= Neutrino-flux sensitivity inversion
{title2=$\mathbf y=A\boldsymbol\phi$}
Why compare inferred <neutrino fluxes> rather than raw event counts? For channel $j$, exposure time $T$, target population $N_j$, <cross section> $\sigma_{j\alpha}$, and <detection efficiency> $\epsilon_{j\alpha}$ including reconstruction and selection, a forward model is
$$
\mathbb E[N_j]-B_j=T N_j^{\mathrm{target}}\sum_\alpha\int\phi_\alpha(E)\sigma_{j\alpha}(E)\epsilon_{j\alpha}(E)\,dE.
$$
Here $B_j$ is expected background; $N_j$ on the left counts events, while $N_j^{\mathrm{target}}$ counts targets. Distinct <cross sections>, target numbers, thresholds and <detection efficiencies> prevent a direct CC/NC count ratio from being an <electron neutrino> fraction. Channel populations themselves can be correlated estimates from overlapping event patterns.
An original reduced <linear algebra> model isolates the inference. Assume specified spectral shapes and divide each background-corrected channel rate by its own calibrated <electron neutrino> response. Put $\phi_x=\phi_\mu+\phi_\tau$ and let $r$ be the spectrum- and response-weighted non-electron/electron ES sensitivity ratio. Then
$$
\begin{pmatrix}y_{\mathrm{CC}}\\y_{\mathrm{NC}}\\y_{\mathrm{ES}}\end{pmatrix}
=\begin{pmatrix}1&0\\1&1\\1&r\end{pmatrix}
\begin{pmatrix}\phi_e\\\phi_x\end{pmatrix}.
$$
The equal $\mu$ and $\tau$ responses determine their sum, not their separate <neutrino fluxes>. Changing the assumed spectra generally changes the integrated coefficients; with arbitrary unknown spectra this small <matrix> is insufficient.
Choose \b[illustrative inputs], not historical measurements: $r=0.15$, $\phi_e=2$ and $\phi_x=3$, with every <neutrino flux> in the same arbitrary unit. The forward prediction is $(y_{\mathrm{CC}},y_{\mathrm{NC}},y_{\mathrm{ES}})=(2,5,2.45)$. CC and NC independently invert to $\phi_e=y_{\mathrm{CC}}=2$, $\phi_x=y_{\mathrm{NC}}-y_{\mathrm{CC}}=3$; substituting into ES returns $2+0.15(3)=2.45$. Alternatively, CC and ES give $\phi_x=(y_{\mathrm{ES}}-y_{\mathrm{CC}})/r=3$. Thus the third channel tests the response model and the inferred mixture.
\Image[/neutrino-survival-and-flux-inversion.png]
{title=Chosen vacuum survival curve and complementary <neutrino flux> constraints}
{description=Original calculations only. Left: stipulated maximal two-flavour mixing and squared-mass splitting, with chosen paths at one GeV. Right: exact normalized-channel equations for chosen fluxes and ES sensitivity; lines are not experimental confidence bands.}
CC alone admits any nonnegative $\phi_x$. CC and ES lose the second independent constraint when $r=0$; for small positive $r$, an ES-minus-CC error is amplified by $1/r$. With chosen independent errors $\sigma_{\mathrm{ES}}=\sigma_{\mathrm{CC}}=0.1$, their propagated <standard deviation> is $\sqrt{0.1^2+0.1^2}/0.15\simeq0.943$. If CC and NC instead have those independent errors, the difference has <standard deviation> $\sqrt{0.1^2+0.1^2}\simeq0.141$. Actual analyses must retain <covariances>, including those introduced by statistical channel separation. Complementarity supplies information; calibration establishes what information each channel carries.
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