Calculation for the number events is taken from Asaka et al. (p. 11, eq. 10/12).

Observed events is equal to difference between numbers of neutrino at end of detector and number of neutrino at the start. The number of events in the MicroBoone detector is calculated with:

$$ N_{Events} = A \ POT \ \int_{M_N}^\infty dE_N \frac{1}{\lambda (M_N,\Theta_\mu,E_N)} \ \int_{x_0}^{x_1} \phi_N(E_N) \ e^{- \frac{x}{\Lambda _N}} $$

$$ N_{Events} = V_d \ \text{POT} \ \int_{M_N}^\infty dE_N \ \frac{\phi_{\nu_h}(E_N)}{\lambda (M_N,\Theta_\mu,E_N)}$$

Where:

$V_d$ is the fiducial volume of the detector $d$ (must be in $\text{m}^3$).

$\text{POT}$ is the total number of protons on target (from BNB).

$|\Theta_\mu|^2$ is the active-heavy mixing parameter for $\mu$ (the others assumed null).

$\phi_{\nu_h}$ is the flux of HSN neutrinos ( must be in $\frac{\nu}{\text{POT} \ \text{MeV} \ \text{m}^2}$).

$\lambda (M_N,\Theta_\mu,E_N)$ is the partial decay length for the channel (must be in $\text{m}$).

$M_N$ and $E_N$ are the mass and the total energy of the heavy sterile neutrino, respectively.

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The formula for the decay width ($\Gamma$) is from Atre et al. (p. 14, eq. 3.1), then using $\tau = \frac{\hbar}{\Gamma}$ and decay length in lab frame is $\lambda = \gamma \beta c \tau$.

$$ \Gamma ^{l^- \pi^+} = \frac{G_F^2}{16 \pi} f^2_{\pi} |V_{q \bar{q}'}|^2 |\Theta_l|^2 m^3_N \ \cdot $$ extra terms.

Where:

$f_{\pi}$ is the meson decay constant.

$|V_{q \bar{q}'}|$ is the CKM matrix element.

The extra terms can be found in appendix C of the paper.