# Design and Diagnostics of High-Precision Accelerator Neutrino Beams
## 2 Accelerator Neutrino Beams
### Narrow and Wide-Band Conventional Beams  
The vast majority of neutrino beams designed in the past [15], together with those that are currently in operation, are optimized to ensure the largest neutrino _flux_ in a broad energy range. These beams are referred-to as _wide-band beams_, where the term \"band\" refers to the width of the momentum spectrum of the hadrons and therefore, produced neutrinos (see Figure 1). These beams have been the workhorse of major discoveries in oscillation physics because they compensate the tiny cross sections of the neutrinos at the GeV scale (\(\sigma\simeq 10^{-38}\) cm\({}^{2}\)) with a significant flux enhancement. The maximum bandwidth of a neutrino beam can be calculated from a _bare target experiment_, where all produced secondary particles and the tertiary neutrinos freely stream towards the decay volume or the detector. As an illustration, for the case of the pions, the Feynman parton model [3, 16] predicts that the average angle of emission of the secondaries is given by:  
\[\theta\simeq\frac{\langle p_{T}\rangle}{p_{\pi}}\simeq\frac{280\text{ MeV}}{p_{\pi}}=\frac{2}{\gamma} \tag{2}\]  
where \(p_{T}\) and \(p_{\pi}\) is the transverse and total momentum of the pion. This is because the parton momentum inside a nucleus is different from zero and the transverse momentum is Lorentz invariant. Therefore, \(p_{T}\) traces the momentum distribution of the parton inside  
Figure 1: **(top)** A typical layout for a narrow-band beam. After the target, a set of quadrupoles, dipoles and collimators (slits) selects the sign and momentum and steers the secondaries toward the decay volume. All charged particles but neutrinos are stopped at the hadron dump located before the neutrino detector. The _baseline_ of the facility is the target to neutrino detector distance. (**bottom**) A wide-band beam where the focusing and momentum selection is performed by a set of horns (in the figure, “Horn 1” and “Horn 2”). The second horn is often called “reflector”. Note that a horn can be inserted also in a narrow-band beam, generally between the target and the first quadrupole. the nucleus independently of the Lorentz boost along the proton direction. Experimental data [3] suggest \(\langle p_{T}\rangle\simeq 280\) MeV and this quantity is scale-independent, i.e., it does not depend on the longitudinal momentum of the secondary particle produced by the parton. The longitudinal particle momentum is described by:  
\[\frac{d^{2}N}{dx_{F}dp_{T}}\simeq f(x_{F})g(p_{T}) \tag{3}\]  
where \(f\) and \(g\) are empirical functions depending only on the Feynman's \(x\) (\(x_{F}\simeq p_{z}/p\)) and the transverse momentum \(p_{T}\), respectively. \(p\) is the momentum of the secondary particle (e.g., the pion) and \(p_{z}\) its longitudinal component. Equation (3) shows that the longitudinal effects are factorized with respect to the transverse momentum. \(g(p_{T})\) then dominates the spread of the secondary beam after the target. A wide-band beam (Figure 1, bottom) employs the focusing given by the horns, described in Section 6.1, to produce a large acceptance beam. The subsequent decays of the pions and the kaons produce neutrinos over a broad energy range. The pion angle decreases with \(\gamma\) and, in turn, the corresponding \(\
u_{\mu}\) is collinear with the pion within an angle \(\simeq 1/\gamma\). As a consequence, if the focusing selects high-energy particles, the neutrinos are well-collimated by the Lorentz boost but transverse focusing is very important for low energy wide-band beam. A _narrow-band beam_ (Figure 1, top) offers a higher degree of precision at the expenses of a strong reduction of the flux. In this case, the momentum and charge of the hadrons (and, in turn, the produced neutrinos) are selected employing magnetic dipoles and slits. The typical narrow-band beam spectrum is called \"dichromatic\" [14] because two distinct peaks are visible: the peak of the neutrinos coming from the selected pions and the peak of the neutrinos from the selected kaons (see Figure 2).