Search for solar Kaluza-Klein axion by annual modulation with the XMASS-I detector

In theories with the large extra dimensions beyond the standard 4-dimensional spacetime, axions could propagate in such extra dimensions, and acquire Kaluza-Klein (KK) excitations. These KK axions are produced in the Sun and could solve unexplained heating of the solar corona. While most of the solar KK axions escape from the solar system, a small fraction is gravitationally trapped in orbits around the Sun. They would decay into two photons inside a terrestrial detector. The event rate is expected to modulate annually depending on the distance from the Sun. We have searched for the annual modulation signature using $832\times 359$ kg$\cdot$days of XMASS-I data. No significant event rate modulation is found, and hence we set the first experimental constraint on the KK axion-photon coupling of $4.8 \times 10^{-12}\, \mathrm{GeV}^{-1}$ at 90% confidence level for a KK axion number density of $\bar{n}_\mathrm{a} = 4.07 \times 10^{13}\, \mathrm{m}^{-3}$, the total number of extra dimensions $n = 2$, and the number of extra dimensions $\delta = 2$ that axions can propagate in.

In theories with large extra dimensions beyond the standard 4-dimensional spacetime, axions could propagate in such extra dimensions, and acquire Kaluza-Klein (KK) excitations. These KK axions are produced in the Sun and could solve the unexplained heating of the solar corona. While most of the solar KK axions escape from the solar system, a small fraction is gravitationally trapped in orbits around the Sun. They would decay into two photons inside a terrestrial detector. The event rate is expected to modulate annually depending on the distance from the Sun. We have searched for the annual modulation signature using 832 × 359 kg·days of XMASS-I data. No significant event rate modulation is found, and hence we set the first experimental constraint on the KK axion-photon coupling of 4.8 × 10 −12 GeV −1 at 90% confidence level for a KK axion number density ofn a = 4.07 × 10 13 m −3 , the total number of extra dimensions n = 2, and the number of extra dimensions δ = 2 that axions can propagate in.

Introduction
In quantum chromodynamics (QCD) there is a problem known as the strong CP problem. The QCD Lagrangian has a CP-violating term, but this violation has not been observed experimentally. Peccei and Quinn introduced a new global symmetry U (1) PQ to solve this problem [1]. This newly introduced symmetry spontaneously breaks at an energy scale of f PQ , which predicts the existence of a pseudo-Goldstone boson, namely the axion. From the experimental results, f PQ is found to be far larger than the electro-weak scale [2]. The mass of the axion is characterized by f PQ and calculated to be: On the other hand, large extra dimensions are also proposed to solve the gauge hierarchy problem [3,4]. Motions of a particle in the extra dimensions could be observed as new particles with heavier masses in the standard 4-dimensional spacetime. These particles are called Kaluza-Klein (KK) particles. Although the PQ model and large extra dimensions are independently motivated, studies to link them emerged in the late 1990s. It was pointed out that large extra dimensions can explain the largeness of f PQ [2]. Following this possible close connection between the PQ model and extra dimensions, studies of axions in extra dimensions, or KK axions, started [5,6].
This opened a possibility of the detection of KK axions produced in the Sun, also called the solar KK axion, in terrestrial detectors [7]. It was also pointed out as an independent motivation for the existence of KK axions that solar KK axions may solve some astrophysical observational problems such as solar coronal heating and X-rays from the dark side of the Moon [8]. In this scenario, a small fraction of the solar KK axions is gravitationally trapped in orbits around the Sun and accumulated over the age of the Sun. These KK axions then decay into two photons, which become a source of the lunar X-rays. The X-rays from KK axions trapped near the solar surface on the other hand may explain the observed coronal heating. It is argued that a gas time projection chamber (TPC) for direct dark matter searches in an underground laboratory is one of the most suitable instruments to detect the decay of KK axions because of its large volume (∼ 1 m 3 ) and the ability to distinguish two photons by recording their tracks [9]. No experimental result on solar KK axions has, however, been reported to date.
XMASS-I is a large-volume liquid xenon scintillation detector designed for multiple physics targets [10]. A variety of searches had been performed using both nuclear and electron channels [11][12][13][14][15]. Thanks to the high photoelectron (PE) yield (about 15 PE/keV), the energy threshold is low enough to search for solar KK axions. XMASS-I data accumulated over more than one year was used for a dark matter direct search in an annual modulation analysis [16]. Annual modulation is expected also for the solar KK axion signal due to the seasonal change in the distance between the Sun and the Earth. Annual modulation searches have the advantage of being robust against most backgrounds. In this study, we conducted the first direct search for solar KK axions by exploiting this advantage.

The XMASS-I detector
The XMASS-I detector is located 1000 m (2700 m water equivalent) underground at the Kamioka Observatory in Japan. Its detailed design and performance were described elsewhere [10]. The inner detector (ID) consists of 832 kg of liquid xenon surrounded by 642 inward-looking 2-inch photomultiplier tubes (PMTs) arranged on a pentakis-dodecahedronshaped copper holder. The photocathode coverage of the detector's inner surface is about 62%. The sensitive volume of the liquid xenon is 0.288 m 3 . This ID is surrounded by an outer detector (OD), which is a water tank 10.5 m in height and 10 m in diameter. The OD has 72 20-inch PMTs and works as a Cherenkov veto counter and a passive shield. The threshold for the ID PMTs is equivalent to 0.2 PE. A trigger is issued when four or more ID PMTs have signals exceeding the threshold within 200 ns. The signals from the inner detector PMTs are recorded by CAEN V1751 waveform digitizers with a 1 GHz sampling rate.
Radioactive sources can be inserted into the ID for calibration. The energy scale is obtained from various sources with energies between 5.9 keV and 122 keV measured, specifically with 55 Fe, 57 Co, 109 Cd, and 241 Am sources [17], and its time variation is traced by weekly 57 Co calibrations. In this paper, the keV ee energy scale is used, which reflects the electron equivalent energy.
A detailed Monte Carlo (MC) simulation based on Geant4 [18] has been developed. It simulates particle tracks, the scintillation process, photon tracking, PMT response, and the readout electronics and is used to calculate the expected signal. The non-linearity of the energy scale is taken into account in the MC using the calibration results and the nonlinearity model from Doke et al. [19]. The energy scale below 5.9 keV is estimated by an extrapolation based on Doke model.

Expected signal
Solar KK axions would be produced thermally in the Sun via the Primakoff effect (γ + Ze → Ze + a, where Ze represents the charge of a nucleus) and a photon coalescence mechanism (γ + γ → a). Since there are an infinite number of KK excitations, KK axions with a spectrum of masses would be produced according to the temperature of the Sun. Most of the produced KK axions escape from the solar system, but a small fraction is trapped by the gravity of the Sun. Figure 1 shows the expected number density of trapped KK axions against the distance from the Sun. This distribution is taken from L. Di Lella and K. Zioutas [8] who calculated up to 200R and it is fitted well by r −4 . Here, R represents the radius of the Sun. The KK axion-photon coupling g aγγ = 9.2 × 10 −14 GeV −1 is fixed so that axion decay can explain the X-ray surface brightness of the quiet Sun. Also, the total number of extra dimensions n = 2, the number of extra dimensions δ = 2 that axions can propagate in, and the fundamental scale M F = 100 TeV are assumed. The mass dependence of the production rate and the rate of KK axion trapping are also calculated in Ref. [8]. Since the number of trapped KK axions produced by the Primakoff effect is three orders of magnitude smaller than those produced by the photon coalescence mechanism, we concentrate on the 3/11 Distance from the Sun (solar radii) contribution from photon coalescence. Since the trapped KK axions are non-relativistic, the two photons from their decay have an energy of m a /2 each. Heavier KK axions are likely to be trapped and would have a shorter life time. From the simulation in Ref. [8], which considers the mass dependencies of the trapping rate and the decay rate, the peak of the decay spectrum of the trapped KK axion near the Earth is about 9 keV. The expected annual modulation signal is calculated as follows: First, the distance between the Earth and the Sun as a function of time r(t) is denoted as where a = 1.496 × 10 8 km = 215.0R and e = 0.0167 are the semi-major axis and the eccentricity of the Earth's orbit, respectively. t is a date in one year, and T represents one year. t 0 represents the date when the Earth is at perihelion. Then, since the number density of the KK axion (n a ) can be fitted with a r −4 curve, the time dependence of the number density n a (t) is written as

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Here,n a is the KK axion number density when r(t) = a. From Eq. (3) and Fig. 1, the number density of trapped KK axions on the Earth is calculated as 4.36 × 10 13 m −3 at perihelion and 3.81 × 10 13 m −3 at aphelion. The expected KK axion decay rate R is proportional to the square of the KK axion-photon coupling g aγγ and the number density [9]: The upper graph of Fig. 2 shows the expected energy spectra of trapped solar KK axions at perihelion and aphelion. The shape of the spectrum is assumed to be the same throughout the year, and using the spectrum shown in Fig. 2 as input, the expected energy spectrum is calculated by a Monte Carlo simulation, which is shown in Fig 3.

Data analysis
Data from November 2013 to March 2015 with a total live time of 359 days is used for this study. The data set is divided into 33 time-bins (t bins ) with about 15 live-days each. Four criteria are applied for event selection: (1) The event is triggered only by the ID. (2) The time difference from the previous event is more than 10 ms and the root mean square of hit timings in the event is less than 100 ns. This cut is applied to reject events caused by afterpulses following bright events. (3) The ratio of the number of hits in the first 20 ns to the total number of hits is less than 0.6. This cut is applied only to the events with the total number of PE less than 200 in order to remove Cherenkov events originating from 40 K decays in the photocathodes of the PMTs [10]. (4) The ratio of the largest number of PE detected by a single PMT to the total number of PE in the event has to be smaller than a certain value. This cut is applied to remove background events occurring in front of a PMT window. The cut value is a function of the total number of PE in the event and varies from about 0.2 at 8 PE to about 0.07 at 50 PE. Details are described in [16]. Figure 4 shows the energy spectra after each selection step. Most of the remaining events originate from radioactivity in the aluminum seal of the PMTs [10]. Seasonal backgrounds, such as radon in the OD water and cosmogenics are found to be negligible. According to our weekly 57 Co calibration, up to 10% variation of the PE yield was observed throughout the measurement period. This variation is understood to be due to the change of the liquid xenon absorption length which changed from about 4 m to 11 m [16]. The cut efficiency is affected by the variation of this PE yield, and changed up to 10%. We corrected this effect by accounting for relative cut efficiencies as evaluated from MC simulation. The uncertainty of this cut efficiency is found to be the largest systematic error (∼ ±5% on the event rate), and enters as terms K ij into our χ 2 evaluation (see Eq. (5)). Due to the normalization of the overall cut efficiency, the systematic uncertainty of the relative cut efficiency becomes 0 at an absorption length of 8 m, and K ij becomes very small around March 2014. The second largest contribution to the systematic error is the non-linearity of the scintillation yield (∼ ±10% on the event rate). This uncertainty is taken into account in calculating the signal expectation, and is introduced as L i in Eq. (6) below. Between April and September 2014, a different way of calibrating electronics gains led to an additional 0.3% uncertainty in the absolute energy scale of the experiment. This uncertainty is represented as σ 2 sys;i,j in Eq. (5). Other systematic uncertainties were found to be negligible.

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Energy (keV) The energy spectra are taken from Ref. [9] and scaled according to the density at the Earth's position. A KK axion-photon coupling constant g aγγ = 9.2 × 10 −14 GeV −1 is assumed and KK axion number densities n a = 4.36 × 10 13 m −3 for perihelion and n a = 3.81 × 10 13 m −3 for aphelion are assumed. The vertical axis on the right shows the expected event rate in liquid xenon.
The final data sample shown by the magenta spectrum in Fig. 4 is evaluated for the KK axion search with an annual modulation method. To this end an annual modulation amplitude is extracted from the data by a least Chi-squares fit. The data in each time-bin are divided into 16 energy-bins (E bins ) with a width of 1 keV ee each. We used two pull terms [20] of α and β in the χ 2 which is defined as: where R data i,j , R ex i,j , σ stat;i,j , σ sys;i,j are the observed event rate, the expected event rate, and the statistical and uncorrelated systematic errors for each bin, respectively. The subscripts i and j denote the respective energy and time bin. K i,j represents the 1σ correlated systematic 6/11 error based on the relative cut efficiency for each period. α is the penalty term associated with K i,j . Based on Eq. (3) and (4), the expected event rate is defined as: where ∆t j is the bin width of the j-th time bin. C i and A i are the constant term and the expected amplitude of the event rate in the i-th energy bin, respectively. A i corresponds to half of the residual event rate in Fig. 2. L i , which is associated with the penalty term β, accounts for the uncertainty stemming from the non-linearity of the scintillation efficiency on A i . ξ is defined as ξ =  Fig. 4 Energy spectra of the observed data after each reduction step. The histogram with the black solid line shows the events after cut (1), the red dotted line shows the events after cut (2), the blue dash-dotted line represents the events after cut (3), and the magenta solid line with filled circle shows the events after cut (4), which is the final sample.
effects associated with the end of the range over which the Cherenkov cut is applied, which can be seen in Fig. 3. Figure 5 shows the event rate modulation and the best fit result for the expected event rate. As a result, the fit obtains ξ = 8.2 × 10 2 with χ 2 /ndf = 522.4/492. We evaluated the significance of this result by using 10,000 no modulation dummy samples which have the same statistical and systematic errors as the data [16]. This evaluation yields a p-value of 0.62. Since no significant excess in amplitude is found, a 90% confidence level (CL) upper limit is set on the KK axion-photon coupling g aγγ as a function of the KK axion number density. We use the likelihood ratio L defined as
This limit can be re-calculated for different KK axion densities and the obtained limit line is shown in Fig. 6. As a benchmark, the assumed solar KK axion model (g aγγ = 9.2 × 10 −14 GeV −1 ,n a = 4.07 × 10 13 m −3 ) [8] is shown in Fig. 6. Note that the tension from the solar neutrino measurements as a consequence of the luminosity limit is L a < 0.1L [21] which corresponds ton a < 2 × 10 13 m −3 , however, there still remains allowed parameter space for solar KK axion models with different values of M F and δ as discussed in Ref. [8]. 10 XMASS Solar KK axion model Fig. 6 The obtained 90% CL excluded region from this work is shown by the black solid slope and the red hatched area. The model assumed in this study based on Ref. [8] is indicated by the blue point.

Conclusion
We searched for the decay of solar KK axions by annual modulation using 832 × 359 kg·days of XMASS-I data. No significant event rate modulation matched to the solar KK axion hypothesis (n = δ = 2) is found, and a 90% CL upper limit on the KK axion-photon coupling of 4.8 × 10 −12 GeV −1 is obtained forn a = 4.07 × 10 13 m −3 . This is the first experimental constraint for KK axions.