Study of a tri-direct littlest seesaw model at MOMENT

Jian Tang1∗ and Tse-Chun Wang1† School of Physics, Sun Yat-Sen University, Guangzhou 510275, China Abstract The tri-direct littlest seesaw model (TDLS) has been proposed, and succeeds in explaining the current global fit results with only four degrees of freedom. This model can be tested by the future experiments that improve the precision of neutrino mixing parameters, such as the MuOn-decay MEdium baseline NeuTrino beam experiment (MOMENT). In this work, we study how much MOMENT can extend our knowledge on the TDLS model. We find that measurements of θ23 and δ are crucial for MOMENT to exclude the model at more than 5σ confidence level, if the best fit values in the last global analysis result is confirmed. Moreover, the 3σ precision of model parameters can be improved at MOMENT by at least a factor of two. Finally, we project the surface at the 3σ confidence level from the model-parameter space to the oscillation-parameter space, and find the potential of MOMENT to observe the sum rule between θ23 and δ predicted by TDLS.


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 (θ 12 , θ 13 , θ 23 ), two mass-square splittings (∆m 2 21 , ∆m 2 31 ), and one Dirac CP phase (δ) [1][2][3][4]. 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 ∆m 2 31 , to measure the value of sin θ 23 , to discover the potential CP violation in the leptonic sector and even to constrain the size of δ [4]. For these purposes, the on-going long baseline experiments (LBLs), such as the NuMI Off-axis ν e Appearance experiment (NOνA) [5] and the Tokai-to-Kamioka experiment (T2K) [6], can answer these questions with the statistical significance 3σ in most of the parameter space. Based on the analysis with their data, the normal mass ordering (∆m 2 31 > 0), the higher θ 23 octant (θ 23 > 45 • ), and δ ∼ 270 • are preferred so far [4]. The future LBLs, Deep Underground Neutrino Experiment (DUNE) [7], Tokai to Hyper-Kamiokande (T2HK) [8], and the medium baseline reactor experiment, the Jiangmen Underground Neutrino Observatory (JUNO) [9,10] 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 electronflavor neutrino contaminations in the reconstructed oscillation signals from the source. In addition, MOMENT [11] is likely to use a Gd-doped water Cherenkov detector capable of detecting multiple channels. MOMENT is understood to have excellent properties to study BSM physics, e.g. the invisible ν 3 decay [12], NSIs [13][14][15] and sterile neutrinos [16][17][18][19].
Though the current studies on MOMENT have mainly focused on other BSM physics, it is also necessary for studying physics related to the standard neutrino oscillation, e.g. the flavour symmetry...etc.
The symmetry of discrete groups, preserved at the high energy but partly broken at the lower energy, predicts the neutrino mixing, mass-square splittings, and the CP violation phase, with reduced degrees of freedom (some of useful review articles are [20][21][22][23][24][25][26]). As a result, these models do not only simplify the theoretical framework for neutrino oscillations, but also provide a theoretical reason for this phenomenon. 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 solarmass term [27][28][29]. The littlest seesaw model in the tri-direct approach (TDLS) has been proposed and succeeds in describing the current global-fit results [30,31]. In this model, four parameters x, η, r, m a are used to describe neutrino oscillations. This model has been studied with simulated data at NOνA, T2K, DUNE, T2HK and JUNO [32]. 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 χ 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 3σ sphere from the modelparameter 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 [30]. In this model, the atmospheric and solar flavon vacuum alignments are φ atm ∝ (1, ω 2 , ω) T and φ sol ∝ (1, x, x) T , where ω = e 2πi/3 stands for a cube root of unity and the parameter x is real because of the imposed model parameters x, η, r, m a combinations of model parameters oscillation parameters  [30]. Two requirements are imposed by TDLS: the smallest mass state m 1 = 0 and the normal mass ordering. The sign of sin δ depends on the sign of x cos ψ: "+" ("−") is for x cos ψ > 0 (< 0).
CP symmetry. As a result, the Dirac neutrino mass matrix reads as follows: (1) The right-handed neutrino Majorana mass matrix is diagonal Under the littlest seesaw mechanism, the light left-handed Majorana neutrino mass matrix is given by where m a = |y 2 a /M atm |, m s = |y 2 s /M sol |, and the only physically important phase η depends on the relative phase between y 2 a /M atm and y 2 s /M 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 Tab. I. Ref. [30] further predicts the sum rule for TDLS, Parameter   We use the best fit value and the 3σ uncertainty of NuFit4.0 [4] (shown in Tab. II), we find the best fit results for TDLS models in Tab. III. The 3σ uncertainty is given as −5.475 < x < −3.37, 0.455 < η/π < 1.545, 0.204 < r < 0.606, 3.343 < m a /meV < 4.597.
Notable between Tabs. II and III is that the most inconsistent oscillation parameters are θ 23 and δ. The others are placed within the 1σ error, or even at the best-fit value (e.g. ∆m 2 21 and ∆m 2 31 ). As a result, we are now looking forward to see the improvement of the precision on θ 23 and δ helps us to further understand this model.

A. Statistics Method
The statistical study on the TDLS model at MOMENT can be understood in Fig. 1 The error propagation is implemented in the simulation code up to the spectra analysis.
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 masssquare splittings and one Dirac-CP phase: 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 to quantify the sensitivity: where µ i is the number rate of bin i predicted by the hypothesis.
• We consider the following parameters from TDLS: − → M = {x , η , 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 with the PMNS parameters as a function of model parameters To describe the impact of systematic uncertainties, we adopt the following modification: where p(ξ, σ) = ξ 2 /σ 2 is a Gaussian prior on the nuisance parameter ξ with the uncertainty σ (subscripts s and b denote signal and background respectively) and is predicted event rate for bin i with the signal rate µ s,i and the background rate µ b,i for each energy bin i.
To include the currently constraints for the neutrino oscillation parameters, we finally use

B. Experiment Setting
We summarize the simulation details for MOMENT in Table IV [33,34]. MOMENT, as a medium muon decay accelerator neutrino experiment, has been originally proposed as MOMENT Fiducial mass Gd-doping Water cherenkov(500 kton)  a future experiment to measure the leptonic CP-violating phase, though it also has good sensitivities on θ 13 , θ 23 and ∆m 2 31 [35]. The neutrino fluxes are kindly provided by the MOMENT working group [11]. The events are taken from 100 to 800 MeV. We assume 5-year data taken at the µ and µ + mode, respectively. Eight oscillation channels (ν e → ν e , ν e → ν µ , ν µ → ν e , ν µ → ν µ 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 oscillated events from different 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 chargedcurrent interactions are used to identify neutrino signals: ν e + n → p + e − ,ν µ + p → n + µ + , ν e + p → n + e + , and ν µ + n → p + µ − , with the new technology using Gd-doped water to separate both Cherenkov and coincident signals from capture of thermal neutrons [36,37].
The energy resolution is assumed 12%/E for all channels. For the systematic uncertainties, we assume σ s = 2.5% for signal normalizations and σ 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 [38]. while the lower left (right) panel is for P (ν µ → ν µ ) (P (ν µ →ν µ )).
In Figs. 2 and 3, we present the variation of probabilities for MOMENT with the 3σ 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 Tab. III. In Fig. 2, we see the variation of ν µ andν µ disappearance channels is much larger than those in the electron neutrino disappearance channels. As a result, ν µ andν µ 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 0 to 1 for the probability within 0.1 GeV ≤ E ν ≤ 0.8 GeV. The second largest effect comes from the model parameter r. It also ranges from 0 to 1, yet the trend is different. For the higher energy (E ν > 0.45 GeV), the lower bound of the probability is getting larger, and it is ∼ 0.45 at E ν = 0.8 GeV for both channels. For the model parameter m a , the probability is changing with ∆P ∼ 0.2 along with the probability for the best fit value in Tab. III. The similar feature is seen for the parameter η; yet the variation of probability is smaller ∆P ∼ 0.05. It seems that η is the distinctive parameter not to be measured by ν µ andν µ disappearance channels as easily as the other three model parameters. Eventually, we find that ν e andν e disappearance channels are more sensitive to the variation of η than the other parameters, where ∆P can approach ∼ 0.1 around the first minimum E ν ∼ 0.3 GeV.  65, 1.13π, 0.511, 3.71 meV). The upper left (right) panel is for P (ν µ → ν e ) (P (ν µ →ν e )), while the lower left (right) panel is for P (ν e → ν µ ) (P (ν e →ν µ )).
The behaviours in four panels are almost the same. The largest variation is given by the impact of η: ∆P ∼ 0.06 around the first maximum E ν ∼ 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 0.
For both parameters, the variation of probability is around ∆P ∼ 0.03. The variation for m a is the smallest around 0.01. We observed that the lower limits reach 0 in a wide range of E ν for most of channels, except ν e andν 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 ν with x or r, as we see in Fig. 4, in which we use P (ν µ → ν 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 0 in a wide E ν region.
To sum up, we see that ν µ andν µ 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 η. 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 θ 23 and δ 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 θ 12 . We also project the ∆χ 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.

A. Model Exclusion
To give the model exclusion curves, we study the minimum of χ 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 χ 2 ex.
as follows: We adopt Wilk's theorem [39]. When comparing nested models, the ∆χ 2 test statistics is a random variable asymptotically distributed according to the χ 2 -distribution with the number of degrees of freedom equal to the difference in the number of free model parameters.
We present our result in Figs ex. can climb up to ∼ 160 (∼ 120) at the upper bound, and reach ∼ 90 (∼ 180) at the lower bound. For ∆m 2 31 , the exclusion level χ 2 ex. at both bounds is close to 8. The worst one among these four parameters is θ 13 , and it cannot even reach 2σ exclusion level at the 3σ uncertainty of NuFit4.0. In Fig. 6, we show 2-dimension contours at 1σ (gray), 2σ (red), 3σ (green), 4σ (blue), and 5σ (magenta) on a combination of two parameters from θ 13 , θ 23 , δ, and ∆m 2 31 . The range we show is the 3σ uncertainty in NuFit4.0. In all panels, the black dot denotes the best fit of

B. Model parameter constraint
We study how model parameters can be constrained by MOMENT. For this purpose, we study the statistics quantity,   3.5meV < m a < 3.85meV. Compared to the result shown in Eq. (5), we see the parameter with the least improvement is r, for which the 3σ uncertainty is improved by a factor of 2.
In Fig. 8, we show 1σ (gray), 2σ (light-orange) and 3σ (black) contours on the plane spanned by any two of model parameters. We see a strong correlation among x, η 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 θ 12 or solar mass-square splitting ∆m 2 21 . 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 θ 12 [32].

C. Projection on the standard-parameter space
In Fig. 9, we project points inside the 3σ sphere from the 4-dimension model-parameter space on each oscillation parameters with their ∆χ 2 values (y-axis). Though MOMENT is not sensitive to θ 12 , we see that this parameter is well constrained to be better than that of NuFit4.0 result. The uncertainty for θ 13 and ∆m 2 21 are almost the same as the 3σ errors NuFit4.0. The asymmetry for θ 12 , θ 23 and ∆m 2 31 is passed by the same feature of x, η, and m a .
In Fig. 10, we project the 3σ 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, δ and ∆m 2 31 are constrained better than those without assuming TDLS models by about a factor of 2. The uncertainty for θ 23 is slightly better when TDLS is assumed.
We have studied the exclusion ability to TDLS models for MOMENT. We found that θ 23 and δ are the most important parameters to exclude this model, though some contributions from θ 13 and ∆m 2 31 are also seen. We noticed that the precision measurement in MOMENT of θ 23 and δ can exclude this model with more than 5σ 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 3σ uncertainty by at least a factor of 2, 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 θ 12 . This degeneracy problem has been addressed in Ref. [32]. We projected the 3σ sphere from the model-parameter space to the oscillation-parameter space. We see the sum rule between θ 23 and δ: tan δ ∝ 1/ cos 2θ 23 (for θ 23 ∼ 45 • ) predicted by Eqs. (13) and (14).
Finally, we come to the conclusion that θ 23 and δ 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 1σ, but also because they can exclude this model at the 5σ 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 θ 23 and δ.
This conclusion can be generalized for any flavour symmetry model that do not get along well with NuFit4.0 results for model parameters in the similar manner.

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.