# Design and Diagnostics of High-Precision Accelerator Neutrino Beams
## 10 Mitigation Techniques of Beam Systematics at the Neutrino Detectors
### Flux Measurement at the Detector  
As mentioned in Section 10.3, the ND cannot measure the flux in a direct manner because it cannot disentangle the product \(\phi(E)\epsilon(E)\sigma(E)\) neither for \(\
u_{e}\) nor for \(\
u_{\mu}\). A possible way out that can be implemented in high-intensity wide-band beam is the use of the neutrino elastic scattering on the electrons of the detector medium [157; 158]:  
\[\
u_{\mu}+e^{-}\rightarrow\
u_{\mu}+e^{-} \tag{17}\]  
The corresponding cross section is purely leptonic and free of QCD/nuclear effects. It is one of the cleanest observables of the Standard Model because is produced by the exchange of a \(Z\) boson and only depends on the couplings between the \(Z\) and the electron. It is, therefore, known with outstanding precision and can be used as a \"standard candle\" to estimate the integrated flux and, with poorer precision, the spectral flux \(\phi_{\
u_{\mu}}\). The signature of Equation (17) is an isolated electromagnetic shower appearing in the same direction as the neutrino beam. For MINERvA [157], no background mips must be found in a 10\({}^{\circ}\) cone around the candidate electron. The dominant background in these analyses is always the \(\
u_{e}\) and \(\overline{\
u}_{e}\) CC events with a missing neutron. This background is mitigated by cuts in the \((E_{e},\theta_{e})\) plane, where \(E_{e}\) is the electron energy and \(\theta_{e}\) is the electron angle with respect to the beam axis. The selected sample in MINERvA is shown in Figure 16.  
Using this method, MINERvA reduced the integrated flux uncertainty from 7.6% to 3.9% and similar performance are expected in DUNE [158]. The electron-scattering technique has three important drawbacks. Firstly, the cross section is extremely small: 10\({}^{-3}\) smaller than the nucleon scattering cross section. This  
Figure 16: Reconstructed electron energy in \(\
u_{\mu}\)\(e^{-}\rightarrow\
u_{\mu}\)\(e^{-}\) after background subtraction and efficiency correction in data and simulation (above) and the ratio of data to simulation (below). The error bars on the data include both statistical and systematic uncertainties. Reproduced under CCA4.0-I licence with permission from the American Physical Society [157]. Figure 15: Black points: neutrino energy resolution, corresponding to the 1\(\sigma\) beam width, versus the distance of the \(\
u_{\mu}\) interaction vertex to the beam axis (“R @ far”). Red points: mean neutrino energy (in GeV) versus R @ far. implies that the measurement does not work for low-intensity beams but is an asset for DUNE and HyperK. For cross section measurements, it can be used only as an ancillary measurement to be combined with the flux constraints of monitored neutrino beams. Secondly, the topology is just an isolated electron around the axis of the beam: background soon becomes intractable if the pile-up rate is high or the \(\
u_{\mu}\) background is not known with very high precision. In turn, this background introduces additional systematics that come from nuclear effects in neutrino-nucleon scattering. Finally, it is worth mentioning that conventional beams are completely dominated by the \(\
u_{\mu}\) flux and, therefore, this technique does not provide information on the \(\
u_{e}\) flux, which must be inferred either by hadron production+simulation or by lepton monitoring as in ENUBET. Another technique, developed by the CCFR collaboration in the 80s, is based on the selection of \(\
u_{\mu}\) CC events (\(\
u_{\mu}+A\rightarrow\mu^{-}+X\)) where the energy of the system \(X\) recoiling against the muon is very small. These events are called \"low-\(\
u\)\" interactions because the kinematic variable \(\
u\) is defined as the sum of the energy of all particles belonging to \(X\) (\(\
u\) is then the hadronic energy \(E_{X}\)).