# Design and Diagnostics of High-Precision Accelerator Neutrino Beams
## 1 Introduction  
Accelerator neutrino beams played a pivotal role in unveiling the electroweak sector of the Standard Model and in the discovery of Neutrino Oscillations [1]. Unlike natural neutrino sources, they offer a superior degree of control on the momentum, direction and flavor of the neutrinos at the source and therefore, are the ideal facilities for the precision era of neutrino oscillation physics [2]. Before the discovery of neutrino oscillations, hadron beam diagnostics was a useful tool to determine the beamline performance, the integrated intensity and the flux variation over time. After the discovery of non-zero \(\theta_{13}\) in 2012 [3], it is evident that these facilities are the optimal tools to investigate lepton mixing with unprecedented precision. Up to now, accelerator neutrino experiments provided a wealth of information on neutrino mixing and masses but they face unprecedented challenges in terms of precision: the next generation of experiments for the study of the missing parameters (the CP violating phase, the mass hierarchy and the octant of \(\theta_{23}\)[4]) employs multi-MW proton beams and very massive detectors (>40,000 tonnes) located in underground laboratories, hundreds of km far from the beam source. These experiments are called _long-baseline_ experiments because the beam-to-detector distance is very large (\(\gg\)10 km). For the first time in the history of neutrino physics, these facilities will be no more limited by the event statistics-thanks to the detector mass and beam power-but by a large set of systematic uncertainties that must be controlled at the per-cent level to achieve our new physics goals [5, 6]. Beam diagnostics is then prominent for the success of the next generation of experiments and, in particular, for DUNE [7] and Hyper-Kamiokande [8] (HK) that will be in data taking in less than a decade. Similarly, novel designs and advanced diagnostics are important tools to reduce the systematic uncertainties of neutrino cross-sections, which are still at 10-30% level in the few-GeV neutrino energy range [9]. The cross sections are measured by dedicatedexperiments where, unlike DUNE or HK, the neutrino detector is located at a short distance from the source to ensure that no oscillation takes place (_short-baseline_ experiments). The neutrino cross sections are extracted by the number of interacting neutrinos in the detector, which is given by:  
\[N(E_{R})\sim\int dE\ \sigma_{f}(E)\epsilon_{f}(E\to E_{R})\phi_{f}(E) \tag{1}\]  
where \(N\) is the number of neutrino interactions observed in the detector for a reconstructed neutrino energy \(E_{R}\), \(\sigma_{f}(E)\) is the neutrino cross section at the true energy \(E\) for a given flavor (\(f=v_{e}\,,v_{\mu}\,,v_{\tau}\,,\bar{v}_{e}\,,\bar{v}_{\mu}\,,\bar{v}_{\tau}\)), \(\epsilon_{f}(E\to E_{R})\) is the probability to detect a neutrino with energy \(E\) and reconstruct it at the energy \(E_{R}\), and \(\phi_{f}(E)\) is the neutrino flux of that flavor expressed in neutrinos/m\({}^{2}\)s. The coefficient that links the left and right side of Equation (1) is the number of scattering centers crossed by the neutrinos and is, therefore, very well known from the detector mass. Despite the complexity of the reconstruction of neutrino interactions, the leading source of systematics on the cross-sections is the flux \(\phi_{f}(E)\), which is generally known with a precision worse than 10%. Beam diagnostics is nearly the only tool we have at our disposal to reduce substantially such uncertainty. In general, precision neutrino physics is driving a revision of beam designs and boosting the field of high-precision beam diagnostics [10], a branch of applied physics that has been perfected in colliders but overlooked in neutrino beams. This review focuses on the design and control of accelerator neutrino beams looking at the three most important applications of these facilities:  
* Neutrino beams with unprecedented precision on \(\phi_{f}(E)\) for the measurement of the \(\
u_{e}\) and \(\overline{\
u}_{e}\) cross sections.