# Design and Diagnostics of High-Precision Accelerator Neutrino Beams
## 10 Mitigation Techniques of Beam Systematics at the Neutrino Detectors
### Appearance Measurements  
If the facility is aimed at measuring \(\
u_{\mu}\to\
u_{e}\) probabilities or its CP conjugate \(\overline{\
u}_{\mu}\to\overline{\
u}_{e}\), substantial changes are necessary to the near/far technique due to the presence of multiple cross sections and efficiencies. In this case, we must either improve or cancel out the errors on the \(\
u_{e}\) cross section, the flux and the intrinsic \(\
u_{e}\) contamination. The simplest way is to employ the near detector to measure the beam contamination. This includes both the intrinsic contamination of \(\
u_{e}\) produced at source (typically a few %) and the mis-identification background. If the near detector were _identical_ to the far detector, it would be optimized for \(\
u_{e}\) detection providing:\[N_{ND}^{e}=\tilde{M}\int dE\Big{[}\epsilon_{e}(E)\ \phi(\
u_{e})\ \sigma_{\
u_{e}}(E)\ +e_{\mu\to e}\phi(\
u_{\mu})\sigma_{\
u_{e}}(E)\Big{]} \tag{8}\]  
where \(\epsilon_{\mu\to e}\) is the mis-identification probability of tagging a \(\
u_{\mu}\) as a \(\
u_{e}\), e.g., due to neutral currents (NC) events. The FD would provide, in this case,  
\[N_{FD}^{e}=\tilde{M}^{\prime}\int dE\Big{[}\epsilon_{e}(E)\ \phi(\
u_{\mu})\ \sigma_{\
u_{e}}(E)P(\
u_{\mu}\to\
u_{e})\ +e_{\mu\to e}\phi(\
u_{\mu})\sigma_{\
u_{e}}(E)\Big{]} \tag{9}\]  
and the \(N_{FD}^{e}/N_{ND}^{e}\) ratio does not provide any more direct access to the oscillation probability. Note in particular that the \"identical\" near detector has no more access to the product \(\phi_{\mu}\epsilon_{e}\sigma_{e}\), which is essential to estimate the signal at the far detector:  
\[\phi(\
u_{\mu})\epsilon_{e}\sigma_{\
u_{e}}P(\
u_{\mu}\to\
u_{e}) \tag{10}\]  
The simplest fix is to have identical detectors that are able to measure both \(\
u_{\mu}\) and \(\
u_{e}\), gaining two additional observables:  
\[N_{ND}^{\mu}=\tilde{M}\int dE\Big{[}\epsilon_{\mu}(E)\ \phi(\
u_{\mu})\ \sigma_{\
u_{e}}(E)\ +e_{e\to\mu}\phi(\
u_{e})\sigma_{\
u_{e}}(E)\Big{]}\simeq \tilde{M}\int dE\Big{[}\epsilon_{\mu}(E)\ \phi(\
u_{\mu})\ \sigma_{\
u_{\mu}}(E)\Big{]} \tag{11}\]  
The last approximation works because \(\phi(\
u_{\mu})\gg\phi(\
u_{e})\) in any conventional beam. The second observable is:  
\[N_{FD}^{\mu}=\tilde{M}^{\prime}\int dE\Big{[}\epsilon_{\mu}(E)\ \phi(\
u_{\mu})\ \sigma_{\
u_{\mu}}(E)P(\
u_{\mu}\to\
u_{\mu})\ +e_{e\to\mu}\phi(\
u_{e})\sigma_{\
u_{e}}(E)P(\
u_{e}\to\
u_{\mu})\Big{]}\simeq\] \[\tilde{M}^{\prime}\int dE\Big{[}\epsilon_{\mu}(E)\ \phi(\
u_{\mu})\ \sigma_{\
u_{\mu}}(E)P(\
u_{\mu}\to\
u_{\mu})\Big{]} \tag{12}\]  
In particular, if the mis-identification probability \(\epsilon_{\mu\to e}\) can be established independently using beam test, cosmic ray runs, control samples, etc., or the background can just be neglected, then \(N_{e}^{ND}\) provides a pure measurement of the beam contamination as seen by the far detector. \(N_{\mu}^{ND}\) measures \(\epsilon_{\mu}(E)\ \phi(\
u_{\mu})\ \sigma_{\
u_{\mu}}(E)\) as seen by the far detector and the main systematics are now on the ratio:  
\[\frac{\epsilon_{e}\sigma_{\
u_{e}}}{\epsilon_{\mu}\sigma_{\
u_{\mu}}} \tag{13}\]  
This is the reason why the current generation of oscillation experiments use the near/far technique mostly to evade the large uncertainties on the flux, relying on the lepton universality (\(\sigma_{\
u_{e}}\simeq\sigma_{\
u_{\mu}}\) ) and the measurements with cosmics or particle beams to mitigate the uncertainties on \(\epsilon_{e}/\epsilon_{\mu}\). This technique has been perfected by T2K and NO\(\
u\)A reaching a systematic budget of 8-5%.  
Clearly, the near/far technique cannot be used for cross-section experiments, where the detector is always located at a short distance and the (poorly known) flux just scales as \(L^{2}\) at large distances.