# Design and Diagnostics of High-Precision Accelerator Neutrino Beams
## 10 Mitigation Techniques of Beam Systematics at the Neutrino Detectors
### The Near-Far Detector Technique  
As mentioned in Section 1, the most powerful decoupling mechanism conceived for the study of oscillations is the construction of two detectors: a smaller detector located in the proximity of the source (Near Detector, ND) and a larger detector located at an optimal baseline to study neutrino oscillations (Far Detector, FD). In its classical implementation, the near detector should be identical to the far detector but located in a position where  
Figure 14: Spectra of muons at the ENUBET muon monitors.  
oscillation effects are negligible. In this way, the Near Detector provides a measurement of \(\
u_{\mu}\) at \(t\simeq 0\) and an estimate of the \(\
u_{e}\) contamination of the beam. The use of identical detectors cancels out systematic differences in the detection efficiency between the ND and the FD and is the technique of choice for moderate-precision disappearance experiments. For instance, if we consider the integral spectrum of \(\
u_{\mu}\) detected at the ND [5], the total number of neutrino interactions will be:  
\[N_{ND}=\tilde{M}\phi^{ND}_{\
u_{\mu}}e_{\
u_{\mu}}\sigma_{\
u_{\mu}} \tag{4}\]  
where \(\tilde{M}\) is the number of scattering centers, which is proportional to the mass and material of the detector, \(\phi^{ND}_{\
u_{\mu}}\) is the total flux integrated during the data taking at the ND location, and \(e_{\
u_{\mu}}\sigma_{\
u_{\mu}}\) is the visible cross section, i.e., the cross-section corrected by the finite efficiency of the detector. For two identical detectors located sufficiently far from the source (In this case, the width of the detector is much smaller than the beam width and the flux decreases as the square of the distance):  
\[\frac{\phi^{ND}_{\
u_{\mu}}}{\phi^{FD}_{\
u_{\mu}}}=\frac{L^{2}_{ND}}{L^{2}_{ FD}} \tag{5}\]  
where \(L\) is the source-to-detector distance. In such an ideal condition, the oscillation probability can be inferred straightforwardly by the far-over-near event ratio because:  
\[N_{FD}=\tilde{M}^{\prime}\phi^{ND}_{\
u_{\mu}}\frac{L^{2}_{FD}}{L^{2}_{ND}} \epsilon_{\
u_{\mu}}\sigma_{\
u_{\mu}}P(\
u_{\mu}\to\
u_{\mu}) \tag{6}\]  
and  
\[\frac{N_{FD}}{N_{ND}}=\frac{\tilde{M}^{\prime}}{\tilde{M}}\frac{L^{2}_{ND}}{L^ {2}_{FD}}P(\
u_{\mu}\to\
u_{\mu}) \tag{7}\]  
where \(\tilde{M}^{\prime}\) are the scattering centers of the FD. \(\tilde{M}^{\prime}\) is generally \(\gg\tilde{M}\) but \(L_{FD}\gg L_{ND}\) and the number of events at the ND is much larger than the FD in long-baseline experiments. Using this simple technique, we can measure \(P(\
u_{\mu}\to\
u_{\mu})\) with <10% precision without relying on sophisticated beam diagnostics. In addition, if the energy of the neutrino can be reconstructed by the final state particles, like in quasi-elastic events where the energy is uniquely determined by the lepton momentum and the scattering angle, the near/far cancellation works for each energy bin and provides the oscillation probability as a function of the energy. This method was mastered by [PERSON] [148] at the time of the discovery of neutrino oscillations and provided some evidence of the oscillation pattern and a solid measurement of the oscillation parameters \(\theta_{23}\) and \(\Delta m^{2}_{23}\).  
The near/far cancellation technique dominated the physics of neutrino oscillation in the 90s but has been challenged over the years, when the need for high-precision measurements has emerged. In particular, after the discovery of \(\theta_{13}\), this technique turned out to be too coarse for the search of CP violation due to the tiny size of the effects to be observed and the need to control with per-cent precision the entire spectrum where oscillations take place. We will describe this paradigm shift in the sections below.