KYUSHU-HET-366 KUNS-3112
Revisiting One-Zero and Two-Zero Neutrino Mass Textures in Light of Recent Oscillation and Cosmological Data
Haruto Kitagawa1 , Coh Miyao2 , Satsuki Nishimura3 , and Hajime Otsuka1,4
arXiv:2607.08384v1 [hep-ph] 9 Jul 2026
1
Department of Physics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan 2 Department of Physics and Astronomy, Faculty of Science and Technology, Tokyo University of Science, Yamazaki, Noda, Chiba 278-8510, Japan 3 Department of Physics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan 4 Quantum and Spacetime Research Institute (QuaSR), Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We revisit one-zero and two-zero textures of the neutrino mass matrix under current experimental and cosmological constraints. We identify the phenomenologically viable texture structures using the latest results on neutrino oscillation parameters, the cosmological bound on the sum of neutrino masses, the kinematic bound on the effective electron-neutrino mass, and limits from neutrinoless double-beta decay. For two-zero textures, several structures are still allowed if only the CMB bound on the neutrino mass sum is imposed. Among them, the B-series textures show a characteristic prediction for the Dirac CP phase, with δCP lying around π/2 and 3π/2, and are within the reach of future neutrinoless double-beta decay searches. When the stronger CMB+BAO constraint is included, however, only the A-series textures remain viable. Therefore, we also analyze one-zero textures by using machine learning techniques, particularly flow matching. It turns out that some of the texture structures are already excluded by current data, while the alP lowed ones give distinct predictions for i mi , meff νe , ⟨mee ⟩, and δCP . We further discuss how the one-zero texture structures can arise from non-invertible selection rules.
Contents 1 Introduction
1
2 Two-zero textures 2.1 Analytical method for two-zero textures 2.2 Cosmological constraints on neutrino masses 2.3 Constraints from kinematics of weak decay 2.4 Neutrinoless double-beta decay 2.5 Prediction from analysis
4 5 7 7 7 8
3 One-zero textures 3.1 Predictions from machine learning 3.1.1 Setup of flow matching 3.1.2 Results from flow matching 3.2 Analytical method for one-zero textures 3.3 Constraints on one-zero texture structures
11 11 12 14 20 21
4 Realization of one-zero neutrino mass textures 4.1 Non-invertible selection rules 4.2 One-zero textures H1,2,3 4.3 One-zero textures G1,2,3
24 24 25 26
5 Conclusions
27
A Two-zero textures
29
B Two-zero minors B.1 Analytical method for two-zero minors B.2 Results
32 32 34
C One-zero minors
40
1
Introduction
The Standard Model (SM) of particle physics, completed by the discovery of the Higgs boson in 2012 [1, 2], is a remarkably successful theory that describes most current experimental results. However, the SM cannot explain neutrino masses. Neutrino oscillations, which imply nonzero neutrino masses, therefore provide one of the clearest motivations for exploring physics beyond the Standard Model (BSM).
–1–
Experimentally, neutrino oscillation parameters have been measured with increasing precision by experiments such as SK [3], T2K [4], KamLAND [5–7], and JUNO [8]. These and other experimental results are combined in the NuFIT global analysis [9], which provides the experimentally favored values of the mixing angles θij (ij = 12, 13, 23), the CP phase δCP , and the neutrino mass-squared differences ∆m221 and ∆m23ℓ (ℓ = 1 for normal ordering and ℓ = 2 for inverted ordering), as summarized in Table 1. In the future, nextgeneration experiments such as HK [10] and DUNE [11] are expected to further improve the precision of neutrino oscillation measurements. From a theoretical perspective, these experimental results can be used to probe BSM physics. In the simplest low-energy effective theory, neutrino masses are described by the dimension-five Weinberg operator [12], cij (1.1) (Li H̃)(Lj H̃), Λ with H̃ = iσ 2 H ∗ . After the Higgs field H acquires a vacuum expectation value (VEV), this operator generates neutrino masses. In many cases, the structure originating from highenergy physics is encoded in the coefficients cij . Therefore, the structure of the neutrino mass matrix can provide insight into the underlying BSM physics. A well-known example is the study of two-zero textures and two-zero minors of the neutrino mass matrix [13]. A two-zero texture has two vanishing elements in the symmetric neutrino mass matrix, whereas a two-zero minor corresponds to two vanishing elements in its inverse. The two zeros yield two complex equations that lead to predictions for neutrino oscillation parameters and neutrino masses [13–24]. This framework enables a model-independent classification of viable structures consistent with current neutrino oscillation data. Indeed, such two-zero structures have been shown to arise in models with Table 1: Results of the NuFIT 6.0 global fit [9] for the neutrino oscillation parameters (IC24 with SK atmospheric data). Observables 2
sin θ12 θ12 /° 2
sin θ13 θ13 /° 2
sin θ23 θ23 /° δCP /π δCP /° ∆m221 2 10−5 eV ∆m23l 10−3 eV2
Normal Ordering (NO) best fit ±1σ
0.308+0.012 −0.011 33.68+0.73 −0.70 0.02215+0.00056 −0.00058 8.56+0.11 −0.11 0.470+0.017 −0.013 43.3+1.0 −0.8 +0.14 1.18−0.23 212+26 −41
3σ range 0.275 → 0.345 31.63 → 35.95
0.02030 → 0.02388 8.19 → 8.89
0.435 → 0.585 41.3 → 49.9 0.69 → 2.02 124 → 364
Inverted Ordering (IO) best fit ±1σ
0.308+0.012 −0.011 33.68+0.73 −0.70 0.02231+0.00056 −0.00056 8.59+0.11 −0.11 0.550+0.012 −0.015 47.9+0.7 −0.9 +0.12 1.52−0.14 274+22 −25
3σ range 0.275 → 0.345 31.63 → 35.95
0.02060 → 0.02409 8.25 → 8.93
0.440 → 0.584 41.5 → 49.8 1.12 → 1.86 201 → 335
7.49+0.19 −0.19
6.92 → 8.05
7.49+0.19 −0.19
6.92 → 8.05
+2.513+0.021 −0.019
+2.451 → +2.578
−2.484+0.020 −0.020
−2.547 → −2.421
–2–
a U(1)Lµ −Lτ gauge symmetry [25–27], indicating that they are well-motivated from the viewpoint of model building. The precision of neutrino oscillation data has now reached a level at which some of these structures are severely constrained. Indeed, Refs. [28, 29] reported that certain two-zero textures and minors are now strongly constrained by the experimental results. Following the analysis of Ref. [28], we revisit all two-zero textures in light of recent experimental data, including cosmological measurements from Planck, the Atacama Cosmology Telescope (ACT), and the Dark Energy Spectroscopic Instrument (DESI). Our comprehensive analysis shows that eight two-zero textures are already inconsistent with current experimental results, while the viability of the remaining textures depends on the neutrino mass ordering, normal (NO) or inverted (IO). This situation suggests that nature may favor less restrictive structures than two-zero textures. Motivated by this possibility, we also analyze one-zero textures and minors. A one-zero texture has a single vanishing element in the neutrino mass matrix, while a one-zero minor has a single vanishing element in its inverse. These structures are of particular interest because they can be realized both in models based on non-invertible selection rules [30, 31], which have recently received much attention, and in models with a U(1)Lµ −Lτ gauge symmetry. Although several studies of these structures have appeared in recent years [32–35], a comprehensive investigation of the current constraints on all one-zero textures and minors has not yet been performed. In this work, we aim to derive comprehensive constraints on one-zero texture and minor structures using a new analysis method. We also use recently developed machine-learning techniques to explore the parameter space and identify regions favored by experimental data. The resulting distributions reveal characteristic features of each one-zero structure and allow us to derive predictions for the corresponding model parameters. We further discuss how one-zero textures can be realized by non-invertible selection rules. The organization of this paper is as follows. In Sec. 2, we analyze all two-zero textures, following the approach of Refs. [28, 29]. The current constraints on two-zero textures are summarized in Table 3. In Sec. 3, we investigate one-zero textures. First we apply machine learning techniques to analyze the one-zero textures in Sec. 3.1.1. Specifically, we utilize flow matching, known as one of the generative artificial intelligence (generative AI) frameworks. We find that viable parameter regions exhibit distinct patterns in the neutrino observables for different one-zero textures, and it turns out that some one-zero textures are disfavored by current data, as summarized in Table 7. In addition, we analyze the onezero textures analytically and discuss the results from the machine learning techniques. In Sec. 4, we discuss a realization of one-zero neutrino mass textures based on non-invertible selection rules. Finally, Sec. 5 is devoted to the conclusion. In Appendix A, we present the distributions of δCP /π and Σ mi as functions of θ23 for the viable two-zero textures other than A1 and A2 . The results for two-zero and one-zero minors are summarized in Appendices B and C, respectively.
–3–
2
Two-zero textures
In this section, we consider neutrino mass matrices with two zero entries. These structures are classified as shown in Table 2. When the neutrino mass matrix exhibits one of these patterns, it is referred to as a twozero texture structure. On the other hand, when the inverse neutrino mass matrix has one of these patterns, it is called a two-zero minor structure. Details of the two-zero minor case are provided in Appendix B. By investigating such structures, we can obtain predictions for neutrino oscillation parameters that remain experimentally uncertain, following the methodology introduced in Ref. [13]. Therefore, we revisit this analytical approach and update the predictions using the latest global analysis results for neutrino oscillation data from NuFIT [9]. Table 2: Classification of two-zero textures. 00∗ A1 : 0 ∗ ∗ ∗∗∗ ∗∗0 B1 : ∗ 0 ∗ 0∗∗ ∗∗∗ C: ∗ 0 ∗ ∗∗0 ∗∗∗ D1 : ∗ 0 0 ∗0∗ 0∗∗ E1 : ∗ 0 ∗ ∗∗∗ ∗00 F1 : 0 ∗ ∗ 0∗∗
0∗0 A2 : ∗ ∗ ∗ 0∗∗ ∗0∗ B2 : 0 ∗ ∗ ∗∗0
∗0∗ B3 : 0 0 ∗ ∗∗∗
∗∗∗ D2 : ∗ ∗ 0 ∗00 0∗∗ E2 : ∗ ∗ ∗ ∗∗0 ∗0∗ F2 : 0 ∗ 0 ∗0∗
0∗∗ E3 : ∗ ∗ 0 ∗0∗ ∗∗0 F3 : ∗ ∗ 0 00∗
–4–
∗∗0 B4 : ∗ ∗ ∗ 0∗0
2.1
Analytical method for two-zero textures
First, we review the analysis method. Conventionally, the neutrino mass matrix in the flavor basis is diagonalized by using the PMNS matrix, which is parametrized as UPMNS c13 0 s13 e−iδCP 1 0 0 c12 s12 0 1 0 0 −s12 c12 0 0 e iα22 0 = 0 1 0 0 c23 s23 iα3 iδ 0 −s23 c23 −s13 e CP 0 0 0 1 0 0 e 2 c13 α3 α2 s13 ei( 2 −δCP ) c12 c13 c13 s12 ei 2 α3 α iδCP (c c − s s s eiδCP )ei 22 = c13 s23 ei 2 12 23 12 13 23 −c23 s12 − c12 s13 s23 e α3 α2 i i iδ iδ s12 s23 − c12 c23 s13 e CP (−c12 s23 − c23 s12 s13 e CP )e 2 c13 c23 e 2 α2 α3 V11 V12 ei 2 V13 ei 2 α α i 3 i 2 ≡ V21 V22 e 2 V23 e 2 . α2 α3 V31 V32 ei 2 V33 ei 2
(2.1)
(2.2)
Here, we define cos θij ≡ cij and sin θij ≡ sij . The parameters δCP , α2,3 and θij denote the Dirac CP phase, the Majorana phases, and the mixing angles, respectively. The quantity ∆m2ij ≡ m2i −m2j represents the neutrino mass-squared difference. Using this PMNS matrix, , is related to the diagonal mass matrix, the neutrino mass matrix in the flavor basis, Mflavor ν Mdiag , by following transformation: ν T Mdiag = UPMNS Mflavor UPMNS . ν ν
(2.3)
Two-zero texture Here, we investigate the two-zero texture case. From Eqs. (2.2) and (2.3), the components of the mass matrix in the flavor basis can be written as ∗ Mflavor = Vi1 Vj1 m1 + eiα2 Vi2 Vj2 m2 + eiα3 Vi3 Vj3 m3 . (2.4) ν ij
Assuming a two-zero texture, we impose ∗ Mflavor = 0, ν ij ∗ Mflavor = 0, ν ρσ
(2.5) (2.6)
with ij ̸= ρσ. Solving these complex equations for the Majorana phases, we find m1 −Vi3 Vj3 Vρ1 Vσ1 + Vi1 Vj1 Vρ3 Vσ3 m1 ≡ R2 (θ12 , θ13 , θ23 , δCP ), m2 Vi3 Vj3 Vρ2 Vσ2 − Vi2 Vj2 Vρ3 Vσ3 m2 m1 Vi2 Vj2 Vρ1 Vσ1 − Vi1 Vj1 Vρ2 Vσ2 m1 eiα3 = ≡ R3 (θ12 , θ13 , θ23 , δCP ). m3 Vi3 Vj3 Vρ2 Vσ2 − Vi2 Vj2 Vρ3 Vσ3 m3
eiα2 =
–5–
(2.7) (2.8)
The quantities R2,3 are functions of θ12 , θ13 , θ23 and δCP . Since the magnitudes of eiα2,3 are unity, we obtain the following relations: m2 = |R2 (θ12 , θ13 , θ23 , δCP )|, m1 m3 = |R3 (θ12 , θ13 , θ23 , δCP )|. m1
(2.9) (2.10)
For the normal ordering (NO), using these relations, we can rewrite the neutrino masssquared differences in terms of |R2,3 | as ∆m221 ≡ m22 − m21 = m21 (|R2 |2 − 1),
∆m231 ≡ m23 − m21 = m21 (|R3 |2 − 1).
(2.11) (2.12)
Solving these equations for m21 and combining them, we finally obtain the equation: ∆m221 ∆m231 = . |R2 |2 − 1 |R3 |2 − 1
(2.13)
This equation includes six parameters: θ12 , θ13 , θ23 , δCP , ∆m221 , ∆m231 . By fixing the four most precisely measured parameters, θ12 , θ13 , ∆m221 , ∆m231 , to the best-fit values in Table 1, we can determine δCP as a function of θ23 by solving Eq. (2.13). After obtaining the θ23 dependence of δCP , we can predict the individual neutrino masses as s s 2 ∆m21 ∆m231 m1 = = , (2.14) |R2 |2 − 1 |R3 |2 − 1 s ∆m221 |R2 |2 , (2.15) m2 = |R2 |2 − 1 s ∆m231 |R3 |2 m3 = . (2.16) |R3 |2 − 1 The sum of these masses is constrained by cosmological limits, as discussed in Sec. 2.2. Hence, we can classify the viable texture structures. Moreover, using these results, we can express the Majorana phases α2,3 in terms of θ23 . The same analysis can be applied to the inverted ordering (IO). In this case, the masssquared differences are written as ∆m221 ≡ m22 − m21 = m21 (|R2 |2 − 1),
(2.17)
∆m232 ≡ m23 − m22 = m21 (|R3 |2 − |R2 |2 ),
(2.18)
∆m221 ∆m232 = . |R2 |2 − 1 |R3 |2 − |R2 |2
(2.19)
and we obtain
–6–
By fixing θ12 , θ13 , ∆m221 , ∆m232 to the best-fit values in Table 1, we can obtain the predictions. Note that the mass formulas for IO are s s ∆m221 ∆m232 m1 = = , (2.20) |R2 |2 − 1 |R3 |2 − |R2 |2 s ∆m221 |R2 |2 m2 = , (2.21) |R2 |2 − 1 s (∆m232 + ∆m221 )|R3 |2 m3 = . (2.22) |R3 |2 − 1 2.2
Cosmological constraints on neutrino masses
Cosmological constraints on the sum of each neutrino masses have been reported by DESI [36]. Using only the Planck CMB and assuming flat ΛCDM, three degenerate neutrino P masses, and mν > 0, the sum of neutrino mass is constrained as X mν < 0.21 eV (95% C.L., CMB) . (2.23)
Including CMB lensing data from Planck and ACT together with the DESI Baryon Acoustic Oscillation (BAO) data, and assuming NO with the lower bound on the mass sum imposed, the constraint becomes X X mν < 0.113 eV (95% C.L., DESI BAO + CMB, mν > 0.059 eV) , (2.24) while for the IO, one finds X X mν < 0.145 eV (95% C.L., DESI BAO + CMB, mν > 0.10 eV) .
2.3
(2.25)
Constraints from kinematics of weak decay
The KATRIN experiment [37] sets a constraint on the effective mass of electron neutrino through measurements of 3 H beta decay. The current limit is sX eff 0.45 eV ≥ mνe ≡ m2i |Uei |2 . (2.26) i
2.4
Neutrinoless double-beta decay
If neutrinos are Majorana particles, neutrinoless double beta decay can occur. This process is characterized by the effective Majorana mass defined by ⟨mee ⟩ =
3 X i=1
–7–
2 mi Uei .
(2.27)
Neutrinoless double beta decay has not yet been observed despite ongoing experimental searches. Consequently, upper limits on the effective neutrino mass have been set by KamLAND-Zen [38] and GERDA [39] respectively as below; ⟨mee ⟩ < 36 − 156 meV,
(2.28)
⟨mee ⟩ < 79 − 180 meV.
(2.29)
In the future, the nEXO experiment [40] is expected to reach a sensitivity to the effective neutrino mass in the range: ⟨mee ⟩ > 4.7 − 20.3 meV. 2.5
(2.30)
Prediction from analysis
In this subsection, we present the results of the analysis for each texture structure. The results are summarized in Table 3, where ⃝ and × denote viable and non-viable structures, respectively. When applying the neutrino mass sum constraint in Eq. (2.23), we find that six structures (namely A1 , A2 , B1 , B2 , B3 , and B4 ) are viable for NO within the 3σ range of θ23 . For IO, three structures (B1 , B3 , and C) are viable within the 3σ range of θ23 . On the other hand, when the neutrino mass sum constraints in Eq. (2.24) and Eq. (2.25) are imposed, only two structures, A1 and A2 , remain viable for NO. Table 3: Summary of the two-zero texture analysis. A1
A2
B1
B2
B3
B4
C
⃝
⃝
⃝ ×
⃝
⃝
⃝ ×
×
IO
×
⃝
⃝
NO
⃝
⃝
×
×
×
×
×
×
×
D1
D2
E1
E2
E3
F1
F2
F3
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
Structure CMB
CMB+BAO
NO
IO
Structure CMB
CMB+BAO
NO
×
× ×
⃝ ×
IO
×
×
×
NO
×
×
×
IO
×
×
×
–8–
×
×
×
×
×
×
×
As phenomenologically viable examples, we present the results for the A1 and A2 textures in NO in Figs. 1 and 2. The left panel shows the θ23 dependence of δCP , where the green (yellow) band indicates the 1σ (3σ) range of δCP given in Table 1. The red line represents the prediction obtained from our analysis. The blue solid (dashed) line indicates the best-fit (1σ) value of θ23 from Table 1. The horizontal axis range corresponds to the 3σ range of θ23 given in Table 1. The right panel shows the θ23 dependence of the sum of neutrino masses, where the region below the horizontal blue dash-dotted line is allowed by the neutrino mass sum constraint in Eq. (2.23). The blue band represents the viable region under the neutrino mass sum constraints given in Eq. (2.24) and Eq. (2.25). When calculating the sum of neutrino masses, we use values of δCP within the 3σ range listed in Table 1, obtained by solving Eq. (2.13) and Eq. (2.19). The red dashed line indicates the prediction derived from the analysis. The blue solid and dashed lines respectively represent the best-fit value and 1σ range of θ23 given in Table 1. As in the left panel, the horizontal axis range corresponds to the 3σ range of θ23 from Table 1. Results for the other two-zero textures are summarized in Appendix A. Furthermore, the Majorana phases α2 , α3 are expressed in terms of θ23 by substituting oscillation parameters θ12 , θ13 , together with the predicted parameters δCP , m1 , m2 , m3 into Eqs. (2.7) and (2.8). Using these Majorana phases, the effective electron neutrino mass mνe and the effective neutrino mass ⟨mee ⟩ can be evaluated. Predictions for viable structures are summarized in Table. 4 for NO and Table 5 for IO. These tables show the allowed P regions of ⟨mee ⟩, meff νe , i mi and δCP /π for each viable structure. We find characteristic predictions for δCP . In particular, the B-series structures predict values round δCP ∼ 1.5π and δCP ∼ 0.5π. C structure predicts 1.3π ≲ δCP ≲ 1.6π and 0.4π ≲ δCP ≲ 0.6π. In addition, these structures will be tested by future experiments searching for neutrinoless double-beta decay such as nEXO. When the minimum value of the sum of neutrino masses is considered, they suggest that even the surviving two-zero texture structures are driven into narrow corners of the allowed parameter space by neutrino oscillation data. Thus, it is evident that these textures are now tightly constrained in light of improving experimental precision.
–9–
Table 4: Summary of the effective neutrino mass for neutrinoless double beta decay ⟨mee ⟩, P the effective electron neutrino mass meff νe , the sum of the predicted neutrino masses i mi , and the Dirac CP phase δCP /π for the viable two-zero textures in NO. P
Structure
⟨mee ⟩ [eV]
meff νe [eV]
A1 texture (NO)
∼0
0.065 − 0.068
A2 texture (NO)
∼0
∼ 0.021
0.065 − 0.068
0.0 − 0.5, 1.5 − 2.0
B1 texture (NO)
> 0.058
> 0.064
> 0.19
B2 texture (NO)
> 0.048
> 0.054
> 0.16
∼ 0.5, ∼ 1.5
B3 texture (NO)
> 0.061
> 0.067
> 0.20
B4 texture (NO)
> 0.051
> 0.057
> 0.17
∼ 0.021
i mi [eV]
δCP /π 0.6 − 1.4
∼ 0.5, ∼ 1.5 ∼ 0.5, ∼ 1.5 ∼ 0.5, ∼ 1.5
Table 5: Summary of the effective neutrino mass for the neutrinoless double beta decay ⟨mee ⟩, the effective electron neutrino mass meff νe , the sum of the predicted neutrino masses P i mi , and the Dirac CP phase δCP /π for the viable two-zero textures in IO. P
Structure
⟨mee ⟩ [eV]
meff νe [eV]
B1 texture (IO)
> 0.068
> 0.070
B3 texture (IO)
> 0.070
> 0.073
> 0.19
C texture (IO)
> 0.044
> 0.067
> 0.17
i mi [eV]
> 0.19
δCP /π
∼ 0.5, ∼ 1.5 ∼ 0.5, ∼ 1.5 0.4 − 0.6, 1.3 − 1.6
Figure 1: Distribution of observables for the A1 structure with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
– 10 –
Figure 2: Distribution of observables for the A2 structure with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
3
One-zero textures
In the previous section, we revisited the analysis of neutrino mass matrices with two-zero texture structures. However, as shown there, these structures are severely constrained. This motivates us to consider the next simplest class of structures, namely those with one-zero entry. Such structures are classified in Table 6. In this section, we discuss the general constraints on the neutrino mass matrix (or its inverse) with one-zero entry. We then examine the viable sets of neutrino parameters that satisfy these constraints using machine learning techniques. Table 6: Classification of one-zero structures.
0∗∗ G1 : ∗ ∗ ∗ ∗∗∗ ∗0∗ H1 : 0 ∗ ∗ ∗∗∗ 3.1
∗∗∗ G2 : ∗ 0 ∗ ∗∗∗ ∗∗0 H2 : ∗ ∗ ∗ 0∗∗
∗∗∗ G3 : ∗ ∗ ∗ ∗∗0 ∗∗∗ H3 : ∗ ∗ 0 ∗0∗
Predictions from machine learning
Recent applications of machine learning techniques to flavor physics (Refs. [41–52]) have demonstrated the effectiveness of such approaches. In particular, generative artificial intelli-
– 11 –
gence, i.e., generative AI, excels at producing new data based on a learned data distribution. For example, diffusion models have been employed to generate phenomenologically viable parameter points that satisfy experimental constraints, as shown in Refs. [46, 47]. Under these developments, we apply generative AI to generate data for the analysis of texture structures. Among various techniques, in this paper we adopt flow matching, which was proposed in Ref. [53] as a simulation-free framework for training continuous normalizing flows. Flow matching learns a vector field that transports samples from a simple source distribution, such as a Gaussian distribution, to the target data distribution. This vector field is conditioned on label information, enabling the generated samples to be guided toward desired physical observables. The implementation of flow matching consists of two phases: the training phase and the generation phase. During training, each data point G with its label L is connected to a randomly sampled point from the source distribution by a continuous path. The neural network is trained to learn the velocity field that moves points along these paths toward the data distribution. In the generation phase, a new point is first sampled from the noise distribution. This point is then transported by solving the learned flow equation under the desired label condition, producing a new sample that follows the learned data distribution. A useful feature of flow matching is the flexibility in choosing the probability path used for training. In particular, paths based on optimal transport displacement interpolation can produce straighter and simpler flows than standard diffusion-type paths. This can reduce the number of integration steps required during generation and improve sampling efficiency. Therefore, in this work, flow matching offers an efficient conditional generative method for producing data samples consistent with specified labels. 3.1.1
Setup of flow matching
In this subsection, we describe the flow matching setup used in our numerical analysis. We employ flow matching as a conditional generative sampler for the parameter space of texture structures. The physical input to the neural network is prepared as a pair consisting of the generation target G and the corresponding label L, where G specifies the independent parameters of the mass matrix, while L is calculated from the mass matrix. As a representative example, we explain the construction by using the one-zero texture G1 . Note that the same procedure applies to the other choices of the zero entry. For the G1 texture, the mass matrix is written as 0 αβ (3.1) Mflavor = Λν ν α γ δ , β δ ϵ where {α, β, γ, δ, ϵ} are complex parameters and Λν denotes the overall scale. We parameterize the latter by s = log10 (Λν /meV). Then, the generation target is defined as G = {Re α, Im α, . . . , Re ϵ, Im ϵ, s} .
– 12 –
(3.2)
Thus, G has 11 real components. In preparing the training data, the real and imaginary parts of {α, β, γ, δ, ϵ} are sampled in the range −1 ≤ {Re α, Im α, . . . , Re ϵ, Im ϵ} ≤ 1,
(3.3)
while the scale parameter is taken as −1 ≤ s ≤ 3. This range corresponds to 10−1 meV ≤ Λν ≤ 103 meV. For each sampled G, we construct Mflavor and diagonalize it by the PMNS matrix as in ν Eq. (2.3). Then, we calculate the label L, which is used as the condition in flow matching. The label L consists of two mass-squared differences and the absolute values of all PMNS matrix elements. For the normal ordering, we use LNO = log10 ∆m221 /meV2 , log10 ∆m231 /meV2 , Vij ,
(3.4)
with i = 1, 2, 3. On the other hand, for the inverted ordering, the second mass-squared difference is replaced by |∆m232 |: LIO = log10 ∆m221 /meV2 , log10 |∆m232 |/meV2 , Vij .
(3.5)
Therefore, L has 11 real components in this setup. The logarithmic preprocessing is applied only to the mass-squared differences, whereas the absolute values of the PMNS matrix elements are used directly. This approach improves training efficiency for quantities that span several orders of magnitude. The simulated dataset consists of pairs (G, L). This dataset is divided into training and validation sets with a ratio of 90% and 10%. For conditional flow matching, we use the simulation-based inference package sbi introduced in Ref. [54]. The flow-matching estimator is trained to learn the conditional distribution of G given a label L. Regarding the training details, we adopt a transformer architecture for the flow-matching estimator, with a hidden dimension of 100 and 5 layers. The conditional labels x = L are standardized using the z-score option “independent”, while no z-score normalization is applied to the generated variables θ = G. The estimator is trained by minimizing the flow-matching loss provided by sbi. We use the Adam optimizer with a learning rate of 5.0 × 10−4 and the OneCycleLR scheduler. The batch size is set to 256, and the network is trained for up to 5 × 104 optimization steps. The validation loss is evaluated every 1,000 steps, and early stopping with a patience of 20 is applied. After training, we fix the labels to the experimental values of the neutrino mass-squared differences and mixing parameters, as specified in Table 1. New candidates for G are then generated from the learned conditional flow. These generated points are mapped back to the neutrino mass matrix, and the corresponding observables are recalculated to verify whether they satisfy the required phenomenological constraints. The CP phases are not imposed as direct labels in this setup; instead, they are evaluated from the generated mass matrices after the sampling procedure.
– 13 –
The accuracy of the generated parameters is evaluated using the chi-squared value, defined as X Pi − µi 2 χ2 = , (3.6) σi i
where {Pi , µi , σi } denotes the model-predicted value of a physical observable, its central value and the corresponding 1σ deviation, respectively. In our analysis, this quantity is calculated using the following five observables: PNO = ∆m221 , ∆m231 , θ12 , θ23 , θ13 , (3.7) PIO = ∆m221 , |∆m232 |, θ12 , θ23 , θ13 .
If the trained network does not achieve the required accuracy, several approaches can be considered to improve its performance. Fine-tuning is one such technique for enhancing the accuracy of generated samples, and its effectiveness has been demonstrated in Refs. [46, 47]. To obtain a sufficient number of viable parameter sets, we also apply fine-tuning to the flow matching estimator. In the actual analysis, fine-tuning is required for G2 of NO and {G2 , G3 , H1 , H2 , H3 } of IO to achieve a sufficiently accurate distribution of viable solutions. For these structures, we prepare 1,000,000 samples using a flow matching model that has been trained once, followed by the fine-tuning procedure. Only G2 requires additional training in the case of NO, and the fine-tuning is conducted over 25 rounds. In the first round, we extract data with χ2 ≤ χ2max from the prepared 1,000,000 samples and retrain the neural network using this data. From the second round onward, 100,000 new samples are generated each round, and those satisfying χ2 < χ2max are added to the accumulated training set. At each round, the estimator is initialized with the weights of the parent estimator, followed by fine-tuning using all accepted samples obtained up to that point. The χ2 threshold is gradually tightened during the procedure: χ2max = 10,000 for rounds 1–10, χ2max = 5,000 for rounds 11–20, and χ2max = 1,000 for rounds 21–25. On the other hand, the criterion for each round is organized as follows in the case of IO. We adopt χ2max = 5,000 for the G3 texture, while the {G2 , H1 , H2 , H3 } textures are analyzed with χ2max = 10,000. Then, 3, 14, 3, 5, and 12 rounds are conducted for {G2 , G3 , H1 , H2 , H3 }, respectively. 3.1.2
Results from flow matching
In this section, we present the numerical results for one-zero texture structures for NO and IO, following the method described in the previous section. The results are summarized in Table 7. Compared to the two-zero texture case, one-zero textures have more degrees of freedom, and hence more structures remain viable. On the other hand, we find that some textures shown in Table 7 are disfavored even in the one-zero texture case.
– 14 –
Table 7: Summary of the one-zero texture analysis. Structure NO
CMB
CMB+BAO
G1
G2
G3
H1
H2
H3
⃝
×
×
⃝
⃝
⃝
×
⃝
⃝
×
IO
×
⃝
⃝
⃝
NO
⃝
×
×
⃝
IO
×
⃝
×
⃝
⃝
×
We plot the distributions of observables generated by flow matching in Figs. 3-5 for the {G1 , H1 , H2 } textures with NO, and in Figs. 6-10 for the {G2 , G3 , H1 , H2 , H3 } textures with IO. In all figures, the red solid and dashed lines represent the experimental best-fit values and 1σ ranges shown in Table 1, respectively. Furthermore, the gray shaded regions are excluded by the CMB constraint, while the blue shaded regions are allowed by the combined CMB+BAO constraints for NO and IO, given in Eqs. (2.24) and (2.25), respectively. For structures other than those listed here, flow matching does not yield parameter points consistent with the observational constraints. The predicted effective neutrino mass for neutrinoless double beta decay ⟨mee ⟩, the P effective electron neutrino mass meff νe , the sum of the predicted neutrino masses i mi , and the Dirac CP phase δCP /π are summarized in Table 8 for NO and Table 9 for IO. For IO, the viable one-zero textures lead to sizable values of ⟨mee ⟩, but their typical ranges are lower than those obtained in the two-zero texture case. A similar tendency is also seen in P the predicted sum of neutrino masses, i mi . Furthermore, H1 and H2 textures with IO prefer values of δCP around π/2 and 3π/2, which can be understood using the analytical methods introduced in the next section. Future cosmological observations and neutrino experiments will therefore be valuable not only for testing these predictions but also for discriminating between one-zero and two-zero texture scenarios. Table 8: Summary of the effective neutrino mass for neutrinoless double beta decay ⟨mee ⟩, P the effective electron neutrino mass meff νe , the sum of the predicted neutrino masses i mi , and the Dirac CP phase δCP /π for the viable one-zero textures with NO.
0.008 − 0.013
P
0.0590 − 0.0721
0.5 − 1.5
0.009 − 0.023
0.064 − 0.098
0.0 − 2.0
Structure
⟨mee ⟩ [eV]
meff νe [eV]
G1 texture (NO)
0.004 − 0.010
H1 texture (NO)
0.007 − 0.022
H2 texture (NO)
0.007 − 0.021
0.010 − 0.022
– 15 –
i mi [eV]
0.065 − 0.094
δCP /π
0.0 − 2.0
Table 9: Summary of the effective neutrino mass for the neutrinoless double beta decay ⟨mee ⟩, the effective electron neutrino mass meff νe , sum of the predicted neutrino masses P i mi , and the Dirac CP phase δCP /π for the viable one-zero textures with IO. P
Structure
⟨mee ⟩ [eV]
meff νe [eV]
G2 texture (IO)
0.047 − 0.053
0.111 − 0.135
G3 texture (IO)
0.052 − 0.061
0.049 − 0.054
H1 texture (IO)
0.047 − 0.059
0.047 − 0.059
0.0975 − 0.1541
H2 texture (IO)
0.047 − 0.063
0.048 − 0.063
0.0997 − 0.1668
H3 texture (IO)
0.049 − 0.056
0.051 − 0.058
0.1228 − 0.1467
i mi [eV]
0.054 − 0.063
0.136 − 0.166
δCP /π
0.0 − 2.0 0.0 − 2.0
0.42 − 0.58 1.4 − 1.6 0.44 − 0.56 1.4 − 1.6 0.0 − 2.0
One-zero texture G1 (NO) 0.010
⟨mee⟩ [eV]
0.012
meff νe [eV]
0.008
0.006
0.011 0.010 0.009
0.004 0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 2.0
0.20
1.5
δCP/π
0.25
0.15
1.0
X
mi [eV]
θ23 [rad]
0.10 0.05
0.5 0.0
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad]
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 1
Figure 3: Distribution of observables for the G1 structure with NO. In the upper row, the left (right) panel shows θ23 vs. ⟨mee ⟩ (meff νe ). In the lower row, the left (right) panel shows θ23 vs. Σ mi (δCP /π). These 1,165 points satisfy the condition χ2 < 45.
– 16 –
One-zero texture H1 (NO) 0.0225 0.020
meff νe [eV]
⟨mee⟩ [eV]
0.0200
0.015
0.0175 0.0150 0.0125
0.010
0.0100 0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 2.0
0.20
1.5
δCP/π
0.25
0.15
1.0
X
mi [eV]
θ23 [rad]
0.10 0.05
0.5 0.0
0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad]
θ23 [rad] 1
Figure 4: Distribution of observables for the H1 structure with NO. The figure layout is the same as in Fig. 3. These 413 points satisfy the condition χ2 < 45. One-zero texture H2 (NO) 0.0200
0.0200
meff νe [eV]
⟨mee⟩ [eV]
0.0175 0.0150 0.0125
0.0175 0.0150 0.0125
0.0100 0.0075
0.0100 0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 2.0
0.20
1.5
δCP/π
0.25
0.15
1.0
X
mi [eV]
θ23 [rad]
0.10 0.05
0.5 0.0
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad]
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 1
Figure 5: Distribution of observables for the H2 structure with NO. The figure layout is the same as in Fig. 3. These 534 points satisfy the condition χ2 < 45.
– 17 –
One-zero texture G2 (IO) 0.054
0.052
meff νe [eV]
⟨mee⟩ [eV]
0.053
0.050
0.048
0.052 0.051 0.050 0.049
0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 2.0
0.20
1.5
δCP/π
0.25
0.15
1.0
X
mi [eV]
θ23 [rad]
0.10 0.05
0.5 0.0
0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad]
θ23 [rad] 1
Figure 6: Distribution of observables for the G2 structure with IO. The figure layout is the same as in Fig. 3. These 401 points satisfy the condition χ2 < 45. One-zero texture G3 (IO) 0.062
meff νe [eV]
⟨mee⟩ [eV]
0.060 0.058 0.056
0.060 0.058 0.056
0.054
0.054
0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 2.0
0.20
1.5
δCP/π
0.25
0.15
1.0
X
mi [eV]
θ23 [rad]
0.10 0.05
0.5 0.0
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad]
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 1
Figure 7: Distribution of observables for the G3 structure with IO. The figure layout is the same as in Fig. 3. These 700 points satisfy the condition χ2 < 45.
– 18 –
One-zero texture H1 (IO) 0.0600 0.0575
⟨mee⟩ [eV]
0.0575
meff νe [eV]
0.0550 0.0525 0.0500
0.0550 0.0525 0.0500
0.0475
0.0475 0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 2.0
0.20
1.5
δCP/π
0.25
0.15
1.0
X
mi [eV]
θ23 [rad]
0.10 0.05
0.5 0.0
0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad]
θ23 [rad] 1
Figure 8: Distribution of observables for the H1 structure with IO. The figure layout is the same as in Fig. 3. These 589 points satisfy the condition χ2 < 45. One-zero texture H2 (IO) 0.060
meff νe [eV]
⟨mee⟩ [eV]
0.060
0.055
0.050
0.055
0.050 0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 2.0
0.20
1.5
δCP/π
0.25
0.15
1.0
X
mi [eV]
θ23 [rad]
0.10 0.05
0.5 0.0
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad]
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 1
Figure 9: Distribution of observables for the H2 structure with IO. The figure layout is the same as in Fig. 3. These 218 points satisfy the condition χ2 < 45.
– 19 –
One-zero texture H3 (IO) 0.058
0.056
0.056
meff νe [eV]
⟨mee⟩ [eV]
0.054 0.052 0.050
0.054 0.052
0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad] 2.0
0.20
1.5
δCP/π
0.25
0.15
1.0
X
mi [eV]
θ23 [rad]
0.10 0.05
0.5 0.0
0.74 0.76 0.78 0.80 0.82 0.84 0.86
0.74 0.76 0.78 0.80 0.82 0.84 0.86
θ23 [rad]
θ23 [rad] 1
Figure 10: Distribution of observables for the H3 structure with IO. The figure layout is the same as in Fig. 3. These 9,078 points satisfy the condition χ2 < 45. 3.2
Analytical method for one-zero textures
In this subsection, we introduce an analytical methodology used to verify the results obtained from flow matching. In the case of one-zero structures, we can use only a single equation: either one of Eqs. (2.4) is set to zero for the one-zero textures, or one of the corresponding minor conditions is set to zero for one-zero minors. However, this is not enough to obtain meaningful predictions due to the large number of unconstrained parameters. Even with the limited number of constraints, we can obtain the viable region by using the one-zero structure together with the cosmological constraints. We then consider how to constrain structures with one-zero components by using inequalities and the cosmological constraints on the sum of neutrino masses. First, we consider the texture case in which a (i, j) component is equal to zero: ∗ Mflavor = 0. (3.8) ν ij
From Eqs. (2.4), this complex equation can be written by −Vi1 Vj1 m1 = eiα2 Vi2 Vj2 m2 + eiα3 Vi3 Vj3 m3 .
(3.9)
Then, applying the triangle inequality to the right-hand side of this equation and taking the absolute value of the left-hand side, we obtain ||Vi2 Vj2 m2 | − |Vi3 Vj3 m3 || ≤ |Vi1 Vj1 m1 | ≤ ||Vi2 Vj2 m2 | + |Vi3 Vj3 m3 ||.
– 20 –
(3.10)
Here, we fix some parameters, θij (ij = 13 for G1 , and ij = 12, 13 for others), ∆m221 , ∆m23ℓ , to the best-fit value of the NuFIT 6.0 global fit [9], as in the analysis in Sec. 2. Since the parameter space remains too vast for an exhaustive analysis of the texture structures, we use the cosmological bounds discussed in Sec. 2.2. Fixing the sum of neutrino masses uniquely determines the individual active neutrino masses via the mass-squared differences ∆m2ij provided by the NuFIT 6.0 global fit [9]. This enables us to scan the parameter space restricted to the cosmologically viable region. Concretely, we explore the P parameter space by varying the mass sum mi in discrete steps within the viable region permitted by the cosmological constraints in Sec. 2.2. For each fixed value of the mass sum, the individual masses are sequentially evaluated using the ∆m2ij values provided by NuFIT 6.0 [9]. Once the individual masses are fixed, Eqs. (3.10) become an inequality involving θ23 (θ12 for G1 ) and δCP for each texture class. Then, by varying δCP in discrete steps over the range of [0, 2π], the inequality finally gives the viable region of θ23 (θ12 for G1 ) which is consistent with the constraints of the mass sum. Combining these results, we derive constraints on the one-zero textures, as shown in the next subsection. 3.3
Constraints on one-zero texture structures
The details of the results are shown in Fig. 11 and Fig. 12. The vertical axis represents δCP /π and the horizontal axis means θ23 (θ12 for G1 ). The blue solid (dashed, dashdotted) line shows the best-fit (1σ, 3σ) value of θ23 (θ12 for G1 ) in Table 1. The light red region denotes P the viable region consistent with the constraint Eq. (2.23) from CMB, mν < 0.21 eV, and the red region indicates the viable region consistent with the constraints Eqs. (2.24) P P and (2.25) from CMB+BAO, 0.059 eV < mν < 0.113 eV for NO and 0.10 eV < mν < 0.145 eV for IO. In the NO case, the G3 texture is not viable with respect to Eq. (2.24), but it is viable under Eq. (2.23). The H3 texture with NO is not viable in either mass range. On the other hand, the other texture structures in NO remain viable under the cosmological mass sum constraints. In the IO case, the G1 structure has no viable region; however, the other structures can be realized consistently with the cosmological limits. In particular, the H1 and H2 textures predict that δCP /π takes values around 0.5 and 1.5. Comparing these analytical results with those obtained from flow matching, we find that they are in good agreement. Indeed, the distribution of δCP for the H1 and H2 textures with IO is consistent between the two approaches.
– 21 –
Figure 11: Viable regions satisfying Eq. (3.10) for NO. The vertical axis shows δCP /π and the horizontal axis shows the θ23 (θ12 for G1 ). The blue solid, dashed and dashdotted lines show the best-fit value, 1σ range, 3σ range of θ23 in Table 1, respectively. The light red P region shows the viable region consistent with the constraint, mν < 0.21 eV, and the red P region shows the viable region consistent with the constraint, 0.059 eV < mν < 0.113 eV. – 22 –
Figure 12: Viable regions satisfying Eq. (3.10) for IO. The vertical axis shows δCP /π and the horizontal axis shows the θ23 (θ12 for G1 ). The blue solid, dashed and dashdotted lines show the best-fit value, 1σ range, 3σ range of θ23 in Table 1, respectively. The light red P region shows the viable region consistent with the constraint, mν < 0.21 eV, and the red P region shows the viable region consistent with the constraint, 0.10 eV < mν < 0.145 eV. – 23 –
4
Realization of one-zero neutrino mass textures
In this section, we propose a scenario to realize the one-zero textures of the neutrino mass matrix. Some neutrino masses discussed in the previous section cannot be realized based on group-theoretical symmetries. To illustrate this point, let us discuss whether one can realize the G1 texture based on a U (1) symmetric model in which the neutrinos carry U (1) charges qi with i = 1, 2, 3. Note that the top-left 2 × 2 submatrix of (mν )ij with i, j = 1, 2 has a specific structure in the G1 texture, and it can only be achieved if the following charge conservation conditions hold: q1 + q1 ̸= 0,
q2 + q1 = 0,
q2 + q2 = 0.
(4.1)
However, no viable charge assignments exist within U (1) symmetric models that can successfully reproduce the G1 texture. This constraint remains equally valid for non-Abelian symmetries. Here, we focus only on the neutrino mass matrix, but the same statement holds even when we consider the Weinberg operator generating the neutrino mass matrix. Furthermore, this statement applies to the other textures including G2,3 and H1,2,3 because they include the same submatrix discussed above. 4.1
Non-invertible selection rules
To realize these neutrino mass textures, we deal with non-invertible selection rules realized by Z2 gauging of ZN [30, 31]. Let g denote a generator of the ZN group, and let a Z2 outer automorphism r be defined via the relations: r2 = e ,
rg k r−1 = g −k ,
(4.2)
where e represents the group identity. Under this setup, the operator r implements the outer automorphism by mapping each group element g k to g −k , thereby allowing us to define the following equivalence classes: [g k ] = {hg k h−1 |h = e, r} = {g k , g −k } ,
(4.3)
where k ranges over 0, 1, ..., ⌊ N2 ⌋ with ⌊·⌋ being the standard floor function. These structures are identical to the Z2 -invariant conjugacy classes of the Dihedral group DN ∼ = ZN ⋊ Z2 . The physical motivation for considering such classes comes from string compactifications, particularly type IIB string theory compactified on toroidal orbifolds with magnetic fluxes [30].1 In the magnetized compactifications, the Kaluza-Klein reduction of a higher-dimensional Yang-Mills theory on a torus typically yields a ZN flavor symmetry within the four-dimensional massless sector [57–59]. The introduction of a Z2 orbifold projection, however, explicitly breaks this flavor symmetry, leaving the Z2 -even modes ϕk to be classified by the specific equivalence classes [g k ] from Eq. (4.3). These modes subsequently follow the fusion rule [30]: [g k1 ] · [g k2 ] = [g k1 +k2 ] + [g k1 −k2 ] . 1
For alternative realizations of non-invertible selection rules, see also Refs. [55, 56].
– 24 –
(4.4)
Since the fields are characterized by the classes rather than discrete individual group elements, the resulting selection rules depart significantly from conventional group-like symmetry patterns. More precisely, if g̃ k represents an arbitrary element within the class [g k ], a bare interaction term involving the 4D fields ϕk1 · · · ϕkn (where each field carries a class [g ki ]) is non-vanishing provided that at least one combination of representative elements g̃ ki ∈ [g ki ] can be found to satisfy g̃ k1 · · · g̃ kn = e .
(4.5)
This mechanism contrasts with standard group-theoretic frameworks, where every field possesses a unique charge and a coupling is allowed if and only if the net charge vanishes. Under the non-invertible selection rule governed by Eq. (4.5), a single field is associated with a collective set of group elements. Consequently, an interaction is allowed as long as the conservation condition holds for any single combination of elements drawn from their corresponding classes. It is worth noting that a field and its charge conjugate fall into the same class [g ki ]. Such non-invertible selection rules find a natural mathematical description within the framework of hypergroups and fusion algebras, which are well-established mathematical concepts known to be compatible with both quantum field theory and string theory (see, e.g., Ref. [60]). In addition, phenomenological applications have been explored for realizing non-trivial mass matrices in the Standard Model [61, 62], the minimal supersymmetric Standard Model [63] and grand unified theories [64–66]. 4.2
One-zero textures H1,2,3
Let us consider a case where the neutrino mass is generated from the Weinberg operator: cij (Li H̃)(Lj H̃) Λ
(4.6)
with i, j = 1, 2, 3, where Li and H̃ = iσ 2 H ∗ respectively denote the left-handed leptons and the Higgs field. In the case of Z2 gauging of ZN , both Li Li and HH include the identity class [g 0 ]. Since the selection rule associated with Z2 gauging of ZN always allow the diagonal entries of cij , one cannot realize the textures G1,2,3 . Hence, we discuss the realization of the other textures H1,2,3 . As discussed in Ref. [62], let us focus on Z2 gauging of Z5 which includes three distinct classes {[g 0 ], [g 1 ], [g 2 ]}. They obey the following commutative fusion rule: [g 0 ] · [g m ] = [g m ] ,
[g 1 ] · [g 1 ] = [g 0 ] + [g 2 ] ,
[g 1 ] · [g 2 ] = [g 1 ] + [g 2 ] .
[g 2 ] · [g 2 ] = [g 0 ] + [g 1 ] ,
(4.7)
with m = 0, 1, 2. Under the non-invertible selection rule, one cannot find the H1,2,3 textures within the Standard Model such that cij has one of the H1,2,3 textures and the Yukawa matrix of charged leptons is diagonal, i.e., Yij L̄i Hej ,
– 25 –
(4.8)
with Yij = diag(∗, ∗, ∗), where ∗ denotes the nonvanishing entry. Here, ej denotes the right-handed leptons. Similar things happen for the other cases such as Z2 gauging of Z3,4 . Hence, we consider type II non-supersymmetric two Higgs doublet models or minimal supersymmetric Standard Model, where Hu and Hd appear in the Weinberg operators and Yukawa couplings of charged leptons, respectively. Then, the H1,2,3 textures can be realized under the class assignments of matter fields shown in Table 10: Table 10: Class assignments of matter fields realizing the H1 , H2 and H3 textures.
H1
H2
H3
4.3
Li = Ei
Hu
Hd
{[g 0 ], [g 1 ], [g 2 ]}
[g 1 ]
[g 0 ]
{[g 0 ], [g 2 ], [g 1 ]}
[g 2 ]
[g 0 ]
{[g 1 ], [g 0 ], [g 2 ]}
[g 1 ]
[g 0 ]
{[g 2 ], [g 0 ], [g 1 ]}
[g 2 ]
[g 0 ]
{[g 0 ], [g 1 ], [g 2 ]}
[g 2 ]
[g 0 ]
{[g 0 ], [g 2 ], [g 1 ]}
[g 1 ]
[g 0 ]
{[g 1 ], [g 2 ], [g 0 ]}
[g 1 ]
[g 0 ]
{[g 2 ], [g 1 ], [g 0 ]}
[g 2 ]
[g 0 ]
{[g 1 ], [g 0 ], [g 2 ]}
[g 2 ]
[g 0 ]
{[g 1 ], [g 2 ], [g 0 ]}
[g 2 ]
[g 0 ]
{[g 2 ], [g 0 ], [g 1 ]}
[g 1 ]
[g 0 ]
{[g 2 ], [g 1 ], [g 0 ]}
[g 1 ]
[g 0 ]
One-zero textures G1,2,3
As mentioned before, one cannot realize the G1,2,3 textures under the non-invertible selection rule arising from Z2 gauging of ZN . Hence, we move to a different gauging scenario, specifically Z3 gauging of Z7 [56].2 When we denote a by the generator of Z7 , the Z3 automorphism b of the Z7 group acts as follows: b−1 ab = a2 ,
b3 = e,
a7 = e.
(4.9)
b2 ak b−2 = a16k ,
(4.10)
Since b acts on a generic element ak as bak b−1 = a4k , one can define the following classes: l
[ak ] ≡ {a2 k |l = 0, 2, 4} = {ak , a2k , a4k }. 2
For a realization of two-zero textures based on the Z3 gauging of Z13 and Z19 , see Ref. [67].
– 26 –
(4.11)
Specifically, for the Z3 gauging of Z7 , there are three distinct classes: C 0 ≡ [a0 ] = {e},
C 1 ≡ [a1 ] = [a2 ] = {a, a2 , a4 },
C 2 ≡ [a3 ] = {a3 , a5 , a6 },
(4.12)
which obey the following commutative fusion rules: C0 · Cm = Cm ,
C1 · C1 = C1 + C2 ,
C1 · C2 = C0 + C1 + C2 ,
C2 · C2 = C1 + C2 ,
(4.13)
with m = 0, 1, 2.3 It includes a S2 ∼ = Z2 permutation symmetry associated with C 1 ↔ C 2 . For the same reason as in the realization of the H1,2,3 textures, we consider type II nonsupersymmetric two Higgs doublet models or minimal supersymmetric Standard Model, where Hu and Hd appear in the Weinberg operators and Yukawa couplings of charged leptons, respectively. We find that the neutrino mass textures G1,2,3 in the diagonal basis of charged leptons can be realized under the class assignments of matter fields shown in Table 11: Table 11: Class assignments of matter fields realizing the G1 , G2 and G3 textures.
G1
Li
Ei
Hu
Hd
{C 0 , C 1 , C 2 }
{C 0 , C 2 , C 1 }
C1
C0
C2
C0
{C 0 , C 2 , C 1 }
{C 0 , C 2 , C 1 } {C 0 , C 1 , C 2 }
C1
C0
{C 0 , C 1 , C 2 }
C2
C0
{C 2 , C 0 , C 1 }
C1
C0
{C 2 , C 0 , C 1 }
C2
C0
{C 1 , C 0 , C 2 }
C1
C0
{C 1 , C 0 , C 2 }
C2
C0
{C 2 , C 1 , C 0 }
C1
C0
{C 2 , C 1 , C 0 }
C2
C0
{C 1 , C 2 , C 0 }
C1
C0
{C 1 , C 2 , C 0 }
C2
C0
{C 0 , C 1 , C 2 } {C 0 , C 2 , C 1 } {C 1 , C 0 , C 2 }
G2
{C 1 , C 0 , C 2 } {C 2 , C 0 , C 1 } {C 2 , C 0 , C 1 } {C 1 , C 2 , C 0 }
G3
{C 1 , C 2 , C 0 } {C 2 , C 1 , C 0 } {C 2 , C 1 , C 0 }
5
Conclusions
We have revisited one-zero and two-zero textures of the neutrino mass matrix using current experimental and cosmological constraints. We have examined which textures remain compatible with neutrino oscillation data, the cosmological bound on the sum of neutrino masses, the kinematic bound on the effective electron-neutrino mass, and limits from neutrinoless double-beta decay. 3
Here, we suppress the multiplicities for simplicity.
– 27 –
For two-zero textures, we updated the conventional analysis using recent oscillation data and cosmological bounds. If only the CMB bound on the sum of neutrino masses is imposed, several textures remain viable: A1 , A2 , and the B-series textures for NO, and B1 , B3 , and C for IO. The results are summarized in Table 3. The B-series textures are particularly interesting, since they prefer δCP around π/2 and 3π/2 and predict relatively large values of ⟨mee ⟩. Consequently, they are within the reach of future neutrinoless doublebeta decay searches. Once the stronger CMB+BAO bound is included, however, the allowed possibilities are significantly reduced, leaving only the A1 and A2 textures for NO. These textures will also predict specific values of δCP once the mixing angle θ23 is measured precisely, as shown in Figs. 1 and 2. We have also conducted a comprehensive analysis of one-zero textures. Due to their greater number of free parameters, one-zero textures are less constrained than two-zero textures; however, not all remain viable. Utilizing flow matching as a conditional generative AI, we explored the parameter space and identified the structures capable of reproducing the observed neutrino oscillation parameters while satisfying experimental and cosmological constraints. When imposing only the CMB bound on the sum of neutrino masses, several textures remain viable: the G1 , H1 , and H2 textures for NO, and G2 , G3 , and the H-series textures for IO. These results are summarized in Table 7. Remarkably, the viable IO textures predict relatively large values of ⟨mee ⟩, although these values are relatively smaller than those predicted by the two-zero textures for IO. The same tendency is observed for the P sum of neutrino masses i mi . These predictions can be tested through future improvements in cosmological observations and neutrino experiments, potentially distinguishing one-zero textures from two-zero textures. We have mainly focused on the texture of the neutrino mass matrix, but one can also analyze one-zero and two-zero minor structures. The analytical methods and results are summarized in Appendix B for two-zero minors and Appendix C for one-zero minors. As shown in Table 12, most two-zero minor structures are disfavored once the CMB+BAO bound is imposed, whereas one-zero minors are still allowed by current cosmological constraints. Finally, we have discussed how one-zero texture structures can arise from non-invertible selection rules. This provides a possible theoretical origin for texture zeros and connects the phenomenological classification to an underlying structure in the lepton sector. Future data will further test these possibilities and may clarify which texture structures are realized in the neutrino mass matrix.
Acknowledgments This work was supported in part by JSPS KAKENHI Grant Numbers JP26KJ0318 (C.M.), JP25KJ1927 (S.N.), JP25H01539 (H.O.) and JP26K07087 (H.O.).
– 28 –
A
Two-zero textures
In this appendix, we present the distributions of δCP /π and Σ mi as functions of θ23 for the viable two-zero textures other than A1 and A2 . The figure layout is the same as A1 and A2 shown in the main text. Normal ordering
Figure 13: Distribution of observables for the B1 structure with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
Figure 14: Distribution of observables for the B2 structure with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
– 29 –
Figure 15: Distribution of observables for the B3 structure with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
Figure 16: Distribution of observables for the B4 structure with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
– 30 –
Inverted ordering
Figure 17: Distribution of observables for the B1 structure with IO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
Figure 18: Distribution of observables for the B3 structure with IO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
– 31 –
Figure 19: Distribution of observables for the C structure with IO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
B
Two-zero minors
In this appendix, we first present the analytical method for two-zero minors in Sec. B.1. In Sec. B.2, we present the distributions of δCP /π and Σ mi as functions of θ23 for the viable two-zero minor structures listed in Table 12. B.1
Analytical method for two-zero minors
The analysis of two-zero minor structures is similar to that of two-zero texture structures, except that the zero conditions are imposed on the inverse neutrino mass matrix. The components of the inverse mass matrix are given by −1 V V Vi2 Vj2 Vi3 Vj3 i1 j1 Mflavor = + eiα2 + eiα3 . (B.1) ν m1 m2 m3 ij Imposing the two-zero minor conditions, −1 −1 flavor Mflavor = 0, M = 0, (B.2) ν ν ij
ρσ
with (i, j) ̸= (ρ, σ), we obtain
m2 −Vi3 Vj3 Vρ1 Vσ1 + Vi1 Vj1 Vρ3 Vσ3 m2 ≡ R2 (θ12 , θ13 , θ23 , δCP ), m1 Vi3 Vj3 Vρ2 Vσ2 − Vi2 Vj2 Vρ3 Vσ3 m1 m3 Vi2 Vj2 Vρ1 Vσ1 − Vi1 Vj1 Vρ2 Vσ2 m3 eiα3 = ≡ R3 (θ12 , θ13 , θ23 , δCP ). m1 Vi3 Vj3 Vρ2 Vσ2 − Vi2 Vj2 Vρ3 Vσ3 m1 eiα2 =
Since the magnitudes of eiα2 and eiα3 are unity, the mass ratios are given by m1 = |R2 (θ12 , θ13 , θ23 , δCP )|, m2 m1 = |R3 (θ12 , θ13 , θ23 , δCP )|. m3
– 32 –
(B.3) (B.4)
(B.5) (B.6)
For NO, the mass-squared differences can be written as 1 2 2 2 2 ∆m21 ≡ m2 − m1 = m1 −1 , |R2 |2 1 2 2 2 2 ∆m31 ≡ m3 − m1 = m1 −1 . |R3 |2
(B.7) (B.8)
Combining these equations, we obtain ∆m221 ∆m2 = 1 31 . 1 −1 −1 |R2 |2 |R3 |2
(B.9)
By fixing θ12 , θ13 , ∆m221 , ∆m231 to the best-fit values in Table 1, we can obtain predictions for the remaining neutrino parameters, including the individual neutrino masses: s s ∆m221 ∆m231 m1 = = , (B.10) 1 1 −1 −1 |R2 |2 |R3 |2 s ∆m221 , (B.11) m2 = 1 − |R2 |2 s ∆m231 m3 = . (B.12) 1 − |R3 |2 On the other hand, for IO, the mass-squared differences are given by 1 − 1), |R2 |2 1 1 ∆m232 ≡ m23 − m22 = m21 ( − ). 2 |R3 | |R2 |2 ∆m221 ≡ m22 − m21 = m21 (
(B.13) (B.14)
Then the equation can be written by ∆m221 ∆m232 . = 1 1 −1 − |R12 |2 |R2 |2 |R3 |2
(B.15)
As in the two-zero texture case, by fixing θ12 , θ13 , ∆m221 , ∆m232 to the best-fit values in Table 1, we obtain the predictions. The concrete form of the each neutrino mass for the two-zero minor structures in IO are given by s s ∆m221 ∆m232 m1 = = , (B.16) 1 1 −1 − |R12 |2 |R2 |2 |R3 |2 s ∆m221 m2 = , (B.17) 1 − |R2 |2 s ∆m232 + ∆m221 m3 = . (B.18) 1 − |R3 |2
– 33 –
Taking into account the neutrino mass sum constraint in Sec. 2.2, the results of the analysis for each minor structure are summarized in Table 12, where ⃝ and and × denote viable and non-viable structures, respectively. Table 12: Summary of the two-zero minor analysis. Structure CMB
CMB+BAO
NO
B2
B3
B4
C
×
×
⃝ ×
⃝
⃝
⃝
⃝
⃝
×
×
×
×
×
×
×
×
×
×
NO
×
×
×
⃝
×
×
×
×
D1
D2
E1
E2
E3
F1
F2
F3
⃝
×
×
×
×
×
×
×
×
IO
×
⃝
NO
⃝
⃝
×
×
×
×
×
×
NO
IO B.2
B1
×
Structure
CMB+BAO
A2
IO
IO
CMB
A1
×
× ×
× ×
× ×
×
×
× ×
× ×
Results
In this section, we present the analysis results for two-zero minor structures. Predictions for viable structures are summarized in Table 13 for NO and Table 14 for IO. These tables P show the allowed regions of ⟨mee ⟩, meff νe , i mi and δCP /π for each viable structure. We find characteristic predictions for δCP . In particular, the B-series structures predict values around δCP ∼ 1.5π and δCP ∼ 0.5π. C structure predicts 1.3π ≲ δCP ≲ 1.6π and 0.4π ≲ δCP ≲ 0.7π. Furthermore, we present the distributions of δCP /π and Σ mi as functions of θ23 for the viable two-zero minors, using the same figure layout as that employed for the two-zero textures.
– 34 –
Table 13: Summary of the effective neutrino mass for the neutrinoless double beta decay ⟨mee ⟩, the effective electron neutrino mass meff νe , the sum of the predicted neutrino masses P i mi , and the Dirac CP phase δCP /π for the viable two-zero minors with NO. P
Structure
⟨mee ⟩ [eV]
meff νe [eV]
B1 minor (NO)
> 0.048
> 0.054
B2 minor (NO)
> 0.058
> 0.064
> 0.19
B3 minor (NO)
> 0.051
> 0.057
> 0.17
B4 minor (NO)
> 0.061
> 0.067
> 0.20
C minor (NO)
> 0.029
> 0.049
> 0.15
0.4 − 0.7, 1.3 − 1.6
D1 minor (NO)
∼0
∼ 0.020
0.065 − 0.068
0.0 − 0.5, 1.5 − 2.0
D2 minor (NO)
∼0
∼ 0.021
i mi [eV]
> 0.16
0.065 − 0.068
δCP /π
∼ 0.5, ∼ 1.5 ∼ 0.5, ∼ 1.5 ∼ 0.5, ∼ 1.5 ∼ 0.5, ∼ 1.5
0.6 − 1.4
Table 14: Summary of the effective neutrino mass for the neutrinoless double beta decay ⟨mee ⟩, the effective electron neutrino mass meff νe , the sum of the predicted neutrino masses P m , and the Dirac CP phase δ /π for the viable two-zero minors with IO. CP i i Structure
⟨mee ⟩ [eV]
meff νe [eV]
B2 texture (IO)
> 0.068
> 0.071
B4 texture (IO)
> 0.070
> 0.073
P
i mi [eV]
> 0.19 > 0.19
δCP /π
∼ 0.5, ∼ 1.5 ∼ 0.5, ∼ 1.5
Normal ordering
Figure 20: Distribution of observables for the B1 minor with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
– 35 –
Figure 21: Distribution of observables for the B2 minor with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
Figure 22: Distribution of observables for the B3 minor with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
– 36 –
Figure 23: Distribution of observables for the B4 minor with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
Figure 24: Distribution of observables for the C minor with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
– 37 –
Figure 25: Distribution of observables for the D1 minor with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
Figure 26: Distribution of observables for the D2 minor with NO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
– 38 –
Inverted ordering
Figure 27: Distribution of observables for the B2 minor with IO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
Figure 28: Distribution of observables for the B4 minor with IO. The left and right panels show θ23 vs. δCP /π and θ23 vs. Σ mi , respectively.
– 39 –
C
One-zero minors
In this appendix, we present the analytical method for one-zero minors. The analysis of one-zero minor structures is similar to that of one-zero texture structures, except that the zero condition is imposed on the inverse neutrino mass matrix. Here, we consider the case in which the (i, j) component of the inverse mass matrix vanishes
Mflavor ν
−1 ij
= 0.
(C.1)
The analysis proceeds in the same way as in the one-zero texture case. From Eqs. (2.4), we obtain −
Vi2 Vj2 Vi3 Vj3 Vi1 Vj1 = eiα2 + eiα3 . m1 m2 m3
(C.2)
This leads to the following inequality: Vi2 Vj2 Vi3 Vj3 − m2 m3
≤
Vi1 Vj1 ≤ m1
Vi2 Vj2 Vi3 Vj3 + . m2 m3
(C.3)
From this inequality, we can obtain the viable region of θ23 for each value of δCP by fixing θ12 , θ13 , ∆m221 , ∆m23ℓ and by imposing the cosmological constraints on the sum of neutrino masses. Results We next present the analysis results for one-zero minor structures for NO and IO. The results are summarized in Table 15. We find that some structures are excluded even in the one-zero minor case: G1 for NO and G2 , G3 , H3 for IO. The characteristic results are shown in Fig. 29, where the figure layout is the same as Figs. 11 and 12. We find that H1 minor in the NO case predicts 0.25π ≲ δCP ≲ 1.75π and H2 minor in the NO case predicts 0 ≲ δCP ≲ 0.75π, 1.3π ≲ δCP ≲ 2.0π. We cannot find a peculiar pattern for the distribution of δCP for the other cases. Table 15: Summary of the one-zero minor analysis. Structure CMB
CMB+BAO
NO
G1
G2
G3
H1
H2
H3
×
⃝
⃝
⃝
⃝
⃝
⃝
⃝
⃝
⃝
⃝
IO
⃝
⃝
⃝
NO
×
⃝
⃝
IO
⃝
×
– 40 –
×
⃝
⃝
×
×
Figure 29: Viable regions in Eq. (C.3) for NO. The vertical axis shows δCP /π, and the horizontal axis shows θ23 , except for G1 , where θ12 is shown instead. The blue solid, dashed, and dash-dotted lines indicate the best-fit value, the 1σ range, and the 3σ range of θ23 given in Table 1. The light-red regions show the viable regions consistent with the P CMB constraint, mν < 0.21 eV, while the red regions indicate those consistent with the P CMB+BAO constraint, 0.059 eV < mν < 0.113 eV.
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