Limits on Dark Matter Annihilation in the Sun using the ANTARES Neutrino Telescope

A search for muon neutrinos originating from dark matter annihilations in the Sun is performed using the data recorded by the ANTARES neutrino telescope from 2007 to 2012. In order to obtain the best possible sensitivities to dark matter signals, an optimisation of the event selection criteria is performed taking into account the background of atmospheric muons, atmospheric neutrinos and the energy spectra of the expected neutrino signals. No significant excess over the background is observed and $90\%$ C.L. upper limits on the neutrino flux, the spin--dependent and spin--independent WIMP-nucleon cross--sections are derived for WIMP masses ranging from $ \rm 50$ GeV to $\rm 5$ TeV for the annihilation channels $\rm WIMP + WIMP \to b \bar b, W^+ W^-$ and $\rm \tau^+ \tau^-$.


Introduction
A number of independent observations in cosmology and astrophysics point to the existence of large amounts of nonbaryonic matter in the Universe [1,2]. These observations indicate that there is approximately five times more of this dark matter than of ordinary baryonic matter.
A well-motivated hypothesis is that dark matter is composed of weakly interacting massive particles (WIMPs) that form halos in which galaxies are embedded. There are different candidates for these WIMPs, amongst which, those provided by supersymmetric models are currently the focus of the attention of a large variety of searches. In the case of the minimal supersymmetric extension of the Standard Model (MSSM), the lightest new particle is stable due to the conservation of a quantum number, the R-parity, that prevents its decay to ordinary particles [3]. If this lightest supersymmetric particle is also electromagnetically neutral, it is a natural WIMP candidate for dark matter. This lightest particle can annihilate into pairs of standard model particles. Neutrinos, in particular, are the final product of a large variety of decay processes, being therefore a good candidate for an indirect search for dark matter. WIMPs tend to accumulate in celestial objects due to scattering with ordinary matter and the gravitation pull of these objects. This is why indirect searches for dark matter concentrate on massive astrophysical bodies such as the Earth, the centre of our Galaxy, galaxy clusters or, as in this case, the Sun.
In this letter, an indirect search for neutrinos coming from WIMP annihilations in the Sun is presented, using data recorded by the ANTARES neutrino telescope from 2007 to 2012. Different quality cuts on the data have been used to reduce the atmospheric background and optimise the sensitivity of the analysis. Sensitivities to the signal neutrino flux, Φ ν , and the spin-dependent and spin-independent WIMP-nucleon cross-sections, σ p SD and σ SI , are derived using three different annihilation channels.

The ANTARES neutrino telescope
The ANTARES detector [4,5] is an undersea neutrino telescope anchored 2475 m below the surface of the Mediterranean Sea and 40 km offshore from Toulon (France) at 42 • 48 ′ N and 6 • 10 ′ E. ANTARES consists of 12 detection lines with 25 storeys per line and 3 optical modules with 10 ′′ photomultipliers per storey. The detection lines are 450 m long and 60-75 m apart horizontally. Data taking started in 2007, when the first five lines of ANTARES were installed. The detector installation was completed in May 2008.
The main channel through which neutrinos are detected is via the muons produced from high-energy muon neutrinos interacting inside, or in the vicinity of, the detector. These muons move at relativistic velocities and induce the emission of Cherenkov 1 Institut Universitaire de France, 75005 Paris, France 2 Also at INFN-Bari 3 Also at APC light that is then detected by the optical modules. In this analysis, only muon neutrinos detected this way will be considered. In the following any mention of 'neutrinos' will refer to muon neutrinos and muon antineutrinos.
The flux of atmospheric muons from above the detector comprises the largest part of the background, with fluxes several orders of magnitude larger than any expected signal. In order to reduce the number of atmospheric muons, a cut on the elevation of reconstructed muon tracks is applied, ensuring that only events that have been reconstructed as upgoing are used. Since muons cannot cross the entire Earth, this cut rejects all atmospheric muons except for a small fraction of misreconstructed events. The atmospheric neutrinos represent the irreducible background for this search.
The expected neutrino energy spectra from WIMP annihilations in the Sun are calculated with the WIMPSim simulation package [12]. The code takes into account the absorption of neutrinos in the solar plasma and the neutrino oscillation inside the Sun and on their way from the Sun to the detector. Neutrino spectra are calculated for 15 WIMP masses ranging from 50 GeV to 5 TeV and three annihilation channels assuming a branching ratio of 100%: As shown in [13], the distribution of the number of muon neutrinos arriving at the Earth per pair of WIMPs selfannihilating in the Sun's core provides hard spectra for the τ + τ − and W + W − and a soft spectrum for bb. Limits calculated for dark matter candidate models will lie between these three channels, depending on their branching ratios. The energy spectrum of each channel (see Figure 2 in [13]) is used to calculate the acceptance for the particular annihilation channel in Equation 1. The acceptance is the neutrino effective area convoluted with the energy spectrum corresponding to a given WIMP mass (see Section 3).
Two reconstruction algorithms are used in this paper. The first one is based on the minimisation of a χ 2 -like quality parameter, Q, of the reconstruction which uses the difference between the expected and measured times of the detected photons, taking into account the effect of light absorption in the water [14]. The second algorithm consists of a multistep procedure to fit the direction of the muon track by maximising a likelihood ratio, Λ, which describes the quality of the reconstruction [15]. In addition to the Λ parameter, the uncertainty of the muon track angle, β, is used for the track selection. These two algorithms are respectively called here QFit and ΛFit. In order to reach the best efficiency of reconstruction in the entire neutrino energy range QFit is used for muon events reconstructed in a single detection line (single-line events), and ΛFit for muon events reconstructed on more than one detection line (multi-line events).
Extensive comparisons between data and simulations have been made elsewhere [16].

Analysis strategy
The search for WIMP annihilation in the Sun is performed based on a maximum-likelihood analysis method. The maximisation of this likelihood function, which is fed with the known information about the characteristics of the expected background and signal, provides an estimate of the amount of signal in the data. The extended likelihood function used for ΛFit is where N bg is the expected number of background events, N tot is the total number of reconstructed events, n s (the variable that changes during the maximisation process) is the number of signal events in the likelihood function, S and B are functions that calculate the likelihood of an event to be either signal or background, ψ i is the angular distance of the i-th event to the Sun, N hit,i is the number of hits used in the reconstruction of the i-th event, which is used as an energy estimate and β i is the value of the angular error estimate for the i-th event. S is calculated from the simulation and B is calculated from time-scrambled data.
For the QFit analysis the likelihood function looks different since for that analysis only single-line events have been used. For these events the azimuth angle can not be determined, so that the difference between the zenith angle of the events and the Sun has to be used instead of ψ: whereN hit,i is the number of hits summed up per storey used for the reconstruction and θ i is the difference in zenith angle between the i-th event and the Sun.S andB are analogous to S and B in the likelihood function used for the ΛFit data.
The angular resolution, which is used in S , is limited by the kinematic angle between neutrino and outgoing muon [16].
In this analysis a blinding protocol is applied for optimising the event selection. Blinding is achieved by using simulations to calculate the sensitivities, and time-scrambled data for calculating the background estimate.
In order to compute sensitivities and limits, 10 4 pseudoexperiments are performed for each combination of WIMP mass, annihilation channel and reconstruction strategy and for each considered value of n s (n s ∈ {0, 1, 2...20}). In a pseudoexperiment, a random distribution of background events is simulated according to the features of the recorded data by randomising the right ascension of the events. Simulated signal events are introduced into these pseudo-experiments. These events are generated using the PSF and the signal characteristics for a given reference flux, which are also used in the likelihood function. For each pseudo-experiment, n s is varied to maximise the likelihood function (when n s = n max ). The test statistic (TS) is then calculated as TS = log 10 L(n max ) Distributions of TS values are generated for different numbers of injected signal events. The overlap of TS distributions with inserted signal events and the TS distribution with only background is a measure of the likelihood to mistake pure background for an event distribution with a certain amount of signal in it. From this, the 90% C.L. sensitivities in terms of detected signal events, µ 90% , are obtained using the Neyman method for generating limits [17]. The so-defined µ 90% quantity corresponds to the lowest number of signal events so that 90% of pseudo-experiments provide TS values above the median of the TS distribution of the pure background case.
Event selection consists of cuts on the quality parameters Λ and Q of the two reconstructions that are used in this analysis. These cuts are optimised with respect to the sensitivities (i.e. the model rejection factor). The optimum cuts for the relevant mass ranges are Λ > −5.4 and β < 1 • for ΛFit and Q < 0.8 for the QFit analysis.
The sensitivities in terms of neutrino fluxes are calculated using the acceptance, defined as where A j eff (E ν µ ) and A j eff (Eν µ ) are the effective areas for the j-th detector configuration period (see below) as a function of the muon neutrino energy, E ν µ , or muon antineutrino energy, Eν µ , dΦ νµ dE νµ ch is the signal neutrino spectrum at the position of the detector for the annihilation channel ch (see Equation 1), E th is the energy threshold of the detector, M WIMP is the WIMP mass and T j eff is the effective live time for the j-th detector configuration period. The effective area is defined as a 100% efficient equivalent area which would produce the same event rate as the detector. It is calculated from simulation. Throughout the lifetime of ANTARES the number of available detector lines has changed. The acceptance for the whole lifetimeĀ is calculated as the sum over the acceptances for all detector configuration periods.
The 90% C.L. sensitivities on the neutrino fluxes are then calculated asΦ whereμ 90% is the 90% C.L. sensitivity obtained from the likelihood function.

Results and discussion
In Figure 1 it can be seen that there is no excess of events large enough to be identified as signal by the likelihood func-tion. The median of the PSF used in the likelihood function is for most masses below 2 degrees. The observed TS is used to extract 90% C.L. upper limits from the absence of signal. However, since the observed value of the TS turns out to be smaller than the median of the TS distribution of pure background for all masses and channels, the sensitivity has been considered as the limit.  The limit on the total number of neutrinos from WIMP annihilations in the sun per unit of time C n is calculated by Sun,rms Φ ν µ +ν µ ,90% , where Φ ν µ +ν µ ,90% is the limit on the neutrino flux and d 2 Sun,rms is the mean squared distance from the detector to the Sun. From this, the annihilation rate is calculated by dividing C n by the average number of neutrinos per annihilation, as obtained by WIMPSim. The sensitivities on the spin-dependent and spinindependent scattering cross-sections are calculated from this annihilation rate assuming an equilibrium between annihilation and capture via scattering [18]. This means that the capture rate is twice as high as the annihilation rate. For the calculation of the capture rate a Maxwellian velocity distribution of the WIMPs with a root mean square velocity of 270 m · s −1 and a local dark matter density of 0.4 GeV · cm −3 is assumed [19]. Therefore, once the average number of neutrinos per annihilation is known, the annihilation rate and consequently the capture rate and the scattering cross-sections can be calculated.
All results are shown in comparison to the results of other experiments in Figures 3 and 4 and summarised for reference in Table 1. Recently an update on the spin-dependent crosssection limits from the IceCube collaboration has been released [20]. These new limits show an improvement of up to a factor of 4 with respect to the previous limits by using the energy information of the events in the likelihood function. In the analysis presented here the inclusion of further event parameters (e.g. N hit , β and Q in Equations 2 and 3) leads to an improvement of a factor of up to 1.7. At WIMP masses of up to a few 100 GeV, the consistent strengthening of the flux limit with increasing WIMP mass (see Figure 2) determines the behaviour of the cross-section limits. Above a WIMP mass of a few 100 GeV the factor of M −2 WIMP in the conversion from neutrino flux to the scattering cross-sections dominates the behaviour of the crosssection limits and causes a rise with the WIMP mass. As a result, the cross-section limits show a minimum at a few 100 GeV.  Figure 3: Limits on the spin-dependent WIMP-nucleon scattering crosssection as a function of WIMP mass for the bb, τ + τ − and W + W − channels. Limits given by other experiments are also shown: IceCube [20], PICO-60 [21], PICO-2L [22], SuperK [23], XENON100 [24].  Figure 4: Limits on the spin-independent WIMP-nucleon scattering crosssection as a function of WIMP mass for the different channels considered. Limits given by other experiments are also shown: IceCube [25], SuperK [23], LUX [26], XENON100 [27].
The possible uncertainties on the background have been circumvented by using time-scrambled data for generating the background function B in the likelihood function. The largest systematic error is an uncertainty of 20% on the angular acceptance of the PMTs [28] and leads to a degradation of the detector efficiency (i.e. the acceptance) of 6% [13]. This effect has been taken into account for the limits presented here.