We propose a two-loop neutrino mass model where fermionic and bosonic dark matter (DM) candidates are simultaneously connected to the neutrinos. However, the fermionic DM candidate is favored compared to the bosonic one due to generating the fermionic DM mass at the one-loop level. In order to obtain our desired Lagrangian and Higgs potential, we introduce a <inline-formula><tex-math id='TM0002' notation='LaTeX'>$Z_3$</tex-math></inline-formula> gauging Tambara&#8211;Yamagami noninvertible fusion rule with the assistance of a nonholomorphic modular <inline-formula><tex-math id='TM0003' notation='LaTeX'>$A_4$</tex-math></inline-formula> symmetry. The fusion rule forbids the mass of the DM candidate at tree level but its mass is generated at the one-loop level where the DM mass term dynamically violates the fusion rule. After that, the neutrino mass matrix is induced at the one-loop level where a remnant <inline-formula><tex-math id='TM0004' notation='LaTeX'>$Z_2$</tex-math></inline-formula> symmetry still remains. The symmetry assures the stability of our DM candidate. The nonholomorphic modular <inline-formula><tex-math id='TM0005' notation='LaTeX'>$A_4$</tex-math></inline-formula> symmetry plays a role in forbidding the interactions between the SM particles and heavier fermions <inline-formula><tex-math id='TM0006' notation='LaTeX'>$X_R$</tex-math></inline-formula> and an isospin singlet inert scalar boson <inline-formula><tex-math id='TM0007' notation='LaTeX'>$S_0$</tex-math></inline-formula> that run in the DM mass loop, in addition to reducing our free parameters that lead to our predictions for the lepton sector. We perform a <inline-formula><tex-math id='TM0008' notation='LaTeX'>$\chi ^2$</tex-math></inline-formula> numerical analysis for the lepton masses, mixing angles, and phases, and we show several predictions for the normal and inverted hierarchies. Then, we demonstrate our lepton-flavor violations (LFVs), muon anomalous magnetic dipole moment, and the relic density of the DM candidate fixing the best-fit point of the lepton sector.

China

Hiroshi Okada

India

Shilpa Jangid

A Radiative Seesaw Model in a Noninvertible Selection Rule with the Assistance of a Nonholomorphic Modular <italic>A</italic> 4 Symmetry

We propose a new mechanism wherein the electron and muon masses are generated at the one-loop level following the dynamical violation of the Ising fusion rule symmetry. This rule, being the minimal symmetry required to realize our model, also plays a crucial role in the neutrino sector. There, while neutrino masses are likewise generated at one loop, the Ising fusion rule stabilizes the particles inside the loop, effectively functioning as an unbroken <italic>Z</italic> <inf>2</inf> symmetry to all orders. We subsequently investigate lepton flavor violating processes, the muon&#8217;s anomalous magnetic dipole moment, and the relic density of dark matter. For the latter, we identify a singlet boson as our dark matter candidate and provide an initial estimate of its relic density via the annihilation cross section. Finally, we present results for both the dark matter annihilation cross section and the muon <math altimg='si2.svg'><mrow><mi>g</mi><mo linebreak='goodbreak'>&#8722;</mo><mn>2</mn></mrow></math> that are consistent with neutrino oscillation data and satisfy the constraints from lepton flavor violation.

Jingqian Chen

Chao-Qiang Geng

Jia-Jun Wu

A radiative lepton model in a non-invertible fusion rule

We study a quasi-two-zero neutrino texture based on a type-II seesaw model with modular <inline-formula><tex-math id='TM0002' notation='LaTeX'>$A_4$</tex-math></inline-formula> symmetry to evade the cosmological bound on the sum of neutrino masses while keeping some predictability in the neutrino sector. Working on three fixed points for the modulus, we discuss predictions of the model and show the allowed points satisfying the cosmological bound on neutrino mass from both cosmic microwave background (CMB) and CMB + baryon acoustic oscillation data.

Takaaki Nomura

Quasi-Two-Zero Texture in Type-II Seesaw at Fixed Points from Modular <italic>A</italic> 4 Symmetry

We study a three-loop induced neutrino mass scenario from a non-holomorphic modular $\textit{A}$ $_{4}$ flavor symmetry and obtain the minimum scenario leading to predictions of the lepton masses, mixing angles, and Dirac and Majorana phases, which are shown through chi square analyses. In addition, we discuss the lepton flavor violations, muon anomalous magnetic moment, lepton universality, and relic density of the dark matter candidate. Moreover, we show that our model can be extended to satisfy the observed relic density of dark matter within the limit of perturbation by adding one singlet scalar boson without changing predictions in the neutrino sector.

Neutrino mass model at a three-loop level from a non-holomorphic modular $\textit{A}$ $_{4}$ symmetry * Takaaki Nomura is supported by the Fundamental Research Funds for the Central Universities. Hiroshi Okada is supported by Zhongyuan Talent (Talent Recruitment Series) Foreign Experts Project

We apply non-invertible selection rules coming from a fusion algebra to radiative neutrino mass models where fields are labeled by the elements in the algebra. Since non-invertible selection rules only hold at tree level, radiative corrections naturally explain the origin of tiny neutrino masses. Furthermore, a remnant symmetry of the fusion algebra protects the stability of dark matter, which is conventionally imposed in radiative neutrino models. We also find that interesting neutrino mass textures are realized by assigning fields to family-dependent elements in the algebra.

Japan

Tatsuo Kobayashi

Hajime Otsuka

Radiative neutrino mass models from non-invertible selection rules

We propose a new inverse seesaw model based on hidden local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> symmetry framework where the inverse seesaw mechanism is induced at one loop level. A Majorana mass term of singlet fermion is forbidden by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> symmetry and it is generated at one-loop level by introducing relevant particle contents to get loop diagram, inducing inverse seesaw mechanism. The same particle contents also contribute to lepton magnetic(electric) dipole moment and lepton flavor violating decays without chiral suppression. We can then obtain sizable muon anomalous magnetic dipole moment that accommodate with deviation from the standard model prediction. The constraints from lepton flavor violating decays and electron magnetic(electric) dipole moment are also discussed to explore testability of the model.

Radiative inverse seesaw model with hidden <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> gauge symmetry enhancing lepton <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mo>−</mo><mn>2</mn></math>

We investigate a radiative seesaw model with two leptoquarks under non-holomorphic modular $\textit{A}$ symmetry. The leptons and quarks belong to non-trivial representations of the modular $\textit{A}$ and the structures of their mass matrices are restricted. Neutrino masses are generated at one-loop level via leptoquark inside loop diagram where structures of relevant Yukawa interactions are determined by the modular $\textit{A}$ symmetry. We scan the free parameters in the model and try to fit all the observed data for both lepton and quark sectors. For allowed parameters, we show some predictions regarding neutrino observables such as sum of neutrino mass and neutrinoless double beta decay.

Xing-Yu Wang

A radiative neutrino mass model with leptoquarks under non-holomorphic modular $\textit{A}$

We propose a novel framework that simultaneously addresses three critical issues: tiny neutrino masses and their mixing patterns, dark matter, and the strong CP problem. Our model extends the Peccei–Quinn (PQ) symmetry by incorporating modular $$S_3$$ symmetry, which plays a central role in explaining the observed neutrino mixing structure. The field content includes two vector-like colored fermions and three colored scalars as $$S_3$$ singlets, an isospin doublet inert scalar and a singlet PQ scalar, each assigned appropriate modular weights. We show that such an extension, together with a suitable assignment of modular weights to the fields, can lead to holomorphic modular forms of Yukawa interactions, which can be derived from a superpotential. Furthermore, we explore an extension of the model to include non-holomorphic Yukawa interactions in the non-supersymmetric framework and show that the results are distinct from the holomorphic case. Tiny neutrino masses are generated radiatively through colored mediators, while the KSVZ-type axion appears to dynamically resolve the strong CP problem. We investigate the phenomenology of lepton flavor violation and the muon $$g-2$$ anomaly within this framework. Additionally, we explore the axion’s properties and its role as dark matter.

South Korea

Sin Kang

Neutrino masses and mixing in an axion model

We propose a new type of lepton seesaw model introducing a modular <italic>A</italic> 4 flavor symmetry in which isospin doublet vector fermions play an important role in constructing seesaw mechanisms for both the charged-lepton mass matrix and the neutrino one. The charged-lepton mass matrix is induced via the Dirac seesaw with a two-by-two block mass matrix. On the other hand, the neutrino mass matrix is generated via a seesaw with four-by-four block mass matrix where the right-handed neutral fermions are also added. Remarkably, the neutral fermion mass matrix provides us with a new type of seesaw formula for the active neutrino mass. Through our chi-squared analysis, we discuss some tendencies of observables, focusing on two fixed points <inline-formula><tex-math id='TM0001' notation='LaTeX'>$\tau =i,\ \omega$</tex-math></inline-formula>.

A New Type of Lepton Seesaw Model in a Modular <italic>A</italic> 4 Symmetry

We search for predictability of lepton masses and mixing patterns of type-II seesaw scenario in a non-holomorphic modular <math altimg='si1.svg'><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub></math> symmetry recently proposed by Qu and Ding. We propose three types of minimum predictive models with different assignments of modular weight, satisfying the neutrino oscillation data in Nufit 5.2. The cosmological bound on the sum of neutrino mass is stringent to our models and CMB bound <math altimg='si2.svg'><mo>&#8721;</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>&#957;</mi></mrow></msub><mo>&#8804;</mo><mn>0.12</mn></math> eV can be satisfied by one of three models playing an important role in discriminating them.