# Design and Diagnostics of High-Precision Accelerator Neutrino Beams
## 6 Focusing of Secondary Mesons
### Horns
Developing horns with significantly improved operational specifications than any of the reported configurations of Table 1 goes beyond the state-of-the-art, and requires dedicated studies or alternative solutions to the problem [115]. This has been the case for many proposed next-generation neutrino beams. For instance, a dedicated study of a \(\sim\)300 kA horn with a 12.5 Hz repetition rate had been carried out for the European Neutrino Superbeam (EURO\(\
u\)SB) [114; 116; 117]. More recently, the European Spallation Source Neutrino Superbeam (ESS\(\
u\)SB) [31; 118], which would operate a proton extraction of 2.68 ms pulses of \(\sim\)10\({}^{15}\) protons repeated at 14 Hz, has chosen to develop a dedicated \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Beamline** & **Current** & **Pulse Length** & **Repetition** & **Intensity** & **Momentum** & **Extraction** \\  & **(kA)** & **(ms)** & **(Hz)** & **(protons)** & **(GeV/c)** & **Extraction** \\ \hline CNGS [112; 113; 114] & 150 & 3.8 & 2 by 50 ms, 0.3 Hz & \(2.4\times 10^{13}\) & 400 & Fast \\ MiniBooNE [30] & 170 & 0.143 & 5 & \(5\times 10^{12}\) & 8.9 & Fast \\ NuMI [29] & 205 & 2.3 & 0.5 & \(3\times 10^{13}\) & 120 & Fast \\ T2K [26; 111] & 320 & 2.4 & 0.5 & \(3\times 10^{14}\) & 30 & Fast \\ WANF [46; 78] & 100 & \(\gtrsim\)7 & 2 by 2.7 s, 0.07 Hz & \(2.4\times 10^{13}\) & 330–450 & Slow (6 ms) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the main horn and extraction parameters for different neutrino beams. The repetition rates of CNGS and WANF consisted in two separate proton pulses, separated by the time interval reported in the table. “Pulse length” is the time duration of the current pulse from the power supply, while “Current” refers to the peak current value of the pulse. accumulator ring to compress the proton pulses to some \(\upmu\)s of temporal length [32; 119], in order to exploit standard horn technologies. The burst-mode slow extraction developed at the SPS as an option for the ENUBET monitored beam (Section 3, Figure 3) also requires dedicated horn studies. Aside from the hardware limitations and constraints, selecting the optimal horn geometry and implementation for a particular neutrino beam is a complex task. As the horn is a non-linear magnet (field \(\propto\,1/R\)), its longitudinal profile plays an important role in the final focusing, and-by design-particles need to cross its surface to be focused. Extensive numerical simulations based on particle tracking and interaction codes (e.g., GEANT4, FLUKA, MARS [120], G4 beamline) are mandatory to quantify the focusing effectiveness. Each horn geometry has a quite high number of degrees of freedom (typically higher than 10) which play a role in its performance, and numerous constraints coming from the hardware side. For this reason, horn designers perform multi-parameter numerical optimizations to find the best configuration, recently also exploiting meta-heuristic global optimization methods (e.g., genetic algorithms) [49; 121; 122; 123]. Horn diagnostics is pivotal not only for safety and reliability reasons. Variations in the current flowing through the horn and permanent deformations of the conductor generate systematic biases in the beam focusing, which ultimately affects the neutrino flux. Deformations change the beam optics from the design one, thus reducing the flux directed toward the Far Detector and increasing the production of background tertiaries. This background originates in unfocused particles hitting the downstream beamline elements. The horn currents are typically measured with a precision of 1%, and give a very small contribution to the flux systematic budget. For experiments where the target is located next to the horn, the systematics are driven by uncertainties in the relative target-to-horn distance [61; 62].