Uncovering Hidden Leptonic Correlations with Flow Matching and Autoencoders

Haruto Kitagawa; Satsuki Nishimura; Hajime Otsuka

hep-phAug 15, 2026

Uncovering Hidden Leptonic Correlations with Flow Matching and Autoencoders

Authors: Haruto Kitagawa, Satsuki Nishimura, Hajime Otsuka

Organizations: Department of Physics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan · Department of Physics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan · Quantum and Spacetime Research Institute (QuaSR), Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

Abstract

We perform a global search for values of the Yukawa matrices and Majorana masses in the Type-I seesaw mechanism. Using flow matching, which is a generative artificial intelligence (generative AI) method, we generate a broad set of solutions reproducing the experimentally measured values of the neutrino mass-squared differences and the mixing angles. Then, a machine learning method known as an autoencoder is applied to uncover non-trivial correlations among physical quantities in the lepton sector. Our analysis reveals new non-linear relations involving neutrino masses and CP phases. These findings may contribute to elucidating the origins of the mass hierarchies and mixing patterns among generation structure.

Figures & tables

Observables	Normal Ordering (NO)		Inverted Ordering (IO)
Observables	best fit $\pm 1\sigma$	$3\sigma$ range	best fit $\pm 1\sigma$	$3\sigma$ range
$\sin^{2}\theta_{12}$	$0.308_{-0.011}^{+0.012}$	$0.275\rightarrow 0.345$	$0.308_{-0.011}^{+0.012}$	$0.275\rightarrow 0.345$
$\theta_{12}/\tcdegree$	$33.68_{-0.70}^{+0.73}$	$31.63\rightarrow 35.95$	$33.68_{-0.70}^{+0.73}$	$31.63\rightarrow 35.95$
$\sin^{2}\theta_{13}$	$0.02215_{-0.00058}^{+0.00056}$	$0.02030\rightarrow 0.02388$	$0.02231_{-0.00056}^{+0.00056}$	$0.02060\rightarrow 0.02409$
$\theta_{13}/\tcdegree$	$8.56_{-0.11}^{+0.11}$	$8.19\rightarrow 8.89$	$8.59_{-0.11}^{+0.11}$	$8.25\rightarrow 8.93$
$\sin^{2}\theta_{23}$	$0.470_{-0.013}^{+0.017}$	$0.435\rightarrow 0.585$	$0.550_{-0.015}^{+0.012}$	$0.440\rightarrow 0.584$

Table 1: The experimental results adopted in this work are taken from the NuFIT 6.0 global fit incorporating the Super-Kamiokande atmospheric-neutrino data [ 20 ] . For neutrino masses,

\Delta m_{3l}^{2}\equiv\Delta m_{31}^{2}=m^{2}_{3}-m^{2}_{1}>0

for NO and

\Delta m_{3l}^{2}\equiv\Delta m_{32}^{2}=m^{2}_{3}-m^{2}_{2}<0

for IO.

Panel of Figure 1 — Figure 1: Transformer architecture for vector field estimation.

Figure 2: Results generated using flow matching. The solid red line represents the best-fit value, and the dashed line represents the 3 $\sigma$ range.

Stage	Layer	Operation	Output dimension
Input	–	Input vector	$9$
	Encoder input	$\operatorname{Linear}(9,64)+\tanh$	$64$
	Down projection	$\operatorname{Linear}(64,32)+\tanh$	$32$
Encoder	Normalization	$\operatorname{LayerNorm}(32)$	$32$
	Residual blocks	$\operatorname{ResBlock}(32)\times 4$	$32$
	Latent projection	$\operatorname{Linear}(32,2)$	$2$

Table 2: Autoencoder architecture used to compress the observables into a two-dimensional latent space and reconstruct the original input vector.

Panel of Figure 3 — Figure 3: The left panel shows the visualization of the latent space, and the right panel shows the clustering result using HDBSCAN. Clusters 0, 1, 2, and 3 contain 105, 141, 177, and 217 points, respectively.

Figure 4: Distribution of observables in each cluster.

Cluster	$m_{1}\,[\mathrm{meV}]$	$m_{2}\,[\mathrm{meV}]$	$m_{3}\,[\mathrm{meV}]$	$\delta_{\mathrm{CP}}\,(\simeq\alpha_{31}/2)\,[\mathrm{rad}]$	$\alpha_{21}\,[\mathrm{rad}]$
0	$0.0144$ – $6.90$	$7.99$ – $10.8$	$48.5$ – $52.1$	$-0.473$ – $+1.92$	$-0.692$ – $+0.733$
1	$0.0133$ – $7.36$	$8.01$ – $11.4$	$48.7$ – $51.7$	$+1.43$ – $+3.14$	$-0.706$ – $+0.727$
2	$0.00147$ – $3.86$	$7.99$ – $9.70$	$48.4$ – $51.5$	$-1.14$ – $+1.17$	$-0.752$ – $+0.705$
3	$0.00559$ – $5.36$	$7.92$ – $10.3$	$48.7$ – $51.5$	$-3.13$ – $-0.642$	$-0.703$ – $+0.719$

Table 3: Range of observables in each cluster.

	Cluster 0	Cluster 1	Cluster 2	Cluster 3
$m_{1}$	0.955	0.978	0.793	0.958
$m_{2}$	0.612	0.697	0.144	0.493
$m_{3}$	0.335	0.697	0.476	0.707
$s_{12}$	0.070	0.017	0.014	0.035
$s_{13}$	0.041	0.024	0.043	0.043
$s_{23}$	0.038	0.030	0.066	0.042

Table 4: Coefficients of determination

R^{2}

for quadratic regressions of the physical observables in terms of the latent variables

(z_{1},z_{2})

, evaluated separately for each cluster. The bold entries indicate three representative results in each cluster.

Figure 5: Parallelogram approximations of the four cluster domains in the two-dimensional latent space for samples satisfying $\chi^{2}<45$ . The dashed lines indicate the linear fits used to determine the orientation of each domain.

Figure 6: The orange surfaces are obtained by mapping the parallelogram domains shown in Fig. 5 through the quadratic regressions of Eqs. ( 4.7 )–( 4.10 ), and are truncated to the data ranges shown in Fig. 4 . The black points denote the generated samples. The red plane corresponds to the adopted prior to obtain Eq. ( 2.11 ), which is the constraint for the NO case from CMB+BAO.

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The shape of quark flavors

Shinsuke Kawai, Nobuchika Okada

Faculty of Science, Yamagata University, 1-4-12 Kojirakawa-machi, Yamagata, 990-8560 Japan · Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama, AL35487 USA

High-Energy Physics

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We construct the Yukawa couplings of the quark sector as overlap integrals of Gaussian wave functions in extra spatial dimensions. Assuming that the wave function of each chiral fermion is localised at a point in the extra dimensions, while that of the Higgs doublet is flat, the Yukawa matrix elements are determined by the relative positions of the three left-handed quark doublets,

q_1

q_2

q_3

, and the six right-handed quark singlets,

u_1

u_2

u_3

d_1

d_2

d_3

. We employ numerical optimisation techniques from machine learning to identify field configurations that satisfy all current experimental constraints on the quark Yukawa sector. In this framework, the large hierarchy of quark masses arises naturally, since the overlap integrals of Gaussian wave functions are exponentially sensitive to

{\mathcal O}(1)

separations in the extra dimensions.

Abstract

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