# Statistical Constraints on the Error of the Leptonic CP Violation of
Neutrinos

Kai Fu, Ying<EMAIL_ADDRESS>Corresponding author.
School of Science, Xi’an Jiaotong University, Xi’an, 710049, China

###### Abstract

A constraint on the error of leptonic CP violation, which require the phase
$\delta_{CP}$ to be less than $\pi/4$ for it to be distinguishable on a $2\pi$
cycle, is presented. Under this constraint, the effects of neutrino detector
’s distance, beam energy, and energy resolution are discussed with reference
to the present values of these parameters in experiments. Although an
optimized detector performances can minimize the deviation to yield a larger
distinguishable range of the leptonic CP phase on a $2\pi$ cycle, it is not
possible to determine an arbitrary leptonic CP phase in the range of $2\pi$
with the statistics from a single detector because of the existence of two
singular points. An efficiency factor $\eta$ is defined to characterize the
distinguishable range of $\delta_{CP}$. To cover the entire possible
$\delta_{CP}$ range, a combined efficiency factor $\eta^{*}$ corresponding to
multiple sets of detection parameters with different neutrino beam energies
and distances is proposed. The combined efficiency factors $\eta^{*}$ of
various major experiments are also presented.

Key words: neutrino, leptonic CP violation, statistical constraints

The neutrino CP phase, $\delta_{CP}$, is an important parameter in the PMNS
parameterization. It is related to neutrino CP violation effects and can be
observed from Jarlskog invariant the $J_{CP}$ BrancoReview . It also plays a
key role in astronomy and cosmology. This phase was predicted using several
differently motivated models in NuCPVmodel1 ; NuCPVmodel2 . Recently, a
$\delta_{CP}$ that is consistent with T2K data has been obtained from the
Yukawaon model, which replaces the SM Yukawa couplings with the VEVs of
Yukawaon scalars YukawaonModel . It has also been predicted using various
newly proposed new mechanisms, such as the $SU(3)$ gauge family symmetry
proposed in SU3FamilySym  and the $\Delta(27)$ flavour symmetry proposed in
Delta27Sym . Moreover, the leptonic CP phase has also been inferred through
global fits to data from neutrino experiments by various research groups
GlobalFitOld1 ; GlobalFitOld2 ; GlobalFitOld3 . More recently updated
optimized values can been found in Refs.GlobalFit20141 ; GlobalFit20142 .
However, to data, it has not been directly measured in experiments. Future
neutrino experiments including long-baseline facilities, superbeams and
neutrino factories, are planned to measure the phase angle. From the
statistical point of view, a central value of the phase alone is not
sufficient to determine its distribution on a cycle. Its error must also be
considered. In this paper, a constraint on the deviation of the leptonic CP
violation angle is proposed based on statistics to ensure that it is
determinate on a cycle. Our motivation is to investigated the properties of
this statistical constraint and its implications for neutrino experiments,
especially reactor experiments.

The paper is organized as follows. Firstly, neutrino oscillation and decay
probabilities, including the Earth matter effect, are reviews. The error on
$\delta_{CP}$ is derived from the neutrino transformation probability. All
contributions from phenomenological parameters are investigated. Particular
attention is paid to the neutrino beam energy $E$ and detecting distance $L$
of the reactor from the reactor to achieve a suppression of the error on
$\delta_{CP}$. The results require a set of $E$ and $L$ values that is
different from that required to achieve significant probability. A statistical
constraint, i.e. the error on $\delta_{CP}$ must be less than $\pi/4$, is
proposed as a requirement for the distinguishability of $\delta_{CP}$. Then,
we recommend a definition of efficiency factor $\eta$ to describe the range
below $\pi/4$ on a $2\pi$ cycle of $\delta_{CP}$. The values of $\eta$ that
correspond to current experiments are calculated, and the corresponding
distinguishable ranges of the leptonic CP phase are also shown. To obtain full
coverage of all possible $\delta_{CP}$ values, a combined $\eta$ based on
multiple detectors is considered. Finally, a short summary is given.

Let us review neutrino oscillations as a starting point. Neutrino oscillation
experiments can be understood in terms of rotations between the mass
eigenstates $\nu_{i}$ ($i=1,2,3$) and the weak gauge eigenstates
$\nu_{\alpha}$ ($\alpha=e,\mu,\tau$) as follows

$\displaystyle\nu_{\alpha}=U_{\alpha i}\nu_{i}$

Here, $U$ is the PMNS matrix and can be written in the form of $VK$.
Generally, $V$ is parameterized as three 1-dimensional rotations

$\displaystyle V=\left(\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{array}\right)\left(\begin{array}[]{ccc}c_{13}&0&s_{13}e^{-i\delta_{CP}}\\\
0&1&0\\\
-s_{13}e^{i\delta_{CP}}&0&c_{13}\end{array}\right)\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{array}\right)$

with the Majorana phase $K=\text{diag}(1,e^{i\delta_{1}},e^{i\delta_{2}})$.
The mixing angles $\theta_{12}$ and $\theta_{23}$ have been precisely measured
by solar and atmospheric oscillation experiments. $\theta_{13}$ is also known
from Daya Bay, RENO and T2K experiments. At present, the latest global fit
results corresponding to the NH and the IH are given in GlobalFit20141  (as
listed in Tab.1).

Table 1: Neutrino oscillation parameters for NH and IH respectively. parameter | NH | IH
---|---|---
$\sin^{2}\theta_{12}/10^{-1}$ | $3.23\pm 0.16$ | $3.23\pm 0.16$
$\sin^{2}\theta_{23}/10^{-1}$ | $5.67^{+0.32}_{-1.28}$ | $5.73^{+0.25}_{-0.43}$
$\sin^{2}\theta_{13}/10^{-2}$ | $2.34\pm 0.20$ | $2.40\pm 0.19$
$\Delta m_{21}^{2}[{10^{-5}\text{eV}^{2}}]$ | $7.6^{+0.19}_{-0.18}$ | $7.6^{+0.19}_{-0.18}$
$|\Delta m_{31}^{2}|[{10^{-3}\text{eV}^{2}}]$ | $2.48^{+0.05}_{-0.07}$ | $2.38^{+0.05}_{-0.06}$

However, the leptonic CP violation parameter is still unknown. With the
increasingly precise measurement capabilities of upcoming neutrino
experiments, the effect of the leptonic CP phase must be considered. When the
matter effect must be considered for long-distance detection experiments in
which the neutrinos pass through the Earth, the oscillation probability
becomes

$\displaystyle
P_{\nu_{\mu}\rightarrow\nu_{e}}\simeq\sin^{2}\theta_{23}\sin^{2}2\theta_{13}\frac{\sin^{2}[(1-A)\Delta]}{(1-A)^{2}}$
$\displaystyle~{}~{}~{}~{}+\alpha\sin 2\theta_{13}\sin 2\theta_{12}\sin
2\theta_{23}\cos\theta_{13}\frac{\sin[(1-A)\Delta]\sin[A\Delta]}{A(1-A)}\cos(\delta_{CP}+\Delta)$
$\displaystyle~{}~{}~{}~{}+\alpha^{2}\cos^{2}\theta_{23}\sin^{2}2\theta_{12}\frac{\sin^{2}[A\Delta]}{A^{2}}$
(2)

with $A=2EV/\Delta m^{2}_{31}$, $\alpha=\Delta m_{21}^{2}/\Delta m_{31}^{2}$
and $\Delta=\Delta m^{2}_{31}L/(4E)$ MatterEff ; WPneutrino . Here, $E$ and
$L$ respect the neutrino beam energy and the distance between the source and
the detector, respectively.

With the aid of the error transfer formula, the standard deviation of
$\Delta\delta_{CP}$ can be calculated using Eq.(2), such that it depends on
the errors of all phenomenological parameters, including the beam energy $E$
and the distance $L$, as follows:

$\displaystyle\left[\delta(\delta_{CP})\right]^{2}$ $\displaystyle=$
$\displaystyle
F_{1}\left[\delta(\sin^{2}\theta_{12})\right]^{2}+F_{2}\left[\delta(\sin^{2}\theta_{13})\right]^{2}+F_{3}\left[\delta(\sin^{2}\theta_{23})\right]^{2}$
(3) $\displaystyle+F_{4}\left[\delta(\Delta
m_{31}^{2})\right]^{2}+F_{5}\left[\delta(\Delta
m_{21}^{2})\right]^{2}+F_{6}\left[\delta E\right]^{2}$

with the abbreviated function $F_{i}=F_{i}(\theta,\Delta
m^{2},E,L,\delta_{CP})$. The errors on the probability
$P_{\nu_{\mu}\rightarrow\nu_{e}}$ and the distance $L$ are treaded ideally.

In Fig. 1, the contributions to $\Delta\delta_{CP}$ for various experimental
scenarios are analyzed on a $2\pi$ cycle of $\delta_{CP}$ for select detector
parameters, $E$ and $L$. For convenience, the energy resolution is set to
zero.

Figure 1: Contributions to the error of leptonic CP phase from
phenomenological parameter measurements and detector performances at fixed
beam energy and detecting distance (a) $L=1$km and $E=5$MeV; (b)$L=300$km and
$E=0.5$GeV; (c) $L=1$km and $E=10$GeV; (d)$L=700$km and $E=10$GeV. Red dotted
line labels statistical limit $\Delta\delta_{CP}=\pi/4$.

The deviation of the CP phase directly affects the means of its measured
values in terms of its statistical distribution. To obtain a separable
$\delta_{CP}$ on a $2\pi$ cycle, the maximum of the standard deviation must be
limited such that $\Delta\delta_{CP}=\pi/4$, which corresponds to $95\%$ C.L.
($2\sigma$) on a half of cycle. In Fig. 1, a total deviation of the CP phase
of below $\pi/4$ means that $\delta_{CP}$ can be distinguished when it is
locateed in the corresponding interval. Otherwise, it will be statistically
uncertain. Fig. 1 exhibits two notable characteristics. The first is the
existence of singular points. There are always two singular points where
$\Delta\delta_{CP}$ tends toward infinity. They originate from the vanishing
sine function in the denominator

$\displaystyle\sin(\delta_{CP}+\Delta)=0$ (4)

that occurs when differentiating the second term of Eq.(2)

$\delta[\cos(\delta_{CP}+\Delta)]=\sin(\delta_{CP}+\Delta)\delta(\delta_{CP}).$

Obviously, the distance between these two singular points is $\pi$. The second
notable characteristics of the figure is the variability in which parameter’s
error dominates the uncertainty, which differ for different detection
parameters. The regions of the $E$-$L$ plane corresponding to parameter error
contributions of more than 30% (and 70%) are depicted in Fig. 2.

Figure 2: Dominant contributions to $\Delta\delta_{CP}$ from two errors of
$\Delta m_{31}^{2}$ and $\Delta\sin^{2}\theta_{23}$.

From the above results, it is apparent that there are two competing
requirements on the beam energy $E$ and distance $L$ for suppressing the error
on the leptonic CP violation and increasing the transformation probability.
Fig. 3 illustrates the corresponding different regions in the plane of $E$ and
$L$. It is possible to satisfy both requirements by choosing $E$ and $L$ from
an area of intersection.

Figure 3: Two competitive requirements to $E$ and $L$: significant probability
vs. suppressed error of $\delta_{CP}$.

A matter of concern is how to describe the discrimination powers of a detector
with respect to an arbitrary leptonic CP violation on a $2\pi$ cycle. One can
introduce an efficiency factor $\eta$ to characterise the effect:

$\displaystyle\eta(E,L)=\frac{\text{the total of intervals below
}\Delta\delta_{CP}=\pi/4\text{on a circle}}{2\pi}.$

A larger value of $\eta$ indicates a larger range in which $\delta_{CP}$ can
be determined. A vanishing $\eta$ indicates that the CP violating phase is
indistinguishable CP violation on a $2\pi$ cycle. When $\eta$ tends toward
$1$, the CP violatingphase can be statistically determined regardless of its
measured value. Because of the singular points, it is not possible to achieve
an efficiency factor $\eta$ that is equal to 1, even for a detector with
perfect energy resolution $\Delta E=0$ that is located at the optimum distance
from the source. Fig. 4 shows that the efficiency factor can be determined
based on the beam energy $E$ and the distance $L$ of a detector. In the low
energy region, the efficiency factor $\eta$ appears to oscillate because of
the bottom-lifting effect when $E$ and $L$ depart from their optimum values.
As the flat bottom of the curve approaches the constraint line
$\Delta\delta_{CP}=\pi/4$, the interval below the constraint line will vary
rapidly.

Figure 4: Efficiency factor vs. beam energy at fixed distance $L$ (1)
$L=0.5km$; (2) $L=1km$; (3) $L=50km$; (4) $L=500km$.

In practical terms, neutrino detection experiments always span a finite energy
region, and the effect of this range of investigated energies should be
expressed in terms of a combined efficiency factor $\eta^{*}$ that represents
the union of all distinguishable intervals for energies $E_{i}$ running over
all possible detection regions:

$\displaystyle\eta^{*}=\bigcup_{E_{i}}\left(\text{intervals below
}\Delta_{CP}=\pi/4\text{ at fixed }E_{i}\right).$ (5)

In Fig. 5, the combined efficiency factors of several major neutrino
experiments are plotted. The efficiency factor at a given energy $E_{i}$ can
be read out from the sum of the horizontal segments. The projection onto the
$\delta_{CP}$ axis of all areas for a given detector represents its maximum
distinguishable range. We find that, at present, there is a blind zone around
$\delta_{CP}=3\pi/3$. If the leptonic CP phase is located in this blind zone,
then these neutrino experiments will not be able to determine it
statistically. To fill in this zone, a new detector must be considered.

Figure 5: Complexed efficiency factors of neutrino experiments for NH and IH.
Black segments at the bottom of each subfigures show all possible
distinguishable region of $\delta_{CP}$. Energy resolutions of detectors has
been considered actually.

In summary, we has investigated the properties of the error on the leptonic CP
violation under a proposed statistical constraint that requires $\delta_{CP}$
to be less than $\pi/4$ for its value to be distinguishale on a cycle. In
particular, the energy and distance for the neutrino detector were considered.
An efficiency factor $\eta$, was defined to describe the discrimination power
of a detector for CP violation. The combined $\eta^{*}$ values for major
neutrino experiments were calculated, and these values provide a clear picture
of the distinguishable range of $\delta_{CP}$. In the future, the $\eta^{*}$
value will be studied further when the relevant phenomenological parameters
can be measured more exactly.

## acknowledgement

We thank our colleague, Dr. Rong Li for helpful discussions.

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