# Design and Diagnostics of High-Precision Accelerator Neutrino Beams
## 11 From Monitored to Tagged Neutrino Beams  
The possibility of measuring the time of neutrino interactions at the \(\mathcal{O}(100\,\mathrm{ps})\) level would open up new opportunities to further improve the precision of neutrino beams. The _stroboscopic beam_ is a proposal that goes in this direction [161]. The arrival time of neutrinos inherits the radio-frequency bunch time-structure of the protons with smearing due to the spectrum of the energies of secondary mesons: in practice, neutrinos generated from low-energy hadron parents will arrive later at the detector and high-energy parents will produce neutrinos that arrive before. We can select samples with markedly different energy spectra by picking-up neutrino interactions according to their arrival time. The fractions of events of each flavor have some dependence on the arrival time, too, as well as the fraction of events originating from different parents (pions, kaons and muon DIF). The achievable performance in an upgraded FNAL Main Injector ring with superconductingradio-frequency cavities capable to re-bunch the present 53.1 MHz by a factor of 10 is discussed in [161]. Improving the time tagging accuracy of monitored neutrino beams down to \(\mathcal{O}(100\) ps) in the decay volume would represent a major breakthrough. The reconstruction of leptons on an event-by-event basis gives us the possibility to time-correlate neutrino interactions with the decay products, i.e., associate every neutrino interaction seen in the neutrino detector to the corresponding charged-lepton seen in the decay volume. This facility was envisaged in the 60s and is called a _tagged neutrino beam_[162; 163]. A precise time-coincidence resolves the flavor of the neutrino a priori, i.e., without inferring it from the reconstruction of the lepton produced in CC interaction (Non-standard neutrino interaction effects could be studied in this way by comparing the lepton emitted at the production and interaction stage). If we are able to reconstruct also the other decay products of the parent (\(\pi\) or \(K\)), we can constrain the energy of the neutrino with even higher precision than NBOA. The time-tagging would, hence, raise the purity of the selected sample of neutrino interactions to an unprecedented level. Once a neutrino interaction is observed in the neutrino detector, we can select all lepton candidates whose recorded time is compatible with the neutrino time within the resolution of the detectors. Neglecting the neutrino mass, the time of the lepton time-tag (\(t_{l}\)) and of the neutrino interaction (\(t_{\
u}\)) are linked by:  
\[c(t_{\
u}-t_{l})=d_{di}-d_{dt}\frac{E_{l}}{p_{l}} \tag{20}\]  
The distance between the position of the decay and the lepton interaction in the decay volume (\(d_{dt}\)) and the distance between the decay and the neutrino interaction (\(d_{di}\)) depend on the radial position of the decay and the interaction in the transverse plane and on the emission angles of the lepton and the neutrino. In a two-body decay, where the lepton and the neutrino are coplanar, we have:  
\[d_{dt} =(r_{t}-r_{d})/\tan\theta_{t}\] \[d_{di} =(r_{i}-r_{t})/\tan\theta_{v}\]  
where \(d\), \(i\), \(t\) indicate the decay, the neutrino interaction and the lepton tag, respectively. For 3-body decays, the azimuthal angles also play a role. Since the neutrino production vertex is unknown, the correction term given by \(d_{di}-d_{dt}\frac{E_{l}}{p_{l}}\) in Equation (20) can only be approximated by \(z_{i}-z_{t}\) i.e., the distance of the neutrino interaction vertex (\(z_{i}\)) and the lepton tag (\(z_{t}\)) projected along the axis of the decay volume. This approximation causes a systematic time-shift due to the fact that the lepton and the neutrino are not perfectly collinear. This term introduces an \"irreducible\" time-spread (\(\delta_{irr}\)), which can exceed 100 ps in some facilities. This is the reason why 100 ps is a natural goal for timing in monitored neutrino beams.