While the search presented specifically addresses two benchmark models described in the paper, its results may be applied to other models predicting long-lived particles that decay into a pair of oppositely charged muons. In Figs. A14--A16 we provide a set of generator-level efficiency maps that approximate the reconstruction-level efficiency of this analysis and allow for reinterpretation of the results in the framework of other models.

We provide signal efficiencies as a function of the smaller of the two values of generated muon $p_{\mathrm{T}}$, $\mathrm{min}\left(p_{\mathrm{T}}\right)$, and of generated muon $d_{0}$, $\mathrm{min}\left(d_{0}\right)$, in three intervals of the generated transverse decay length $L_{\mathrm{xy}}$ of the signal dimuon, $L_{\mathrm{xy}} < $ 20 cm, 20 $ < L_{\mathrm{xy}} < $ 70 cm, and 70 $ < L_{\mathrm{xy}} < $ 320 cm. These 3D efficiency maps $\epsilon \left( \mathrm{min}\left(p_{\mathrm{T}}\right); \mathrm{min}\left(d_{0}\right); L_{\mathrm{xy}} \right)$ are provided separately for each of the three dimuon categories, STA-STA, STA-TMS, and TMS-TMS, and are shown in Fig. A14, Fig. A15, and Fig. A16 for dimuons with $L_{\mathrm{xy}} < $ 20 cm, 20 $< L_{\mathrm{xy}} < $ 70 cm, and 70 $< L_{\mathrm{xy}} < $ 320 cm, respectively. Each figure contains two sets of maps, for 2016 and 2018 data sets, corresponding to integrated luminosities of 36.3 $ \pm$ 0.4 and 61.3 $\pm$ 1.5 fb$^{-1}$, respectively.

The efficiency in each ($\mathrm{min}\left(p_{\mathrm{T}}\right)$; $\mathrm{min}\left(d_{0}\right)$; $L_{\mathrm{xy}}$) bin of the 3D efficiency map is computed as the ratio of the number of simulated signal dimuons in that bin that pass the trigger requirements and selection criteria applied for a given dimuon category (Table A6) to the total number of simulated signal dimuons in that bin and within the geometric acceptance. The computation is performed using an ensemble of all generated $\phi \to \mathrm{XX} \to \mu\mu +$anything signal samples. The geometric acceptance is defined as $\mathrm{min}\left(d_{0}\right) < $ 300 cm, $L_{\mathrm{xy}} < $ 320 cm, generated longitudinal decay length $L_{\mathrm{z}}$ smaller than 800 cm, and $|\eta|$ of both generated muons forming the dimuon smaller than 2.0. The efficiencies obtained from simulation were further corrected by the data-to-simulation scale factors described in the paper.

When applied to the previously untested models, the signal efficiency in each dimuon category $j$, $\epsilon_j$, can be obtained from the 3D efficiency maps using the $p_{\mathrm{T}}$, $d_{0}$, and $L_{\mathrm{xy}}$ at generator level: