Long-term monitoring of the ANTARES optical module efficiencies using $^{40}\mathrm{K}$ decays in sea water

Cherenkov light induced by radioactive decay products is one of the major sources of background light for deep-sea neutrino telescopes such as ANTARES. These decays are at the same time a powerful calibration source. Using data collected by the ANTARES neutrino telescope from mid 2008 to 2017, the time evolution of the photon detection efficiency of optical modules is studied. A modest loss of only 20% in 9 years is observed. The relative time calibration between adjacent modules is derived as well.


Introduction
The ANTARES neutrino telescope [1] aims at the exploration of the high-energy Universe by using neutrinos as cosmic probes. Three main search methods are exploited. The first one refers to searches for steady neutrino sources, through the identification of an excess of events from a given sky direction [2]. The second one searches for an excess of high-energy cosmic neutrinos over the background of atmospheric events [3]. The third method looks for temporal and/or spatial correlations with transient events observed through other probes. An example of this last method is offered by the searches for neutrinos in concomitance with GW170817 [4]. In addition, the ANTARES detector is used for particle physics studies, such as neutrino oscillations [5], search for magnetic monopole [6], and indirect searches for dark matter [7,8].
For all these studies, a continuous monitoring of the positioning of all parts of the detector is needed in order to preserve the precision on the reconstruction of neutrino-induced events. Moreover, it is necessary to check the stability of the optical sensors as a function of time, in order to guarantee an equable estimate of the energy released by charged particles. Studies on the performance as well as on the stability of optical sensors of this kind have been discussed also by other Collaborations (e.g. [9,10]). The originality of the present paper relies on the use of a natural calibration tool offered by the presence of radioactive elements dissolved in sea water.
The ANTARES neutrino telescope is located in the Mediterranean Sea, 40 km off the coast of Toulon, France, at a mooring depth of about 2475 m. The detector was completed in 2008. It is composed of 12 detection lines, each one equipped with 25 storeys of 3 optical modules (OMs), except line 12 with only 20 storeys of OMs, for a total of 885 OMs. The horizontal spacing among the lines is ∼60 m, while the vertical spacing between the storeys is 14.5 m. Each OM hosts a photomultiplier tube (PMT), whose axis points 45 • downwards (see Figure 1). The PMTs are 10-inch tubes from Hamamatsu [11]. The relative position of each OM in the detector is monitored in real time by a dedicated positioning system [12]. All signals from the PMTs that pass a threshold of 0.3 single photoelectrons (hits) are digitised and sent to the shore station [13,14].
The decay products of radioactive elements dissolved in sea water constitute the principal source of background light for deep-sea neutrino telescopes. Among these isotopes, potassium-40 ( 40 K) is the most abundant. Other radioactive decays (mainly from the U/Th chain) induce Cherenkov photons on the permille level compared to 40 K decays and can be neglected. This process constitutes an important calibration tool as well. If a 40 K nucleus decays near a storey, the resulting Cherenkov light can be recorded by two OMs almost simultaneously. Such coincidences are used to derive the relative photon detection efficiencies [15] and for time calibration between OMs in the storey.
The document is organised as follows: in Section 2 the 40 K decay rate in sea water is computed and its stability is discussed; the method for the OM photon detection efficiency computation using 40 K decays is described in Section 3, together with the data sample employed, while the results of the analysis are presented in Section 4. A brief explanation on the use of the 40 K decay measurements for the time calibration of the detector is given in Section 5, while in Section 6 conclusions are presented.
2 40 K decay rate at the ANTARES site The main decay channels of 40 K are: → 40 Ar + γ The electron produced in the β-decay channel, with an energy up to 1.3 MeV, leads to the production of Cherenkov light when traveling in water. In the electron capture channel, fast electrons with subsequent Cherenkov light emission are produced by Compton scattering of the 1.46 MeV photon, released by the excited Ar nuclei.
The detection rate, R s , of Cherenkov photons from products of 40 K decays on an ANTARES optical module can be factorised as: with B q the 40 K decay rate per unit volume and V s the effective volume around a single OM for detecting a single photon from a 40 K decay. As there is no preferred decay direction, V s can be calculated in a semi-analytical way, by integrating over the decay positions around the OM. One obtains, for a given wavelength λ, a factorisation into a wavelength-dependent effective area A(λ) and the corresponding photon absorption length L abs (λ) of sea water. V s can then be written as: with A s = A(λ)Φ(λ)L abs (λ)dλ/Φ tot L 0 abs , with Φ(λ) containing the -arbitrarily normalised -λ dependence of the Cherenkov photon flux (close to 1/λ 2 ), Φ tot its integral over a given wavelength range and L 0 abs the absorption length at some reference wavelength. A(λ) is determined by simulating an isotropic photon flux around an OM and depends on the OM properties such as the size of its photocathode, its quantum efficiency, and its angular acceptance. These simulations yield A s = 420 ± 50 cm 2 , where the error is dominated by uncertainties in the OM properties. The absorption length at the ANTARES site has been measured to L 0 abs = 60 ± 8 m for λ = 470 nm [16]. If a 40 K nucleus decays near a storey, the associated Cherenkov light can be recorded by two OMs almost simultaneously: this kind of signal is referred to as genuine coincidence. Its rate can be written as: where V c is an effective volume around a pair of OMs for the detection of a coincident signal from a single 40 K decay. The value of V c is derived from simulating 40 K decays around a pair of OMs and propagating the resulting Cherenkov photons through the sea water. These simulations yield V c = 1100 ± 370 cm 3 . Whereas 40 K decays can contribute to the rate of a single OM, R s , up to a distance of ∼ L 0 abs , contributions to the rate of genuine coincidences, R c , are confined to a small volume in the vicinity of the storey, with 90% occuring within 3 m. At such a distance, the effect of absorption is below 5% and the associated uncertainties become negligible. The dominating error for R c originates from the uncertainties on the OM properties.
Both R s and R c depend on B q , with: where N A is the Avogadro number, A = 39.96 [17] and τ 1/2 = 1.25×10 9 years [18] are the atomic mass and lifetime of 40 K, and ρ = 1.038 g/cm 3 is the density of deep-sea water at the ANTARES site, which is derived from in situ measurements of pressure, temperature and salinity, r s . The parameter r K is the potassium fraction in Mediterranean Sea salt and r I is the isotope fraction of 40 K. The last three quantities can be considered stable over the lifetime of ANTARES with variations smaller than 1%. The salinity is monitored directly with the ANTARES instruments and found to be r s = 3.844%, while r K = 1.11% from [19] and r I = 1.17 · 10 −4 from [20].
With this value, R s = 35 ± 8 kHz and R c = 15 ± 5 Hz are the single and genuine coincidence rates obtained from simulations. The observed single photon rates in ANTARES OMs are about 55 kHz, due to additional light from bioluminescence that cannot be filtered out at the single photon level. Therefore single photon rates are difficult to exploit as calibration source and the rest of the paper is devoted to the analysis of the coincident signal.
3 Detection efficiency calibration using 40 K Data collected from mid 2008 to December 2017 have been analysed in this work. From the completion of the detector in May 2008 until November 2009, dedicated runs with 40 K triggers have been taken once per week, with a downscaling factor of 4 in order not to saturate the readout data acquisition system. The 40 K trigger has then been integrated into the standard data-taking setup, with a down-scaling factor of 200. This trigger selects coincident photons in adjacent OMs if they are detected within a narrow time window of typically 50 ns. In order to get similar statistics, the dedicated runs belonging to the same month have been merged together. After the integration of the 40 K trigger into the standard run setup, instead, each month of data taking has been divided into five periods.
Both the genuine coincidences induced by the same 40 K decay process and random background coincidences, which are due to distinct 40 K decays as well as to bioluminescence effects, contribute to hit pairs on adjacent OMs with small time differences ∆t. Figure 2 shows an example of the distribution of the time difference between hits on two adjacent OMs. A Gaussian peak from genuine coincidences is clearly visible on top of a flat pedestal from uncorrelated coincidences.
The distribution of the coincidence signals is fitted with a Gaussian distribution added to a constant: where p is the baseline, a the amplitude of the Gaussian peak due to genuine coincidences, σ is the peak width and t 0 the residual time offset between the hits on adjacent OMs. A value of σ ∼ 4 ns is expected, mainly due to the distance between the OMs and the spatial distribution of detected 40 K coincidences around the storey. A simple estimation can be obtained by considering the difference in the distance traveled by two photons emitted at the same position and detected by two different PMTs. In ANTARES the distance between the centers of photocathodes of two PMTs of the same storey is l = 1.0 m (see Figure 1), while the photocathode diameter d is about 25 cm. Therefore, neglecting light scattering, the maximum traveled path difference for two photons is l + d =  For perfectly calibrated OMs, t 0 would be expected at 0 ns. Deviations from the expected value of t 0 are mainly due to different PMT transit times. This makes the 40 K method an integral part of the time calibration procedure, as will be discussed in more detail in Section 5.
The fit parameters can be used to estimate the number of events corresponding to the peak area, as: where ∆τ = 0.4 ns is the bin width used for the histogram. For each storey three coincidence rates are measured (R 01 , R 12 and R 20 ). These quantities are directly related to the photon detection efficiency of the three OMs ( 0 , 1 and 2 ): where R * c is the rate for two nominal OMs with efficiencies equal to 1. A value of R * c = 15 Hz is used. This value is obtained as an average detector coincidence rate at the beginning of the analysed data set, and coincides with what is found in simulation. Solving the system of three equations, the corresponding efficiencies are derived: When an OM in a storey is not working for a given period, only one coincidence rate is measured, which is not sufficient to determine the two efficiencies. In this case, equal efficiencies for the two working OMs are assumed, namely: If two OMs in a storey are inactive no coincidences can be measured. In this case the average efficiency value of the line hosting that particular storey is assigned to the working OM for this period.
All coincidence histograms for all periods have been fitted according to Equation 5. Data quality selection criteria have been applied, to ensure stable and reliable input for the subsequent efficiency calculation. A cut on the number of entries on each histogram, N entr , excludes from the analysis all those cases for which the fit fails due to lack of statistics. This includes the cases, for instance, in which one of the two OMs is only partially active. Taking into account that there are four fitted parameters and 120 bins in each histogram, a χ 2 of 116 is expected for a good fit. Histograms with χ 2 200 are excluded. Additional cuts on the fitted amplitude value and its uncertainty, ∆a, have been applied to ensure a clear signal above the background. Furthermore, expected values of the Gaussian mean and width are known, thus cuts on these parameters have been applied. All the selection criteria applied are reported in Table 1.
Accepted Values N entr > 2000 χ 2 < 200 a > 0.1 Hz ∆a/a < 0.1 1.5 ns < σ < 6.5 ns |t 0 | < 20.0 ns Table 1: Values of data selection criteria applied in the analysis. The parameter nomenclature corresponds to the one used in Equation 5.
After applying these cuts, the efficiency can be determined on average for 77% of those OMs which are active in a given period. The efficiency of the remaining working OMs is derived from adjacent OMs or from the average of the corresponding detection line as described above.

Results
Histograms passing the quality criteria are then used to compute the OM photon detection efficiency, as described in Section 3. Figure 3 shows the photon detection efficiency as a function of time for one OM as an example. The uncertainty on the resulting photon detection efficiency of each OM has been calculated from the uncertainty on the fitted parameters and it is found to be around 3% for all analysed periods. The total uncertainty is dominated by the statistical error. The systematic uncertainties (discussed later in this section) contribute on average only about 20% of the total uncertainty.  In order to monitor the status of the whole detector over several years, the average photon detection efficiency as a function of time has been determined. For each period, the average is computed over all OMs with non-zero efficiency. The result is shown in Figure 4. It can be seen that, after a decrease over several years, the OM photon detection efficiency has finally stabilised over the last years. An overall modest detection efficiency loss of 20% is observed over the whole analysed time period.
The distribution of the detected charge is regularly monitored for all PMTs and if either a significant broadening or a shift from the nominal position of the peak due to a single photoelectron is noticed a recalibration is performed by means of high voltage tuning (HVT). The HVT procedure adjusts the effective gain of individual PMTs to the nominal one; the procedure thus prevents detection effeciency losses due to a low gain and bias on the trigger logics. The effects of the HVT procedure, which is performed once or twice per year, are evident on the time dependence of shown in Figure 4. When averaging the efficiencies of individual PMTs over the whole detector, two statistical quantities are considered: the standard deviation and the error on the mean. The standard deviation is given by: where N is the total number of considered OMs for a given period and i is the efficiency of OM i in that particular period. Thanks to the HVT procedure, the σ std remains stable at the order of 10%, as can be seen in Figure 5, where this quantity is shown for all analysed periods. This justifies the fact that we used the average efficiency of the OMs on a given line for those working OMs whose efficiency cannot be computed through the 40 K calibration. The statistical error shown in Figure 4 is the error on the mean, defined as: This quantity is typically of the order of 1%. Possible systematic uncertainties on the individual OM efficiencies, based on the assumption on the Gaussian shape for the distribution of the coincidence rates, have been considered as well. In addition to the intrinsic width of the coincidence peak due to the arrival of photons from a 40 K decay process, the shape of the distribution is also affected by photon scattering and the time response of the PMTs. In order to account for these effects, the analysis considers an additional Gaussian term with a larger width compared to that already shown in Equation 5. Using as genuine coincidence rate the sum of the areas under the two Gaussians, the systematic uncertainty has been evaluated as the difference between the resulting efficiencies from the two procedures. Generally it is found that the area under the leading Gaussian is equal to the one of the one Gaussian fit, and the area under the second Gaussian is significantly smaller than the one under the leading Gaussian.
It is worth mentioning that the effects due to some known artefacts from a typical PMT response [11], such as delayed hits, remain undetectable in the narrow time window used in this work. The calculated efficiencies exclude these hits, which are later added in the ANTARES simulation chain.

Time calibration
In order to meet the target angular resolution of the detector, a time syncronisation between all detector components better than 1 ns is required [21]. To achieve such precision a master clock system, located onshore, provides a common reference time to all the offshore electronics, via a network of optical fibers. Further calibration is needed for the delay between the time when the hit is detected and the photon arrival time at the photocathode of the PMT. These time calibrations were performed during the detector construction and are continuously monitored on a weekly basis [21]. Two systems of external light sources are used: LED beacons located on four storeys on each detector line, and laser beacons located on the base, at the bottom of several detection lines. They allow to perform both inter-line and inter-storey time calibrations [22]. Reconstructions of tracks from downward going muons created in cosmic-ray interactions in the atmosphere are used as well to determine inter-line and inter-storey time calibrations [23]. The 40 K method completes the time calibration chain by providing intra-storey timing. All three methods combined assure a precision level of 0.5 ns for each individual PMT.
In Figure 6 the distribution of the fitted time offset, t 0 , obtained from the 40 K coincidence histogram of one OM pair is shown as a function of time.  It can be seen that between subsequent HVT procedures the time difference between two adjacent OMs is stable and its value can be monitored to better than 0.5 ns. These values, which are produced for each OM pair, serve as input for the intra-storey calibration procedure.
The standard deviation of the time offset distribution, averaged over the whole detector, is illustrated in Figure 7. The apparent trend is an increase of the standard deviation as a function of time (from ∼ 2 to ∼ 4.5 ns in 9 years). This increase could be correlated to the HVT procedure, which is performed in order to keep the OM detection efficiency at their best.

Conclusions and outlook
Using data collected by the ANTARES neutrino telescope with a dedicated 40 K trigger, the photon detection efficiencies for all OMs have been computed from mid 2008 to the end of 2017. The paper presents the stability of a PMT based detector in the hostile environment of the deep-sea, for the longest period ever recorded. It demonstrates that future underwater experiments can remain in operation for timescale of at least a decade without major efficiency degradation. An average decrease of the OM efficiency by 20%, as observed from 2008 to 2017, implies a loss of only 15% in the detection efficiency of an astrophysical signal with a full sky E −2 spectrum.
The results of this study serve as input for detailed Monte Carlo simulations of the ANTARES detector, which include a realistic simulation of the OM efficiencies in each data taking run. The 40 K method is also part of the time calibration of the detector.
This procedure can be also exploited in KM3NeT, the next-generation neutrino telescope in the Mediterranean Sea [24]. KM3NeT will consist of two main detectors, ARCA in Sicily, devoted to high energy astroparticle physics, and ORCA in France, focused on few-GeV atmospheric neutrino studies. They both use a configuration similar to the one of ANTARES, but with 31 instead of three PMTs on each storey. This will allow to collect not only double coincidences from 40 K decays, but also higher multiplicities, improving the technique to determine the photon detection efficiencies.