⚛️ phenomenology

Taming lepton portal dark matter by a non-invertible selection rule

This paper proposes a model where a non-invertible selection rule, realized through the $\mathbb{Z}_2$ gauging of a $\mathbb{Z}_5$ symmetry, suppresses flavor-violating portal couplings in lepton portal dark matter to achieve one-flavor dominance while remaining consistent with the PMNS mixing matrix and offering testable predictions for lepton flavor non-universality in $Z$ boson decays.

Original authors: Junichiro Kawamura

Published 2026-09-01

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Junichiro Kawamura

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is filled with invisible matter that holds galaxies together, yet we cannot see it, touch it, or directly detect it. Scientists call this dark matter, and for decades, they have searched for it by looking for tiny, rare collisions between these invisible particles and the atoms in detectors buried deep underground. While many theories suggest dark matter should crash into ordinary matter frequently enough to be found, the most sensitive experiments have come up empty-handed. This silence has forced physicists to rethink their ideas, leading them to consider a more elusive candidate: a type of dark matter that interacts almost exclusively with the lightest particles in the atom, the electrons and their heavier cousins, the muons and taus, while ignoring the protons and neutrons that make up our bodies and the Earth. This scenario, known as lepton portal dark matter, offers a clever way to hide from current detectors, but it introduces a new, stubborn problem: why does this dark matter seem to pick only one type of lepton to talk to, when the laws of physics suggest it should be able to talk to all of them equally?

A recent study by Junichiro Kawamura addresses this puzzle by proposing a new kind of rule that governs how particles interact. In the standard view of particle physics, the universe follows strict symmetry rules, much like a lock that only opens with a specific key. These rules usually prevent dark matter from causing chaos by interacting with the wrong particles. However, when scientists tried to apply these standard rules to the lepton portal model, they found a contradiction. The model requires dark matter to interact with only one specific lepton flavor to avoid detection, but the same rules that allow the dark matter to exist also force the lepton flavors to mix together in a way that would cause the dark matter to interact with all of them, creating a mess of forbidden signals that experiments should have already seen. The standard rules simply cannot have it both ways; they cannot allow the dark matter to hide while also allowing the known mixing of particles that we observe in nature.

Kawamura's solution involves stepping outside the familiar world of standard symmetries and into a more abstract mathematical landscape known as non-invertible selection rules. Imagine a set of instructions for a game where, instead of every move having a clear "undo" button that returns you to the exact previous state, some moves are permanent or transform the game board in a way that cannot be simply reversed. In this new framework, particles are not assigned simple labels or charges that can be added and subtracted like numbers. Instead, they are grouped into families based on how they transform under these complex rules. The researcher constructed a specific model using a mathematical structure based on the number five, where the dark matter and the heavy partner particles it interacts with are assigned to specific families. In this setup, the rules naturally forbid the dark matter from interacting with two of the three lepton flavors, while still allowing the necessary mixing between all three flavors to happen in the neutrino sector, which is where the mixing we observe in nature actually takes place.

The result is a model where the dark matter is "tamed." It interacts strongly enough with a single type of lepton to create the correct amount of dark matter in the universe through a process called thermal freeze-out, where particles in the early universe cooled down and stopped annihilating each other. At the same time, the new rules prevent the dark matter from leaking into the other lepton flavors, which would have triggered alarms in experiments looking for rare decays. The study shows that this approach works without needing to invent artificial, ad-hoc fixes or extra fields that feel forced. It provides a consistent picture where the dark matter remains hidden from direct detection experiments, yet the universe retains the complex flavor structure we see in the particles around us.

The paper also looks ahead to how this idea could be tested. Because the dark matter interacts with leptons, it leaves subtle fingerprints on the behavior of the Z boson, a heavy particle that decays into pairs of leptons. The model predicts that the Z boson will decay into different types of leptons at slightly different rates than the standard model predicts, a difference known as lepton flavor non-universality. While current experiments are not sensitive enough to see this tiny deviation, which is expected to be about one part in one hundred thousand, the next generation of particle colliders, such as the proposed Future Circular Collider, will be able to measure these rates with the precision needed to confirm or rule out this theory. Additionally, the study examines the magnetic properties of leptons, finding that if the dark matter interacts with electrons or muons in a specific way, it would create a magnetic shift that is already too large to be consistent with current measurements, effectively ruling out those specific scenarios. However, if the dark matter interacts primarily with the heaviest lepton, the tau, the effects are small enough to remain hidden for now.

This work represents a significant step in understanding how dark matter might hide in plain sight. By showing that ordinary symmetries are insufficient to explain the observed silence of dark matter detectors, the study points toward a more exotic mathematical structure as the key to unlocking the mystery. It suggests that the universe operates on selection rules that are more complex and less reversible than previously thought, allowing dark matter to coexist with the known particles without causing the catastrophic flavor violations that would have been detected long ago. The findings do not prove that this specific model is the final answer, but they demonstrate that a viable path exists where the dark matter is both abundant and invisible, waiting to be discovered not by its collisions with our atoms, but by the subtle, precise ways it alters the behavior of the lightest particles in the cosmos.

Technical Summary: Taming Lepton Portal Dark Matter by a Non-Invertible Selection Rule

1. Problem Statement

Lepton Portal Dark Matter (LPDM) is a well-motivated Weakly Interacting Massive Particle (WIMP) scenario where a real scalar dark matter (DM) candidate, $X$ , couples to Standard Model (SM) leptons via vector-like leptons ( $L, E$ ). This setup naturally suppresses direct detection rates (occurring only at the 2-loop level for nucleon scattering), allowing it to evade stringent constraints from experiments like XENON and LUX.

However, LPDM generically suffers from a severe flavor problem. To reproduce the observed relic density ( $\Omega h^2 \simeq 0.12$ ) via thermal freeze-out, the portal Yukawa couplings ( $\lambda_L, \lambda_E$ ) must be of order $\mathcal{O}(1)$ . If these couplings connect to more than one SM lepton flavor, they induce large Lepton Flavor Violation (LFV) processes, such as $\mu \to e\gamma$ and $\tau \to \ell\gamma$ , which are experimentally excluded.

A common phenomenological workaround is to assume "one-flavor dominance," where the DM couples exclusively to a single lepton flavor (e.g., only electrons or only muons). The paper argues that ordinary invertible symmetries (such as Abelian discrete symmetries $\mathbb{Z}_N$ or continuous $U(1)$ ) cannot justify this one-flavor dominance while simultaneously accommodating the large mixing angles observed in the PMNS matrix, unless artificial spurion fields are introduced.

2. Methodology

The author proposes utilizing non-invertible symmetries to enforce selection rules on the Lagrangian. Specifically, they employ the non-invertible selection rule obtained by gauging a $\mathbb{Z}_2$ automorphism of a $\mathbb{Z}_N$ symmetry.

Model Construction

Field Content: The model includes the SM particles, a real scalar DM $X$ , and vector-like leptons $L$ (doublet) and $E$ (singlet).
Symmetry Mechanism: Instead of assigning fields to ordinary group charges, fields are assigned to conjugacy classes of the $\mathbb{Z}_N$ $Z_{N}$ group after gauging the $\mathbb{Z}_2$ $Z_{2}$ automorphism.
- The $\mathbb{Z}_2$ automorphism acts as a reflection: $r g_k r^{-1} = g_{-k}$ .
- Fields are assigned to classes $[g_k] = \{g_k, g_{-k}\}$ .
- Interactions are allowed if the fusion product of the classes contains the trivial class $[g_0]$ . The fusion rule is $[g_{k_1}][g_{k_2}] = [g_{k_1+k_2}] \oplus [g_{-k_1+k_2}]$ .
Specific Realization: The author constructs a specific model based on the $\mathbb{Z}_2$ $Z_{2}$ gauging of a $\mathbb{Z}_5$ symmetry.
- Assignments: $X \in [g_0]$ , $L \in [g_1]$ , $E \in [g_2]$ , $H \in [g_2]$ .
- Lepton assignments: $\vec{\ell} = ([g_1], [g_2], [g_2])$ and $\vec{e} = ([g_2], [g_0], [g_0])$ .

Analysis

The author analyzes the resulting textures of the mass matrices and Yukawa couplings:

Portal Couplings: The selection rules forbid flavor-violating terms in the portal couplings ( $\lambda_L$ and $\lambda_E$ ), enforcing one-flavor dominance in the gauge basis.
Charged Lepton Masses: The resulting texture for the charged lepton mass matrix $M_e$ is block-diagonal, ensuring that the flavor basis of the charged leptons aligns with the flavor basis of the portal couplings. This prevents the rotation to the mass basis from reintroducing flavor violation.
Neutrino Sector: The Weinberg operator (responsible for neutrino masses) is shown to be fully allowed (all elements non-zero). This allows the PMNS matrix to be generated entirely by the diagonalization of the neutrino mass matrix ( $U_{PMNS} = V_L^\dagger U_n$ ), consistent with observed large mixing angles.

3. Key Contributions and Results

3.1 Impossibility of Ordinary Symmetries

The paper demonstrates that for ordinary $\mathbb{Z}_N$ symmetries, the conditions required to forbid flavor-violating portal couplings (e.g., $q_1^\ell + q_L = 0$ while $q_{2,3}^\ell + q_L \neq 0$ ) inevitably force the charged lepton mass matrix to be block-diagonal in a way that prevents mixing between the first flavor and the others. Consequently, the large mixing angles required for the PMNS matrix cannot be realized without introducing additional spurion fields, which the author argues is artificial.

3.2 Success of Non-Invertible Selection Rules

The $\mathbb{Z}_5$ non-invertible construction successfully achieves:

One-flavor dominance: The portal couplings are restricted to a single flavor (e.g., $\lambda_L \propto (1, 0, 0)^T$ ) in the gauge basis.
Consistency with PMNS: The charged lepton mass texture preserves this dominance in the mass basis, while the neutrino sector allows for general mixing, reproducing the observed PMNS matrix.
Minimality: The author notes that $N=5$ is the minimal integer allowing this specific combination of textures.

3.3 Flavor Leakage in Type-I Seesaw

The author investigates potential "flavor leakage" (violation of the selection rule) arising from quantum corrections, specifically in a Type-I Seesaw UV completion.

At the 1-loop level, the renormalization group (RG) evolution of the portal coupling $\lambda_L$ is driven by the neutrino Yukawa matrix $y_n$ .
The induced flavor-violating coupling at the seesaw scale is estimated as:
$\lambda_L(\Lambda_n) \sim \left( 1 + \frac{y_n^T y_n^*}{32\pi^2} \ln \frac{\Lambda_n}{\Lambda} \right) \lambda_L(\Lambda)$
The magnitude of this leakage is correlated with the right-handed neutrino Yukawa couplings. For typical seesaw scales ( $M_N \sim 10^{14}$ GeV), the leakage is estimated to be $\mathcal{O}(10^{-3})$ , potentially observable via future LFV experiments.

3.4 Phenomenological Probes

The paper discusses experimental signatures of the flavor-conserving LPDM:

Anomalous Magnetic Moments ( $g-2$ ):
- If the annihilation is $d$ -wave dominated ( $\lambda_L \lambda_E = 0$ ), the contribution to $\Delta a_\ell$ is small ( $\sim 10^{-10}$ for muons), below current sensitivity but potentially reachable by future improvements.
- If $s$ -wave dominates ( $\lambda_L \lambda_E \neq 0$ ), chiral enhancement leads to large contributions. The paper finds that electron- and muon-philic cases are excluded by current $g-2$ data in the absence of co-annihilation, while tau-philic cases remain viable.
$Z$ Boson Flavor Universality:
- Loop corrections induce lepton flavor non-universality in $Z$ decays ( $A_Z$ ).
- The predicted deviation is $\mathcal{O}(10^{-5})$ .
- Current experimental uncertainties are $\mathcal{O}(10^{-3})$ , but future $Z$ factories (FCC-ee, CEPC) with projected sensitivities of $\mathcal{O}(10^{-5})$ could probe this model.

4. Significance and Claims

The paper claims to provide a theoretical mechanism to resolve the flavor problem in Lepton Portal Dark Matter without resorting to ad-hoc assumptions or artificial spurion fields. By leveraging the recently developed framework of non-invertible symmetries, the author demonstrates that it is possible to:

Suppress dangerous flavor-violating portal couplings to satisfy experimental bounds.
Simultaneously accommodate the large mixing angles of the PMNS matrix.
Maintain the viability of the LPDM scenario as a thermal WIMP candidate.

The author emphasizes that this is the first realization of such a selection rule in the context of LPDM, offering a new direction for model building in dark matter physics. They conclude that while the model is currently consistent with data, it predicts specific, testable deviations in lepton flavor universality at future high-precision colliders and potential signals in LFV processes linked to the neutrino sector.