Study of a tri-direct littlest seesaw model at MOMENT

TL;DR

This paper evaluates how the MOMENT neutrino experiment can test and refine the tri-direct littlest seesaw (TDLS) model, highlighting its potential to exclude the model and improve parameter precision.

Contribution

The study analyzes the impact of MOMENT on the TDLS model, demonstrating its ability to exclude the model at high confidence and enhance parameter measurement accuracy.

Findings

MOMENT can exclude the TDLS model at over 5σ if current best-fit values are confirmed.

MOMENT improves the 3σ precision of model parameters by at least a factor of two.

The experiment can observe the predicted sum rule between heta_{23} and \delta$.

Abstract

The flavour symmetry succeeds in explaining the current global fit results. Flavour-symmetry models can be tested by the future experiments that improve the precision of neutrino oscillation parameters, \textit{such as} the MuOn-decay MEdium baseline NeuTrino beam experiment (MOMENT). In this work, we consider tri-direct littlest seesaw (TDLS) models for a case study, and analyze how much MOMENT can extend our knowledge on the TDLS model. We find that measurements of $\theta_{23}$ and $\delta$ are crucial for MOMENT to exclude the model at more than $5\sigma$ confidence level, if the best fit values in the last global analysis result is confirmed. Moreover, the $3\sigma$ precision of model parameters can be improved at MOMENT by at least a factor of two. Finally, we project the surface at the $3\sigma$ confidence level from the model-parameter space to the oscillation-parameter space, and find the potential of MOMENT to observe the sum rule between $\theta_{23}$ and $\delta$ predicted by TDLS.

Tables

Table 1. Table 1: A summary of the relation between oscillation parameters and TDLS model parameters  Ding et al. ( 2018 ) . Two requirements are imposed by TDLS: the smallest mass state m 1 = 0 subscript 𝑚 1 0 m_{1}=0 and the normal mass ordering. The sign of sin ⁡ δ 𝛿 \sin\delta depends on the sign of x ​ cos ⁡ ψ 𝑥 𝜓 x\cos\psi : “ + + ” (“ − - ”) is for x ​ cos ⁡ ψ > 0 𝑥 𝜓 0 x\cos\psi>0 ( < 0 absent 0 <0 ).

model parameters	$x$, $\eta$, $r$, $m_a$
combinations of model parameters	$y=\frac{5x^2+2x+2}{2\left(x^2+x+1\right)}\left(m_a+e^{i\eta }{m}_s\right)$,
	$z=-\frac{\sqrt{5x^2+2x+2}}{2\left(x^2+x+1\right)}\left[\left(x+2\right)m_a-x\left(2x+1\right)e^{i\eta }{m}_s\right]$,
	$w=\frac{1}{2\left(x^2+x+1\right)}\left[{\left(x+2\right)}^2{m}_a+x^2{\left(2x+1\right)}^2{e}^{i\eta }{m}_s\right]$,
	$\sin \psi =\frac{\Im \left(y^{\ast }z+w{z}^{\ast }\right)}{\left|y^{\ast }z+w{z}^{\ast }\right|},\cos \psi =\frac{\Re \left(y^{\ast }z+w{z}^{\ast }\right)}{\left|y^{\ast }z+w{z}^{\ast }\right|}$.
	$\sin 2\theta =\frac{2\left|y^{\ast }z+w{z}^{\ast }\right|}{\sqrt{{\left({\left|w\right|}^2-{\left|y\right|}^2\right)}^2+4{\left|y^{\ast }z+w{z}^{\ast }\right|}^2}},$
	$\cos 2\theta =\frac{{\left|w\right|}^2-{\left|y\right|}^2}{\sqrt{{\left({\left|w\right|}^2-{\left|y\right|}^2\right)}^2+4{\left|y^{\ast }z+w{z}^{\ast }\right|}^2}}$.
oscillation parameters	$\Delta m_{21}^2=m_2^2=\frac{1}{2}\left[{\left|y\right|}^2+{\left|w\right|}^2+2{\left|z\right|}^2-\frac{{\left|w\right|}^2-{\left|y\right|}^2}{\cos \theta }\right]$,
	$\Delta m_{31}^2=m_3^2=\frac{1}{2}\left[{\left|y\right|}^2+{\left|w\right|}^2+2{\left|z\right|}^2+\frac{{\left|w\right|}^2-{\left|y\right|}^2}{\cos \theta }\right]$,
	${\sin }^2{\theta }_{12}=1-\frac{3x^2}{3x^2+2\left(x^2+x+1\right){\cos }^2\theta }$,
	${\sin }^2{\theta }_{13}=\frac{2\left(x^2+x+1\right){\sin }^2\theta }{5x^2+2x+2}$,
	${\sin }^2{\theta }_{23}=\frac{1}{2}+\frac{x\sqrt{3\left(5x^2+2x+2\right)}\sin 2\theta \sin \psi }{2\left[3x^2+2\left(x^2+x+1\right){\cos }^2\theta \right]}$,
	$\cos \delta =\frac{\cot 2{\theta }_{23}\left[3x^2-\left(4x^2+x+1\right){\cos }^2{\theta }_{13}\right]}{\sqrt{3}\left|x\right|\sin {\theta }_{13}\sqrt{\left(5x^2+2x+2\right){\cos }^2{\theta }_{13}-3x^2}}$,
	$\sin \delta =\pm \csc 2{\theta }_{23}\sqrt{1+\frac{{\left(x^2+x+1\right)}^2{\cot }^2{\theta }_{13}{\cos }^{2}2{\theta }_{23}}{3x^2\left[3x^2{\tan }^2{\theta }_{13}-2\left(x^2+x+1\right)\right]}}$.

Table 2. Table 2: The best fit and 3 ​ σ 3 𝜎 3\sigma uncertainty, in the results of NuFit4.0 Esteban et al. ( 2019 ) .

Parameter	${\theta }_{12}{/}^{\circ }$	${\theta }_{13}{/}^{\circ }$	${\theta }_{23}{/}^{\circ }$	$\delta {/}^{\circ }$	$\Delta m_{21}^2/{10}^{-5}{\text{eV}}^2$	$\Delta m_{31}^2/{10}^{-3}{\text{eV}}^2$
best fit	$33.82$	$8.61$	$49.6$	$215$	$7.39$	$2.525$
$3\sigma$ Range	$31.61-36.27$	$8.22-8.99$	$40.3-52.4$	$125-392$	$6.79-8.01$	$2.47-2.625$

Table 3. Table 3: The best fit for x 𝑥 x , η 𝜂 \eta , r 𝑟 r , m a subscript 𝑚 𝑎 m_{a} with the result of NuFit4.0 Esteban et al. ( 2019 ) , and the corresponding oscillation parameters.

$\Delta {\chi }^2$	$x$	$\eta /\pi$	$r$	$m_a/$ meV	${\theta }_{12}{/}^{\circ }$	${\theta }_{13}{/}^{\circ }$	${\theta }_{23}{/}^{\circ }$	$\delta {/}^{\circ }$	$\Delta m_{21}^2/{10}^{-5}{\text{eV}}^2$	$\Delta m_{31}^2/{10}^{-3}{\text{eV}}^2$
$4.98$	$-3.65$	$1.13$	$0.511$	$3.71$	$35.25$	$8.63$	$46.98$	$278.96$	$7.39$	$2.525$

Table 4. Table 4: Assumptions for the source, detector and the running time at MOMENT in the simulation.

MOMENT
Fiducial mass00	0Gd-doping Water cherenkov(500 kton)
Channels00	0${\nu }_e\left({\overline{\nu }}_e\right)\to {\nu }_e\left({\overline{\nu }}_e\right)$, ${\nu }_{\mu }\left({\overline{\nu }}_{\mu }\right)\to {\nu }_{\mu }\left({\overline{\nu }}_{\mu }\right)$,
	${\nu }_e\left({\overline{\nu }}_e\right)\to {\nu }_{\mu }\left({\overline{\nu }}_{\mu }\right)$, ${\nu }_{\mu }\left({\overline{\nu }}_{\mu }\right)\to {\nu }_e\left({\overline{\nu }}_e\right)$
Energy resolution0	$12\%/E$
Runtime	${\mu }^{-}$ mode 5 yrs+ ${\mu }^{+}$ mode 5 yrs
Baseline	150 km
Energy range	100 MeV to 800 MeV
Normalization	appearance channels: $2.5\%$
(error on signal)	disappearance channels: 5$\%$
Sources of	Neutral current, Atmospheric neutrinos
Background	Charge misidentification

Equations

$$m_D=\left(\begin{array}{cc}{y}_a& y_s\\ \omega y_a& x{y}_s\\ {\omega }^2{y}_a& x{y}_s\end{array}\right).$$

$$m_N=\left(\begin{array}{cc}{M}_{\text{atm}}& 0\\ 0& M_{\text{sol}}\end{array}\right).$$

$$m_{\nu }=m_a\left(\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& 1& \omega \end{array}\right)+e^{i\eta }{m}_s\left(\begin{array}{ccc}1& x& x\\ x& x^2& x^2\\ x& x^2& x^2\end{array}\right),$$

$${\cos }^2{\theta }_{12}{\cos }^2{\theta }_{13}=\frac{3x^2}{5x^2+2x+2}.$$

$$\begin{array}[]{c}-5.475<x<-3.37,~{}0.455<\eta/\pi<1.545,\\ ~{}0.204<r<0.606,~{}3.343<m_{a}/\text{meV}<4.597.\end{array}$$

$${\chi }_{st.}^2(\overrightarrow{\mathcal{O}})=\sum _{i=1}^N{\left[\frac{{\mu }_i(\overrightarrow{\mathcal{O}})-n_i}{{\sigma }_i}\right]}^2,$$

$${\chi }_{st.}^2(\overrightarrow{\mathcal{M}})=\sum _{i=1}^N{\left[\frac{{\mu }_i(\overrightarrow{\mathcal{O}}(\overrightarrow{\mathcal{M}}))-n_i}{{\sigma }_i}\right]}^2,$$

$${\chi }_{sys.}^2(\overrightarrow{\mathcal{O}}\text{or}\overrightarrow{\mathcal{M}})=\underset{\{{\xi }_s,{\xi }_b\}}{\min }\sum _{i=1}^N{\left[\frac{{\mu }_i((\overrightarrow{\mathcal{O}}\text{or}\overrightarrow{\mathcal{O}}(\overrightarrow{\mathcal{M}});{\xi }_s,{\xi }_b)-n_i}{{\sigma }_i}\right]}^2+p({\xi }_s,{\sigma }_s)+p({\xi }_b,{\sigma }_b).$$

$${\mu }_i((\overrightarrow{\mathcal{O}}\text{or}\overrightarrow{\mathcal{O}}(\overrightarrow{\mathcal{M}});{\xi }_s,{\xi }_b)=(1+{\xi }_s)\times {\mu }_{s,i}+(1+{\xi }_b)\times {\mu }_{b,i},$$

$${\chi }^2(\overrightarrow{\mathcal{O}}\text{or}\overrightarrow{\mathcal{M}})=\underset{\overrightarrow{\mathcal{O}}\text{or}\overrightarrow{\mathcal{M}}}{\min }{\chi }_{sys.}^2(\overrightarrow{\mathcal{O}}\text{or}\overrightarrow{\mathcal{M}})+\sum _{i}p({\overrightarrow{\mathcal{O}}}_i(\overrightarrow{\mathcal{M}}),{\overrightarrow{\mathcal{O}}}_{cen.,i},{\overrightarrow{\sigma }}_i),$$

$${\chi }_{ex.}^2=\sum _i\underset{\overrightarrow{\mathcal{M}}}{\min }{\chi }^2({\mu }_i(\overrightarrow{\mathcal{M}}),n_i({\overrightarrow{\mathcal{O}}}_{true})).$$

$$\Delta {\chi }^2=\sum _i{\chi }^2({\mu }_i({\overrightarrow{\mathcal{M}}}_{hyp.}),n_i({\overrightarrow{\mathcal{M}}}_{true}))-\sum _i{\chi }^2({\mu }_i({\overrightarrow{\mathcal{M}}}_{b.f.}),n_i({\overrightarrow{\mathcal{M}}}_{true})),$$

$$\cos \delta =\frac{\cot 2{\theta }_{23}\left[3x^2-\left(4x^2+x+1\right){\cos }^2{\theta }_{13}\right]}{\sqrt{3}\left|x\right|\sin {\theta }_{13}\sqrt{\left(5x^2+2x+2\right){\cos }^2{\theta }_{13}-3x^2}},$$

$$\sin \delta =\pm \csc 2{\theta }_{23}\sqrt{1+\frac{{\left(x^2+x+1\right)}^2{\cot }^2{\theta }_{13}{\cos }^{2}2{\theta }_{23}}{3x^2\left[3x^2\tan 2{\theta }_{13}-2\left(x^2+x+1\right)\right]}},$$

$$\begin{array}[]{c}\cos\delta\propto\cot 2\theta_{23}=\frac{\cos 2\theta_{23}}{\sin 2\theta_{23}},\\ \sin\delta\propto\pm\csc 2\theta_{23}=\pm\frac{1}{\sin 2\theta_{23}}.\end{array}$$

$$\tan \delta \propto 1/\cos 2{\theta }_{23}.$$

Full text

Keywords: Neutrino Oscillations, Leptonic Flavour Symmetry

I Introduction

The discovery of neutrino oscillations points out the fact that neutrinos have mass, and provides evidence beyond the Standard Model (BSM). This phenomenon is successfully described by a theoretical framework with the help of three neutrino mixing angles ($\theta_{12}$, $\theta_{13}$, $\theta_{23}$), two mass-square splittings ($\Delta m_{21}^{2}$, $\Delta m_{31}^{2}$), and one Dirac CP phase ($\delta$) Pontecorvo (1968); Maki et al. (1962); Pontecorvo (1958); Esteban et al. (2019). Thanks to the great efforts in the past two decades, we almost have a complete understanding of such a neutrino oscillation framework. More data in the neutrino oscillation experiments is needed to determine the sign of $\Delta m_{31}^{2}$, to measure the value of $\sin\theta_{23}$, to discover the potential CP violation in the leptonic sector and even to constrain the size of $\delta$ Esteban et al. (2019). For these purposes, the on-going long baseline experiments (LBLs), such as the NuMI Off-axis $\nu_{e}$ Appearance experiment (NO$\nu$A) Ayres et al. (2007) and the Tokai-to-Kamioka experiment (T2K) Abe et al. (2011), can answer these questions with the statistical significance $\gtrsim 3\sigma$ in most of the parameter space. Based on the analysis with their data, the normal mass ordering ($\Delta m_{31}^{2}>0$), the higher $\theta_{23}$ octant ($\theta_{23}>45^{\circ}$), and $\delta\sim 270^{\circ}$ are preferred so far Esteban et al. (2019). The future LBLs, Deep Underground Neutrino Experiment (DUNE) Acciarri et al. (2015), Tokai to Hyper-Kamiokande (T2HK) Abe et al. (2014), and the medium baseline reactor experiment, the Jiangmen Underground Neutrino Observatory (JUNO) Djurcic et al. (2015); An et al. (2016) will further complete our knowledge of neutrino oscillations.

The MuOn-decay MEdium baseline NeuTrino beam experiment (MOMENT) has been proposed and is under consideration. Apart from superbeam neutrino experiments like DUNE or T2HK, it is planned to be at muon-decay accelerator neutrino experiments. In such experiments, neutrinos come from a three-body decay process, avoiding intrinsic electron-flavor neutrino contaminations in the reconstructed signals from the source. In addition, MOMENT Cao et al. (2014) is likely to use a Gd-doped water Cherenkov detector, which is capable of detecting multiple channels. MOMENT is understood to have excellent properties to study BSM physics, e.g. the invisible $\nu_{3}$ decay Tang et al. (2019a), NSIs Gavela et al. (2009); Bonnet et al. (2009); Krauss et al. (2011) and sterile neutrinos Gariazzo et al. (2017); Abazajian et al. (2012); Drewes et al. (2017); Minkowski (1977). Though the current studies on MOMENT have mainly focused on other BSM physics Tang and Zhang (2018); Tang et al. (2017), it is also necessary to perform physics study related to the standard neutrino oscillation to test the flavour symmetry models.

The symmetry of discrete groups, preserved at the high energy but slightly broken at the lower energy, predicts the neutrino mixing, mass-square splittings, and the CP violation phase (Dirac and Majorana phases), with reduced degrees of freedom (some of useful review articles are Altarelli and Feruglio (2010); Ishimori et al. (2010); King and Luhn (2013); King et al. (2014); King (2015a, b, 2017)). As a result, these models do not only simplify the theoretical framework for neutrino oscillations, but also provide a theoretical reason for this phenomenon. Many of these models can well describe the current neutrino-oscillation data. One of the most predictive models is the littlest seesaw model (LSS), which includes two massive right-handed neutrinos: one corresponds to the atmospheric-mass term, while the other is included for the solar-mass term King (2013, 2016); King and Luhn (2016). The littlest seesaw model in the tri-direct approach (TDLS) has been proposed and succeeds in describing the current global-fit results Ding et al. (2018, 2019a). In this model, four parameters $x$, $\eta$, $r$, $m_{a}$ are used to describe neutrino oscillations. This model has been studied with simulated data at NO$\nu$A, T2K, DUNE, T2HK and JUNO Ding et al. (2019b). In this work, we study how the next-generation neutrino project using muon-decay beams such as MOMENT can further extend our knowledge on the TDLS model.

This paper is arranged as follows. In Sec. II, we will introduce how TDLS models predict oscillation parameters, before presenting how this model describes the NuFit4.0 result. In Sec. III, we will introduce the statistics and simulation details used in this paper. We will show the definition of $\chi^{2}$, including the way that we implement “the pull method” to estimate the impact of systematic uncertainties, and how we include the current global-fit results by priors. We will then summarize the assumed configurations for the MOMENT experiment, and will show how the probabilities for MOMENT will be changed by varying each of model parameters. The simulation results will be shown in Sec. IV. We will present the model exclusion capability at MOMENT and how model parameters can be constrained by MOMENT data. We will discuss results of projecting the $3\sigma$ sphere from the model-parameter space to the standard-parameter space. Finally, we will close up this paper in Sec.V with our conclusions.

II Model review: littlest Seesaw in the Tri-Direct approach

The littlest seesaw model in the tri-direct approach is currently proposed, and succeeds in describing the current neutrino-oscillation data Ding et al. (2018). In this model, the atmospheric and solar flavon vacuum alignments are $\langle\phi_{\text{atm}}\rangle\propto\left(1,\omega^{2},\omega\right)^{T}$ and $\langle\phi_{\text{sol}}\rangle\propto\left(1,x,x\right)^{T}$, where $\omega=e^{2\pi i/3}$ stands for a cube root of unity and the parameter $x$ is real because of the imposed CP symmetry. As a result, the Dirac neutrino mass matrix reads as follows:

[TABLE]

The right-handed neutrino Majorana mass matrix is diagonal

[TABLE]

Under the littlest seesaw model, the light left-handed Majorana neutrino mass matrix is given by

[TABLE]

where $m_{a}=|y^{2}_{a}/M_{\text{atm}}|$, $m_{s}=|y^{2}_{s}/M_{\text{sol}}|$, and the only physically important phase $\eta$ depends on the relative phase between $y^{2}_{a}/M_{\text{atm}}$ and $y^{2}_{s}/M_{\text{sol}}$. Obviously, from Eq. (3), $m_{1}=0$ and the normal mass ordering are imposed by TDLS. We summarise the dependence of oscillation parameters on model parameters in Table 1. Ref. Ding et al. (2018) further predicts the sum rule for TDLS,

[TABLE]

We use the best fit value and the $3\sigma$ uncertainty of NuFit4.0 Esteban et al. (2019) (shown in Table 2), we find the best fit results for TDLS models in Table 3. The $3\sigma$ uncertainty is given as

[TABLE]

Notable between Tables 2 and 3 is that the most inconsistent oscillation parameters are $\theta_{23}$ and $\delta$. The others are placed within the $1\sigma$ error, or even at the best-fit value (e.g. $\Delta m_{21}^{2}$ and $\Delta m_{31}^{2}$). As a result, we are looking forward to improving precision measurements on $\theta_{23}$ and $\delta$ for further understanding of this model.

III Simulation details

III.1 Statistics Method

The statistical study on the TDLS model at MOMENT can be understood in Fig. 1. The model imposes correlations between or among the standard neutrino oscillation parameters, and predicts the oscillation spectra for MOMENT. In other words, the neutrino spectra of MOMENT can constrain the standard oscillation parameters, and therefore test the TDLS model or constrain the model parameters. Based on this perspective, we use two methods to conduct the numerical analysis with the simulated data:

• •

The standard three neutrino oscillations expressed by three mixing angles, two mass-square splittings and one Dirac-CP phase: $\overrightarrow{\mathcal{O}}=\{\theta_{12}\,,\theta_{13}\,,\theta_{23}\,,\delta_{\mathrm{CP}}\,,\Delta m_{21}^{2}\,,\Delta m_{31}^{2}\}$. We expect that precision measurements of mixing parameters are correlated with uncertainties of current global fit results. We suppose that a given experiment reconstructs neutrino spectra in $N$ bins sequentially. The number of observed events in the bin $i$ is recorded as $n_{i}$, which in our work is predicted by the true model. We can build a $\chi^{2}_{st.}(\overrightarrow{\mathcal{O}})$ to quantify the sensitivity:

[TABLE]

where $\mu_{i}$ is the number rate of bin $i$ predicted by the hypothesis $\overrightarrow{\mathcal{O}}$.

• •

We consider the following parameters from TDLS: $\overrightarrow{\mathcal{M}}=\{x\,,\eta\,,m_{a}\,,r\}$. Other steps in the likelihood analysis will follow the same strategy as the above method, but replace the equation Eq. (6) with

[TABLE]

with standard oscillation parameters as a function of model parameters $\overrightarrow{\mathcal{O}}(\overrightarrow{\mathcal{M}})$.

To describe the impact of systematic uncertainties, we adopt the following modification:

[TABLE]

where $p(\xi,\sigma)=\xi^{2}/\sigma^{2}$ is a Gaussian prior on the nuisance parameter $\xi$ with the uncertainty $\sigma$ (subscripts $s$ and $b$ denote signal and background respectively) and $\mu_{i}((\overrightarrow{\mathcal{O}}~{}\text{or}~{}\overrightarrow{\mathcal{O}}(\overrightarrow{\mathcal{M}});~{}\xi_{s},\xi_{b})$ is predicted event rate for bin $i$

[TABLE]

with the signal rate $\mu_{s,i}$ and the background rate $\mu_{b,i}$ for each energy bin $i$.

To include the currently constraints for the neutrino oscillation parameters, we finally use

[TABLE]

where $\sum_{i}p(\overrightarrow{\mathcal{O}}_{hyp.},\overrightarrow{\mathcal{O}}_{cen.},\overrightarrow{\sigma})$ is the summation of Gaussian priors over all oscillation parameters with two vectors: one includes all central values $\overrightarrow{\mathcal{O}}_{cen.}$ and the other consists of the standard deviation $\overrightarrow{\sigma}$. The values for $\overrightarrow{\mathcal{O}}_{cen.}$ and $\overrightarrow{\sigma}$ are taken from the best-fit value and according to $3\sigma$ uncertainties of the NuFit4.0 result Esteban et al. (2019) (shown in Table 2), respectively. In this work, the values of $\overrightarrow{\mathcal{O}}(\overrightarrow{\mathcal{M}})$ are predicted by the TDLS model.

III.2 Experiment Setting

We summarize the simulation details for MOMENT in Table 4. MOMENT, as a medium muon decay accelerator neutrino experiment, has been originally proposed as a future experiment to measure the leptonic CP-violating phase, though it also has good sensitivities on $\theta_{13}$, $\theta_{23}$ and $\Delta m^{2}_{31}$ Tang et al. (2019b).

The neutrino fluxes are kindly provided by the MOMENT working group Cao et al. (2014). The events are taken from $100$ to $800$ MeV. We assume five-year data taken at the $\mu^{-}$ and $\mu^{+}$ mode, respectively. Eight oscillation channels ($\nu_{e}\rightarrow\nu_{e}$, $\nu_{e}\rightarrow\nu_{\mu}$, $\nu_{\mu}\rightarrow\nu_{e}$, $\nu_{\mu}\rightarrow\nu_{\mu}$ and their CP-conjugate partners) are considered in this work. Multi-channel analyses are helpful in measuring the values of multiple parameters. As a result, the detector design is also crucial to precisely read out the events from different neutrino-oscillation channels. We have to consider flavour and charge identifications to distinguish secondary particles by means of an advanced neutrino detector — a 500 kton Gd-doped water cherenkov detector. The charged-current interactions are used to identify neutrino signals: $\nu_{e}+n\rightarrow p+e^{-}$, $\bar{\nu}_{\mu}+p\rightarrow n+\mu^{+}$, $\bar{\nu}_{e}+p\rightarrow n+e^{+}$, and $\nu_{\mu}+n\rightarrow p+\mu^{-}$, with the new technology using Gd-doped water to separate both Cherenkov and coincident signals from capture of thermal neutrons Campagne et al. (2007); Ishida (2013). The energy resolution is assumed $12\%/E$ for all channels. For the systematic uncertainties, we assume $\sigma_{s}=2.5\%$ for signal normalizations and $\sigma_{b}=5\%$ for background fluctuations.

The major background components come from the atmospheric neutrinos, neutral current backgrounds and charge mis-identifications. They can be largely suppressed with the beam direction and a proper modelling background spectra during the beam-off period, which are to be extensively studied in detector simulations. We consider matter effects during neutrino propagations with the help of the Preliminary Reference Earth Model (PREM) density profile is considered in the numerical calculations Dziewonski and Anderson (1981).

III.3 Neutrino oscillation probabilities in the TDLS model

In Figs. 2 and 3, we present the variation of probabilities for MOMENT with the $3\sigma$ uncertainty for model parameters in terms of NuFit4.0 results given in Eq. (5). We also show the probability with the best fit values as the input Table 3. In Fig. 2, we see the variation of $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ disappearance channels is much larger than those in the electron neutrino disappearance channels. As a result, $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ disappearance channels are two most dominating channels for the TDLS model. In the lower two panels, we see the variation of $x$ in the model has the largest impact, covering the range from [math] to $1$ for the probability within $0.1~{}\text{GeV}\leq E_{\nu}\leq 0.8~{}\text{GeV}$. The second largest effect comes from the model parameter $r$. It also ranges from [math] to $1$, yet the trend is different. For the higher energy ($E_{\nu}>0.45$ GeV), the lower bound of the probability is getting larger, and it is $\sim 0.45$ at $E_{\nu}=0.8$ GeV for both channels. For the model parameter $m_{a}$, the probability is changing with $\Delta P\sim 0.2$ along with the probability for the best fit value in Table 3. The similar feature is seen for the parameter $\eta$; yet the variation of probability is smaller $\Delta P\sim 0.05$. It seems that $\eta$ is the distinctive parameter not to be measured by $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ disappearance channels as easily as the other three model parameters. Eventually, we find that $\nu_{e}$ and $\bar{\nu}_{e}$ disappearance channels are more sensitive to the variation of $\eta$ than the other parameters, where $\Delta P$ can approach $\sim 0.1$ around the first minimum $E_{\nu}\sim 0.3$ GeV.

In Fig. 3, we show variations of $P(\nu_{\mu}\rightarrow\nu_{e})$, $P(\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e})$, $P(\nu_{e}\rightarrow\nu_{\mu})$, and $P(\bar{\nu}_{e}\rightarrow\bar{\nu}_{\mu})$. The behaviours in four panels are almost the same. The largest variation is given by the impact of $\eta$: $\Delta P\sim 0.06$ around the first maximum $E_{\nu}\sim 0.3$ GeV for all panels. The impact of model parameters $x$ and $r$ can reduce the lower bound significantly in the probability plane. From the first minimum to $8$ GeV, the lower bound of probability can even reach [math]. For both parameters, the variation of probability is around $\Delta P\sim 0.03$. The variation for $m_{a}$ is the smallest around $0.01$.

We observed that the lower limits reach [math] in a wide range of $E_{\nu}$ for most of channels, except $\nu_{e}$ and $\bar{\nu}_{e}$ disappearance ones. This happens when we varying the values of $x$ and $r$. The reason for this feature is that the oscillation minimum moves in wide range of $E_{\nu}$ with $x$ or $r$, as we see in Fig. 4, in which we use $P(\nu_{\mu}\rightarrow\nu_{e})$ as an example. We vary $x$ from $-5.5$ to $-3.5$ (left panel), and vary $r$ from $0.2$ to $0.6$ (right panel). The result demonstrates that the horizontal shift of the minimum makes the lower limit of the band to be [math] in a wide $E_{\nu}$ region.

To sum up, we see that $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ disappearance channels are the most important channels to constrain TDLS models, especially for $x$, $r$ and $m_{a}$. However, the other six channels can provide information for $\eta$. Thanks to the multiple channel features, MOMENT can be used to study TDLS models and can even measure model parameters precisely.

IV Results

In this section, we present physics potentials of MOMENT on the TDLS model. We firstly predict the exclusion limit for this model in different scenarios. We will see that $\theta_{23}$ and $\delta$ are key parameters to exclude TDLS models. Then, we study how MOMENT data can be used to constrain model parameters. We will see model-parameter degeneracies due to the poor measurement of $\theta_{12}$. We also project the $\Delta\chi^{2}$ to the standard neutrino mixing parameter space from the model parameter space. This shows an interesting correlation and demonstrate the goodness of fit in the analysis of simulated data.

IV.1 Model Exclusion

To give the model exclusion curves, we study the minimum of $\chi^{2}$ value for the TDLS with a given set of true values for the standard oscillation parameters (three mixing angles, two mass-square splittings, and a Dirac CP phase), and define the statistical quantity $\chi^{2}_{ex.}$ as follows:

[TABLE]

We adopt Wilk’s theorem Wilks (1938). When comparing nested models, the $\Delta\chi^{2}$ test statistics is a random variable asymptotically distributed according to the $\chi^{2}$-distribution with the number of degrees of freedom, which is equal to the difference in the number of free model parameters.

We present our result in Figs. 5 and LABEL:fig:SR_2D. In these figures, we vary true values for each one or two of standard oscillation parameters, while the other standard oscillation parameters are fixed at the TDLS predictions $(\theta_{12},~{}\theta_{13},~{}\theta_{23},~{}\delta,~{}\Delta m_{21}^{2},~{}\Delta m_{31}^{2})\sim(36.25^{\circ},~{}8.63^{\circ},~{}47^{\circ},~{}279^{\circ},~{}7.39\times 10^{-5}~{}\text{eV}^{2},~{}2.525\times 10^{-3}~{}\text{eV}^{2})$. As we do not see any impact on $\theta_{12}$ and $\Delta m_{21}^{2}$, we will simply ignore them in our discussion from now on.

In Fig. 5, we show the $\chi^{2}_{ex.}$ values against various true values for $\theta_{13}$ (upper-left), $\theta_{23}$ (upper-right), $\delta$ (lower-left), $\Delta m_{31}^{2}$ (lower-right). The range we show is given by the $3\sigma$ uncertainty in the NuFit4.0. Strikingly, we see very high exclusion levels for $\theta_{23}$ and $\delta$; for $\theta_{23}$ ($\delta$), $\chi^{2}_{ex.}$ can climb up to $\sim 160$ ($\sim 120$) at the upper bound, and reach $\sim 90$ ($\sim 180$) at the lower bound. For $\Delta m_{31}^{2}$, the exclusion level $\chi^{2}_{ex.}$ at both bounds is close to $8$. The worst one among these four parameters is $\theta_{13}$, and it cannot even reach $2\sigma$ exclusion level at the $3\sigma$ uncertainty of NuFit4.0.

In Fig. LABEL:fig:SR_2D, we show 2-dimension contours at $1\sigma$ (gray), $2\sigma$ (red), $3\sigma$ (green), $4\sigma$ (blue), and $5\sigma$ (magenta) on a combination of two parameters from $\theta_{13}$, $\theta_{23}$, $\delta$, and $\Delta m_{31}^{2}$. The range we show is the $3\sigma$ uncertainty in NuFit4.0. In all panels, the black dot denotes the best fit of NuFit4.0 results ($(\theta_{12},~{}\theta_{13},~{}\theta_{23},~{}\delta,~{}\Delta m_{21}^{2},~{}\Delta m_{31}^{2})=(33.82^{\circ},~{}8.61^{\circ},~{}49.6^{\circ},~{}215^{\circ},~{}7.39\times 10^{-5}~{}\text{eV}^{2},~{}2.525\times 10^{-3}~{}\text{eV}^{2})$), while the star is the prediction by the tri-direct littlest seesaw model with NuFit4.0 results ($(\theta_{12},~{}\theta_{13},~{}\theta_{23},~{}\delta,~{}\Delta m_{21}^{2},~{}\Delta m_{31}^{2})\sim(36.25^{\circ},~{}8.63^{\circ},~{}47.^{\circ},~{}279^{\circ},~{}7.39\times 10^{-5}~{}\text{eV}^{2},~{}2.525\times 10^{-3}~{}\text{eV}^{2})$). Though we do not see any correlations, we find that the black dot is outside of $5\sigma$ contour on the $\theta_{23}$-$\delta$ plane. This tells us that the measurement of $\theta_{23}$ and $\delta$ for MOMENT can exclude the TDLS over $5\sigma$ if NuFit4.0 results are confirmed.

IV.2 Model parameter constraint

We study how model parameters can be constrained by MOMENT. For this purpose, we study the statistics quantity,

[TABLE]

where $\overrightarrow{\mathcal{M}}_{hyp.}$ is the hypothesis, $\overrightarrow{\mathcal{M}}_{true}$ is the true values, and $\overrightarrow{\mathcal{M}}_{b.f.}$ is the best fit. Here $\overrightarrow{\mathcal{M}}_{b.f.}$ is exactly $\overrightarrow{\mathcal{M}}_{true}$. We show our result in Figs. 7 and LABEL:fig:model_2D. We set the true values at the $(x,~{}\eta,~{}r,~{}m_{a})=(-7/2,~{}\pi,~{}0.553,~{}3.72~{}\text{meV})$, which is the best fit with NuFit4.0 results. And the range for each panel is the $3\sigma$ uncertainty with NuFit4.0 results Eq. (5): $-5.475<x<-3.37$ (red band), $0.455<\eta/\pi<1.545$ (dark grey band),  $0.204<r<0.606$ (blue band), $3.343<m_{a}/\text{meV}<4.597$ (yellow band). At $3\sigma$ confidence level, the uncertainty of the model parameter $x$ lies roughly from $-4.25$ to $-3.5$. For the model parameter $\eta$, it ranges from $\sim 0.925\pi$ to $\sim 1.275\pi$. The $3\sigma$ errors for $r$ and $m_{a}$ are about $0.36<r<0.58$ and $3.5\text{meV}<m_{a}<3.85\text{meV}$. Compared to the result shown in Eq. (5), we see the parameter with the least improvement is $r$, for which the $3\sigma$ uncertainty is improved by a factor of $2$.

In Fig. LABEL:fig:model_2D, we show $1\sigma$ (gray), $2\sigma$ (light-orange) and $3\sigma$ (black) contours on the plane spanned by any two of model parameters. We see a strong correlation among $x$, $\eta$ and $r$, which is consistent with Eq. (3). In Eq. (3), we see these three parameters joint in the matrix for the neutrino solar mass. As a result these degeneracies can be resolved by precision measurement of solar mixing angle $\theta_{12}$ or solar mass-square splitting $\Delta m_{21}^{2}$. This degeneracy problem has also addressed by simulation results in other LBL experimental configurations, and is known to be resolved by the precision measurement of $\theta_{12}$ Ding et al. (2019b).

IV.3 Projection on the standard-parameter space

In Fig. 9, we project points inside the $3\sigma$ sphere from the 4-dimension model-parameter space on each oscillation parameters with their $\Delta\chi^{2}$ values (y-axis). Though MOMENT is not sensitive to $\theta_{12}$, we see that this parameter is well constrained to be better than that of NuFit4.0 result. The uncertainty for $\theta_{13}$ and $\Delta m_{21}^{2}$ are almost the same as the $3\sigma$ errors NuFit4.0. The asymmetry for $\theta_{12}$, $\theta_{23}$ and $\Delta m_{31}^{2}$ is passed by the same feature of $x$, $\eta$, and $m_{a}$.

In Fig. 10, we project the $3\sigma$ sphere from the 4-dimension model-parameter space to the two-dimension plane spanned by the standard oscillation parameters. We see that under the TDLS model, $\delta$ and $\Delta m_{31}^{2}$ are constrained better than those without assuming TDLS models by about a factor of $2$. The uncertainty for $\theta_{23}$ is slightly better when TDLS is assumed. The $3\sigma$ uncertainty for $\theta_{13}$ is roughly the same between with and without assuming TDLS models. The band feature in the $\theta_{23}$-$\delta$ panel can be understood by the expansions of $\cos\delta$ and $\sin\delta$ in Table 1:

[TABLE]

and

[TABLE]

with “$+$” for $x\cos\psi>0$ and “$-$” for $x\cos\psi<0$.

Considering $\theta_{23}\sim 45^{\circ}$, we have

[TABLE]

Therefore, we have

[TABLE]

Eq. (16) predicts that if $\theta_{23}=45^{\circ}$, $\delta=90^{\circ}$ or $270^{\circ}$, which is also confirmed in the $\theta_{23}$-$\delta$ panel of Fig. 10. On the other hand, due to the poor sensitivity to the solar angle of MOMENT, we do not see the result reflecting the sum rule Eq. (4).

V Conclusion

We have studied how we can extend our knowledge on the flavor symmetry with MOMENT, using eight channels of neutrino oscillations ($\nu_{e}\rightarrow\nu_{e}$, $\nu_{e}\rightarrow\nu_{\mu}$, $\nu_{\mu}\rightarrow\nu_{e}$, $\nu_{\mu}\rightarrow\nu_{\mu}$ and their CP-conjugate partners) with the help of the following detection processes in a Gd-doped water Cherenkov detector: $\nu_{e}+n\rightarrow p+e^{-}$, $\bar{\nu}_{\mu}+p\rightarrow n+\mu^{+}$, $\bar{\nu}_{e}+p\rightarrow n+e^{+}$, and $\nu_{\mu}+n\rightarrow p+\mu^{-}$. We have analyzed the physics potential of MOMENT on littlest seesaw models in the tri-direct approach given in Eq. (3) as a case study.

We have studied the exclusion ability to TDLS models for MOMENT. We found that $\theta_{23}$ and $\delta$ are the most important parameters to exclude this model, though some contributions from $\theta_{13}$ and $\Delta m_{31}^{2}$ are also seen. We noticed that the precision measurement in MOMENT of $\theta_{23}$ and $\delta$ can exclude this model with more than $5\sigma$ significance, if the best fit of NuFit4.0 is confirmed. We also presented the constraint on model parameters with simulated MOMENT data. We have found MOMENT data can improve the $3\sigma$ uncertainty by at least a factor of two, compared to those by NuFit4.0 results shown in Eq. (5). We have found the degeneracy problem, which is caused by the poor measurement of $\theta_{12}$. This degeneracy problem has been addressed in Ref. Ding et al. (2019b). We projected the $3\sigma$ sphere from the model-parameter space to the oscillation-parameter space. Finally, we have found that the sum rule between $\theta_{23}$ and $\delta$: $\tan\delta\propto 1/\cos 2\theta_{23}$ (for $\theta_{23}\sim 45^{\circ}$) predicted by Eqs. (13) and (14) can be checked by MOMENT.

Finally, we come to the conclusion that $\theta_{23}$ and $\delta$ are the most important parameters in the standard neutrino mixing framework to understand the underlying TDLS model. It is not only because they are the only two parameters, of which the model prediction deviates from the best fit of NuFit4.0 by more than $1\sigma$, but also because they can exclude this model at the $5\sigma$ confidence level as soon as the best fit values are confirmed in the future global analysis. As a result, to optimize the experimental design at MOMENT for the purpose of understanding the TDLS model, we need to aim at precision measurements of $\theta_{23}$ and $\delta$.

VI Acknowledgement

This work is supported in part by the National Natural Science Foundation of China under Grant No. 11505301 and No. 11881240247. We appreciate Gui-Jun Ding’s great help in understanding the tri-direct symmetry models. We would like to thank the accelerator working group of MOMENT for useful discussions and for kindly providing flux files for the MOMENT experiment. We finally acknowledge Dr. Sampsa Vihonen’s help to improve the readability of this paper.