# Muon Beam for Neutrino CP Violation: connecting energy and neutrino
frontiers

Alim Ruzi<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and
Technology, School of Physics, Peking University, Beijing, 100871, China
Tianyi Yang<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and
Technology, School of Physics, Peking University, Beijing, 100871, China
Dawei Fu<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and
Technology, School of Physics, Peking University, Beijing, 100871, China
Sitian Qian<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and
Technology, School of Physics, Peking University, Beijing, 100871, China
Leyun Gao  State Key Laboratory of Nuclear Physics and Technology, School of
Physics, Peking University, Beijing, 100871, China  Qiang Li
<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and Technology,
School of Physics, Peking University, Beijing, 100871, China

###### Abstract

We propose here a proposal to connect neutrino and energy frontiers, by
exploiting collimated muon beams for neutrino oscillations, which generate
symmetric neutrino and antineutrino sources: $\mu^{+}\rightarrow
e^{+}\,\bar{\nu}_{\mu}\,\nu_{e}$ and $\mu^{-}\rightarrow
e^{-}\,\nu_{\mu}\,\bar{\nu}_{e}$. Interfacing with long baseline neutrino
detectors such as DUNE and T2K, this experiment can be applicable to measure
tau neutrino properties, and also to probe neutrino CP phase, by measuring
muon electron (anti-)neutrino mixing or tau (anti-)neutrino appearance, and
differences between neutrino and antineutrino rates. There are several
significant benefits leading to large neutrino flux and high sensitivity on CP
phase, including 1) collimated and manipulable muon beams, which lead to a
larger acceptance of neutrino sources in the far detector side; 2) symmetric
$\mu^{+}$ and $\mu^{-}$ beams, and thus symmetric neutrino and antineutrino
sources, which make this proposal ideally useful for measuring neutrino CP
violation. More importantly, $\bar{\nu}_{e,\mu}\rightarrow\bar{\nu}_{\tau}$
and $\nu_{e,\mu}\rightarrow\nu_{\tau}$, and,
$\bar{\nu}_{e}\rightarrow\bar{\nu}_{\mu}$ and $\nu_{e}\rightarrow\nu_{\mu}$
oscillation signals can be collected simultaneously, with no needs for
separate specific runs for neutrinos or antineutrinos. Furthermore, it is
possible to exchange $\mu^{+}$ and $\mu^{-}$ flying routes, thus further
reducing possible bias or systematic uncertainties. In an optimistic way, we
estimate $10^{4}$ tau (anti-) neutrinos can be collected per year thus this
proposal can serve as a brighter tau neutrino factory. Moreover, 5 standard
deviations of sensitivity can be easily reached for CP phase as $|\pi/2|$,
with only 1–2 years of data taking, by combining tau and muon (anti-) neutrino
appearances. With the development of a more intensive muon beam targeting
future muon collider, the neutrino potential of the current proposal will
surely be further improved.

Novel collision methods and rich phenomena are crucial to keeping high-energy
collision physics more robust and attractive [1]. Recent years have witnessed
vast development towards next generation high energy colliders, including
various proposals on Higgs factory [2, 3], revived interest in Muon collider
[4, 5, 6, 7, 8], etc.

As for the muon collider design, we take positron on target method (LEMMA) as
an example, which has been proposed for high quality muon beam production [9,
10] (Earlier studies using proton on target muon beams can be found in
Refs.[11, 12, 13] ) Although it is still quite challenging to achieve enough
high luminosity for muon beam collisions [6, 7], we find it quite promising
for neutrino oscillation studies, with comparable or even larger neutrino flux
than other long baseline neutrino experiments, with more details to be
discussed below. In the LEMMA approach, the incident positron energy is around
45 GeV, producing collimated muon pairs with opening angles of around 0.005
rad. and a large boost about $\gamma\sim 200$, which extends the muon lifetime
by the order of $\mathcal{O}(10^{2})$. Generally, the number of muon pairs
produced per positron bunch on target can be expressed as

$n(\mu^{+}\mu^{-})=n^{+}\rho_{e^{-}}l\sigma(\mu^{+}\mu^{-})$ (1)

where $n^{+}$ is the number of $e^{+}$ in each positron bunch, $\rho_{e^{-}}$
is the electron density in the medium, $l$ is the thickness of the the target,
and $\sigma(\mu^{+}\mu^{-})$ being the cross section of the muon pair
production. The number of muon pairs per positron bunch on target can be
maximally estimated as $n(\mu^{+}\mu^{-})_{\mathrm{max}}\approx n^{+}\times
10^{-5}$.

On the other hand, neutrinos are among the most abundant and least understood
of all particles in the SM that make up our universe. The history of neutrino
physics was full of novel discoveries. One of them is the observation of
neutrino oscillations, confirming that at least two types of SM neutrinos have
a tiny, but strictly nonzero mass. Nowadays, the neutrino system is described
by nine parameters, the masses $m_{1}$, $m_{2}$ and $m_{3}$ of the three mass
eigenstates, three mixing angles, $\theta_{12}$, $\theta_{23}$, and
$\theta_{13}$, and three phases, one Dirac phase, $\delta_{\rm CP}$ and two
Majorana phases. The Majorana phase only plays a part in neutrinoless double
beta decay [14]. The mixing angles and the phases are the elements of a
unitary matrix called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [15, 16].
The available experiments on neutrino oscillations to date have measured five
of the neutrino mixing parameters, three mixing angles $\theta_{12}$,
$\theta_{23}$, $\theta_{13}$, and the two squared-mass differences $\Delta
m_{21}^{2}$, $\Delta m_{32}^{2}$ within 3$\sigma$ range [17, 18, 19, 20, 21].
Neutrino oscillation probabilities from one flavor into another are functions
of these mixing angles and the squared-mass differences. The mass of the
individual neutrino and the mass hierarchies are not known.

The determination of the CP-violating phase, the Dirac phase, has been the
core research program in neutrino physics for decades because it provides a
potential source of CP violation in the SM lepton sector. It has been known
that the leptonic CP violation could generate the matter-antimatter asymmetry
through leptogenesis [22]. CP violation in neutrino oscillation can be
measured through the difference between the oscillation probability of the
neutrino and antineutrino, expressed as $\Delta P_{\alpha\beta}^{\rm
CP}=P_{\alpha\beta}-\overline{P}_{\alpha\beta}$, which is well quantified by
$\delta_{\rm CP}$. There are several experiments worldwide dedicated to the
measurements of the neutrino parameters, especially the CP phase, performing
searches of short-baseline and long-baseline neutrino oscillation. To ensure
that there are enough neutrino flavors oscillated from source neutrino and to
be detected by Far Detector (FD), a long-baseline neutrino oscillation
experiment is preferable rather than a short-baseline. Recently, the long-
baseline experiments, T2K (Tokai to Kamioka) [23, 24] and NOvA [25] report
their results. T2K reports a measured value for CP phase, $\delta_{\rm
CP}=-1.97_{-0.70}^{+0.97}$ while excluding $\delta_{\rm CP}$ = 0 and $\pi$ at
90% CL, indicating CP violation in the lepton sector. The FD in this case is
the Super-Kamiokande, a 50 Kton water Cherenkov detector. A narrow band
neutrino beam is produced at an angle of $2.5^{\circ}$ by a 30 GeV proton beam
hitting on graphite target. With this off-axis method, the narrow band
neutrino energy has a peak at 0.6 GeV. The secondary neutrino produced from
decays of Kaon or Pion travels a distance of 295 Km to reach the Super-
Kamiokande detector. T2K plans to extend its term to 2026, followed by the
Hyper-K project with the mass of the far detector to be increased by a factor
of 10, and will offer a broad science program [26].

On the other hand, the NOvA experiment [25] in the US is also a long-baseline
accelerator-based neutrino oscillation experiment. It uses the upgraded
Fermilab NuMI beam and measures electron-neutrino appearance and muon-neutrino
disappearance at its far detector in Ash River, Minnesota. The reported NOvA
result shows no strong preference for any particular value of the neutrino CP
phase and has a slight tension with T2K’s.

Another promising long-baseline neutrino experiment under construction is DUNE
(Deep Underground Neutrino Experiment), whose goals are the determination of
the neutrino mass ordering, observation of CP violation (up to 50% level), and
precise measurements of oscillation parameters, such as $\delta_{\rm CP}$,
$\sin^{2}(2\theta_{21})$. The idea is to send a wide-band high-intensity muon
neutrino beam from Fermilab to the Sanford Underground Facility in Homestake
at the 1300 Km distance. The detector technology of DUNE experiment is based
on building liquid argon time projection chambers (LArTPC). Unlike the T2K
experiment, the neutrino beam energy has a peak at 2.5 GeV with a broad range
of neutrino energies. The neutrino beam is produced from proton collision on
the graphite target. In the corresponding DUNE TDR report, it is shown that
favorable values for $\delta_{\rm CP}$ with $3\sigma$ $(5\sigma)$ can be
achieved after five (ten) years of running.

In this letter, we are interested in applying collimated muon beams into
neutrino mixing and CP phase measurements. Although the beam density is lower
than the proton-on-target scenario, there are several significant benefits
leading to large neutrino flux and high sensitivity on CP phase, including 1)
collimated and manipulable muon beams, which lead to a larger acceptance of
neutrino sources in the far detector side; 2) symmetric $\mu^{+}$ and
$\mu^{-}$ beams, and thus symmetric neutrino and antineutrino sources, which
make this proposal ideally for measuring neutrino CP violation. What is more
important is that antineutrino and neutrino flux here are similar, and thus
for example, $\bar{\nu}_{e}\rightarrow\bar{\nu}_{\mu}$ and
$\nu_{e}\rightarrow\nu_{\mu}$ oscillation signals can be collected
simultaneously, with no needs for separate specific runs for neutrinos or
antineutrinos as done in the DUNE experiment [27, 28].

As to be discussed below, the estimated neutrino flux in our proposal is
comparable to or even larger than the DUNE experiment [27, 28]. The neutrino
energy has wide distributions in the 1–15 GeV region, and peaks at around 7
GeV (neutrino energy can be further tuned with on-axis and off-axis
techniques), suggesting our proposal is also suitable for tau neutrino
studies. Taking into account both muon and electron neutrinos and
antineutrinos, the signal yields indeed can be doubled or more. Finally, we
point out that it is possible to exchange $\mu^{+}$ and $\mu^{-}$ flying
routes, thus further reducing possible bias or systematic uncertainties.

Figure 1: A proposed neutrino oscillation experiment to probe neutrino
violation CP phase by measuring muon electron (anti-) neutrino oscillation and
the differences of resulted $\nu_{e,\mu}\rightarrow\nu_{\tau}$,
$\nu_{e}\rightarrow\nu_{\mu}$, and their antineutrino correspondents. This
proposal is based on collimated muon beams achieved from e.g., Positron on
Target method, where 45 GeV positron beams are fired. Dipoles are used to
separate $\mu^{+}$ and $\mu^{-}$ with an angle around 0.01 rad., with
direction changeable. Muon beams fly about 10 Km and radiate neutrinos before
being swept away. Neutrinos then further fly 1300 Km to reach DUNE type of
detectors. Figure 2: A proposed neutrino oscillation experiment to probe tau
neutrino physics by measuring tau (anti-)neutrino appearance. This proposal is
based on collimated muon beams achieved from e.g., Positron on Target method,
where 45 GeV positron beams are shed. Neutrinos are radiated along the muon
beam line and fly 100 to 1300 Km to reach DUNE type of detectors. Quadrupoles
can be applied to keep muon beams more collimated.

Fig. 1 shows a proposed neutrino oscillation experiment to probe neutrino CP
phase by measuring muon electron (anti-)neutrino mixing and their differences.
We are especially interested in the oscillation modes of
$\nu_{e,\mu}\rightarrow\nu_{\tau}$, $\nu_{e}\rightarrow\nu_{\mu}$ and their
antineutrino correspondents, with more details to be given later in this
paper. This proposal is based on collimated muon beams achieved from e.g.,
Positron on Target method, where 45 GeV positron beams are shed on the target.
Dipoles are used to separate $\mu^{+}$ and $\mu^{-}$ with an angle around 0.01
rad., with direction changeable. Muon beams fly about 10 Km and radiate
neutrinos before being swept away. Neutrinos then further fly e.g., 1300 Km to
reach DUNE or T2K type of detectors [29]. Fig. 2 shows a similar but simpler
proposal focusing mainly on tau (anti-)neutrino appearance and related physics
studies.

  * •

Muon production rates $n(\mu^{+}\mu^{-})_{max}\approx n^{+}\times 10^{-5}$
[10]. Assuming positron bunch density as $10^{12}$/bunch and bunch crossing
frequency as $10^{5}$/sec, we get muon production rates $dN_{\mu}/dt\sim
10^{12}$/sec (or $10^{19}$/year). (Notice the future TeV scale muon collider
is indeed targeting a much larger intensity beam by more than 1-2 orders of
magnitudes [5, 6, 7].)

  * •

For muons with the energy around 20 GeV, the mean flying distance is around
100 Km. If there is a straight tube with a length around 5-10 Km to let muons
go through with quadrupoles to keep them merged, the decayed fraction can
reach $10^{-1}$ in realistic. On the other hand, we can also refer to a muon
complex as discussed in Ref. [30, 8], where the muon beam is accelerated in a
circular section and then extracted into the rectangular section for decays.
The intensity of the neutrino beam compared with the incoming muon beam is
suppressed by a ratio around $10^{-1}$, i.e., the fraction of the collider
ring circumference occupied by the production straight section.

  * •

The opening angles between muons and decay products are around 0.005 rad. as
shown in Fig. 3 and may be kept smaller with quadrupoles. For neutrinos
traveling 1300 Km to reach far detectors, the spread size can be around 1-5
Km. For a DUNE-like detector with a cubic size of about 20 m [29], the
neutrino acceptance is then $10^{-4}$.

  * •

Muon/electron neutrinos and antineutrinos interacting with detectors. With a
$L=20$ m long detector (DUNE far detector indeed has a length around 50m
[29]), the expected event yield rate can be roughly estimated with:
$dN_{\mu}/dt\times L\times\sigma_{n\nu}\times\rho N_{A}\cdot dE$ $\sim
10^{-9}\times dN_{\mu}/dt$, where $N_{A}$ is the Avogadro constant, $\rho\sim$
2 $\rm{g}/\rm{cm}^{3}$, $\sigma_{n\nu}$ symbols the neutrino nucleon cross
sections and is around $10^{-37}\rm{cm}^{2}$ for a 10 GeV neutrino [31, 32].

Combining all above numbers, with the conservative option (e.g., 20-meter
cubic size detector), we get the muon/electron (anti-)neutrino Charged Current
(CC) events per year as

$N^{cc}_{\nu_{\mu,e}}\sim 10^{19}\times 10^{-1}\times 10^{-4}\times
10^{-9}=10^{5}/{\rm year},$ (2)

Figure 3: 2D distributions of energy and angle in respect to muon flying
direction, for muon and electron neutrinos from 22.5 GeV $\mu^{+}\rightarrow
e^{+}\,\bar{\nu}_{\mu}\,\nu_{e}$ (similarly for $\mu^{-}$ decay).

Figure 4: The oscillation probability of $\nu_{\mu}\to\nu_{\tau}$,
$\nu_{\mu}\to\nu_{e}$, $\nu_{e}\to\nu_{\tau}$, and $\nu_{e}\to\nu_{\tau}$ at
three $\delta_{\rm CP}$ angles as function of $\nu_{\mu}$ energy, for a long
baseline of L =1300 Km.

On top of measuring muon (anti-)neutrino rates, our proposal can also be used
to probe tau neutrino appearance. One additional factor needs to be
considered, i.e., the fraction of muon neutrino oscillated into tau neutrino
[33]:

$\displaystyle\begin{split}P(\nu_{\mu}\to\nu_{\tau})&\simeq\sin^{2}\left(2\theta_{23}\right)\cos^{4}(\theta_{13})\sin^{2}\left(1.27\frac{\Delta
m_{32}^{2}L}{E_{\nu}}\right)\\\ &\pm 1.27\Delta
m_{21}^{2}\frac{L}{E_{\nu}}\sin^{2}\left(1.27\frac{\Delta
m_{32}^{2}L}{E_{\nu}}\right)\times 8J_{\rm CP}\end{split}$ (3)

where the “Jarlskog invariant” [34, 35],

$\begin{split}J_{\mathrm{CP}}&\equiv\sin\theta_{13}\cos^{2}\theta_{13}\sin\theta_{12}\cos\theta_{12}\sin\theta_{23}\cos\theta_{23}\sin\delta_{\textrm{CP}}\\\
&=0.03359\pm 0.0006(\pm 0.0019)\sin\delta_{\textrm{CP}}\end{split}$ (4)

is a function of $\sin\delta_{\textrm{CP}}$. The oscillation probability
difference between $P(\nu_{\mu}\rightarrow\nu_{\tau})$ and
$P(\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\tau})$ reads as

$\Delta P(\nu_{\mu}\rightarrow\nu_{\tau})=16J_{\rm CP}\times 1.27\Delta
m_{12}^{2}\frac{L}{E_{\nu}}\sin^{2}\left(\frac{\Delta
m_{32}^{2}L}{E_{\nu}}\right)$ (5)

Thus we can estimate tau neutrino CC events per year as (for simplicity we
ignore the cross section difference of neutrino and antineutrino nucleon cross
sections)

$N^{cc}_{\nu_{\tau}}\sim[(3\times 10^{4})\pm(2.6\times 10^{2})]/{\rm year},$
(6)

where $(2.6\times 10^{2})$ corresponds to the CP violation term. Notice the
yearly expected tau neutrino yields is already comparable or even surpass the
rates at the SHiP experiment at CERN [36]. Thus our proposal can serve as a
‘brighter’ factory for tau neutrinos.

The oscillation probability of $P(\nu_{e}\rightarrow\nu_{\tau})$ can be
estimated as

$\displaystyle\begin{split}P(\nu_{e}\rightarrow\nu_{\tau})&\simeq\sin^{2}(2\theta_{13})\cos^{2}(\theta_{23})\sin^{2}\left(1.27\Delta
m^{2}_{32}\frac{L}{E_{\nu}}\right)\\\ &\mp 1.27\Delta
m^{2}_{21}\frac{L}{E_{\nu}}\sin^{2}\left(1.27\Delta
m^{2}_{32}\frac{L}{E_{\nu}}\right)\times 8J_{\textrm{CP}}\end{split}$ (7)

Because of its heavy mass and very short lifetime, the tau neutrino production
in abundance in conventional accelerator is almost not possible. On the
contrary, we have rich tau neutrino flux because of the higher
$P(\nu_{\mu}\to\nu_{\tau})$ oscillation. The tau neutrino flux can be further
strengthened by $P(\nu_{e}\rightarrow\nu_{\tau})$ oscillation . With this
promising feature in this proposal, some of the new physics models, such as
charged Higgs doublet [37] or leptoquarks [38] maybe tested through neutrino-
type collision [39].

Apart from tau neutrino appearance, the oscillation rates with CP phase
dependence for $\nu_{\mu}\rightarrow\nu_{e}$ is shown as below [23, 24]:

$\displaystyle\begin{split}P(\nu_{\mu}\rightarrow\nu_{e})&\simeq\sin^{2}(2\theta_{13})\sin^{2}(\theta_{23})\sin^{2}\left(1.27\Delta
m^{2}_{32}\frac{L}{E_{\nu}}\right)\\\ &\mp 1.27\Delta
m^{2}_{21}\frac{L}{E_{\nu}}\sin^{2}\left(1.27\Delta
m^{2}_{32}\frac{L}{E_{\nu}}\right)\times 8J_{\textrm{CP}}\end{split}$ (8)

The corresponding oscillation probability for $\nu_{e}\to\nu_{\mu}$ only
differs a minus sign from $\nu_{\mu}\to\nu_{e}$ oscillation in CP violating
term:

$\displaystyle\begin{split}P(\nu_{e}\rightarrow\nu_{\mu})&\simeq\sin^{2}(2\theta_{13})\sin^{2}(\theta_{23})\sin^{2}\left(1.27\Delta
m^{2}_{32}\frac{L}{E_{\nu}}\right)\\\ &\pm 1.27\Delta
m^{2}_{21}\frac{L}{E_{\nu}}\sin^{2}\left(1.27\Delta
m^{2}_{32}\frac{L}{E_{\nu}}\right)\times 8J_{\textrm{CP}}\end{split}$ (9)

Using the current measured values of the mixing angles and squared mass
differences [31] and taking the distance of neutrino propagation as L =1300
Km, we have the numeric values for the neutrino oscillations at
$E_{\nu}=7\,(5)~{}\rm{GeV}$ (see more in Fig. 4):

$\displaystyle P(\nu_{\mu}\to\nu_{\tau})$ $\displaystyle=0.2916\pm
0.0026\sin\delta_{\rm CP}\,(0.5093\pm 0.0048\sin\delta_{\rm CP}),$ (10)
$\displaystyle P(\nu_{\mu}\rightarrow\nu_{e})$ $\displaystyle=0.0151\mp
0.0026\sin\delta_{\rm CP}\,(0.0264\mp 0.0048\sin\delta_{\rm CP}),$ (11)
$\displaystyle P(\nu_{e}\rightarrow\nu_{\mu})$ $\displaystyle=0.0151\pm
0.0026\sin\delta_{\rm CP}\,(0.0264\pm 0.0048\sin\delta_{\rm CP}),$ (12)
$\displaystyle P(\nu_{e}\rightarrow\nu_{\tau})$ $\displaystyle=0.0119\mp
0.0026\sin\delta_{\rm CP}\,(0.0209\mp 0.0048\sin\delta_{\rm CP}).$ (13)

We will now evaluate the sensitivities on neutrino CP violation, taking
$\delta_{\textrm{CP}}=\pm\pi/2$ and $E_{\nu}=7~{}\rm{GeV}$ as benchmark
parameters:

  * •

1) Firstly, we consider the tau (anti-) neutrino appearance from muon and
electron neutrino oscillations:

$\displaystyle\mu^{+}\rightarrow e^{+}\,\bar{\nu}_{\mu
1}\,\nu_{e1}\,\,\Longrightarrow\,\,\,\,$ $\displaystyle\bar{\nu}_{\mu
1}\rightarrow\bar{\nu}_{\tau 1},\,\,\nu_{e1}\rightarrow\nu_{\tau 1},$ (14)
$\displaystyle\mu^{-}\rightarrow e^{-}\,\nu_{\mu
2}\,\bar{\nu}_{e2}\,\,\Longrightarrow\,\,\,\,$ $\displaystyle\nu_{\mu
2}\rightarrow\nu_{\tau 2},\,\,\bar{\nu}_{e2}\rightarrow\bar{\nu}_{\tau 2},$
(15)

where ‘1’ and ‘2’ symbol the two far detectors as shown in Fig. 1.

Notice that the CP dependence of $P(\nu_{e}\to\nu_{\tau})$ and
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{\tau})$ as shown in Eq. 10 vary in the same
direction. If we count on tau-related events in the far detector inclusively,
this means our signal doubles. The sensitivity can be estimated then as

$\displaystyle\frac{\bar{\nu}_{\tau 2}+\nu_{\tau 2}-\bar{\nu}_{\tau
1}-\nu_{\tau 1}}{\sqrt{\bar{\nu}_{\tau 2}+\nu_{\tau 2}+\bar{\nu}_{\tau
1}+\nu_{\tau 1}}}$ (16)

which is around $4\times 260/\sqrt{60000}\sim 4.2$ standard deviations
($\sigma$) in one year, and can reach near $13.4\sigma$ in 10 years. Although
only statistics are taken into account here, systematics should be able to be
reduced efficiently due to the symmetric property of the proposed device.
Furthermore, it is possible to exchange $\mu^{+}$ and $\mu^{-}$ flying routes,
thus further reducing possible bias or systematic.

  * •

2) Secondly, if the far detector can distinguish tau neutrino from
antineutrino such as the CERN SHiP experiment [36], then with only
$P(\nu_{e}\to\nu_{\tau})$, we can already have higher CP sensitivity. The
sensitivity can be estimated then as

$\displaystyle\frac{\bar{\nu}_{\tau 2}-\nu_{\tau 1}}{\sqrt{\bar{\nu}_{\tau
2}+\nu_{\tau 1}}}$ (17)

which is around $2\times 260/\sqrt{2200}\sim 11$ $\sigma$ in one year.

  * •

3) Finally, one can also exploit electron to muon oscillation which has also
clear sensitivity on neutrino CP phase, if the far detector can distinguish
muon neutrino from antineutrino, possibly can be achieved with moderate
magnets. The sensitivity can be estimated then as $2\times 260/\sqrt{3000}\sim
9.5$ $\sigma$ in one year.

By combing all three options and taking into account statistic errors, one can
achieve a sensitivity of far more than 5 $\sigma$ in one year, for
$\delta_{\textrm{CP}}=\pm\pi/2$. Option 1) corresponds to a most conservative
scenario, where one can reuse such as the DUNE-like detectors. Options 2) and
3) lead to much larger sensitivities on neutrino CP violation, yet putting
additional requirements on the experimental side, such as to distinguish
$\mu^{+}$ from $\mu^{-}$, or $\tau^{+}$ from $\tau^{-}$ under the help of
moderate magnet field.

In Summary, we propose here a new idea to exploit collimated muon beams which
generate symmetric neutrino and antineutrino sources: $\mu^{+}\rightarrow
e^{+}\,\bar{\nu}_{\mu}\,\nu_{e}$ and $\mu^{-}\rightarrow
e^{-}\,\nu_{\mu}\,\bar{\nu}_{e}$. Interfacing with long baseline neutrino
detectors such as in DUNE or T2K, this experiment can be useful to measure tau
neutrino properties, and also to probe neutrino CP phase, by measuring muon
(anti-) neutrino disappearance or tau (anti-)neutrino appearance, and
differences between neutrino and antineutrino rates.

There are several significant benefits leading to large neutrino flux and high
sensitivity on CP phase, including 1) collimated and manipulable muon beams,
which lead to a larger acceptance of neutrino sources in the far detector
side; 2) symmetric $\mu^{+}$ and $\mu^{-}$ beams, and thus symmetric neutrino
and antineutrino sources, which make this proposal ideally for measuring
neutrino CP violation. More importantly,
$\bar{\nu}_{e,\mu}\rightarrow\bar{\nu}_{\tau}$ and
$\nu_{e,\mu}\rightarrow\nu_{\tau}$, and,
$\bar{\nu}_{e}\rightarrow\bar{\nu}_{\mu}$ and $\nu_{e}\rightarrow\nu_{\mu}$
oscillation signals can be collected simultaneously, with no needs for
separate specific runs for neutrinos or antineutrinos. It is also possible to
exchange $\mu^{+}$ and $\mu^{-}$ flying routes, thus further reducing possible
bias or systematic. In an optimistic way, we estimate $10^{4}$ tau (anti-)
neutrinos can be collected per year thus our proposal can serve as a brighter
tau neutrino factor. Moreover, 5 standard deviations of sensitivity can be
reached easily for CP phase as $|\pi/2|$, with only 1 year of data taking, by
combining tau and muon (anti-) neutrino appearances.

Notice for tau (anti-)neutrino appearance, the CP dependence of
$P(\nu_{e}\to\nu_{\tau})$ and $P(\bar{\nu}_{\mu}\to\bar{\nu}_{\tau})$, or
$P(\bar{\nu}_{e}\to\bar{\nu}_{\tau})$ and $P(\nu_{\mu}\to\nu_{\tau})$ vary in
the same direction, thus signal rates double and one can reuse directly DUNE-
like far detector.

In this draft, we mainly provide a preliminary estimation of the feasibility
study. A detailed study is surely necessary to follow up. On the other hand,
there exist also rich potential to be further explored with such a proposal
that connects energy and neutrino frontiers. Especially, one can imagine a
post-DUNE (or in parallel to DUNE as the probe channels are indeed orthogonal
and thus complementary) experiment with neutrinos from a strong muon source
located at the Fermilab site. This connection between energy and neutrino
frontiers can also serve as a precursor for future high-energy muon colliders.
Notice that a muon collider requires a 1–2 orders of magnitude more intense
beam as compared with the number ($dN_{\mu}/dt\sim 10^{12}$/sec ) listed above
as our benchmark. Thus with the development of a more intensive muon beam
targeting future muon colliders, it surely will improve further the neutrino
potential of the current proposal.

###### Acknowledgements.

This work is supported in part by the National Natural Science Foundation of
China under Grants No. 12150005, No. 12075004, and No. 12061141002, by MOST
under grant No. 2018YFA0403900.

## References

  * [1] M. Lu et al. [PKU SYSU Novel Collider Group], [arXiv:2210.06690 [hep-ph]].
  * [2] R. K. Ellis, B. Heinemann, J. de Blas, M. Cepeda, C. Grojean, F. Maltoni, A. Nisati, E. Petit, R. Rattazzi and W. Verkerke, et al. [arXiv:1910.11775 [hep-ex]].
  * [3] J. A. Bagger, B. C. Barish, S. Belomestnykh, P. C. Bhat, J. E. Brau, M. Demarteau, D. Denisov, S. C. Eno, C. G. R. Geddes and P. D. Grannis, et al. [arXiv:2203.06164 [hep-ex]].
  * [4] C. Aime, A. Apyan, M. A. Mahmoud Mohammed, N. Bartosik, F. Batsch, A. Bertolin, M. Bonesini, S. Bottaro, D. Buttazzo and R. M. Capdevilla Roldan, et al. [arXiv:2203.07256 [hep-ph]].
  * [5] D. Stratakis et al. [Muon Collider], [arXiv:2203.08033 [physics.acc-ph]].
  * [6] S. Jindariani et al. [Muon Collider], [arXiv:2203.07224 [physics.ins-det]].
  * [7] J. de Blas et al. [Muon Collider], [arXiv:2203.07261 [hep-ph]].
  * [8] C. Quigg, AIP Conf. Proc. 435, no.1, 242-257 (1998) doi:10.1063/1.56235 [arXiv:hep-ph/9803326 [hep-ph]].
  * [9] M. Antonelli, M. Boscolo, R. Di Nardo and P. Raimondi, Nucl. Instrum. Meth. A 807, 101-107 (2016) doi:10.1016/j.nima.2015.10.097 [arXiv:1509.04454 [physics.acc-ph]].
  * [10] D. Alesini, M. Antonelli, M. E. Biagini, M. Boscolo, O. R. Blanco-García, A. Ciarma, R. Cimino, M. Iafrati, A. Giribono and S. Guiducci, et al. [arXiv:1905.05747 [physics.acc-ph]].
  * [11] J. Tang, S. Vihonen and Y. Xu, Commun. Theor. Phys. 74, no.3, 035201 (2022) doi:10.1088/1572-9494/ac5245
  * [12] J. Burguet-Castell, M. B. Gavela, J. J. Gomez-Cadenas, P. Hernandez and O. Mena, Nucl. Phys. B 608, 301-318 (2001) doi:10.1016/S0550-3213(01)00248-6 [arXiv:hep-ph/0103258 [hep-ph]].
  * [13] J. Cao, M. He, Z. L. Hou, H. T. Jing, Y. F. Li, Z. H. Li, Y. P. Song, J. Y. Tang, Y. F. Wang and Q. F. Wu, et al. Phys. Rev. ST Accel. Beams 17, 090101 (2014) doi:10.1103/PhysRevSTAB.17.090101 [arXiv:1401.8125 [physics.acc-ph]].
  * [14] J. D. Vergados, H. Ejiri and F. Simkovic, Rept. Prog. Phys. 75, 106301 (2012) doi:10.1088/0034-4885/75/10/106301 [arXiv:1205.0649 [hep-ph]].
  * [15] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 870-880 (1962) doi:10.1143/PTP.28.870
  * [16] B. Pontecorvo, Zh. Eksp. Teor. Fiz. 53, 1717-1725 (1967)
  * [17] F. P. An et al. [Daya Bay], [arXiv:2211.14988 [hep-ex]].
  * [18] P. Adamson et al. [NOvA], Phys. Rev. Lett. 118, no.15, 151802 (2017) doi:10.1103/PhysRevLett.118.151802 [arXiv:1701.05891 [hep-ex]].
  * [19] K. Abe et al. [T2K], Phys. Rev. Lett. 121, no.17, 171802 (2018) doi:10.1103/PhysRevLett.121.171802 [arXiv:1807.07891 [hep-ex]].
  * [20] K. Abe et al. [Super-Kamiokande], Phys. Rev. D 97, no.7, 072001 (2018) doi:10.1103/PhysRevD.97.072001 [arXiv:1710.09126 [hep-ex]].
  * [21] M. A. Acero et al. [NOvA], Phys. Rev. Lett. 123, no.15, 151803 (2019) doi:10.1103/PhysRevLett.123.151803 [arXiv:1906.04907 [hep-ex]].
  * [22] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45-47 (1986) doi:10.1016/0370-2693(86)91126-3
  * [23] K. Abe et al. [T2K], Nature 580, no.7803, 339-344 (2020) [erratum: Nature 583, no.7814, E16 (2020)] doi:10.1038/s41586-020-2177-0
  * [24] J. G. Walsh [T2K], [arXiv:2208.01164 [hep-ex]].
  * [25] M. A. Acero et al. [NOvA], Phys. Rev. D 106, no.3, 032004 (2022) doi:10.1103/PhysRevD.106.032004 [arXiv:2108.08219 [hep-ex]].
  * [26] K. Abe et al. [Hyper-Kamiokande], [arXiv:1805.04163 [physics.ins-det]].
  * [27] B. Abi et al. [DUNE], [arXiv:2002.03005 [hep-ex]].
  * [28] B. Abi et al. [DUNE], [arXiv:2103.04797 [hep-ex]].
  * [29] B. Abi et al. [DUNE], JINST 15, no.08, T08009 (2020) doi:10.1088/1748-0221/15/08/T08009 [arXiv:2002.03008 [physics.ins-det]].
  * [30] B. J. King, AIP Conf. Proc. 530, no.1, 142-164 (2000).
  * [31] P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020) doi:10.1093/ptep/ptaa104
  * [32] J. A. Formaggio and G. P. Zeller, Rev. Mod. Phys. 84, 1307-1341 (2012) doi:10.1103/RevModPhys.84.1307 [arXiv:1305.7513 [hep-ex]].
  * [33] A. De Gouvêa, K. J. Kelly, G. V. Stenico and P. Pasquini, Phys. Rev. D 100, no.1, 016004 (2019) doi:10.1103/PhysRevD.100.016004
  * [34] C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985) doi:10.1103/PhysRevLett.55.1039
  * [35] R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097
  * [36] C. Ahdida et al. [SHiP], Eur. Phys. J. C 82, no.5, 486 (2022) doi:10.1140/epjc/s10052-022-10346-5 [arXiv:2112.01487 [physics.ins-det]].
  * [37] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. 516, 1-102 (2012) doi:10.1016/j.physrep.2012.02.002 [arXiv:1106.0034 [hep-ph]].
  * [38] I. de Medeiros Varzielas and J. Talbert, Eur. Phys. J. C 79, no.6, 536 (2019) doi:10.1140/epjc/s10052-019-7047-2 [arXiv:1901.10484 [hep-ph]].
  * [39] S. Qian, T. Yang, S. Deng, J. Xiao, L. Gao, A. M. Levin, Q. Li, M. Lu and Z. You, [arXiv:2205.15350 [hep-ph]].