The experiment with a 12 l liquid scintillation detector was conducted at the Soudan Underground Laboratory with a live-time of 982.1 days over 4 years. The detector is calibrated from \(\sim \)1 MeV to \(\sim 20\) MeV by using gamma ray sources \(^{22}\hbox {Na}\) (1.275 MeV), AmBe (4.4 MeV), and the minimum ionization peak from cosmic muons (20.4 MeV) [16]. The energy response to the entire energy range is accumulated and shown in Fig. 1. To maintain a stable energy scale over the entire experimental period, the peak position of the muon minimum ionization is closely monitored. Energies are re-calibrated on weekly basis according to the variation of the peak position from the muon minimum ionization along with time. The pedestal value is also monitored and used in the correction of the pulse shapes when calculating energies. The \(\gamma \) rays from radioactive decays (\(^{40}\hbox {K}\), \(^{232}\hbox {Th}\) and \(^{238}\hbox {U}\)) and (\(\alpha \), \(\gamma \)) reactions induced by radon decays are also recorded and analyzed [18]. The energies of \(\gamma \) rays are significantly below 20 MeV. Muon-induced neutrons and (\(\alpha \),n) neutrons have been reported in an earlier paper [17] and the event rates are significantly smaller than the muon rate reported in this work.
Muons detected in our detector are largely suppressed by the overburden of the rock. The variation of muon intensity is believed to correlated with the seasonal temperature variation in the stratosphere of atmosphere above the ground. An effective temperature \(T_{eff}\) is defined using a weighted average over the atmospheric depth[11]:
$$\begin{aligned} T_{eff} = \frac{\int _{0}^{\infty }\mathrm {d}XT(X)W(X)}{\int _{0}^{\infty }\mathrm {d}XW(X)}. \end{aligned}$$
(1)
Where T(X) is atmospheric temperature in the stratosphere at a given atmospheric depth X, and the weight W(X) is the temperature dependence of the production of mesons and their decay into muons that can be observed in our detector. The variation of atmospheric temperature in the stratosphere results in a change of the air density. Consequently, the change of the air density modifies the ratio of meson decays to hadronic interaction and the hence changing the muon flux observed underground. An effective temperature coefficient \(\alpha _{T}\) can be defined as:
$$\begin{aligned} \frac{\varDelta I_{\mu }}{< I_{\mu }> } = \alpha _{T}\frac{\varDelta T_{eff}}{ < T_{eff} > }. \end{aligned}$$
(2)
Figure 2 shows the variation of amplitude along with time. The formula we use to determine the fractional modulation amplitude \(\delta I / \overline{ I }\) and the period T is described in Eq. (3).
$$\begin{aligned} I = \overline{I} + \varDelta I = \overline{I} + \delta I \cos \left( \frac{2\pi }{T}(t-t_{0})\right) . \end{aligned}$$
(3)
where \(\overline{I}\) is the mean value and \(\delta I \) is the variation amplitude. The phase \(t_{0}\) is the time when the signal reaches its maximum. The top plot in Fig. 2 represents the effective temperature variation of the atmosphere above the ground of the Soudan site. The atmospheric temperature data is obtained from Ref. [21]. A fixed period of 365.1 days is applied to fit the variation pattern. The fitted variation amplitude is found to be (\(2.87\pm 0.08\))% with the phase at Jul \(12 \pm 3.4\) days. The bottom plot in Fig. 2 is the muon variation curve. Data points with energy greater than the muon minimum ionization peak are collected to avoid the gamma ray contamination and any potential energy shift. With the fixed period of 365.1 days, the fitted result gives the variation amplitude of (\(2.64\pm 0.07\))% with the phase at Jul \(22 \pm 36.2\) days.
The correlation of the percentage variation in the observed muon rate \(\varDelta I_{\mu }/< I_{\mu } > \) correlates with the change in effective temperature \( \varDelta T_{eff}/< T_{eff} > \) is shown in Fig. 3. The fitting result determines the value of \(\alpha _{T} = 0.898 \pm 0.025\). The error is dominated by the statistical uncertainty, since the systematical uncertainty is negligible. This is because the systematical uncertainty was carefully avoided using weekly calibration and the muon events were selected with energy greater than 20 MeV, which largely excludes gamma-ray (\(E_{\gamma } < 20\hbox { MeV}\)) contamination. In addition, the event rate from the muon-induced neutrons is much smaller than the muon event rate detected in the detector.
The remaining uncertainty associated with the value of \(\alpha _{T}\) measured in this work is therefore governed by the limited statistical error per bin (64 days with 1836 muon events per bin).
Figure 4 summarizes the measured values for \(\alpha _{T}\) from various underground depths. The reported values at different underground sites agree with the predicted \(\alpha _{T}\) (red curve in Fig. 4) well. Our detector is adjacent to the MINOS far detector at the same depth level of the Soudan Underground Laboratory. Both results show a good agreement with the prediction.
The charged \(K/\pi \) ratio, \(r(K/\pi \)), can be determined using the relation below [14]:
$$\begin{aligned} r(K/\pi ) = \frac{(\alpha _T)_\pi /\alpha _T - 1}{ 1- (\alpha _T)_{K}/\alpha _T}, \end{aligned}$$
(4)
where \((\alpha _{T})_{K,\pi }\) can be obtained using the theoretical prediction [14]:
$$\begin{aligned} (\alpha _T)_{K,\pi } = 1/\Bigg [\frac{\gamma }{\gamma +1} \cdot \frac{\epsilon _{K,\pi }}{1.1E_{th}cos\theta } + 1\Bigg ]. \end{aligned}$$
(5)
Utilizing the muon spectrum index, \(\gamma =1.7\pm 0.1\), kaon critical energy \(\epsilon _{k} = 0.851\pm 0.014\,\hbox { TeV}\), and pion critical energy \(\epsilon _{\pi } = 0.114\pm 0.003\,\hbox { TeV}\) given by the Particle Data Group [22] and \(\hbox {E}_{th}\hbox {cos}\theta = 0.795\pm 0.14\hbox { TeV}\) from MINOS [11], we can obtain \((\alpha _T)_{K} = 0.620^{+0.029}_{-0.037}\) and \((\alpha _{T})_{\pi } = 0.924^{+0.008}_{-0.011}\). Plugging the values of \((\alpha _{T})_{K,\pi }\) and the measured \(\alpha _{T = 0.898\pm 0.025}\) into Eq. 4, we obtain \(r(K/\pi ) = 0.094^{+0.044}_{-0.061}\). This is consistent with \(r(K/\pi ) = 0.12^{+0.07}_{-0.05}\) determined by MINOS [11]. To further constrain the uncertainty of \(r(K/\pi \)) from the measurements, we conduct a Geant4 simulation to study the correlation between the temperature variation in the stratosphere and the muon rate annual modulation underground at the Soudan Underground Laboratory for a given \(r(K/\pi \)).