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# Neutrino Mixing Matrix with SU(2)4 Anyon Braids

Michel Planat

► To cite this version:

Michel Planat. Neutrino Mixing Matrix with SU(2)4 Anyon Braids. Quantum Reports, 2025, 7 (3), pp.30 (9). 10.3390/quantum7030030. hal-05325870

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Submitted on 22 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
NEUTRINO MIXING MATRIX WITH SU(2)₄ ANYON BRAIDS

MICHEL PLANAT†

ABSTRACT. We recently classified baryonic matter in the ground and first excited states thanks to the discrete group of braids inherent to SU(2)$_2$ Ising anyons. Remarkably, the braids of SU(2)$_4$ anyons allow to generate the neutrino mixing matrix with an accuracy close to measurements. This is an improvement over the model based on tribimaximal neutrino mixing which predicts a vanishing solar neutrino angle $\theta_{13}$ which is now ruled out. The discrete group of braids for SU(2)$_4$ anyons is isomorphic to the small group (162, 14) generated by a diagonal matrix $\sigma_1 = R$ and a symmetric complex matrix $\sigma_2 = FRF^{-1}$, where the (3×3) matrices F and R correspond to the fusion and exchange of anyons, respectively. We make use of the Takagi decomposition $\sigma_2 = UDU^T$ of $\sigma_2$, where U is the expected PMNS unitary matrix and D is real and diagonal. We get agreement with the experimental results in about the 3$\sigma$ range for the complex entries of the PMNS matrix with the angles $\theta_{13} \sim 10^\circ$, $\theta_{12} \sim 30^\circ$, $\theta_{23} \sim 38^\circ$ and $\delta_{CP} \sim 240^\circ$. Potential physical consequences of our model are discussed.

Keywords: Neutrino mixing matrix; SU(2)$_4$ anyons; Takagi decomposition.

## 1. INTRODUCTION

The discovery of neutrino oscillations at the turn of the 21st century marked a major breakthrough in particle physics, definitively establishing that neutrinos are massive. As a result, a neutrino of one flavour converts into a different flavour, causing the number of each type of neutrino not to be conserved. Thus the notion of lepton flavour conservation does not hold in the neutral lepton sector. This phenomenon is conventionally described by a unitary matrix, now known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [1,2]. Analogous to the CKM matrix in the quark sector, the PMNS matrix encodes the mismatch between the mass and flavor eigenstates of neutrinos.

The PMNS matrix is crucial for understanding the structure of the Standard Model and potential physics beyond it. It provides three mixing angles ($\theta_{12}$, $\theta_{23}$, $\theta_{13}$) and a CP-violating phase $\delta_{CP}$, which are constrained by various neutrino oscillation experiments. While $\theta_{12}$ and $\theta_{23}$ were measured to be relatively large early on, the angle $\theta_{13}$ was long believed to be very small or zero, a belief embedded in the so-called tribimaximal (TBM) model of the PMNS matrix [3]
$$ (1) \qquad U_{\text{TBM}} = \begin{pmatrix} \frac{\sqrt{2}}{\sqrt{6}} & \frac{1}{\sqrt{3}} & 0 \\ -\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} \end{pmatrix}, $$

which predicts the mixing angles

$$ \theta_{12} = \arcsin\left(\frac{1}{\sqrt{3}}\right) \approx 35.3^{\circ}, \quad \theta_{23} = 45^{\circ}, \quad \theta_{13} = 0^{\circ}. $$

The TBM matrix was consistent with early oscillation data and is symmetric and aesthetically pleasing. As such, it motivated a variety of flavor symmetry models based on finite groups such as $A_4$ [4], $S_4$ [5], and $T'$ [6], where discrete non-Abelian symmetries were imposed on the lepton sector to enforce TBM mixing.

However, the discovery of a nonzero reactor angle $\theta_{13} \approx 8.6^{\circ}$ by the Daya Bay [7], RENO [8], and Double Chooz [9] experiments ruled out the TBM ansatz. This forced the community to either:

* introduce perturbations to the TBM form (e.g., charged lepton corrections), or

* search for alternative structures and underlying symmetries beyond traditional flavor groups.

Despite intense theoretical efforts, a compelling derivation of the PMNS matrix from first principles remains elusive. Models based on continuous and discrete flavor symmetries [10–13], extra dimensions [14, 15], grand unification [16, 17], and string theory [18, 19] have been proposed, each aiming to explain the observed pattern of neutrino mixing. Yet none have achieved a universally accepted explanation that naturally accommodates the experimental data, especially the large mixing angles and CP violation.

In this work, we propose a novel topological model where the PMNS matrix arises from the representation theory of the braid group associated to the modular tensor category (MTC) of type $SU(2)_4$, as explored by Freedman, Bauer, and Levaillant in the context of topological quantum computation [20, 21]. We show that the non-diagonal braid generator $\sigma_2$ of the group $D(9; 1; 1; 2; 1; 1)$—which is isomorphic to the small finite group (162, 14)—encodes the structure of the PMNS matrix through a Takagi factorization $\sigma_2 = FRF^{-1}$, where F and R correspond to the fusion and exchange of anyons, respectively. This approach naturally generates a non-zero $\theta_{13}$ and a CP phase $\delta_{CP}$ close to the current experimental central values, without invoking flavor symmetries ad hoc.

Our result opens a new direction in the quest to understand neutrino mixing, connecting deep mathematical structures such as modular tensor categories, braid group representations, and finite group theory to phenomenological observables in particle physics.

## 2. STANDARD PARAMETRIZATION OF THE PMNS MATRIX

The PMNS matrix $U_{\text{PMNS}}$ is a unitary $3 \times 3$ matrix that describes the mixing between neutrino flavor eigenstates ($\nu_e, \nu_\mu, \nu_\tau$) (of electron neutrino, muon neutrino and tau neutrino, respectively) and mass eigenstates ($\nu_1, \nu_2, \nu_3$).
Its entries $U_{\alpha i}$ are the amplitudes of mass eigenstates $i = 1, 2, 3$ in terms of flavors $\alpha = e, \mu, \tau$. It can be parametrized in terms of three Euler mixing angles $(\theta_{12}, \theta_{23}, \theta_{13})$ and a CP-violating phase $\delta_{CP}$ as follows:

$$ (2) \qquad U_{\text{PMNS}} = R_{23}(\theta_{23}; 0) R_{13}(\theta_{13}; \delta_{CP}) R_{23}(\theta_{12}; 0) P, $$

$R_{ij}(\theta; \phi)$ being a rotation around the $ij$-axis and $P$ containing Majorana phases, if any. As a result:

$$ (3) \qquad U_{\text{PMNS}} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{pmatrix} \begin{pmatrix} c_{13} & 0 & s_{13}e^{-i\delta_{CP}} \\ 0 & 1 & 0 \\ -s_{13}e^{i\delta_{CP}} & 0 & c_{13} \end{pmatrix} \begin{pmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{pmatrix}, $$

where $s_{ij} = \sin \theta_{ij}$ and $c_{ij} = \cos \theta_{ij}$ for $ij = 12, 13, 23$.

Multiplying the matrices yields the explicit form:

$$ (4) \qquad U_{\text{PMNS}} = \begin{pmatrix} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta_{CP}} \\ -s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i\delta_{CP}} & c_{12}c_{23} - s_{12}s_{23}s_{13}e^{i\delta_{CP}} & s_{23}c_{13} \\ s_{12}s_{23} - c_{12}c_{23}s_{13}e^{i\delta_{CP}} & -c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i\delta_{CP}} & c_{23}c_{13} \end{pmatrix}. $$

This matrix governs neutrino oscillation probabilities and is constrained by experimental data from solar, atmospheric, reactor, and accelerator neutrino experiments. The current best-fit values and the values in the ranges $3\sigma$ ans $1\sigma$ (from NuFIT.org [22]) are given in Table 1.

But the position of the Dirac phase in (2) is pure convention. Rewriting (2) in a more symmetrical form, PMNS matrix read [23, Equ. 5]

$$ (5) \qquad U_{\text{PMNS}} = R_{23}(\theta_{23}; \phi_{23}) R_{13}(\theta_{13}; \phi_{13}) R_{23}(\theta_{12}; \phi_{12}), $$

$$ (6) \qquad U_{\text{PMNS}} = \begin{pmatrix} c_{12}c_{13} & s_{12}c_{13}e^{-i\phi_{12}} & s_{13}e^{-i\phi_{13}} \\ -s_{12}c_{23}e^{i\phi_{12}} - c_{12}s_{23}s_{13}e^{-i(\phi_{23}-\phi_{13})} & c_{12}c_{23} - s_{12}s_{23}s_{13}e^{-i(\phi_{12}+\phi_{23}-\phi_{13})} & s_{23}c_{13}e^{-i\phi_{23}} \\ s_{12}s_{23}e^{i(\phi_{12}+\phi_{23})} - c_{12}c_{23}s_{13}e^{i\phi_{13}} & -c_{12}s_{23}e^{i\phi_{23}} - s_{12}c_{23}s_{13}e^{-i(\phi_{12}-\phi_{13})} & c_{23}c_{13} \end{pmatrix}. $$

In this form the CP phase reads

$$ (7) \qquad \delta_{CP} = \phi_{13} - \phi_{12} - \phi_{23}. $$

### 3. MODULAR TENSOR CATEGORIES AND ANYON MODELS

Modular tensor categories (MTCs) are rich algebraic structures that arise naturally in the mathematical formulation of topological quantum field theory (TQFT) and rational conformal field theory (RCFT). An MTC consists of a braided, balanced, and semisimple ribbon category with finitely many simple objects, endowed with fusion and braiding rules that satisfy the pentagon and hexagon identities. These structures provide a consistent framework for modeling non-Abelian anyons, which are quasiparticles exhibiting exotic exchange statistics in two spatial dimensions.
In the context of topological quantum computation (TQC), MTCs play a central role. They describe the topological degrees of freedom that can be manipulated by braiding anyons to implement quantum gates [24]. Each MTC defines a unitary representation of the braid group, with generators acting on the fusion spaces of anyons. The matrices associated with these representations—the *F*-symbols (for associativity of fusion) and *R*-symbols (for braiding)—encode the fundamental algebraic data of the theory.

Among the simplest yet physically relevant MTCs are those associated with the quantum group $SU(2)_k$, where $k$ is a positive integer known as the level. Recent work claimed the possible use of such MTCs in the context of explainable large language models [25]. The case $k=2$ corresponds to the Ising anyon theory, which supports Majorana fermions and has been extensively studied both theoretically and experimentally. The author recently found that braids of Ising anyons may be seen as corresponding to baryon families in their ground and first excited states [26]. In contrast, the case $k=4$ gives rise to a richer set of non-Abelian anyons with more intricate fusion and braiding properties [27–29].

Freedman, Bauer, and Levaillant investigated the computational power of the $SU(2)_4$ MTC and classified its finite image braid representations [20, 21]. The representation of the braid group on three anyons in the $SU(2)_4$ theory yields a finite group of type $D(9; 1; 1; 2; 1; 1)$ in the Conway–Atlas notation, which is isomorphic to the small group (162, 14). This group is a valid candidate for understanding CKM matrix for the mixing of quarks as well as PMNS matrix [30, Tables 3 & A1]. Group (162, 14) may also be generated by two braid matrices $\sigma_1 = R$ and $\sigma_2 = FRF^{-1}$, where $F$ and $R$ are the aforementioned fusion and braiding matrices [25, Sect. 3.6]. Interestingly, while $\sigma_1$ is diagonal, $\sigma_2$ is complex and symmetric, making it an ideal candidate for a physical observable such as the PMNS matrix.

In this work, we focus on the Takagi decomposition of $\sigma_2$ as a route to extracting the PMNS matrix. This approach connects the representation theory of braid groups, modular tensor categories, and particle physics through the language of topological phases and quantum symmetries. The accurate prediction of mixing angles and the CP phase from a topological origin suggests a deep relationship between neutrino phenomenology and low-dimensional quantum topology.

## 4. BRAIDING IN THE $SU(2)_4$ MODULAR TENSOR CATEGORY

The modular tensor category $SU(2)_k$ for integer level $k$ encodes the fusion and braiding properties of anyonic particles with spin labels $j = 0, \frac{1}{2}, 1, \dots, \frac{k}{2}$. For $k=4$, the simple objects of the category are labeled by $j = 0, \frac{1}{2}, 1, \frac{3}{2}, 2$, with fusion rules subject to the truncation $j_1 \otimes j_2 = |j_1 - j_2| \oplus \cdots \oplus \min(j_1 + j_2, k - j_1 - j_2)$. The fusion rules are associative but nontrivial due to the presence of a nontrivial *F*-symbol associator.

In Reference [20], an irreducible braid group representation $B_4 \to U_3$ of the 4-strand braid group $B_4$ on the 3-dimensional unitary group $U(3)$ is derived. It is obtained by braiding the four $SU(2)_4$ anyons of topological charge 2 on a fusion tree of total topological charge 0 following the Jones-Kauffman approach of Chern-Simons theory [31, 32].
Braiding matrices for the SU(2)₄ anyons are obtained as

$$ \sigma_1^{(4)} = \begin{pmatrix} \exp(\frac{7i\pi}{9}) & 0 & 0 \\ 0 & -\exp(\frac{4i\pi}{9}) & 0 \\ 0 & 0 & -\exp(\frac{7i\pi}{9}) \end{pmatrix}, $$

$$ \sigma_2^{(4)} = \begin{pmatrix} -\frac{1}{2} \exp(\frac{4i\pi}{9}) & \frac{1}{\sqrt{2}} \exp(\frac{7i\pi}{9}) & \frac{1}{\sqrt{2}} \exp(\frac{4i\pi}{9}) \\ \frac{1}{\sqrt{2}} \exp(\frac{7i\pi}{9}) & 0 & \frac{1}{\sqrt{2}} \exp(\frac{7i\pi}{9}) \\ \frac{1}{2} \exp(\frac{4i\pi}{9}) & \frac{1}{\sqrt{2}} \exp(\frac{7i\pi}{9}) & -\frac{1}{2} \exp(\frac{4i\pi}{9}) \end{pmatrix}. $$

The matrix $\sigma_2$ turns out to be symmetric but non-diagonal and complex. This makes it a candidate observable for unitary diagonalization. Our central claim is that the Takagi factorization of $\sigma_2$,

$$ \sigma_2 = UDU^T, $$

with D real diagonal and U unitary, yields a unitary matrix U that is close to the PMNS mixing matrix. Importantly, the phases and moduli of the entries of U obtained from this decomposition are numerically close to the experimentally measured values of neutrino mixing parameters, including a nonzero $\theta_{13} \sim 10^\circ$ and a sizable CP phase $\delta_{CP} \sim 240^\circ$.

This perspective is novel in that it does not require a postulated flavor symmetry group acting on the lepton families. Instead, the structure of mixing arises naturally from the braiding of anyons in a topologically ordered phase described by SU(2)₄ MTC, where the fusion channel corresponds to lepton generation entanglement.

In the next section, we provide the Takagi decomposition of $\sigma_2$, analyze the resulting mixing matrix U, and compare the resulting mixing angles and CP phase with current experimental constraints.

## 5. TAKAGI FACTORIZATION AND THE PMNS MATRIX

The Takagi factorization is a canonical decomposition of complex symmetric matrices. Given a complex symmetric matrix $A = A^T$, the Takagi factorization expresses A as

$$ (8) \qquad A = UDU^T, $$

where U is a unitary matrix and D is a real, non-negative diagonal matrix. This is analogous to diagonalizing a Hermitian matrix using a unitary transformation, but it applies to symmetric (not necessarily Hermitian) complex matrices.

In contrast to the standard eigenvalue decomposition, where a matrix $A$ is written as $A = V\Lambda V^{-1}$ with $\Lambda$ diagonal, the Takagi decomposition is unique up to diagonal phase ambiguities in $U$ and is always possible for complex symmetric matrices. It plays a key role in quantum information theory and the theory of complex normal modes.

In our context, the Takagi factorization is applied to the braid generator $\sigma_2 = \text{FRF}^{-1}$ of the SU(2)₄ modular tensor category. This matrix is complex and symmetric, and therefore admits a Takagi decomposition.
Under high-precision Takagi-Autonne decomposition [33], followed by the exchange of the first two columns (corresponding to the exchange of flavour states $\nu_e$ and $\nu_\mu$), we obtain

$$ (9) \qquad \sigma_2 = UDU^T, \quad D \sim I, $$

where $I$ is the identity matrix and $U$ approximates the PMNS mixing matrix. For the modulus we obtain

$$ |U|_{\text{braid}} \sim \begin{pmatrix} 0.852 & 0.491 & 0.176 \\ 0.359 & 0.706 & 0.609 \\ 0.278 & 0.509 & 0.773 \end{pmatrix}, $$

leading to the values of angles $\theta_{13} \sim 10^\circ$, $\theta_{12} \sim 30^\circ$, $\theta_{23} \sim 38^\circ$ that match the global-fit PMNS values within about $3\sigma$ [22].

To get a value of the CP phase we need to output the argument of the unitary matrix

$$ |\text{Arg}(U)|_{\text{braid}} \sim \begin{pmatrix} -44 & 60 & 28 \\ 171 & 77 & 166 \\ 117 & 65 & -37 \end{pmatrix}, $$

where the entries are given in degrees.

To approach the matrix (6) an extra shift angle $\sim 40^\circ$ may be introduced so that

$$ |\text{Arg}(U)|_{\text{braid}} \sim \begin{pmatrix} -4 & 100 & 68 \\ 211 & 117 & 206 \\ 157 & 105 & 3 \end{pmatrix}, $$

Up to the approximation $\pm 4^\circ$ we obtain the CP phase $\delta_{CP} = \phi_{13} - \phi_{12} - \phi_{23} \sim (-68 + 100 + 206)^\circ = 238^\circ$, which falls in the range of current experimental values [22]. Table 1 summarizes the results.

<table><caption>TABLE 1. Model predictions vs. NuFIT 6.0 (normal ordering). Ranges are $1\sigma$ and $3\sigma$ limits quoted in Ref. [22]. In the NuFIT global-fit plots the joint $(\theta_{23}, \delta_{CP})$ likelihood surface exhibits two almost-degenerate local minima, often called “islands.” (p) is for the primary island and (s) for the second island.</caption><thead><tr><th>Angles</th><th>This work</th><th>NuFIT best fit</th><th>NuFIT 1σ</th><th>NuFIT 3σ</th></tr></thead><tbody><tr><td>θ<sub>12</sub></td><td>30°</td><td>33.4°</td><td>32.7°–34.2°</td><td>31.0°–36.0°</td></tr><tr><td>θ<sub>13</sub></td><td>10°</td><td>8.57°</td><td>8.50°–8.74°</td><td>8.27°–8.95°</td></tr><tr><td>θ<sub>23</sub></td><td>38°</td><td>49.2°</td><td>48.2°–50.2°</td><td>41.0°–52.4°</td></tr><tr><td>δ<sub>CP</sub></td><td>240°</td><td>197° (p) / 259° (s)</td><td>173°–224° (p) / 247°–286° (s)</td><td>116°–345°</td></tr></tbody></table>

This Takagi outcome is sensitive to numerical precision. When computed at standard floating-point resolution without appropriate formatting, the same matrix yields incorrect angles resembling democratic mixing [34]. The accurate result only emerges with a Takagi routine with appropriate threshold.
The success of this minimal braid construction illustrates the nontrivial role of the SU(2)₄ anyonic braid structure in encoding realistic flavor physics, without relying on continuous symmetry assumptions. The matrix also reflects a robust encoding of CP violation via its complex eigenphases, linking categorical gauge to observable leptonic parameters.

## 6. DISCUSSION AND CONCLUSIONS

**Summary of results.** In this work we have shown that the sole non-diagonal braid generator $\sigma_2 = \text{FRF}^{-1}$ arising from the 4-anyon fusion channel of the SU(2)₄ modular tensor category encodes, through its Takagi factorization, a unitary matrix U that numerically reproduces the observed PMNS parameters within $\sim 3\sigma$:

$$ \theta_{13} \approx 10^{\circ}, \quad \theta_{12} \approx 30^{\circ}, \quad \theta_{23} \approx 38^{\circ}, \quad \delta_{CP} \approx 240^{\circ}. $$

This agreement is achieved without invoking extra flavour groups, Froggatt-Nielsen charges, continuous symmetries, or large parameter scans. Instead, it follows from the intrinsic topological data (F- and R-symbols) of a well-studied anyon theory.

**Physical interpretation.**

(1) *Topological origin of leptonic flavour.* The result suggests that lepton-generation mixing may originate from an underlying topological phase whose low-energy effective description is the SU(2)₄ MTC. In this picture, different neutrino flavours correspond to distinct fusion channels, while braiding operations realise basis changes between flavour and mass eigenstates.

(2) *Built-in CP violation.* The complex phases of $\sigma_2$ naturally induce a Dirac phase that emerges from the same braid data that fix the mixing angles.

(3) *Minimality.* Only two generators $\{\sigma_1, \sigma_2\}$ of the small group (162, 14) are required. No additional degrees of freedom beyond those already present in the SU(2)₄ category enter the construction.

**Phenomenological tests.** The framework yields several falsifiable consequences:

* *Predicted Majorana phases.* Although the Takagi decomposition fixes U only up to three diagonal phases, our scheme singles out a definite set via the eigenphases of $\sigma_2$. These Majorana phases can, in principle, be probed in next-generation neutrinoless double-beta decay experiments such as LEGEND [35] and nEXO [36].

* *Correlated angle shifts.* If future long-baseline facilities (DUNE [37], T2HK [38]) narrow the allowed region of $(\theta_{23}, \delta_{CP})$, the model predicts specific correlated shifts in $\theta_{12}$ and $\theta_{13}$, testable at JUNO [39] and IceCube Upgrade [40].

* *Absence of charged-lepton corrections.* Because the PMNS matrix is generated directly from a braid operator, charged-lepton rotations should be small. Observables sensitive to $U_{\nu\tau}$ therefore critically test the proposal.
## 6.1. Open questions.

(1) *Embedding into a full quantum field theory.* A concrete mechanism linking the anyonic sector to Standard-Model leptons remains to be constructed. Possible routes include effective 2D defects in 4D space-time or holographic duals of 3D TQFTs.

(2) *Quark-lepton unification.* The same finite group (162, 14) has appeared in attempts to model the CKM matrix [30]. Whether a single MTC or a larger braided product can generate both CKM and PMNS consistently is an enticing avenue.

(3) *Higher-category generalisations.* Extending the analysis to $SU(2)_k$ with $k > 4$ or to other rank-2 MTCs could reveal a systematic classification of flavour patterns in terms of braid statistics.

## 6.2. Concluding remarks.
Our findings point to an unexpected bridge between the mathematics of low-dimensional topology and the flavour structure of elementary particles. Should future data continue to converge on the parameter values predicted here, the case for a topological origin of neutrino mixing will strengthen considerably. Conversely, precise deviations would illuminate where additional dynamics, perhaps related to symmetry breaking, extra dimensions, or quantum gravity, must be incorporated. Either outcome promises to deepen our understanding of both neutrinos and topological quantum matter.

# APPENDIX: CHECK OF CONSISTENCY

An appendix is created to answer your comment that they may be a wrong numerical result. But it is not the case. It is not needed to add this appendix in the submission. It can be left private. It is enough to refer to the Python script I attach, the same than before but with more outputs. To illustrate these outputs and clarify the “black box” I give some details here. You can see that everything in my submission is consistent. I do not need to do any new change in my paper apart from adding the excellent reference [1] from Ludl et al. I hope you will agree after a further look based on the clarifications below.

The (complex symmetric) braiding matrix $\sigma_2$ may be expressed with numerical entries as

$$ \sigma_2 \sim \begin{pmatrix} -0.08682409 - 0.49240388j & -0.54167522 + 0.45451948j & 0.08682409 + 0.49240388j \\ -0.54167522 + 0.45451948j & 0 & -0.54167522 + 0.45451948j \\ 0.08682409 + 0.49240388j & -0.54167522 + 0.45451948j & -0.08682409 - 0.49240388j \end{pmatrix}, $$

After high precision Autunne-Takagi decomposition of $\sigma_2$ one obtains

$$ U_{\text{braid}} \sim \begin{pmatrix} 0.24533512 + 0.42583594j & 0.61016422 - 0.5959268j & 0.15523615 + 0.08334889j \\ 0.15310808 + 0.68995426j & -0.35477426 + 0.05858513j & -0.5921935 + 0.14328606j \\ 0.21599048 + 0.46080752j & -0.1725632 + 0.33689141j & 0.61656955 - 0.46644684j \end{pmatrix}. $$

It is straightforward to check that $D \sim U_{\text{braid}}^{\dagger}\sigma_2 U_{\text{braid}}^* \sim I$ and $\sigma_2 \sim U_{\text{braid}}U_{\text{braid}}^T$ (with $10^{-16}$ accuracy).

It follows that we get for the modulus
$$|U|_{\text{braid}} \sim \begin{pmatrix} 0.49145251 & 0.85289456 & 0.17619677 \\ 0.70673826 & 0.35957891 & 0.60928158 \\ 0.50891596 & 0.37851537 & 0.77313043 \end{pmatrix},$$

and for the argument (expressed in radians)

$$\operatorname{Arg}(U_{\text{braid}}) \sim \begin{pmatrix} 1.04811631 & -0.77359412 & 0.49274304 \\ 1.35242448 & 2.97793603 & 2.90419697 \\ 1.13248297 & 2.04417369 & -0.6476605 \end{pmatrix},$$

from which it can easily be checked that $U_{\text{braid}} \sim |U|_{\text{braid}} \exp[j\operatorname{Arg}(U_{\text{braid}})]$.

With exchange of the first two columns (corresponding to the exchange of flavour states $\nu_e$ and $\nu_\mu$), we obtain the values of Section 5 for the PMNS matrix.

If you check this consistency with the truncated values in Section 5, we roughly get your Equation (1) (where the exchange of the two first columns is still not applied)

$$U_{\text{trunc}} \sim \begin{pmatrix} 0.2451587 + 0.42541534j & 0.60987782 - 0.59493953j & 0.15512461 + 0.08314057j \\ 0.15324069 + 0.68916855j & -0.35414818 + 0.05882233j & -0.5918916 + 0.14333643j \\ 0.21582387 + 0.45987396j & -0.17226981 + 0.33646264j & 0.61677343 - 0.46596087j \end{pmatrix},$$

Checking the consistency of the product $D \sim U_{\text{trunc}}^\dagger \sigma_2 U_{\text{trunc}}^*$ we obtain

$$D_{\text{trunc}} \sim \begin{pmatrix} 0.997580077 + 0.00072886j & 0.000239137655 + 0.00013277j & 0.000311917143 + 0.00045076j \\ 0.000239263518 + 0.00013329j & 0.997667434 - 0.00057107j & 0.000162417719 + 0.00056213j \\ 0.000311865279 + 0.00045045j & 0.000163024722 + 0.00056199j & 0.999384486 - 0.00059656j \end{pmatrix},$$

which is still close to the identity matrix I and better than your calculation (3). You may have done a different truncation than me.

## ACKNOWLEDGMENTS

The author would like to acknowledge the contribution of the COST Action CA21169, supported by COST (European Cooperation in Science and Technology). He also acknowledges Raymond Aschheim for discussing the present topic at an earlier stage.

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† UNIVERSITÉ MARIE ET LOUIS PASTEUR, INSTITUT FEMTO-ST CNRS UMR 6174, 15 B AVENUE DES MONTBOUCONS, F-25044 BESANÇON, FRANCE.

Email address: michel.planat@femto-st.fr