Why should $\nu_\mu\to \nu_e$ oscillations be expected to be unobservable in short baseline experiments?

Question

The blog here says

On the other hand, several rogue experiments refuse to conform to the standard 3-flavor picture. The most severe anomaly is the appearance of electron neutrinos in a muon neutrino beam observed by the LSND and MiniBooNE experiments.

Within the $3$-flavour mixing and $3$-flavour neutrino oscillation scenario, why is it unlikely to observe $\nu_\mu\to \nu_e$ oscillation in short baseline ($L\leq 1$km) experiments?

In the standard 3-flavour mixing $$P(\nu_\mu\to\nu_e)=s^2_{23}\sin^2 2\theta_{13}\sin^2\Big(\frac{\Delta m^2_{31}L}{4E}\Big)$$

Why should $\nu_\mu\to \nu_e$ oscillations be expected to be unobservable in short baseline experiments?

Answer 1

Plugging in the implicit factor of $\hbar c$ and some conversion factors allows you to express the oscillation formula in units which are easier to calculate with:

$$P(\nu_\mu \rightarrow \nu_e) = \sin^2(\theta_{23})\sin^2(2\theta_{13})\sin^2\left(1.27 \frac{\Delta m_{31}^2}{[\mathrm{eV}^2]} \frac{L}{E} \frac{\mathrm{[km]}}{\mathrm{[GeV]}}\right)$$

The values of the mixing parameters and mass differences squared are approximately:

$$\sin^2(\theta_{23}) \approx 0.4$$ $$\sin^2(2\theta_{13}) \approx 0.08$$ $$\Delta m_{31}^2 \approx 0.002~\mathrm{eV}^2$$.

For Miniboone the distance of the detector from the target is approximately ~1 km and the energy of the neutrinos from the beam are of order 1 GeV. Plugging these in you get a value for the oscillation probability of approximately:

$$P(\nu_\mu \rightarrow \nu_e) \approx 10^{-7}$$

Since they observe ~100s of events, the probability of observing a $\nu_e$ due to neutrino oscillation is extremely unlikely.

Note that these detectors do expect to see some number of electron neutrino events coming from other sources. In Figure 1 from their recent paper here you can see that they do expect a non negligible number of electron neutrino events from kaon and muon decays, but the anomoly referred to in the blog is that they see more of these events than they expect.