11institutetext: CERN, BE Department, Esplanade des Particules 1, Meyrin, 1211
Geneva 23, Switzerland 22institutetext: Aix Marseille Univ, CNRS/IN2P3, CPPM,
Marseille, France

# NuTag: proof-of-concept study for a long-baseline neutrino beam

A. Baratto-Roldán _Corresponding author: A. Baratto-Roldán (CERN).
<EMAIL_ADDRESS>M. Perrin-Terrin 22 E.G. Parozzi 11 M.A.
Jebramcik 11 N. Charitonidis 11

(Received: date / Revised version: date)

###### Abstract

The study of neutrino oscillation at accelerators is limited by systematic
uncertainties, in particular on the neutrino flux, cross-section, and energy
estimates. These systematic uncertainties could be eliminated by a novel
experimental technique: neutrino tagging. This technique relies on a new type
of neutrino beamline and its associated instrumentation which would enable the
kinematical reconstruction of the neutrinos produced in
$\pi^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
and
$K^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
decays. This article presents a proof-of-concept study for such a tagged
beamline, aiming to serve a long baseline neutrino experiment exploiting a
megaton scale natural water Cherenkov detector. After optimizing the target
and the beamline optics to first order, a complete Monte Carlo simulation of
the beamline has been performed. The results show that the beamline provides a
meson beam compatible with the operation of the spectrometer, and delivers a
neutrino flux sufficient to collect neutrino samples with a size comparable
with similar experiments and with other un-tagged long-baseline neutrino
experimental proposals.

###### pacs:

PACS-keydiscribing text of that key and PACS-keydiscribing text of that key

##  1 Introduction

Neutrino tagging is a novel technique being developed for accelerator-based
neutrino experiments hep-ph_Perrin-Terrin_2022 . Usually, the characteristics
of the neutrinos (energy, flavour, direction, and chirality) are estimated
based on the signals they induce when interacting in a neutrino detector. The
properties of these interacting neutrinos are then used to infer the overall
neutrino flux, which is essential to study neutrino oscillations. However, as
only a tiny fraction of neutrinos interact with the detector, the flux
calculation is limited by large uncertainties. In addition, the precision on
the expected neutrino characteristic is strongly limited by the uncertainties
in the models describing the neutrino interactions. While the neutrino
oscillations studies could so far accommodate these uncertainties, they are
becoming the limiting factor for the physics program of the next generations
of experiments hep-ph_BrancaEtAl_2021 ; hep-ph_ChatterjeeEtAl_2021 ; hep-
ph_DeRomeriEtAl_2016 .

The neutrino tagging (NuTag) technique hep-ph_Perrin-Terrin_2022  proposes to
exploit the neutrino production mechanisms to complement the information
obtained using the interactions in neutrino detectors. While neutrinos can
interact with matter through multiple interaction channels, all exhibiting
large experimental signature variability hep-ph_SajjadAtharEtAl_2023 , their
production mechanisms are dominated by two experimentally well controlled
processes: the
$\pi^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
and the
$K^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
decays. Hence, the neutrinos can be kinematically reconstructed from the
$\pi^{\pm}$ and $\mu^{\pm}$ or $K^{\pm}$ and $\mu^{\pm}$ characteristics.
These particles being electrically charged, they can be detected with high
efficiency and precision Knoll . The neutrinos reconstructed as such are
called tagged neutrinos. This reconstruction method offers unmatched precision
on the neutrino characteristics. For example, neutrino energy resolutions
better than 1% hep-ph_Perrin-Terrin_2022  can be obtained with the tagging,
while the typical energy resolutions of neutrino detectors are of the order of
10-30% hep-ph_FriedlandEtAl_2019 ; hep-ph_HyperKamiokande_2018 . Moreover, as
elaborated in hep-ph_Perrin-Terrin_2022 , the resolutions on the direction and
time of arrival of the tagged neutrinos allow associating each interacting
neutrino to its corresponding tagged neutrino. As a result, a tagged neutrino
experiment can exploit the precise information provided by the tagging to
study a wide range of topics both at short and long baseline experiments, such
as neutrino interaction models and cross-sections or, neutrino oscillations.
The former two are crucial inputs for the upcoming long baseline experiments
hep-ph_ESPP_2020 , DUNE hep-ph_DUNE_2020  and T2HK hep-ph_HyperKamiokande_2018
. For the latter, preliminary studies hep-ph_Perrin-Terrin_2022  indicate that
a tagged neutrino beam with a $\mathcal{O}(\mathrm{Mton})$ scale natural-water
Cherenkov detector would reach unmatched precision on the phase controlling
the charge-parity violation in the neutrino sector $\delta_{\textrm{CP}}$.

The concept of neutrino tagging was originally introduced in the 1970s with
various experimental proposals hep-ph_Pontecorvo_1979 ; hep-ph_Nedyalkov_1984
; hep-ph_Kaftanov_1979 ; hep-ph_Hand_1969 ; hep-ph_BoikovEtAl_1980 ; hep-
ph_Bohm_1987 ; hep-ph_BernsteinEtAl_1990 ; hep-ph_Bernstein_1989 . Despite
their differences, all these designs relied on instrumenting the beamline to
detect the meson decay products and, ideally, the mesons themselves. Among
these designs, only one has been implemented: the Tagged Neutrino Facility
(TNF) hep-ph_Bohm_1987  in Serpukhov and it only featured instruments to
characterise the mesons decay products. To the authors’ best knowledge, the
TNF collaboration has not published any results, except preliminary ones
showing one neutrino in coincidence with a $\mu^{+}$ hep-ph_AnikeevEtAl_1998 .

In a tagged neutrino experiment, the neutrino rate is directly limited by the
beam instrumentation performance (e.g. readout capability, time resolution)
hep-ph_Nedyalkov_1984 . Such limitations probably hindered the development of
the tagging technique and led the community to adopt experimental designs with
maximum beam intensities towards the maximum number of neutrinos to be
detected in the neutrino detectors. However, with the recent progress of
silicon pixel technologies, the detection rate capability has strongly
increased. The ongoing developments in the silicon pixel detector technology
field hep-ph_AglieriRinellaEtAl_2019 ; hep-ph_Lai_2018 ; hep-ph_ECFA_2020
will allow to reach readout fluxes as high as
$100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$,
integrated fluences of
$10^{16-17}\text{\,}\mathrm{n}_{eq}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$ and
time resolutions as good as $10\text{\,}\mathrm{ps}$ hep-ph_Lampis_2022 . In
this context, our present work targets the design of a beamline specifically
tailored to exploit the advantages of the NuTag technique.

In this article, we present a conceptual design for a tagged long-baseline
neutrino beam for the study of neutrino oscillations. In such tagged setups,
the neutrino initial flavour, chirality, energy, and direction are determined
by the tagger with high precision. Thus, the purpose of the far detector is
limited to the identification of the neutrinos flavour. As a result, the
granularity of the far detectors is not as crucial as for non-tagged
experiments. Hence, an interesting option is to employ
$\mathcal{O}(\mathrm{Mton})$ scale natural-water Cherenkov detectors which can
attain sizes two orders of magnitude larger than the detectors of the upcoming
experiments (DUNE and T2HK). For example, KM3NeT-ORCA hep-ph_KM3Net_2016 ,
under construction off-shore France, will ultimately instrument
$6.8\text{\,}\mathrm{M}\mathrm{t}\mathrm{o}\mathrm{n}$ of sea water hep-
ph_KM3NeT_2022 . The energy threshold of this detector is
$4\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ hep-ph_KM3NeT_2022 . In this paper,
we have mainly considered the case study of an experiment with a first
oscillation maximum energy ($E^{\rm{vac.}}_{1}$)111 The $E^{\rm{vac.}}_{1}$is
computed for neutrinos in vacuum and using the oscillation parameters values
from hep-ph_ParticleDataGroup_2022  at
$5.0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$, which corresponds to a baseline
length (BL) of $2600\text{\,}\mathrm{k}\mathrm{m}$ similar to P2O hep-
ph_AkindinovEtAl_2019 . As the natural water Cherenkov detection technology
developed for KM3NeT/ORCA hep-ph_KM3NeT_2018  would allow to access both high
and low energy ranges, with energies as low as
$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ hep-ph_HofestaedtEtAl_2020 ,
several other configurations could be envisaged such as:

  * •

CERN to Pylos, Greece (BL of $1700\text{\,}\mathrm{k}\mathrm{m}$ \-
$E^{\rm{vac.}}_{1}$ of $3.3\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$) hep-
ph_BallEtAl_1995 ,

  * •

CERN to Gulf of Taranto, Italy (BL of $1250\text{\,}\mathrm{k}\mathrm{m}$ \-
$E^{\rm{vac.}}_{1}$ of $2.4\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$) hep-
ph_BallEtAl_2007 ,

  * •

CERN to Capo-Pasero (KM3NeT-ARCA site hep-ph_KM3Net_2016 ), Italy (BL of
$1300\text{\,}\mathrm{k}\mathrm{m}$ \- $E^{\rm{vac.}}_{1}$ of
$2.5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$),

  * •

European Spallation Source222Extra accelerating infrastructure would be needed
to bring the protons to an energy sufficient to produce neutrino’s at the
$E^{\rm{vac.}}_{1}$. to La Seyne-Sur-Mer (KM3NeT-ORCA site hep-ph_KM3Net_2016
), France (BL of $1500\text{\,}\mathrm{k}\mathrm{m}$ – $E^{\rm{vac.}}_{1}$ of
$2.9\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$),

  * •

Fermilab to Neptune (BL of $3100\text{\,}\mathrm{k}\mathrm{m}$ \-
$E^{\rm{vac.}}_{1}$ of $6.0\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$)
hep_ph_Vallee .

For this next generation of tagged neutrino experiments to supersede the
upcoming ones hep-ph_DUNE_2020 ; hep-ph_HyperKamiokande_2018 , they have to
provide, first, a better control of the systematical uncertainties on the
neutrino flux and energy estimate and, second, neutrino samples at least as
large. While the tagging is expected to drastically reduce theses systematical
uncertainties, it will also limit strongly the neutrino beam rate which,
together with the detector mass, determines the neutrino sample size. Hence,
we use the number of neutrino interactions,
$n_{\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.75pt}\cr\vskip-0.90417pt\cr\hbox to5.60002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.39998pt}\cr\vskip-0.90417pt\cr\hbox to4.89998pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.20552pt_{e}}$,
as a figure of merit for our beam line study. This number can be expressed as

$n_{\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.75pt}\cr\vskip-0.90417pt\cr\hbox to5.60002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.39998pt}\cr\vskip-0.90417pt\cr\hbox to4.89998pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.20552pt_{e}}(E^{\rm{mat.}}_{1})=\mathcal{F}_{\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.75pt}\cr\vskip-0.90417pt\cr\hbox to5.60002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.39998pt}\cr\vskip-0.90417pt\cr\hbox to4.89998pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.20552pt_{\mu}}(E^{\rm{mat.}}_{1})\cdot
P(\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}\to\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{e})(E^{\rm{mat.}}_{1})\cdot\sigma^{CC}_{\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.75pt}\cr\vskip-0.90417pt\cr\hbox to5.60002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.39998pt}\cr\vskip-0.90417pt\cr\hbox to4.89998pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.20552pt_{e}}\cdot\mathcal{N}_{\rm{nucl}},$
(1)

with $E^{\rm{mat.}}_{1}$ the energy of the first oscillation maximum derived
with the OscProb software OscProb  accounting for matter effects hep-
ph_MikheyevEtAl_1985 ; hep-ph_Wolfenstein_1978  and using the same oscillation
parameters values as hep-ph_DUNE_2020 ,
$\mathcal{F}_{\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.75pt}\cr\vskip-0.90417pt\cr\hbox to5.60002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.39998pt}\cr\vskip-0.90417pt\cr\hbox to4.89998pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.20552pt_{\mu}}$
the neutrino flux,
$P(\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}\to\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{e})$
the oscillation probability,
$\sigma^{CC}_{\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.79996pt}\cr\vskip-0.90417pt\cr\hbox to7.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.75pt}\cr\vskip-0.90417pt\cr\hbox to5.60002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.39998pt}\cr\vskip-0.90417pt\cr\hbox to4.89998pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-0.90417pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.20552pt_{e}}$
the charged current neutrino cross-section and $\mathcal{N}_{nuc}$ the number
of nucleons in the detector. The latter is obtained from the detector
effective mass, the molar masses of its constitutive elements and the Avogadro
number. Hence, using Equation (1), with the number of oscillated
$\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{e}$
expected for DUNE at the first oscillation maximum hep-ph_DUNE_2020 , one can
derive a minimum flux for a given detector mass and $E^{\rm{mat.}}_{1}$. The
minimum fluxes for a 5 and a
$10\text{\,}\mathrm{M}\mathrm{t}\mathrm{o}\mathrm{n}$ detectors are shown in
Figure 1. For neutrinos, both the oscillation probability and cross-section
increase with $E^{\rm{mat.}}_{1}$. Hence, the minimum neutrino flux steadily
decreases with $E^{\rm{mat.}}_{1}$. By contrast, for anti-neutrinos, while the
cross-section increases with $E^{\rm{mat.}}_{1}$, the oscillation probability
decreases with it. As a result, the minimum flux is constant above
$3\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$.

(a)

(b)

Figure 1: Minimum neutrino (a) and anti-neutrino (b) fluxes required to
collect neutrino samples as large as DUNE hep-ph_DUNE_2020  assuming a
$5\text{\,}\mathrm{M}\mathrm{t}\mathrm{o}\mathrm{n}$ and a
$10\text{\,}\mathrm{M}\mathrm{t}\mathrm{o}\mathrm{n}$ far detectors. The
fluxes are shown as a function of the energy of the first oscillation maximum,
calculated accounting for matter effects hep-ph_MikheyevEtAl_1985 ; hep-
ph_Wolfenstein_1978  with the OscProb software OscProb .

##  2 Neutrino beamline design general considerations

The first systematic studies on accelerator neutrino beams were presented in
1965 Frazinetti , where the definition of “conventional” neutrino beams, being
used until today, was coined. In conventional neutrino beams, the neutrinos
originate in the decay of $\pi^{\pm}$ or $K^{\pm}$ typically produced by the
interaction of a high-intensity, high-momentum proton beam with a target
material. The
$\pi^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
process is dominating over the
$K^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
due to the fewer number of kaons produced, and the electron-neutrinos are
always subdominant because they are created either by $K^{\pm}\rightarrow
e^{\pm}\pi^{0}\nu_{e}\ (\overline{\nu}_{e}\ )$ or by the decay in flight of
$\mu^{\pm}$, like $\pi^{+}\rightarrow\mu^{+}\nu_{\mu}\
\rightarrow(e^{+}\nu_{e}\ \overline{\nu}_{\mu})\ \nu_{\mu}\ $. The readers can
find an extensive review of the design on accelerator neutrino beams in
kopp2006  and a more recent discussion in app11041644 . For historical reasons
Dore:2018ldz , the majority of the accelerator-driven systems for neutrino
production are optimized for the maximum neutrino flux rather than for the
precision on the knowledge of the neutrino momentum and rate. These $``$wide-
band” beams compensate the small interaction cross-sections of the low-GeV/c
scale neutrinos with a large flux.

The production of a wide (or narrow) band neutrino beam can be quite different
depending on the physics scope of the experiment being served by it. However,
in general, the following considerations must be taken into account in all
cases:

  * •

A high-intensity proton beam impinging on a target material to produce the
unstable neutrino parent particles is necessary. The primary beam must be of
sufficiently high momentum and power so that a satisfactory number of
secondary particles, i.e. pions and kaons, are produced hep-
ph_ParticleDataGroup_2022 . The proton extraction on the target can be fast
(of the order of a few $\mathrm{ns}$), slow (above $700\text{\,}\mathrm{ms}$)
or fast-slow (of the order of $1\text{\,}\mathrm{ms}$ as done in the now
dismantled West Area Neutrino Facility Acquistapace:286076  at CERN). This may
have implications on the target lifetime and may require explicit cooling or
special design considerations. If the beam is fastly extracted, as was
proposed in the case of the Neutrino Factory IDS-NF:2011swj  for instance,
then the target design becomes very challenging, and only novel target
concepts like W-powder PhysRevAccelBeams.21.033401 ; CAI20222650  or liquid
mercury RADAIDEH202241  have been proposed for such cases. Even in the case of
slowly extracted beams, the radiation damage after an assumed 10 year
operation may be an important concern, especially for graphite-based target
materials (see for example SIMOS2021100028 ).

  * •

Downstream of the target, a focusing section must be installed to transversely
select a satisfactory fraction of the produced secondaries at the target.
Typically either quadrupole magnets or magnetic horns are being employed, with
their respective advantages or disadvantages (discussed extensively in
app11041644 ) and in conjunction with the selected extraction scheme. On top,
most of the horns used or proposed worldwide are optimized for focusing a
single charge polarity, due to the toroidal field of the horn horn1 ; horn2 ;
horn3 .

  * •

If a momentum selection is necessary for the parent hadrons, or if a central
momentum of the pions and therefore the neutrinos is wanted for the
experimental scope, a momentum-selection section of the hadron beamline is
needed. Typically, this involves bending magnets and collimating slits.

  * •

The overall background imperatively needs to be minimized, in order to avoid
the production of secondaries either after the momentum selection section or
in the apertures. Indeed, these secondaries will create pions or muons that
will subsequently decay to neutrinos of different momentum, possibly reaching
also the far detector and generating events that cannot be reconstructed. The
largest challenge in this respect is the primary beam, which needs either to
be dumped outside the axis of the neutrino trajectory or very early upstream
in the line, ideally followed by bending magnets that can remove the particles
with wrong momenta. Another argument for the background minimisation is the
slow read-out rates of the neutrino detectors, e.g. the reader may refer to
the novel NP-04 neutrino detector for the future DUNE experiment, with a
readout rate of the order of a few Hz Abi_2020 .

##  3 The NuTag beamline concept

A schematic of the proposed beamline concept is shown in Figure 2. In the
proposed beamline, an accelerator delivers a high-energy, high-intensity
proton beam to a target. The secondary hadrons emitted are subsequently
focused and momentum selected within a very broad momentum range
($\delta{p}/p$ of the order of 25%), while the primary proton beam is dumped
in the thick material between the holes of a collimator that acts as an
effective beam dump integrated in this momentum slit. Then, the momentum,
direction, and time of arrival of all beam particles are measured by a beam
spectrometer made of four dipole magnets, alternated with tracking stations
and arranged to form an achromat Wiedemann . These particles then drift in a
free space (order of $900\text{\,}\mathrm{m}$) where the majority will decay.
At the end of this drift space, a second spectrometer, made of two pairs of
tracking stations installed before and after a large dipole magnet, measures
the momentum, direction, and time of arrival of the beam particle decay
products, in particular the $\mu^{\pm}$’s from the
$\pi^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
and
$K^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$.
The information provided by the two spectrometers together allows to
reconstruct the tagged neutrino based on the decays kinematics. The second
spectrometer is followed by an absorber that stops the remaining surviving
mesons before they decay. The length of the beamline section upstream of the
first spectrometer and downstream of the second one should be as short as
possible to reduce the number of $\pi^{\pm}$ and $K^{\pm}$ decaying there,
since these cannot be reconstructed333To estimate the fraction of un-taggable
neutrinos, one can conservatively assume that
$\pi^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
decays occurring upstream of the tagger yield as many neutrinos in the far
detector acceptance as the decays occurring downstream of it. Under this
assumption, the fraction of un-taggable neutrinos is $(e^{d/\beta\gamma
c\tau}-1)/(1-e^{-D/\beta\gamma c\tau})$ where $d$ and $D$ are the lengths of
the beamline sections upstream and downstream of the tagger, $\beta$ and
$\gamma$ the reduced velocity and Lorentz boost of the $\pi^{\pm}$, $c$ the
speed of light and $\tau$ the $\pi^{\pm}$ mean lifetime. Finally, as the
neutrino chirality is determined by the charge of the particles in the
spectrometer, the hereby proposed NuTag technique does not require removing
one of the two beam particle polarities. Hence, the beamline should ideally
transport beam particles of both charges. Besides the obvious gain in terms of
running time, collecting both neutrino and antineutrino simultaneously is a
strong asset to reduce systematical uncertainties for key measurements such as
$\delta_{\textrm{CP}}$.

Figure 2: Conceptual scheme of the proposed NuTag beamline. The high energy
beam impinging on the target produces secondary particles (only the $\pi^{+}$
and $\pi^{-}$ beams are shown for simplicity) that are subsequently focused,
momentum selected and transported towards the meson tracking station. After
their tagging at that point, the secondary beam continues straight towards a
decay region where both $\pi^{+}$ and $\pi^{-}$ will decay to neutrinos.
Finally, a second spectrometer measures the momentum, direction and time of
arrival of the $\mu^{\pm}$ from the decay, followed by a dump to absorb the
remaining surviving beam particles.

###  3.1 Proposed NuTag beamline layout

In order to satisfy the physics case discussed in the introduction, the
beamline design hereby is optimized for a long-baseline type of facility. The
primary beam is considered here to be a slowly extracted, high-momentum proton
beam. On one hand, the constraint for the maximum rate that is accepted by the
tracking detectors (of the order of
$10-100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$)
can be only achieved via slow extraction. On the other hand, the high proton
momentum a-priori favors the meson production at the momenta of interest
($p<$$20\text{\,}\mathrm{GeV}$/c), as phenomenologically demonstrated (e.g.
the interested reader can observe the production maxima in hep-
ph_ParticleDataGroup_2022  and Atherton:133786 , where the maximum production
for secondaries of momentum $p$ seems to lie within a fraction of 10-20% of
the primary proton momentum). In our case, and using an existing beam example,
we chose the $400\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$
of the CERN Super Proton Synchrotron (SPS). The assumption of the slowly
extracted beam also defines essentially the focusing mechanism downstream of
the target. In our case, we have based this on large aperture quadrupole
magnets. Magnetic horns have not been considered in this study, given the
complexity of production and optimisation kopp2006 , despite the possible
advantages on the acceptance. Furthermore, these devices cannot easily be used
in a slow extraction scheme since they don’t stand heating beyond a few
millisecond. The magnets employed in our design are existing magnets that have
been used or are currently in use at CERN experimental areas magnetskit , with
known dimensions, properties, and characterized magnetic field gradients.
However, for a real implementation, new or different magnets could of course
be considered. Concerning the transverse optics design, the first-order matrix
approximation is used only as a guide for the beam expected properties. The
first-order optics was developed using TRANSPORT TRANSPORT  and MADX madx ,
while the calculation of the hadronic interaction with the target material was
performed using the well-known Monte-Carlo software FLUKA
10.3389/fphy.2021.788253 ; Boehlen:2109973 . Finally, a custom C++ software
was developed to estimate the neutrino flux at the detector based on the meson
flux simulated with FLUKA.

####  3.1.1 Target and acceptance stage front-end

The high-intensity, high-momentum beam from an accelerator-driven system, will
be slowly extracted on a target to produce the mesons that will then generate
the neutrino beam. The dominant production mode being
$\pi^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
we limited our study to this mode. For this study, a dedicated optimization
effort has taken place in order to identify the best target for the described
physics case. Various target geometries have been studied; literature research
for various other neutrino beams (proposals and existing ones), like CNGS
Gschwendtner:1053097 ; Bruno:note138 , NuMi jyoti2017study ; ADAMSON2016279 ,
T2K ABE2011106  and ENUBET Acerbi:2868571 , have demonstrated that graphite is
the preferred material, due to its (relatively) low radiological impact and
its robustness to high beam powers and temperatures. In our optimization
process, we started from lengths and radii values used or proposed in other
facilities, combining the two to explore the relative effect of each parameter
on the target properties. The different target configurations that
demonstrated the best performance in our optimisation study, are summarized in
Table 1. In our study, all the targets have been simulated in FLUKA as
continuous rods of graphite (density
$1.7\text{\,}\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$). Considering a
right handed Cartesian coordinate system with the beamline aligned along the
$z$ axis, the 4-vectors of produced particles in terms of $x$, $x^{\prime}$,
$y$, $y^{\prime}$, and total momentum $p$ have been scored for all different
targets in a perpendicular plane at $z=$0\text{\,}\mathrm{m}$$, the starting
point of the beamline, coinciding with the end surface of each target. The
primary beam momentum was assumed to be in all cases,
$400\text{\,}\mathrm{GeV}$/c, for reasons discussed in the previous sub-
section.

As discussed in Section 1, different baselines covering different neutrino
$E^{\rm{vac.}}_{1}$ between $2.4\text{\,}\mathrm{GeV}$ and
$6.0\text{\,}\mathrm{GeV}$ could in principle be envisaged, corresponding
roughly to a pion momentum range of $6-$12\text{\,}\mathrm{GeV}$$/c. The
following target studies have been optimized for a reference pion momentum of
$12\text{\,}\mathrm{GeV}$/c (corresponding to a baseline of
$\approx$2600\text{\,}\mathrm{km}$$ with a $E^{\rm{vac.}}_{1}$ at
$5.0\text{\,}\mathrm{GeV}$), however, other momenta have been also taken into
account, as shown later on.

Identifier | Length [cm] | Radius [mm] |  Comment
---|---|---|---
A | 70 | 5 |  Length of ENUBET proposed target Acerbi:2868571
B | 70 | 13 |  Length of ENUBET proposed target Acerbi:2868571  with T2K’s target radius ABE2011106
C | 126.1 | 2.5 |  Length and radius of CNGS target Bruno:note138
D | 126.1 | 13 |  Length of CNGS target Bruno:note138  with T2K’s target radius ABE2011106
E | 94 | 3.7 |  Length and radius of NuMi first design target ADAMSON2016279
F | 91.4 | 13 |  T2K-like target ABE2011106
Table 1: Target configurations studied in the framework of the present study.
Figure 3: $x$ ($y$) distribution for $\pi^{\pm}$’s at $z=0$. On the left,
$\pi^{\pm}$’s covering the full momentum range; on the right, $\pi^{\pm}$’s
having a reference central momentum of $12\text{\,}\mathrm{GeV}$/c assuming a
momentum spread of $\pm 25\%$

Figure 4: $x^{\prime}$ ($y^{\prime}$) distribution for $\pi^{\pm}$’s at $z=0$.
On the left, $\pi^{\pm}$’s covering the full momentum range; on the right,
$\pi^{\pm}$’s having a reference central momentum of
$12\text{\,}\mathrm{GeV}$/c assuming a momentum spread of $\pm 25\%$

.

Figure 3 and Figure 4 show a comparison, among the various targets, of the $x$
($y$) and $x^{\prime}$ ($y^{\prime}$) distributions for $\pi^{\pm}$’s at
$z=0$. In these plots, $x$ ($y$) and $x^{\prime}$ ($y^{\prime}$) distributions
are shown both for $\pi^{\pm}$’s covering the full momentum spectrum and for
$\pi^{\pm}$’s having a reference central momentum of
$12\text{\,}\mathrm{GeV}$/c, considering a momentum spread of $\pm 25\%$. The
dip observed in the $\pi^{\pm}$’s yield in the bottom right plot of Figure 4,
present in the longer and thinner targets identified as C and E in Table 1, is
due to re-absorption of the forward emitted $\pi^{\pm}$’s
($x^{\prime}<\arctan(R/L)$ where $R$ is the radius and $L$ the target length)
by the target material. This is a combined effect of the target radius and
length, as shown in Figure 5. This figure represents the $\pi^{\pm}$’s angular
divergence $x^{\prime}$ as a function of the target depth for the targets C
(5(a)) and D (5(b)), which share the same length but different radius. From
the colour scale in 5(a) emerges that the $\pi^{\pm}$’s in the considered
momentum range are largely produced at angles
$\arctan(R/L)\approx$2\text{\,}\mathrm{mrad}$<x^{\prime}<$20\text{\,}\mathrm{mrad}$$
at the very beginning of target C, and therefore escape transversely through
its sides without being reabsorbed, compared to $\pi^{\pm}$’s emitted forward,
causing the dip in the distribution on the right of Figure 4. Being this an
effect of the radius to length ratio of the target, the dip in the
$x^{\prime}$ distribution disappears as the target radius gets larger and/or
the target length smaller. Indeed, as shown in 5(b), the larger radius of
target D ($13\text{\,}\mathrm{m}\mathrm{m}$) would require much larger
emission angles $x^{\prime}>\arctan(R/L)\approx$10\text{\,}\mathrm{mrad}$$ for
a $\pi^{\pm}$ to escape transversely through the target sides. Since the
probability of emission reduces very quickly for larger angles, the effective
length seen by the $\pi^{\pm}$’s produced at the beginning of the target is
constant and the $x^{\prime}$ distribution at $z=0$ is (approximately)
Gaussian.

(a) Target C

(b) Target D

Figure 5: Angular distribution of $\pi^{\pm}$’s in the $\pm 25\%$ momentum
range around $12\text{\,}\mathrm{GeV}$/c emerging from the target C (a) and D
(b), which share the same length but different radius, as a function of the
target depth. The primary proton beam is impinging on the target front face
from the left.

Taking into account the results of the aforementioned studies, target C has
been chosen as the baseline for the current study. Further target optimization
could of course be envisaged, for example by considering spacing the target in
different sections with gaps in between to reduce re-absorption, as done for
the CNGS Bruno:note138  and NuMi jyoti2017study  targets. Besides our
reference $p=$12\text{\,}\mathrm{GeV}$/c$ $\pi^{\pm}$ beam, in our study we
also considered $\pi^{\pm}$ beams of $8\text{\,}\mathrm{GeV}$/c and
$6\text{\,}\mathrm{GeV}$/c. Figure 6 shows the position and angular
distribution of $\pi^{\pm}$’s emerging from the target for different central
momenta, considering a broad momentum spread of 25%.

Figure 6: $x$ ($y$) distribution (right) and $x^{\prime}$ ($y^{\prime}$)
distribution (left) at $z=0$ of $\pi^{\pm}$’s in the $\pm 25\%$ momentum range
around different central momenta. The overall $\pi^{\pm}$ yield decreases with
the central momentum, while a significantly larger amount of $\pi^{\pm}$’s is
emitted at larger angles.

In Figure 6, the overall $\pi^{\pm}$ yield decreases as the central momentum
decreases, with a significantly larger amount of $\pi^{\pm}$’s emitted at
larger angles for the lowest momentum considered. In all cases, the peaks of
the distributions are included in the [-20,20] $\mathrm{mrad}$ angular range.
Therefore, a critical aspect of the beamline design is the optimization of the
angular acceptance, which needs to be maximized.

Figure 7: First-order optics of the NuTag proposed beamline. The upper part
corresponds to the horizontal plane and the bottom plot to the vertical plane.
The red line corresponds to the sine-like ray, the green line to the cosine-
like ray, and the blue dotted line to the dispersive ray Wiedemann  The
tracking detectors are installed in the second achromat, just before the decay
region as discussed in the text.

####  3.1.2 First-order Optics

The first-order, transverse optics drawing of the proposed beamline is shown
in Figure 7. Downstream of the target, four quadrupoles (notated as _Q1_ ,
_Q2_ , _Q3_ and _Q4_) ensure the acceptance of a large part of the phase space
of particles produced at the target and brings the beam in both the bending
and non bending planes, in our case also in both polarities (after an
180$\degree$ phase-advance), at a double focus in the middle of a first 4-bend
achromat structure. The quadruplet, apart from allowing a large acceptance in
both planes, preserves an absolute symmetry between the R-terms (transport
matrix elements, also called $\mathbf{R}$ matrix) of the horizontal and
vertical plane up to the middle of the first achromat. At this exact position,
the momentum selection takes place, with the $\pi^{+}$’s and the $\pi^{-}$’s
passing through two collimating holes of $10.25\text{\,}\mathrm{cm}$ radius
placed respectively at +$18.25\text{\,}\mathrm{cm}$ and
-$18.25\text{\,}\mathrm{cm}$ from the centre of a shielding block (see Figure
8), which also acts as a primary beam dump as mentioned in Section 3.

Figure 8: Simplified model (visualized with FLAIR FLAIR ) of the custom
collimator structure placed in the middle of the first achromat, acting both
as a momentum slit for the $\pi^{+}$ and $\pi^{-}$ beams and as a primary beam
dump. It consists of a $5\text{\,}\mathrm{m}$ long copper block, with two
parallel holes to allow for the passage of the $\pi^{\pm}$ beams. These holes
have a radius of $10.25\text{\,}\mathrm{cm}$ and a centre to centre distance
of $36.5\text{\,}\mathrm{cm}$.

Downstream of this initial momentum selection, a second quadruplet in series
with the first (same quadrupole names correspond to same power converters),
brings the momentum-selected beam to 360 degrees phase-advance just before the
final focusing stage. At this position, the beam can be cleaned with the use
of collimators from background particles that have interacted with the magnet
apertures. At this stage, a quadrupole triplet transports (with a 90-degree
phase-advance) the double polarity beam towards a second achromat, similar to
the first but vertically oriented. The reason for the inverse orientation
compared with the upstream achormat was a slight reduction in the acceptance
caused by the large spot-size of the beam in the horizontal plane. Also, a
vertical orientation favours the placement of shielding in both sides of the
tunnel in case this was needed for background reduction. Tracking stations are
installed around and inside the second achromat in order to measure the
momentum, direction, position and time of all the charged particles, on an
event-by-event basis. The large R11 and R33 terms in the horizontal plane
along with the $900\text{\,}\mathrm{m}$ length of the decay tunnel, will
inevitably result in a transverse beam size of the order of
$\mathcal{O}($\mathrm{m}$)$ at the final tracking station, taking also into
account the muon decay angles. For this reason, only very large aperture
dipoles, such as GOLIATH Rosenthal:2310483  or MORPURGO MORPURGO1979411 ,
present today in the North Area at CERN, can serve as spectrometers in the
final tracking station, the details of which lie beyond the scope of this
paper.

The described optics predict an angular acceptance of
$\mathcal{O}$(20)$\mathrm{mrad}$ in the horizontal and
$\mathcal{O}$(9)$\mathrm{mrad}$ in the vertical plane, as will be discussed
later in Section 4.

As already mentioned, the part of the primary beam that does not interact with
the target ($<<1\%$), is dumped on the large collimator structure (shown in
Figure 8) between the two bending dipoles of the first achromat. Indeed, a
sufficient separation between the $\pi^{+}$ and $\pi^{-}$ beams (roughly
$27\text{\,}\mathrm{cm}$ for the central momentum) ensures that the
$\pi^{\pm}$ beams pass through the holes and continue downstream, while the
protons and all the neutral or off-momentum charged particles are dumped. The
collimator/dump structure proposed here is purely conceptual, serving as first
assumption, and a proper engineering design would have to follow in future
studies.

###  3.2 Meson tracking station

The beam spectrometer has the purpose of measuring the time, position,
direction, charge and momentum of all charged beam particles. The detector
design is conceptually similar to the NA62 GigaTraKer (GTK) hep-
ph_AglieriRinellaEtAl_2019 ; hep-ph_NA62_2017  and its proposed upgrade for
the HIKE experiment hep-ph_HIKE_2023 . The spectrometer is placed at the
second achromat and is composed of a set of tracking stations made from
silicon pixel detectors which are located before, after and inside this
achromat. The station should be as thin as possible to reduce the multiple
coulomb scattering of particles. A material budget of 0.5% X0 per station was
achieved for the NA62 GTK.

The measurements from all the stations allow to reconstruct the trajectory and
momentum of each beam charged particle traversing the achromat. Indeed, apart
from scattering effects, the trajectories of the particles before and after
the achromat are aligned and parallel to the trajectory described by the
particle in the innermost drift space of the achromat. The distance between
these parallel trajectories is inversely proportional to the particle
momentum. As for the NA62 GTK hep-ph_AglieriRinellaEtAl_2019 , the material
budget of the station is expected to be one of the dominant factors limiting
the precision to which the particle’s direction and momentum can be
determined.

For a configuration with three tracking planes, placed before, after and
inside the achromat, as the initial GTK design, the momentum resolution can be
expressed as hep-ph_NA62_2010 :

$\frac{\delta
p}{p}=\frac{1}{\Delta}\cdot\sqrt{\sigma^{2}_{y}+\left(\frac{\sigma_{y}\cdot
d_{12}}{d_{12}+d_{23}}\right)^{2}+\left(\frac{\sigma_{y}\cdot
d_{23}}{d_{12}+d_{23}}\right)^{2}+\left(\frac{\delta_{\rm{MCS}}\cdot
d_{13}\cdot d_{23}}{d_{12}+d_{23}}\right)^{2}}\,,$ (2)

where:

  * •

$\Delta$ is the distance between the trajectories inside and outside the
achromat for the nominal momentum and is determined by the dipole magnet
bending angle,

  * •

$\delta_{12}$ and $\delta_{23}$ are the distances along the $z$-axis between
the first and second and, second and third tracking planes,

  * •

$\sigma_{y}$ is the resolution on the $y$ coordinate of the particle at the
tracking planes and is proportional to the pixel size,

  * •

$\delta_{\rm{MCS}}$ is the mean scattering angle due to the multiple Coulomb
scattering in the tracking plane.

The momentum resolution is shown as a function of $\Delta$ in Figure 9,
assuming $6\text{\,}\mathrm{m}$ spacing between consecutive tracking stations,
a pixel size of $45\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a tracking
plane thickness equivalent to 0.5% X0. In the proposed beamline,
$\Delta=$11.7\text{\,}\mathrm{cm}$$ for a bending angle of
$45\text{\,}\mathrm{mrad}$, which corresponds to a $\delta p/p$ of 0.21% for a
$12\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ pion beam. The
$\pi^{\pm}$ momentum resolution provides a good estimate of the neutrino
energy resolution for a LBL setup, as the energy of a neutrino emitted
collinear to the $\pi^{\pm}$ is $0.43\cdot E_{\pi}$, where $E_{\pi}$ is the
$\pi^{\pm}$ energy. Hence, as long as $\Delta$ is larger than
$5\text{\,}\mathrm{c}\mathrm{m}$, a tagged neutrino experiment is able to
provide a neutrino energy resolution better than 1%.

Figure 9: Momentum resolution on the $\pi^{+}$ provided by the meson
spectrometer, assuming $6\text{\,}\mathrm{m}$ spacing between consecutive
tracking stations, a pixel size of $45\text{\,}\mathrm{\SIUnitSymbolMicro m}$
and a tracking plane thickness equivalent to 0.5% X0.

The pixel technology considered for the station is similar to the one
envisaged for the high luminosity phase of the LHC experiments LHCbU2_TDR_2022
and is expected to be able to operate at a particle flux as high as
$10-100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$
hep-ph_Lai_2018  and at an integrated fluence of up to
$10^{16-17}\text{\,}\mathrm{n}_{eq}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$. The
former specification sets a strong constraint on the peak flux of charged beam
particles at the second achromat. In addition, the beam transverse size should
remain $<0.1\text{\,}\mathrm{m}^{2}$ as assembling larger pixel modules with a
material budget of 0.5% X0 (i.e. about $500\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ of silicon) would be mechanically challenging.

##  4 Simulated performance

Figure 10: FLUKA model of the proposed beamline, visualized with FLAIR. The
acceptance stage is followed by the collimation and momentum selection
station, while afterwards the beam is transported to the downstream tracking
station. The modelling of the second meson tracking station and dump shown in
Figure 2 is beyond the scope of this study.

In order to validate the operational principle of the line, a full FLUKA model
has been developed up to the second achromat; its geometrical layout is shown
in Figure 10. This model includes a complete description of the magnetic two-
dimensional field map (both in the gap and in the yoke) for each quadrupole
present in the beamline, since the magnets used for this study (QPL and QPS
magnetskit ) are well characterized and currently in use in many secondary
beamlines at CERN. For the bending magnets however, a uniform field throughout
the aperture and with no fields in the yoke is assumed. To transport both
$\pi^{+}$ and $\pi^{-}$ beams with large angular acceptance and ensuring a
satisfactory momentum selection, bending magnets with larger horizontal
apertures than the ones currently present at CERN would be necessary. These
magnets, would be identical to the existing MBPS magnets in use today (and
described also in magnetskit ) just with a larger horizontal aperture of 600
mm. The characteristics of the proposed magnets are summarized in Table 2.

Name |  Type |  Length [m] |  Aperture type |  Aperture Radius [mm] |  Aperture Width [mm] |  Aperture Height [mm] |  Peak Field (Gradient) [T] ([T/m])
---|---|---|---|---|---|---|---
QPS |  Quadrupole |  1 |  Circular |  100 |  — |  — |  10.5
QPL |  Quadrupole |  2 |  Circular |  100 |  — |  — |  10.5
MPBS |  Dipole |  1 |  Rectan. |  — |  300/600 |  140 |  1.84
Table 2: Specifications and characteristics of the magnet types used in this
study magnetskit . A possible adaptation of the MBPS bending magnet, with a
larger aperture width ($600\text{\,}\mathrm{mm}$ compared to the standard
$300\text{\,}\mathrm{mm}$), has been considered in the FLUKA simulations, to
allow for the simultaneous transportation of both pion polarities.

In our study, we employed a simple and robust methodology in order to (a)
precisely calculate the neutrinos expected in the far detector while (b)
avoiding performing computationally expensive Monte-Carlo simulations and/or
variance reduction techniques that could create extra uncertainties in the
final yield. Specifically, we followed the steps listed below:

  1. 1.

Following the target optimisation described in Section 3.1.1, we subsequently
validated the hadron beam optics and the resulting acceptance based on
analytic calculations. At this stage, we made sure that no elements are overly
restricting the acceptance and we confirmed the robustness of our
computational framework (see Section 4.1).

  2. 2.

Subsequently, we developed a detailed FLUKA model for the beamline and used
the produced particle phase space, based on the analysis in Section 4.1 in
order to calculate the expected transmission through the beamline (see Section
4.2). We validated the overall beamline behaviour by running a full FLUKA
simulation of the primary beam, the target and all the magnetic elements down
to the beginning of the decay region.

  3. 3.

Using the results from the step above, we performed analytic kinematic
calculations for the expected number and momenta of neutrinos that will reach
the far detector (see Section 4.4).

###  4.1 Linear acceptance calculation

In order to validate the calculated optics, as a first step, we evaluated the
phase space of accepted $\pi^{\pm}$’s that can be transported from the target
to the end of the beamline. This acceptance depends strongly on the aperture
limitations of the line. These limitations are defined by the magnets
apertures, the collimator holes and the beam pipes. The beam size and angular
distribution in each element of the line is defined by the transport matrix,
that can be easily retrieved via any optics code or analytical calculations.
Therefore, by projecting backwards the individual apertures and mapping the
calculation to the beginning of the line, it is possible to identify the
smallest limiting aperture and thus the fraction of pions emitted from the
target that would eventually be transported to the end of the line.

(a)

(b)

Figure 11: Accepted phase space of the nominal momentum $\pi^{+}$ (a) and
$\pi^{-}$ (b) beam: (left) horizontal plane and (right) vertical plane. Each
of the black lines corresponds to an aperture $``$wall” or limitation present
along the beamline at some point $z$, projected backwards to the start of the
line. The red polygon represents the remaining accepted phase space, obtained
by connecting the intersection points of the most inner aperture walls.
Compared to $\pi^{+}$, the $\pi^{-}$ $y^{\prime}$ acceptance is strongly
constrained by the vertical gap of the first and last bending magnets in the
first achromat.

(a)

(b)

Figure 12: Target sample $\pi^{+}$ (a) and $\pi^{-}$ (b) accepted phase space
in the $\pm 1\%$ momentum range around $12\text{\,}\mathrm{GeV}$/c. The
selection criteria identified in Section 4.1 have been applied to the
$\pi^{\pm}$ yields emitted by the target, obtained from FLUKA simulations, to
identify the phase space theoretically transported through the line. Results
are normalized per primary proton. As expected by the linear optics, the
horizontal plane of the $\pi^{-}$ is very similar with the vertical plane of
the $\pi^{+}$ and vice-versa, as discussed in the text.

11(a) and 11(b) show the accepted phase space for the nominal momentum in the
horizontal and vertical planes for $\pi^{+}$ and $\pi^{-}$ respectively. As
anticipated from the first-order optics calculation shown in Figure 7, the
maximum horizontal and vertical angular acceptances are
$\sim$$20\text{\,}\mathrm{mrad}$ and $\sim$$9\text{\,}\mathrm{mrad}$ for
$\pi^{+}$ and inverted for $\pi^{-}$. On the other hand, the horizontal and
vertical position acceptances are $\sim$$15\text{\,}\mathrm{mm}$ and
$\sim$$25\text{\,}\mathrm{mm}$ for $\pi^{+}$ and the opposite for $\pi^{-}$.
The differences observed in the $\pi^{\pm}$ accepted phase spaces, already at
the first order optics level, are due to the asymmetric aperture of the
bending magnets that constitute the two achromats. Indeed, the small vertical
gap (140 mm) of the MBPS magnets constrains strongly the $y^{\prime}$
acceptance for the $\pi^{-}$ beam, while the acceptance of the $\pi^{+}$ is
constrained by the majority of elements present in the line, in both planes.

Particles with momenta different from the nominal momentum, however, are
subject to chromatic aberrations that become particularly important for large
momentum offsets. For momentum offsets beyond $\left|\delta
p/p_{0}\right|>10\%$, the limit of the applicability of linear optics is
reached, since the first-order approximation (used in the TWISS module of
MAD-X used to calculate the optics) is no longer valid brown . For a more
accurate evaluation of these particles trajectories, a Monte Carlo simulation
was used as will be shown in Section 4.2.

###  4.2 FLUKA simulations and beamline transmission

The acceptance selection criteria depicted by the red lines in 11(a) and 11(b)
have been applied to the $\pi^{\pm}$ yields produced by the target simulations
and scored in FLUKA, as described in Section 3.1.1. The results of this
selection are shown in Figure 12 for $\pi^{\pm}$ in a $\pm 1\%$ momentum range
around $12\text{\,}\mathrm{GeV}$/c, to stay within the validity limits of the
first order optics and avoid second order chromatic effects.

Starting from the identified accepted phase space, and assuming a 1% momentum
acceptance, we performed extensive FLUKA simulations to validate the
transmission of the momentum-selected hadron beams through the beamline (i.e.,
excluding all background), modelling all elements outside the beam-pipe as the
FLUKA “blackhole” i.e, a fully absorbent material that prevents any further
transportation of interacting particles. The result of these FLUKA simulations
is shown in Figure 13. It can be seen that for this small momentum spread, the
beam trajectory is in very good agreement with the linear optics for both
$\pi^{+}$ and $\pi^{-}$. At the momentum-selection position, the slight
astigmatism observed between the horizontal and vertical plane, is due to the
slightly different magnification terms (green dashed lines in Figure 7) along
with the dispersion present in the horizontal plane. The result is that the
beam waist is displaced by a few cm upstream.

Figure 13: FLUKA simulation of accepted $\pi^{+}$ and $\pi^{-}$ phase spaces
identified in Figure 12 transported through the beamline. (Top) horizontal and
(Bottom) vertical planes. All the results are normalized per primary proton.
The trajectories of the particles inside the decay region are not shown.

###  4.3 Full Simulation of the beamline and calculation of momentum
acceptance

To obtain a fully realistic estimation of the number of the $\pi^{\pm}$’s
transmitted through the line and their momentum distribution (that defines the
“momentum acceptance“ of the beamline), a second simulation was performed
using the whole target sample as a source. From this simulation, we derived
the momentum distribution of the transmitted pion beams that we show in Figure
14 for three different reference momenta. It can be seen that the centroid of
the $\pi^{+}$ and $\pi^{-}$ distributions do not coincide exactly, being the
first peaked around $12\text{\,}\mathrm{GeV}$/c and the second around
$13\text{\,}\mathrm{GeV}$/c for a $12\text{\,}\mathrm{GeV}$/c reference
momentum. This effect can be explained by second order aberrations and is
independent on the reference momentum chosen, since the two beams $\pi^{+}$
and the $\pi^{-}$ have waists in different positions along z, compared to the
focus position, and therefore are (slightly) differently momentum-selected.

Figure 14: Momentum spectrum of $\pi^{\pm}$’s at the start of the decay
region, after having been transported through the beamline. The spectrum is
shown for three different beamline momenta: $12\text{\,}\mathrm{GeV}$ (solid
line), $8\text{\,}\mathrm{GeV}$ (dashed line) and $6\text{\,}\mathrm{GeV}$
(dotted line). The individual contribution to the full spectrum (black) of
$\pi^{+}$(red) and $\pi^{-}$beams is also shown for each reference momentum.

The higher momentum particles observed in Figure 14 especially in the
$\pi^{+}$ beam are a consequence of the wider horizontal phase space and the
particular configuration chosen for the collimator apertures. This collimator
is $5\text{\,}\mathrm{m}$ ($\approx 30$ interaction lengths) long to ensure
the complete absorption of the primary proton beam as well as a satisfactory
absorption of the secondaries produced by the primary beam interaction with
it. For this reason, the collimating slits of the $\pi^{+}$ and $\pi^{-}$
beams are effectively much larger than what strictly required by the
$\mathcal{O}($25%) theoretical momentum acceptance
($10.25\text{\,}\mathrm{cm}$ radius compared to $3.43\text{\,}\mathrm{cm}$
radius). Indeed, while the beam has a waist around the slit position, the size
of the accepted beam at its entrance and exit is larger (see Figure 13) and
therefore the momentum selection is less stringent. Furthermore, the wide
horizontal angular acceptance for the $\pi^{+}$ beam in combination with the
large aperture holes, allows for some $\pi^{+}$’s with higher momenta and
large angles to be transported through the beamline, as shown in Figure 15.
The overall content of the high-momentum tail is about 20% of the total, and
has been quantified by comparing the histogram integral in two different
ranges: $[0,16]~{}$\mathrm{GeV}$$/c and $[16,30]~{}$\mathrm{GeV}$$/c.

Figure 15: Two example trajectories of higher momentum $\pi^{+}$’s,
$18\text{\,}\mathrm{GeV}$/c and $20\text{\,}\mathrm{GeV}$/c, emitted at large
angles from the target, $10\text{\,}\mathrm{mrad}$/c and
$5\text{\,}\mathrm{mrad}$/c, respectively, and their transmission through the
beamline. These larger momentum, but large offset particles constitute the
species that populate the peaks averaged around
$8\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$,
$11\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ and
$18\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ that can be
seen on the right of the central momenta peaks of
$6\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$,
$8\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ and
$12\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ in Figure 14.

The simulation also allows to estimate the beam transverse size and particle
flux at the second achromat. As explained in Section 3.2, the limitations of
the tracker technology require the beam size to be
$<0.1\text{\,}\mathrm{m}^{2}$ and the rate
$10-100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$.
Figure 16 shows a two-dimensional map of the beam particle flux at the exit of
the second achromat. The two specifications are fulfilled with a beam size of
about $15\times 50\text{\,}\mathrm{c}\mathrm{m}^{2}$ and a peak flux of
$<15\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$,
considering an impinging primary proton beam of $2.5\times
10^{13}\text{\,}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{l}\mathrm{l}$
with a spill length of $4.8\text{\,}\mathrm{s}$.

(a)

(b)

(c)

Figure 16: $(x,y)$ map of the $\pi^{\pm}$ flux after the second achromat ($z=$
$63\text{\,}\mathrm{m}$) obtained from the FLUKA simulation for a beamline
with a nominal $\pi^{\pm}$ momentum of
$12\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ (a),
$8\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ (b) and
$6\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ (c),
considering an impinging primary proton beam of $2.5\times
10^{13}\text{\,}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{l}\mathrm{l}$
with a spill length of $4.8\text{\,}\mathrm{s}$.

###  4.4 Neutrino yield evaluation at the far detector

Knowing the total number as well as the full 6D phase space ($x$,
$x^{\prime}$, $y$, $y^{\prime}$, $\text{d}p/p$, $z$) of the pions transmitted
through the beamline it is possible to precisely calculate the expected
neutrino yield at the far detector. To do so, a custom calculation code has
been developed.

The software, developed in C++, generates $\pi^{\pm}$’s at the exit of the
second achromat by sampling, for $\pi^{+}$ and $\pi^{-}$ separately, the $x$,
$y$ two-dimensional distribution and $p$, $x^{\prime}$, $y^{\prime}$ three-
dimensional distribution obtained from the FLUKA simulation at this position.
For each $\pi^{\pm}$, the $z$ coordinate of a decay vertex, $z_{\rm{vtx}}$, is
generated on an exponential distribution with a decay constant of
$1/(\beta\gamma c\tau)$ where $\beta$ and $\gamma$ are the reduced velocity
and Lorentz boost of the decaying $\pi^{\pm}$, $c$ the speed of light and
$\tau$ the $\pi^{\pm}$ mean lifetime PhysRevD.98.030001 . Each $\pi^{\pm}$ is
then propagated to this $z_{\rm{vtx}}$ where the
$\pi^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
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301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
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width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
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width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
is generated using standard decay kinematics hep-ph_James_1968 . The resulting
$\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
are then propagated to the far detector, which is assumed to have a transverse
half-size of $100\text{\,}\mathrm{m}$. The neutrino energy distribution at the
far detector is obtained from the distribution of the generated
$\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$’s
which occur to be in the far detector acceptance. The flux per POT is obtained
by scaling the previous distribution by the ratio of $\pi^{\pm}$ per POT at
the second achromat exit (obtained from FLUKA) and the number of generated
$\pi^{\pm}$. The annual fluxes are shown in Figure 17 assuming $1.8\times
10^{19}$ POT per year. This number corresponds to $2.5\times 10^{13}$ protons
per spill, 3000 spills per day, 30 days per month over the course of 8 months
in a year of operation. In such conditions, the radiation resistance of the
silicon pixel detector (see Sec 3.2 would allow the experiment to operate for
about 10 years.

The beamline is thus able to deliver a neutrino flux which would allow to
collect, with megaton scale natural water Cherenkov detectors, neutrino
samples with a size comparable to the ones of the upcoming LBL experiments.
While systematical uncertainties are expected to be a limiting factor for the
latter, they will be marginal for the former due to the tagging technique.
Hence, this new generation of tagged LBL experiments, exploiting the beamline
developed in this study, will be able to improve the knowledge on the neutrino
oscillation in a sustainable way. Indeed, it requires beam powers of
$(10-100)\text{\,}\mathrm{k}\mathrm{W}$ which, first, are much lower than the
$\mathcal{O}(1)\unit{MW}$ beams of the upcoming experiments and of most of the
proposed options for the next generation of LBL neutrino experiments hep-
ph_AlekouEtAl_2022 ; hep-ph_BogaczEtAl_2022  and, second, are partially
available in existing facilities (however, not directly to the required
intensity of $1.8\times 10^{19}$ POT per year).

(a)

(b)

Figure 17: $\nu_{\mu}$(a) and $\bar{\nu}_{\mu}$(b) fluxes from pion decays in
the far detector acceptance for beamlines with nominal energies of
$12\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ (dark blue),
$8\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ (green) and
$6\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ (light blue) overlaid with the
minimal neutrino fluxes required to collect as many detected neutrino as DUNE
hep-ph_DUNE_2020  at the first oscillation maximum for a
$5\text{\,}\mathrm{M}\mathrm{t}\mathrm{o}\mathrm{n}$ (light red) and a
$10\text{\,}\mathrm{M}\mathrm{t}\mathrm{o}\mathrm{n}$ (dark red) detectors.

##  5 Summary and Discussion

The work presented here aimed to introduce a novel beamline concept for
neutrino beam long baseline experiments, allowing to implement the neutrino
tagging technique. This technique relies on the operation of charged particles
spectrometers instrumented with trackers and installed in the beamline to
kinematically reconstruct the
$\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
from
$\pi^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
and
$K^{\pm}\to\mu^{\pm}\mathchoice{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\displaystyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=3.99994pt}\cr\vskip-1.29167pt\cr\hbox to10.00002pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\textstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=2.5pt}\cr\vskip-1.29167pt\cr\hbox to8.00003pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptstyle\nu$\cr}}}{\vbox{\offinterlineskip\halign{\hfill#\hfill\cr\vbox{\hrule
width=1.99997pt}\cr\vskip-1.29167pt\cr\hbox to6.99997pt{$\mathchar
300\relax\hfil\mathchar
301\relax$}\cr\vskip-1.29167pt\cr$\scriptscriptstyle\nu$\cr}}}\kern-1.72218pt_{\mu}$
decays. The operation of these trackers limits the beam particle flux and
size. Indeed, at the tracker location, the charged beam particle flux has to
be
$10-100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$
and its transverse size $<0.1\text{\,}\mathrm{m}^{2}$, constraints that
results in a lower neutrino flux, compared to standard LBL neutrino
experiments. This low neutrino flux is compensated by the use of
$\mathcal{O}(1)\text{\,}\mathrm{M}\mathrm{t}\mathrm{o}\mathrm{n}$ natural
water Cherenkov detectors which are two orders of magnitude larger than the
ones used in standard LBL neutrino experiments but allow for much lower
precision on the neutrino characteristics. In a tagged experiment, this lower
precision is compensated by the one of the kinematical reconstruction, that
allows to estimate the neutrino energy with $<1$% resolution, and to determine
the neutrinos initial flavour and chirality.

The beamline proposed in this work allows to simultaneously transport
$\pi^{+}$and $\pi^{-}$beams, halving the required run time to collect neutrino
and anti-neutrino data while reducing the systematical uncertainties for key
measurements such as the charge-parity violation phase, $\delta_{CP}\,$, where
the oscillation probabilities of neutrinos and anti-neutrino are compared. To
fulfill the constraints on the particle flux, the beamline was designed to
operate with a slow extracted proton beam. The study was performed assuming a
$400\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ proton beam
and, for this momentum, a graphite target of $\approx$1.2\text{\,}\mathrm{m}$$
was found to yield the largest amount of secondary $\pi^{\pm}$’s. In a tagged
beamline the neutrino flux is limited by the sustainable charged particle flux
at the spectrometer which must be
$<100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$.
In the hereby proposed beamline, this flux is only
$20\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$
per $2.5\times
10^{13}\text{\,}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{l}\mathrm{l}$
over a spill of $4.8\text{\,}\mathrm{s}$. Hence, a significant margin is
available to further improve the $\pi^{\pm}$ yield.

With our study and the proposed tagged beamline, we demonstrated that it is
possible to collect neutrino interaction samples with a size comparable to the
one expected for upcoming experiments using beams of much lower power. As the
tagging technique eliminates most of the systematical uncertainties limiting
the sensitivity of these upcoming experiments, our tagged beamline offers a
valuable option to further improve the knowledge on the neutrino oscillation
parameters. At such tagged experiments, the precision on these parameters
would then be limited by the size of the neutrino samples and thus by the
neutrino fluxes. Therefore, any effort in the direction of improving the
beamline performances should aim at maximizing the $\pi^{\pm}$yields and,
consequently, the overall neutrino fluxes. This optimisation process would
target two different objectives: (1) improvement of the target performances
and (2) further improvement of the optics design to increase as much as
possible the accepted phase space, including the acceptance through the final
spectrometer.

Particularly for the first part, the target geometry could be further
optimized, including e.g segmentation that would possibly increase the pion
yield. Also, a more complete optimisation process could be carried out,
eventually considering other materials and graphite densities available in the
market. This process should also include thermo-mechanical calculations to
evaluate in depth the target station exact geometry and target life. Regarding
the second point, various paths could be explored leading to an overall
optimization of the optics and accepted phase space, in a similar manner as in
mussolini:ipac2021-thpab130 . As emerged from 11(a) and 11(b), the vertical
$140\text{\,}\mathrm{mm}$ aperture of the bending magnets composing the first
and second achromat, strongly limits the $\pi^{\pm}$ accepted phase space.
While a very strong field, and therefore small gap, is needed to separate the
two charges at the momentum selection stage in the first achromat, the
constraint on the aperture could be relaxed at the meson tracking station,
where the resolution of the spectrometer is enough to measure $\pi^{\pm}$
momentum with smaller displacements (for a $<1\%$ momentum resolution a
minimum displacement from the beam axis of $5\text{\,}\mathrm{cm}$ is
required, compared to the $\approx$12\text{\,}\mathrm{cm}$$ considered in this
study). This would most likely improve also the $\pi^{\pm}$ beam size and
divergence, without exceeding the limits of the tagger maximum accepted rate.
Another element affecting the phase space and momentum acceptance is the
collimator. Besides momentum selecting the $\pi^{\pm}$ beams, the purpose of
this beamline element is to act as a dump for the primary proton beam. Our
conceptual dump (comprising approximately 30 nuclear interaction lengths) is
very effective in absorbing the primary beam and in cleaning the produced low
energy secondaries from being transported downstream, however, other designs
(e.g with asymmetric or tapered apertures) and a shorter length could be
foreseen, finding the best compromise in the signal to background ratio.

Finally, the analysis of the background and its implications on the
performances of the tagger have not been treated in the current study but are
fundamental to have a complete picture of the proposed beamline capabilities.
Some preliminary simulations have been done to estimate the main sources of
background, which are mainly muons produced by the interaction of the primary
proton beam with the dump and neutrinos produced both in the target and in the
dump. While the background neutrinos are mostly emitted at large angles and
are not likely to reach the far detector, the muon background could be
concerning both for the tagger, in terms of maximum charged particle rate, and
for the neutrino yield measured at the far detector. Preliminary results show
a muon background of
$\approx$0.015\text{\,}\mathrm{m}\mathrm{u}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}$$
in the momentum range
$p<$14\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$$ for a
central momentum of
$12\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$, compared to
the $\approx$ 0.053 pions/primary $\pi^{\pm}$ signal. Further studies will be
conducted in the future to better understand the effects of this background on
the overall performance of the beamline and on the precision on the
measurement of the neutrino flux. In this context, inherent Monte Carlo
uncertainties will also have to be looked at in more detail, including a
respective analysis of fitting benchmark data.

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