study by [16] could yield precious insights into the reliability of the MC models and corroborate the determination of nat K through the coincidence method.
• Likewise, showing the α-peaks in the spectrum above ∼ 2 MeV could reaffirm the average levels of 238 U and 232 Th determined by DAMA and clarify their contributions to the count rates in the DM signal region.