††thanks: 0000-0002-5209-872X

# CP-Violation with Neutrino Disappearance

Peter B. Denton<EMAIL_ADDRESS>High Energy Theory Group, Physics Department,
Brookhaven National Laboratory, Upton, NY 11973, USA

(September 6, 2023)

###### Abstract

The best way to probe CP violation in the lepton sector is with long-baseline
accelerator neutrino experiments in the appearance mode: the appearance of
$\nu_{e}$ in predominantly $\nu_{\mu}$ beams. Here we show that it is possible
to discover CP violation with disappearance experiments only, by combining
JUNO for electron neutrinos and DUNE or Hyper-Kamiokande for muon neutrinos.
While the maximum sensitivity to discover CP is quite modest ($1.6\sigma$ with
6 years of JUNO and 13 years of DUNE), some values of $\delta$ may be
disfavored by $>3\sigma$ depending on the true value of $\delta$.

##  I Introduction

There are three free parameters in our current model of particle physics that
control the size of CP violation in their respective sector: the
$\bar{\theta}$ term in the gluon sector which seems to be either zero or very
small Pendlebury  _et al._ (2015), the CKM matrix Cabibbo (1963); Kobayashi
and Maskawa (1973) describing quark mixing which is known to have some CP
violation Tanabashi  _et al._ (2018); Charles  _et al._ (2015), and the PMNS
matrix Pontecorvo (1957); Maki  _et al._ (1962) describing lepton mixing. It
is unknown if there is CP violation in the lepton mixing matrix de Salas  _et
al._ (2021); Esteban  _et al._ (2020); Capozzi  _et al._ (2021); Denton  _et
al._ (2021) and thus determining if CP is violated in the lepton sector is of
the utmost priority in particle physics.

The best way to probe CP violation in the leptonic sector is by an appearance
measurement of an oscillation maximum Itow  _et al._ (2001); Marciano (2001);
Dick  _et al._ (1999); Cervera  _et al._ (2000); Autiero  _et al._ (2004);
Nunokawa  _et al._ (2008); Coloma  _et al._ (2013); Denton  _et al._ (2022).
That is, the detection of one flavor of neutrino in a source of neutrino that
is predominantly a differently flavor at a baseline and energy that
corresponds to one of the $\Delta m^{2}_{ij}$’s. To date only NOvA Adamson
_et al._ (2016) and T2K Abe  _et al._ (2014) have strong evidence for the
detection of appearance by detecting electron neutrinos in predominantly muon
neutrino sources, but do not yet significantly probe CP violation Acero  _et
al._ (2022a); Ramírez  _et al._ (2023). Atmospheric neutrinos, which are
mostly muon neutrinos but with a significant electron neutrino contribution,
also have some evidence for appearance Li  _et al._ (2018); Jiang  _et al._
(2019); Aartsen  _et al._ (2019) via the detection of all three flavors and
thus a total of six oscillation channels, although the appearance information
is thus somewhat scrambled.

We will show how it is also possible to probe CP violation via neutrino
disappearance measurements only. With good measurements of the disappearance
of two different flavors it is possible to determine a total of four
independent parameters in the PMNS matrix. Since the mixing matrix is only
described by four parameters111Two additional parameters, the so-called
Majorana phases, may also be physical depending on the nature of neutrinos,
but their impact in neutrino oscillation experiments is suppressed by
$(m_{\nu}/E_{\nu})^{2}<10^{-14}$ or smaller in oscillations and thus can be
safely ignored. it is possible to learn about CP violation while only
measuring CP conserving channels. The key physics effect that makes this
possible is unitarity222We note that unitarity is used for most experiment’s
extraction of one or more of the mixing parameters. For example, medium-
baseline reactor neutrino experiments at $L\simeq 1$ km actually determine
$4|U_{e3}|^{2}(|U_{e1}|^{2}+|U_{e2}|^{2})$ but by unitarity this is equal to
$4|U_{e3}|^{2}(1-|U_{e3}|^{2})$ which can be expressed as
$\sin^{2}2\theta_{13}$ under the assumption the matrix is unitary. and thus if
there is new physics in the neutrino sector this story may get more
complicated. Given the sizable expected improvements in disappearance
measurements in the $\nu_{e}$ channel with the Jiangmen Underground Neutrino
Observatory (JUNO) and the $\nu_{\mu}$ channel with the Deep Underground
Neutrino Experiment (DUNE) and Hyper-Kamiokande (HK), such a study is quite
timely. Moreover, disappearance has different (and often cleaner) systematics
than appearance measurements since the neutrinos at the near and far detectors
are the same flavor which means that this can be a valuable cross check of CP
violation probes in the appearance channel.

In this paper, we will briefly review the standard CP violation picture. We
will then develop the theory for where there is information about CP violation
in disappearance measurements. Finally, we will perform numerical studies
indicating the sensitivity to measure $\delta$, and thus determine if CP is
violated or not, via disappearance measurements only.

##  II Conventional CP violation picture

It is true that, consistent with conventional wisdom in the literature,
disappearance channels are CP invariant, see e.g. Dick  _et al._ (1999); Itow
_et al._ (2001); Nunokawa  _et al._ (2008), under the assumption of that CPT
is conserved. That is, by CPT conservation:

$P(\nu_{\alpha}\to\nu_{\alpha})=P(\bar{\nu}_{\alpha}\to\bar{\nu}_{\alpha})\,,$
(1)

in vacuum333The presence of matter modifies neutrino oscillations Wolfenstein
(1978) and induces additional apparent CP (or CPT) violation which has been
discussed extensively in the literature Kuo and Pantaleone (1987); Krastev and
Petcov (1988); Toshev (1989, 1991); Arafune and Sato (1997); Parke and Weiler
(2001); Akhmedov  _et al._ (2001); Xing (2013); Petcov and Zhou (2018);
Bernabéu and Segarra (2019); Schwetz and Segarra (2022), but this is easily
accounted for and is not the focus of this paper.. Thus neutrinos and
antineutrinos act the same in vacuum disappearance experiments.

The CP asymmetry, on the other hand, is only nonzero for appearance and is
proportional to the Jarlskog invariant $J\equiv
s_{12}c_{12}s_{13}c_{13}^{2}s_{23}c_{23}\sin\delta$ Jarlskog (1985). The
difference in probabilities for neutrinos and antineutrinos is,

$P(\nu_{\alpha}\to\nu_{\beta})-P(\bar{\nu}_{\alpha}\to\bar{\nu}_{\beta})\simeq\pm
8\pi J\frac{\Delta m^{2}_{21}}{\Delta m^{2}_{31}}\,,$ (2)

in vacuum near the first oscillation maximum with $\alpha\neq\beta$ where the
sign depends on $\alpha$ and $\beta$. Thus a determination of $J$, which
requires measuring all four parameters of the PMNS matrix, indicates how
neutrinos and antineutrinos behave differently in appearance oscillation
measurements444If neutrinos have decohered or are in an oscillation averaged
regime after traveling many oscillation periods then the term in eq. 2 will
vanish..

New physics such as sterile neutrinos Acero  _et al._ (2022b), non-standard
neutrino interactions Wolfenstein (1978); Pro (2019); Argüelles  _et al._
(2023), or unitarity violation Martinez-Soler and Minakata (2020) could also
modify this picture in non-trivial ways by making fully CP conserving
scenarios appear CP violating or other such nightmare scenarios. It may be
possible to avoid these scenarios via a combination of experiments at
different baselines and energies, see e.g. de Gouvêa and Kelly (2016); Denton
_et al._ (2021); Chatterjee and Palazzo (2021).

There are several other non-conventional means of probing leptonic CP
violation. One is via one-loop corrections to elastic scattering Sarantakos
_et al._ (1983); Marciano and Parsa (2003); Tomalak and Hill (2020); Hill and
Tomalak (2020) in solar neutrinos which may achieve up to $\sim 1\sigma$
sensitivity to CP violation with optimistic experimental assumptions Brdar and
Xu (2023). This process is still an appearance measurement in that it depends
on CP violation via measuring $\nu_{e}\to\nu_{\mu}$. Another is via a one-loop
correction to the standard matter effect for neutrinos propagating inside the
Sun which leads to an additional matter effect term $5\times 10^{-5}$ times
smaller than the standard matter effect Botella  _et al._ (1987) which could
lead to a CP violating effect, but it is $\sim 10^{-5}$ the size of the
standard matter effect and thus beyond the hope even of up-coming experiments
Minakata and Watanabe (1999).

##  III CP violation in disappearance

###  III.1 An understanding

While it is not directly possible to determine if nature prefers neutrinos or
antineutrinos via disappearance measurements alone, it is possible to
determine if nature treats neutrinos and antineutrinos the same or differently
via measurements of these CP conserving disappearance channels. That is,
disappearance measurements cannot provide information on
$\operatorname{sign}(\sin\delta)$ or equivalently on $\operatorname{sign}J$,
but can constrain $\cos\delta$ and thus potentially rule out CP conserving
values of $|\cos\delta|=1$.

The disappearance probability in vacuum for flavor $\alpha$ is

$\displaystyle P(\nu_{\alpha}\to\nu_{\alpha})=1$ $\displaystyle-4|U_{\alpha
1}|^{2}|U_{\alpha 2}|^{2}\sin^{2}\Delta_{21}$ $\displaystyle-4|U_{\alpha
1}|^{2}|U_{\alpha 3}|^{2}\sin^{2}\Delta_{31}$ $\displaystyle-4|U_{\alpha
2}|^{2}|U_{\alpha 3}|^{2}\sin^{2}\Delta_{32}\,,$ (3)

where $\Delta_{ij}=\Delta m^{2}_{ij}L/4E$ is the kinematic term.

To understand how one can determine if CP is conserved or not, we focus on the
four degrees of freedom that describe the mixing matrix. We begin by examining
the PMNS mixing matrix in the usual parameterization555While we are working in
a parameterization dependent framework, the ability to discover CP violation
is not artificially enhanced by this and CP violation can be discovered in any
unitary parameterization of the mixing matrix. Tanabashi  _et al._ (2018);
Denton and Pestes (2021):

$U=\begin{pmatrix}c_{13}c_{12}&c_{13}s_{12}&s_{13}e^{-i\delta}\\\
-c_{23}s_{12}-s_{23}s_{13}c_{12}e^{i\delta}&c_{23}c_{12}-s_{23}s_{13}s_{12}e^{i\delta}&s_{23}c_{13}\\\
s_{23}s_{12}-c_{23}s_{13}c_{12}e^{i\delta}&-s_{23}c_{12}-c_{23}s_{13}s_{12}e^{i\delta}&c_{23}c_{13}\end{pmatrix}\,.$
(4)

Since disappearance measurements only constrain absolute values of elements of
the PMNS matrix, we notice that the measurements of the first row, the
$\nu_{e}$ row, provide no information about $\delta$. It would seem that
measurements of either the $\nu_{\mu}$ or $\nu_{\tau}$ row would provide
information about $\delta$, specifically $\cos\delta$, implying that
$\nu_{e}$’s are somehow special and different from the other two flavors. In
reality, any one row (and any one column) can be made to be “simple”: only a
product of sines, cosines, and $e^{\pm i\delta}$, see e.g. Denton and Pestes
(2021). The remaining four elements must always be “complicated”: the sum or
difference of the products of such terms, one of which always contains $e^{\pm
i\delta}$. This provides one means of understanding why two separate
disappearance measurements are required to probe CP violation. That is, if,
for example, we had an excellent measurement of $\nu_{\mu}$ disappearance but
not $\nu_{e}$ or $\nu_{\tau}$ disappearance, since we could choose to make the
$\nu_{\mu}$ row simple then therefore we cannot learn anything about CP
violation.

Thus the absolute value of the complicated elements contains a $\cos\delta$
contribution which is the focus of this paper. We also note that in the usual
parameterization, the absolute value of the elements in the $\nu_{e}$ row
depends on only two parameters: $\theta_{13}$ and $\theta_{12}$, while the
absolute value of the elements in the other two rows each depend on all four
parameters in the mixing matrix.

A perfect measurement of a disappearance channel allows for the determination
of the coefficients of all three terms, but provides only two constraints on
the mixing matrix due to unitarity. That is, one can always define away one of
the $|U_{\alpha i}|^{2}$ in terms of the other two by $|U_{\alpha
1}|^{2}+|U_{\alpha 2}|^{2}+|U_{\alpha 3}|^{2}=1$. Thus a perfect measurement
of the $\nu_{e}$ disappearance probability can constrain two parameters which,
given how we typically parameterize the mixing matrix, the constrained degrees
of freedom map onto the parameters $\theta_{13}$ and $\theta_{12}$. To date,
Daya Bay Adey  _et al._ (2018) and RENO Seo  _et al._ (2018) provide excellent
constraints on $\theta_{13}$ while KamLAND Gando  _et al._ (2013), SNO Aharmim
_et al._ (2013), Super-Kamiokande Abe  _et al._ (2016), and Borexino Agostini
_et al._ (2018) provide good constraints on $\theta_{12}$. In the future JUNO
Abusleme  _et al._ (2022) will measure $\theta_{12}$ with excellent precision.
Thus the $\nu_{e}$ row is in excellent shape.

The constraint on $\theta_{13}$, which comes from measuring the quantity
$|U_{e3}|^{2}(|U_{e1}|^{2}+|U_{e2}|^{2})=|U_{e3}|^{2}(1-|U_{e3}|^{2})$, is
determined via the kinematic terms $\Delta_{31}$ and $\Delta_{32}$ which has a
maximum effect for reactor neutrinos at a baseline of $\sim 1.5$ km. The
oscillation666Solar neutrinos, which largely do not oscillate Smirnov (2016),
constrain $\simeq|U_{e2}|^{2}(1-|U_{e3}|^{2})$. constraint on $\theta_{12}$,
which comes from measuring the quantity $|U_{e1}|^{2}|U_{e2}|^{2}$, is
determined via the kinematic term $\Delta_{21}$ which has a maximum effect for
reactor neutrinos at a baseline about $\frac{\Delta m^{2}_{31}}{\Delta
m^{2}_{21}}\simeq 30$ times farther of $\sim 50$ km.

For the $\nu_{\mu}$ row, disappearance measurements provide up to two degrees
of freedom that map onto four parameters: $\theta_{23}$, $\theta_{13}$,
$\theta_{12}$, and $\cos\delta$. But since $\theta_{13}$ and $\theta_{12}$ are
or will be well known, then similar measurements of $\nu_{\mu}$ disappearance
will provide information about $\theta_{23}$ and $\cos\delta$.

Disappearance experiments exist that measure $\theta_{23}$ via the quantity
$|U_{\mu 3}|^{2}(|U_{\mu 1}|^{2}+|U_{\mu 2}|^{2})=|U_{\mu 3}|^{2}(1-|U_{\mu
3}|^{2})$ (combined with information about $\theta_{13}$) at baselines from
about 300 km to 10,000 km. Thus one can get at the second piece of information
in the $\nu_{\mu}$ disappearance picture by performing a $\nu_{\mu}$
disappearance oscillation experiment at 30 times the baseline for a fixed
energy. This is not feasible for atmospheric neutrinos and one would need to
imagine an extremely optimistic configuration from e.g. Tokai, the accelerator
neutrino source for T2K and HK, to SURF, the upcoming far detector location
for DUNE, which is a distance of 8235 km, close to the oscillation minimum at
8850 km assuming the same off-axis angle as T2K/HK, although the flux is lower
by a factor of almost 800. See the appendix for a discussion of this scenario.

We instead focus on leveraging data in planned experiments such as DUNE and HK
and a careful spectral measurement to provide information about the beginning
of the $\Delta_{21}$ oscillations in a $\Delta_{31}$ and $\Delta_{32}$
dominated regime. This is similar to the discussed plan for measuring the
solar parameters $\Delta m^{2}_{21}$ and $\theta_{12}$ with Daya Bay data
Forero  _et al._ (2021). The effect of CP violation thus begins to show up at
the low energy side of the $\nu_{\mu}$ disappearance spectrum, and thus DUNE
has an advantage: $\nu_{\mu}$ experience more oscillations before the
$\nu_{\mu}$ charged-current cross section hits the muon threshold. While event
rates and reconstructions are challenging at lower energies, the effect will
impact the rate at which the oscillation maximum decreases where the
probability is near one, so there is no probability suppression, which helps
the rate.

Since the appearance channel essentially constrains $\sin\delta$ (see eq. 2)
while the disappearance channel constrains $\cos\delta$, these two
measurements provide key complementary information. In fact, there will be
sign degeneracies in many regions of parameter space of $\delta$ with either
only appearance or disappearance. Moreover, the precision on $\delta$ near
$\pi/2$ or $3\pi/2$ is determined by the sensitivity to $\cos\delta$ which
comes from this combination of disappearance measurements making this
disappearance based measurement crucial for determining the exact value of
$\delta$ if we are near $|\sin\delta|=1$ as some data Ramírez  _et al._ (2023)
may be indicating.

We now turn to a more quantitative investigation of the size of the effect.

###  III.2 Analytic approximation

We now strive to understand exactly how $\cos\delta$, which can provide key
information about CP violation, appears in the $\nu_{\mu}$ disappearance
probability in matter. First, we note that the $\Delta_{31}$ and $\Delta_{32}$
terms in eq. 3 can be approximately combined as mentioned above, see also
Nunokawa  _et al._ (2005). Thus the $\cos\delta$ dependence in the magnitudes
of these two terms will approximately cancel in vacuum, although the matter
effect will somewhat change this. Second, we focus on the $\Delta_{21}$ term.
In vacuum, to first order in $s_{13}$, the term is

$-4c_{23}^{2}\left(s_{12}^{2}c_{12}^{2}+s_{23}c_{23}s_{13}\sin
2\theta_{12}\cos 2\theta_{12}\cos\delta\right)\sin^{2}\Delta_{21}\,,$ (5)

where the term in parentheses is numerically $\simeq 0.21+0.03\cos\delta$.
Thus we would expect that the maxima will be shifted lower for $\cos\delta=1$
and higher for $\cos\delta=-1$. Third, we include the correction due to the
matter effect. The matter effect has almost no impact on $\theta_{23}$ or
$\delta$ below the atmospheric resonance at $E\simeq 11$ GeV Denton  _et al._
(2016); Xing  _et al._ (2018). In addition, while $\theta_{13}$, $\Delta
m^{2}_{31}$, and $\Delta m^{2}_{32}$ do evolve somewhat in matter, the change
from the vacuum value is $\lesssim 10\%$ and can be safely ignored at this
level of discussion. The solar parameters, $\theta_{12}$ and $\Delta
m^{2}_{21}$, on the other hand, evolve considerably in matter at these
energies. To a sufficient approximation, the matter correction factor for the
solar parameters is Denton  _et al._ (2016); Denton and Parke (2018)

$\mathcal{S}_{\odot}\simeq\sqrt{(\cos 2\theta_{12}-c_{13}^{2}a/\Delta
m^{2}_{21})^{2}+\sin^{2}2\theta_{12}}\,,$ (6)

where $a=2\sqrt{2}G_{F}N_{e}E$ is the contribution from the matter effect. See
the appendix for more discussion of the solar corrections including higher
order terms. Then the solar mixing angle is approximately given by

$\cos 2\widehat{\theta_{12}}\simeq\frac{\cos 2\theta_{12}-c_{13}^{2}a/\Delta
m^{2}_{21}}{\mathcal{S}_{\odot}}\,,$ (7)

where $\widehat{x}$ is the quantity $x$ in matter. We note that past the solar
resonance at $E=0.13$ GeV, $\widehat{\theta_{12}}>\pi/4$ and thus $\cos
2\widehat{\theta_{12}}<0$. Therefore the second term in the parentheses in eq.
5 changes sign when the matter effect is considered. That is, we expect that
the probability in matter should be highest for $\cos\delta=1$ and lowest for
$\cos\delta=-1$ as is confirmed numerically in fig. 1. This measurement
therefore also provides another indirect test of the matter effect if one is
able to compare measurements of $\delta$ between the appearance and the
disappearance channels. Various additional effects are also in play, see the
appendix, but this is sufficient to understand the general role of
$\cos\delta$ in the $\nu_{\mu}$ disappearance channel. The above calculation
applies for both DUNE and HK (see also the appendix) but the effect at HK is
smaller than that for DUNE.

Figure 1: The $\nu_{\mu}$ disappearance probability for DUNE at different
values of $\cos\delta$; see also fig. 7 for the same plot for HK.

For antineutrinos the story is somewhat different. The value of $\cos\delta$
does not change as $\delta\to-\delta$, so it is the same as for neutrinos. In
matter we note that while $\cos 2\widehat{\theta_{12}}<0$ for neutrinos, it
remains positive for antineutrinos, as in vacuum. Thus the impact on the
oscillation maxima for antineutrinos is the same in matter as in vacuum and
the probability is comparatively large for $\cos\delta=-1$ and small for
$\cos\delta=+1$. Due to the lower statistics in $\bar{\nu}_{\mu}$ mode, this
will not contribute as much as the neutrino channel to the total significance
for probing $\cos\delta$ and CP violation.

##  IV Estimated experimental sensitivities

To quantify the magnitude of the effect, we simulate $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ disappearance in DUNE using DUNE’s simulation files Abi  _et
al._ (2021); Huber  _et al._ (2005). We consider 40 kt fiducial volume and 6.5
years in each neutrino and antineutrino mode with 1.2 MW and 56% beam uptime.
We consider priors on the five oscillation parameters other than $\delta$ from
one of 1) our current knowledge of the oscillation parameters Esteban  _et
al._ (2020), 2) the expected improvement on $\theta_{12}$, $\Delta
m^{2}_{21}$, and $\Delta m^{2}_{31}$ from the inclusion of 6 years of JUNO’s
$\bar{\nu}_{e}$ disappearance data Abusleme  _et al._ (2022), and 3) the
hypothetical scenario with perfect knowledge of all five oscillation
parameters. We now perform a statistical test to determine DUNE’s capability
to determine $\cos\delta$ including systematics, efficiency, smearing, and
backgrounds as estimated by DUNE Abi  _et al._ (2021) and show our results in
fig. 2 for each of the three different choices of priors. For $\cos\delta=\pm
1$, the combination of DUNE and JUNO combined can disfavor $\cos=\mp 1$ at
$>3\sigma$ and improvements in JUNO’s measurement can reach close to
$4\sigma$. In addition, for $\cos\delta=0$ (CP violating), $|\cos\delta|=1$
(CP conserving) can be disfavored at $1.6\sigma$, see fig. 3 in the appendix.

Figure 2: The expected sensitivity to $\cos\delta$ using only $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ disappearance from DUNE along with external priors on the
other five oscillation parameters from: the current precision, the expected
improvement with JUNO, or if the other oscillation are known perfectly.

We have confirmed that there is information about $\cos\delta$ in each of
neutrino and antineutrino modes individually, although neutrino mode
contributes more to the information due to higher statistics from the larger
cross section and lower wrong-sign lepton rates. In addition, as suggested by
the theory discussion above, $\cos\delta$ can also be determined if DUNE was
performed in vacuum, although the results would be modified. Additional
numerical results for DUNE as well as HK can be found in the appendix.

We also checked the precision with which $\cos\delta$ can be determined. The
$1\sigma$ uncertainty is essentially independent of the true value and is
$0.63$ and $0.51$ given external information at the level associated with 6
years of JUNO and perfect knowledge, respectively.

One could also consider $\nu_{\mu}$ disappearance with atmospheric neutrinos
at HK Abe  _et al._ (2018), IceCube Abbasi  _et al._ (2023), KM3NeT Adrian-
Martinez  _et al._ (2016), or JUNO Abusleme  _et al._ (2021); Suliga and
Beacom (2023). The expected precision in the standard disappearance
oscillation parameters ($\Delta m^{2}_{31}$ and $\sin^{2}2\theta_{23}$) is
expected to be much better in the upcoming long-baseline accelerator-based
experiments than in the atmospheric experiments. Nonetheless, due to the
different systematics and timelines it may be useful to consider a fit
including atmospheric neutrinos alongside state-of-the-art $\nu_{e}$
disappearance measurements.

In principle one could probe $\cos\delta$ with existing disappearance data.
The current status of the data is that the best $\nu_{\mu}$ disappearance
measurements come from NOvA Acero  _et al._ (2022a) and T2K Ramírez  _et al._
(2023) and the best $\nu_{e}$ disappearance measurements come from Daya Bay
Adey  _et al._ (2018) and KamLAND Gando  _et al._ (2013). The $\nu_{e}$
disappearance data is described by the “current” curves in figs. 2, 3, and 8
which show that even DUNE or HK can only provide at most $\sim 1.4\sigma$
sensitivity to $\cos\delta$; with existing NOvA and T2K data there would not
be any significant $\cos\delta$ information at all.

##  V Conclusion

Determining if CP is violated in the neutrino sector is one of the highest
priorities in particle physics. The best way to do so is with neutrino
oscillations in the appearance channels. As this measurement will face many
significant systematic uncertainties, additional means of probing $\delta$ and
CP violation will be crucial to ensure robustness. While disappearance
channels are fundamentally CP conserving, we have shown that they can still
provide information about $\delta$, specifically $\cos\delta$, which is
sufficient to determine if CP is violated or not. Nonetheless, it cannot be
done with any one disappearance measurement; we require good precision
measurements of the disappearance probability of (at least) two different
flavors.

The matter effect affects the details of this story somewhat, but CP violation
can be determined in vacuum. In addition, neutrinos and antineutrinos behave
somewhat differently in disappearance due to the matter effect, but neutrino
mode alone (or antineutrino mode alone) is sufficient to determine if CP is
conserved or violated.

In the upcoming generation of experiments, JUNO will measure the
$\bar{\nu}_{e}$ disappearance probability with unprecedented precision by
directly observing all three oscillation frequencies. Long-baseline
experiments like DUNE and HK will measure $\nu_{\mu}$ disappearance primarily
focused on the weighted average of the $\Delta m^{2}_{31}$ and $\Delta
m^{2}_{32}$ frequencies, but will also detect at a subleading level the
$\Delta m^{2}_{21}$ frequency Denton and Gehrlein (2023). This is enough to
provide some information about $\delta$. In particular, we find that DUNE and
JUNO combined will be able disfavor some values of $\cos\delta$ at up to
$>3\sigma$ depending on the true value. Since $\nu_{\mu}$ disappearance has
somewhat cleaner and, more importantly, different, systematics from $\nu_{e}$
appearance in long-baseline measurements at DUNE and HK, this channel will
provide a crucial robustness test of CP violation when combined with JUNO
data.

###### Acknowledgements.

The author acknowledges support by the United States Department of Energy
under Grant Contract No. DE-SC0012704.

##  Appendix A Appendix A: CP violation discovery sensitivity

We quantify in fig. 3 the expected sensitivity to discover CP violation (that
is, ruling out $|\cos\delta|=1$) as a function of the true value of $\delta$.
We also include priors of the five oscillation parameters other than $\delta$
constrained at three different levels: perfect knowledge, the expected future
precision including 6 years of JUNO constraining $\theta_{12}$, $\Delta
m^{2}_{21}$, and $\Delta m^{2}_{31}$ Abusleme  _et al._ (2022), and the
current knowledge Esteban  _et al._ (2020). The different colors correspond to
including both appearance and disappearance (the standard DUNE analysis), only
appearance, and only disappearance. The different line styles correspond to
external pulls from the current knowledge of the oscillation parameters, the
expected improvements with JUNO, and hypothetical perfect knowledge. The blue
dotted curve (both channels and current knowledge of the oscillation
parameters) agrees with DUNE’s curve very well.

Figure 3: DUNE’s sensitivity to disfavor CP conservation, broken down by
channel (colors) and the external priors (line styles). For the same plot with
HK, see fig. 8.

While the disappearance channel does not contribute nearly as much information
on CP violation as the appearance channel, it does provide some information
enhancing the overall sensitivity to CP violation beyond appearance alone. It
also provides an important cross-check on CP violation measurements with
different systematics. We also see that the combination of DUNE and JUNO’s
disappearance measurements provides for up to $1.6\sigma$ sensitivity to
discover CP violation. Finally, we note that the combined combined fit with
both appearance and disappearance data yields more information than the naive
sum of the $\Delta\chi^{2}$’s of each separately in the cases with the current
or expected JUNO priors due to the fact that $\nu_{\mu}$ disappearance will
provide world leading measurements of $\Delta m^{2}_{31}$ and $\theta_{23}$,
but with perfect knowledge of the other five oscillation parameters, the
combined fit is the same as the naive sum of $\Delta\chi^{2}$’s.

##  Appendix B Appendix B: Tokai to SURF

We examine the solar $\Delta m^{2}_{21}$ minimum at a hypothetical very long-
baseline $\nu_{\mu}$ disappearance accelerator neutrino experiment from J-PARC
in Tokai, Japan to SURF in Lead, South Dakota, United States at a baseline of
8235 km through the Earth’s mantle.

In vacuum, the oscillation minimum due to $\Delta m^{2}_{21}$ happens at 0.5
GeV and depends on $\cos\delta$ since the amplitude of the oscillation, to
first order in $s_{13}$ is

$|U_{\mu 1}|^{2}|U_{\mu 2}|^{2}\approx
c_{23}^{4}s_{12}^{2}c_{12}^{2}+s_{23}c_{23}^{3}s_{13}\sin 2\theta_{12}\cos
2\theta_{12}\cos\delta\,.$ (8)

While the energies are low, since we are considering oscillations related to
$\Delta m^{2}_{21}$ instead of $\Delta m^{2}_{31}$, the matter effect will
play a role and change the amplitude and shift the location of the minimum.
The matter value of $\theta_{12}$ reaches $\pi/4$ at $2EV_{\rm CC}=\Delta
m^{2}_{21}\cos 2\theta_{12}/c_{13}^{2}$ Denton and Parke (2019, 2018) where
$V_{\rm CC}=\sqrt{2}G_{F}N_{e}$ is the matter potential and $N_{e}$ is the
electron number density. For $\rho=4$ g/cc, a typical mantle density, this
occurs at $0.09$ GeV

For the frequency, the matter correction is

$\Delta m^{2}_{21}\to\widehat{\Delta m^{2}_{21}}\approx\Delta
m^{2}_{21}\mathcal{S}_{\odot}\,,$ (9)

where the hat denotes a quantity in matter and

$S_{\odot}=\sqrt{(\cos 2\theta_{12}-c_{13}^{2}a/\Delta
m^{2}_{21})^{2}+\sin^{2}2\theta_{12}}\,,$ (10)

is the solar correction factor Denton  _et al._ (2016); Denton and Parke
(2018, 2019). We find that the oscillation minimum happens near

$\widehat{\Delta_{21}}=\frac{3}{2}\pi\,.$ (11)

Normally there would be one oscillation minimum at
${\widehat{\Delta_{21}}}=\pi/2$, but ${\widehat{\Delta_{21}}}>\pi/2$ for all
energies at this density and baseline. This is because at higher energies the
matter potential causes $\mathcal{S}_{\odot}\propto E$ so $\widehat{\Delta
m^{2}_{21}}\propto E$ and thus $\widehat{\Delta_{21}}$ is independent of
energy. At lower energies, $\widehat{\Delta m^{2}_{21}}$ reaches a minimum at
the solar minimum and continues to increase somewhat to vacuum (roughly
proportional to the neutrino energy, but the proximity to the resonance
somewhat modifies this dependence), meanwhile the $1/E$ contribution in
$\widehat{\Delta_{21}}$ ensures that $\widehat{\Delta_{21}}$ grows at small
energies as well. Since there is a minimum of $\widehat{\Delta_{21}}$ as a
function of $E$, then for certain baselines and densities it may be the case
that the first (or even higher) oscillation extremum is never reached for any
energies.

There is also an additional shift in the location of the minimum due to the
prefactor in front of the $\sin^{2}\Delta_{21}$ term which is proportional to
$\sin^{2}2\theta_{12}$ in vacuum which is approximately
$\sin^{2}2\theta_{12}/\mathcal{S}_{\odot}^{2}$ in matter, which also depends
on the energy. The shift due to this is small; note that $\theta_{23}$ and
$\delta$ don’t vary much in matter at all, and the impact of matter on $\Delta
m^{2}_{31}$ and $\theta_{13}$ is small enough at these energies Denton  _et
al._ (2016); Xing  _et al._ (2018).

Thus the location of the minimum is well estimated in this region of parameter
space by solving eq. 11 for the energy,

$E_{\rm min}\simeq\frac{(\Delta m^{2}_{21}L)^{2}/2}{c_{13}^{2}\cos
2\theta_{12}\Delta m^{2}_{21}L^{2}V_{\rm CC}+\sqrt{(\Delta
m^{2}_{21}L)^{2}[(3\pi)^{2}-(2c_{13}^{2}\sin 2\theta_{12}LV_{\rm
CC})^{2}]}}=0.16{\rm\ GeV}\,.$ (12)

Since this energy is above $0.09$ GeV, the leading order term proportional to
$\cos\delta$ in eq. 8 is nonzero and $\cos\delta=1$ will decrease eq. 8 and
thus increase the probability relative to $\cos\delta=0$. In addition, since
the $\cos\delta$ dependence in the next $s_{13}$ order correction depends on
$-\cos^{2}\delta$ and since both orders may be similar due to the $\cos
2\theta_{12}$ suppression in the first order term, we see that the probability
should be highest for $\cos\delta=1$ and slightly lower, but comparable, for
other values of $\cos\delta$.

Next, notice that the size of the large and fast $\Delta_{31}$ and
$\Delta_{32}$ amplitudes also depend on $\cos\delta$. This is because
$\Delta_{31}$ and $\Delta_{32}$ are sufficiently different and no longer in
phase that they are to be treated separately. Then we notice that in vacuum
through first order in $s_{13}$ the amplitudes are:

$\displaystyle-4s_{23}^{2}\left(c_{23}^{2}s_{12}^{2}+2s_{23}c_{23}s_{13}s_{12}c_{12}\cos\delta\right)\,,$
(13)
$\displaystyle-4s_{23}^{2}\left(c_{23}^{2}c_{12}^{2}-2s_{23}c_{23}s_{13}s_{12}c_{12}\cos\delta\right)\,,$
(14)

where the first (second) line is for $\Delta_{31}$ ($\Delta_{32}$). Since
$\widehat{s_{12}^{2}}>\widehat{c_{12}^{2}}$ at these energies, it is the
$\Delta_{31}$ term that dominates. So for $\cos\delta=1$ the dominant
amplitude increases in magnitude and the only decrease is in the lesser
amplitude, while for $\cos\delta=-1$ the large amplitude is suppressed and the
increase in the smaller amplitude brings the two amplitudes closer together.
Then since $\Delta_{31}$ and $\Delta_{32}$ are roughly out of phase at these
energies, as expected at the oscillation minimum for $\Delta_{21}$, there is
destructive interference when the amplitudes of each phase are close together.

All of these effects discussed above can be seen in fig. 4. The dashed lines
are the same as the solid lines but with a $10\%/\sqrt{E/{\rm GeV}}$ energy
resolution smearing, see e.g. Friedland and Li (2019). This shows that even
with an optimistic energy resolution, the fast $\Delta_{31}$ and $\Delta_{32}$
oscillations are completely averaged out. First, the large $\Delta_{21}$
minimum happens at $0.16$ GeV as expected. Second, we see that the smeared out
probability is higher for $\cos\delta=1$ than the other cases which are all
similar. Third, the amplitude of the fast oscillations decreases near the
minimum but decrease much more so for $\cos\delta=-1$ than for $\cos\delta=1$.

Experimentally, the low energy of the $\Delta m^{2}_{21}$ minimum would
require the beam be more off-axis than T2K or HK, which further reduces the
flux considerably, and the $\nu_{\mu}$ CC cross section experiences
considerable suppression from the muon mass threshold making an experiment of
this nature extremely unfeasible. An additional issue arises which is one of
energy resolution. Given that the $\Delta m^{2}_{31}$ and $\Delta m^{2}_{32}$
oscillations have large amplitude and oscillate $\sim 35$ times faster than
the larger $\widehat{\Delta m^{2}_{21}}$ oscillation, the effect due to the
differing amplitudes of the $\widehat{\Delta m^{2}_{21}}$ oscillations from
$\cos\delta$ will be rapidly averaged out unless unrealistically exceptional
energy resolution can be achieved. Thus only the effect described after eq. 12
from eq. 8 could be detectable, not the effect described in eqs. 13-14.
Therefore despite its theoretical interest, this determination of $\cos\delta$
is extremely unfeasible, although theoretically possible.

Figure 4: The $\nu_{\mu}$ disappearance probability from Tokai to SURF at
different values of $\cos\delta$. The dashed lines have an additional smearing
to show the probability independent of the fast oscillations.

##  Appendix C Appendix C: DUNE event rates and regions of interest

While the full statistical fits performed for figs. 2 and 3 contain all of the
relevant information, it is useful to gain a physical understanding of where
we can expect the effects to appear in the data. To this end, in fig. 5 we
calculated the expected $\nu_{\mu}$ event rates after 6.5 years in neutrino
mode only including efficiency, smearing, and backgrounds for several key
values of $\cos\delta$. We have defined two interesting regions of interest
(ROIs) separated by the local minimum and constrained by half the event rate
of each local maximum. In the lower energy ROI, ROI 1, we see that the event
rate increases with $\cos\delta$ while in ROI 2 it decreases with
$\cos\delta$. The statistics in each ROI for the different values of $\delta$
are shown in table 1 and it is easy to see that each ROI contributes to the
statistical test to disfavor CP conservation given $\cos\delta=0$ at the level
of $\Delta\chi^{2}\sim 1.4$ based on statistics only leading to a combined
estimate of $\sim 1.7\sigma$ sensitivity from neutrino mode, close to the
correct estimate of $1.6\sigma$ (including neutrino and antineutrino modes)
shown in fig. 3. A realistic analysis needs to include uncertainties on the
oscillation parameters and systematic uncertainties which will decrease the
sensitivity somewhat, but also shape information which will increase it
somewhat.

Figure 5: The expected $\nu_{\mu}$ disappearance event rates after 6.5 years in neutrino mode for various values of $\cos\delta$. We have also defined two regions of interest allowing for a simple statistical understanding of the sensitivity to $\cos\delta$. Table 1: The expected $\nu_{\mu}$ event rates after 6.5 years of neutrino mode at DUNE as a function of $\cos\delta$ in the two ROIs, see fig. 5. $\cos\delta$ | ROI 1 | ROI 2
---|---|---
1 | 5506 | 5038
0 | 5418 | 5115
-1 | 5334 | 5193

##  Appendix D Appendix D: Runtime impact

We also calculate the sensitivity based on the runtime of DUNE and JUNO. For
DUNE we keep the target mass, uptime, and proton power the same and vary the
runtime, split evenly between $\nu_{\mu}$ and $\bar{\nu}_{\mu}$. For JUNO we
assume that the precision on the three parameters that they mainly determine,
$\Delta m^{2}_{21}$, $\Delta m^{2}_{31}$, and $\theta_{12}$, continue to scale
as expected with a variance proportional to the sum of run time and a
systematic term determined by the numbers in Abusleme  _et al._ (2022). We
plot the ability to rule out CP conservation assuming $\delta=3\pi/2$; other
statistical tests such as ruling out $\cos\delta=1$ assuming $\cos\delta=-1$
(see fig. 2) and so on all scale in a similar fashion. The results are shown
in fig. 6. We see that further improvement beyond the benchmark point requires
more time from both experiments. Instead of additional time, the statistics
can also be improved by increasing the beam power (DUNE/HK) or reactor power
(JUNO); all experiments have the possibility of seeing such upgrades. There is
a discussion about upgrading the accelerator for DUNE from 1.2 MW to 2.4 MW
Abed Abud  _et al._ (2022); Denton  _et al._ (2022) and JUNO may benefit from
additional nuclear reactors increasing the power from 26.6 GWth to 35.8 GWth
An  _et al._ (2016). This further highlights the important synergy among long-
baseline reactor and accelerator neutrino experiments as improvement in the
statistics of either experiment alone will not significantly improve the
sensitivity to CP violation, but additional runtime for both will since the
effect is not specific to either the $\nu_{e}$ or the $\nu_{\mu}$ row.

Figure 6: The sensitivity to disfavor $\sin\delta=0$ assuming $\delta=3\pi/2$
as a function of DUNE and JUNO’s runtime. The star denotes the benchmark point
considered elsewhere in the text.

##  Appendix E Hyper-Kamiokande

We repeat the same calculations performed in the main text and elsewhere in
the appendix, but for HK for completeness. First we examine the probability
itself in fig. 7 and find that the impact of varying $\cos\delta$ is much
smaller than for DUNE, see fig. 1, as expected since $\widehat{\theta_{12}}$
is closer to $\pi/4$ and thus the leading order $\cos\delta$ dependence in the
$\Delta_{21}$ term in eq. 5 is closer to zero. Second, we calculate HK’s
sensitivity to discover CP violation as a function of the true value of
$\delta$ broken down by appearance and disappearance as well as the role of
external priors in fig. 8. We assume 1.3 MW, 187 kton fiducial mass, 1:3
neutrino to antineutrino run time ratio, and 10 years of running at 100%
uptime to generally agree with the nominal HK prediction Abe  _et al._ (2018).
Note that we assume that the mass ordering is known which is relevant for HK
and not for DUNE. We find that HK is somewhat less sensitive to discovering CP
violation in the disappearance channel than DUNE since the effect is smaller,
but the larger statistics mostly compensate for the difference.

Figure 7: The $\nu_{\mu}$ disappearance probability for HK at different values
of $\cos\delta$; see also fig. 1 for the same plot for DUNE. Figure 8: HK’s
sensitivity to disfavor CP conservation, broken down by channel (colors) and
the external priors (line styles). For the same plot with DUNE, see fig. 3.

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