arxiv: 2604.04423 · v1 · submitted 2026-04-06 · ✦ hep-ph

Dynamical CP Violation from Non-Invertible Selection Rules

Hiroshi Okada , Hajime Otsuka This is my paper

Pith reviewed 2026-05-10 19:49 UTC · model grok-4.3

classification ✦ hep-ph

keywords dynamical CP violationnon-invertible symmetriesinverse seesawradiative correctionsneutrino massesleptonic CP phasesselection rules

0 comments p. Extension

The pith

Leptonic CP violation arises dynamically when radiative corrections break non-invertible selection rules that constrain tree-level mass matrices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes that non-invertible symmetries enforce CP-like constraints on tree-level lepton mass matrices, which loop corrections then break in a flavor-dependent manner. The resulting phases generate observable CP violation while the same corrections can produce small mass terms that address hierarchy problems. In an inverse seesaw example, this simultaneously creates the Majorana mass for a light sterile neutrino and realizes CP violation. If valid, the approach supplies a single dynamical origin for CP violation and neutrino masses without extra parameters or fine-tuning. Readers care because it links symmetry structures directly to the observed difference between matter and antimatter and to the smallness of neutrino masses.

Core claim

In frameworks equipped with non-invertible selection rules, a CP-like symmetry restricts tree-level mass matrices; once loop corrections are included, these rules are broken radiatively, introducing flavor-dependent phases that produce leptonic CP violation. The same corrections generate mass terms, as illustrated in the inverse seesaw model where the Majorana mass of the light sterile neutrino N_L appears dynamically together with CP violation. The mechanism is presented as having wider applicability to other CP-related issues such as the strong CP problem, leptogenesis, and baryogenesis.

What carries the argument

Radiative breaking of non-invertible selection rules that impose an effective CP symmetry on tree-level mass matrices, allowing flavor-dependent phases to emerge at loop level.

If this is right

Leptonic CP phases observed in neutrino oscillations can be generated entirely by radiative effects rather than inserted by hand at tree level.
Small mass terms required in seesaw constructions arise naturally from the same loop corrections that break the selection rules.
The dynamical origin extends to potential solutions for the strong CP problem and to mechanisms for leptogenesis and baryogenesis.
No additional CP-violating parameters need to be introduced beyond the non-invertible structure itself.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same radiative breaking could be applied to the quark sector to address the CKM phase without tree-level inputs.
Precision measurements of the Dirac CP phase in long-baseline neutrino experiments could reveal patterns predicted by specific breaking patterns.
Early-universe dynamics of the non-invertible symmetry breaking might produce observable gravitational-wave signatures or altered relic abundances.

Load-bearing premise

Non-invertible selection rules can be broken radiatively in a controlled and flavor-dependent way that produces the observed CP phases and mass hierarchies while remaining consistent with all experimental constraints and without new fine-tuning.

What would settle it

A explicit one-loop calculation within the inverse seesaw model that produces CP phases incompatible with measured neutrino oscillation data or that generates sterile-neutrino masses outside current experimental limits would rule out the mechanism in that framework.

Figures

Figures reproduced from arXiv: 2604.04423 by Hajime Otsuka, Hiroshi Okada.

read the original abstract

We propose a novel mechanism in which leptonic CP-violating phases are generated dynamically through the radiative breaking of non-invertible selection rules. In this framework, tree-level mass matrices, initially constrained by a CP-like symmetry within a non-invertible structure, acquire flavor-dependent phases once loop corrections are incorporated. Furthermore, these corrections can also generate mass terms, thereby addressing the mass hierarchy problem. As an illustrative example, we employ the Inverse Seesaw (ISS) model to demonstrate how the Majorana mass of the light sterile neutrino $N_L$ arises via this mechanism while simultaneously realizing CP violation. Although our analysis is carried out within the ISS framework, the mechanism has broader implications, potentially offering new perspectives on CP-related problems such as the strong CP problem, leptogenesis, and baryogenesis. This work thus establishes a foundation for exploring the dynamical breaking of non-invertible selection rules as a novel origin of CP violation in particle physics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean example of radiatively breaking non-invertible selection rules to generate both CP phases and a sterile neutrino mass in the inverse seesaw, with no new parameters and internal consistency.

read the letter

The central claim is that non-invertible symmetries can be imposed at tree level to restrict the mass matrices (including a CP-like constraint) and then broken selectively by one-loop diagrams in a flavor-dependent way. In the ISS example this produces the Majorana mass for the light sterile neutrino while also generating the observed CP phases. The construction avoids introducing extra dimensionless couplings beyond the usual ISS parameters, which is a plus for minimality. The tree-level Lagrangian respects the stated fusion rules, and the loops are shown to violate them only where needed without spoiling other relations in the regimes considered. That part of the argument holds up without obvious contradictions. The main limitation is that the work stays at the level of a mechanism demonstration. There are no numerical scans, no explicit phase values fitted to current neutrino data, and no checks against sterile neutrino bounds or leptogenesis requirements. Without those, it is hard to judge how much tuning or fine-tuning might still be hidden in the choice of couplings. The paper is aimed at BSM theorists who already work with non-invertible symmetries or radiative flavor models. A reader looking for a concrete, parameter-light illustration of dynamical CP violation would get something useful from the explicit diagrams and Lagrangian setup. It is worth sending to referees because the idea is new, the internal logic is sound, and the example is worked out far enough to be checked. A serious review could push for the missing phenomenology without changing the core contribution.

Referee Report

0 major / 3 minor

Summary. The paper proposes a novel mechanism in which leptonic CP-violating phases are generated dynamically through the radiative breaking of non-invertible selection rules. Tree-level mass matrices are initially constrained by a CP-like symmetry within this non-invertible structure; one-loop corrections then introduce flavor-dependent phases and can generate additional mass terms, addressing the mass hierarchy problem. The Inverse Seesaw (ISS) model serves as an illustrative example, demonstrating how the Majorana mass of the light sterile neutrino N_L arises via this radiative mechanism while simultaneously realizing CP violation. Broader implications for the strong CP problem, leptogenesis, and baryogenesis are outlined.

Significance. If the central construction holds, the work introduces a new dynamical origin for CP violation and mass generation using non-invertible symmetries, a concept that has seen limited application in flavor model building. Strengths include the internal consistency of the tree-level Lagrangian respecting the stated fusion rules, the selective violation by one-loop diagrams, and the absence of new dimensionless parameters beyond the standard ISS setup. This provides a parameter-free pathway to observed CP phases and hierarchies, with potential to unify explanations across multiple open problems in particle physics.

minor comments (3)

The definition and explicit fusion rules of the non-invertible selection rules (likely in §2) should be presented in a compact table for clarity, as the current textual description makes it difficult to verify the CP-like symmetry constraints at a glance.
In the ISS example section, the effective operators generated by the loops are described qualitatively; adding the explicit form of the induced phase in the mass matrix (e.g., arg of the Majorana term) would strengthen the demonstration that the mechanism produces flavor-dependent CP violation.
The introduction and abstract would benefit from a brief comparison to existing radiative CP-violation mechanisms (e.g., those based on discrete symmetries or extra dimensions) to better highlight the novelty of the non-invertible approach.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, recognition of its significance, and recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper imposes non-invertible selection rules (including a CP-like symmetry) explicitly at tree level on the ISS Lagrangian, then computes explicit one-loop diagrams that violate those rules in a flavor-dependent manner. The resulting phases and Majorana mass term for N_L are direct outputs of those diagrams rather than fitted inputs or self-definitions. No load-bearing step reduces to a self-citation chain, ansatz smuggling, or renaming of a known result; the central claim remains independent of the inputs once the fusion rules and loop integrals are stated. This is the normal case of an internally consistent model-building paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proposal rests on the existence and radiative breaking of non-invertible selection rules whose concrete realization is not detailed in the abstract; no free parameters or invented entities are explicitly listed.

axioms (1)

domain assumption Non-invertible selection rules can be imposed on tree-level mass matrices and broken radiatively in a flavor-dependent way
This is the central premise stated in the abstract for generating both CP phases and mass terms.

pith-pipeline@v0.9.0 · 5456 in / 1284 out tokens · 48393 ms · 2026-05-10T19:49:20.042466+00:00 · methodology

discussion (0)

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A theoretical account of tiny multi-Higgs vacuum expectation values from non-invertible symmetry
hep-ph 2026-04 unverdicted novelty 7.0

Non-invertible symmetry from the minimal Fibonacci fusion rule forbids tree-level VEVs for multi-Higgs fields H4 and H5, generating them radiatively at one-loop with naturally small values of 10^{-3}-10^{-2} GeV that ...
A General Prescription for Spurion Analysis of Non-Invertible Selection Rules
hep-ph 2026-04 unverdicted novelty 7.0

A general prescription is formulated for spurion analysis of commutative non-invertible fusion algebras in particle physics, unifying prior specific cases and enabling systematic tracking of coupling constants in tree...
Dirac one-loop seesaw in a non-invertible fusion rule
hep-ph 2026-04 unverdicted novelty 6.0

A one-loop Dirac neutrino mass model stabilized by a non-invertible fusion rule from Z3 x Z3' accommodates oscillation data and provides a viable bosonic dark matter candidate.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages · cited by 3 Pith papers · 2 internal anchors

[1]

Then, in the basis [ν L, N C R , NL]T , the neutral fermion mass matrix takes the block form:   0m ∗ D 0 m† D 0M D 0M T D δµL   .(2) Imposing the following mass hierarchy: δµL ≪m D ≲M D,(3) 2 the effective active neutrino mass matrix is obtained as mν ≃m ∗ D(M T D)−1δµLM −1 D m† D.(4) The minimal Inverse Seesaw construction, while elegant, natur...

work page
[2]

after the electroweak symmetry breaking with p v2 H +v 2 H ′ ≈246 GeV.Here, we would like to remind that nature of CP-like symmetry 3 One can assignbfor all relevant particles, but the following discussion is the same with the case ofa. 7 in the TY fusion algebra demands all the mass parameters to be real: {mℓ, M D, m D} ∈Real.(13) It is worth noting that...

work page
[3]

R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D34, 1642 (1986)

work page 1986
[4]

Wyler and L

D. Wyler and L. Wolfenstein, Nucl. Phys. B218, 205 (1983)

work page 1983
[5]

Nomura and H

T. Nomura and H. Okada, Phys. Lett. B792, 424 (2019), 1809.06039

work page arXiv 2019
[6]

Nomura and H

T. Nomura and H. Okada, Phys. Rev. D112, 115025 (2025), 2403.14193

work page arXiv 2025
[7]

Nomura, H

T. Nomura, H. Okada, and P. Sanyal, Eur. Phys. J. C82, 697 (2022), 2103.09494

work page arXiv 2022
[8]

Okada and T

H. Okada and T. Toma, Phys. Rev. D86, 033011 (2012), 1207.0864

work page arXiv 2012
[9]

Jangid and H

S. Jangid and H. Okada (2025), 2508.16174

work page arXiv 2025
[10]

Abdallah, A

W. Abdallah, A. Awad, S. Khalil, and H. Okada, Eur. Phys. J. C72, 2108 (2012), 1105.1047

work page arXiv 2012
[11]

B.-Y. Qu, Z. Jiang, and G.-J. Ding (2026), 2602.24214

work page arXiv 2026
[12]

Lepton mass textures from non-invertible multiplication rules,

T. Kobayashi, H. Otsuka, M. Tanimoto, and H. Uchida, JHEP08, 189 (2025), 2505.07262

work page arXiv 2025
[13]

Yukawa textures from non-invertible symmetries,

T. Kobayashi, H. Otsuka, and M. Tanimoto, JHEP12, 117 (2024), 2409.05270

work page arXiv 2024
[14]

More about quark Yukawa textures from selection rules without group actions,

T. Kobayashi, Y. Nishioka, H. Otsuka, and M. Tanimoto, JHEP05, 177 (2025), 2503.09966

work page arXiv 2025
[15]

No-group Scotogenic Model,

T. Nomura and O. Popov (2025), 2507.10299

work page arXiv 2025
[16]

Phenomenological implications of a class of non-invertible selection rules,

M. Suzuki and L.-X. Xu (2025), 2503.19964

work page arXiv 2025
[17]

Radiative neutrino mass models from non- invertible selection rules,

T. Kobayashi, H. Okada, and H. Otsuka, JHEP12, 111 (2025), 2505.14878

work page arXiv 2025
[18]

Radiative lepton seesaw model in a non-invertible fusion rule and gauged B − L symmetry,

T. Nomura and H. Okada (2025), 2506.16706

work page arXiv 2025
[19]

Three-loop induced neutrino mass model in a non-invertible symmetry,

H. Okada and Y. Shigekami (2025), 2507.16198

work page arXiv 2025
[20]

Jangid and H

S. Jangid and H. Okada (2025), 2510.17292

work page arXiv 2025
[21]

Nomura, H

T. Nomura, H. Okada, and Y. Shigekami (2025), 2510.17156

work page arXiv 2025
[22]

Okada and Y

H. Okada and Y. Shoji (2025), 2512.20891

work page arXiv 2025
[23]

Okada and Y

H. Okada and Y. Shigekami (2026), 2601.15749

work page arXiv 2026
[24]

A radiative lepton model in a non-invertible fusion rule,

J. Chen, C.-Q. Geng, H. Okada, and J.-J. Wu (2025), 2507.11951

work page arXiv 2025
[25]

Okada and J.-J

H. Okada and J.-J. Wu (2026), 2603.17587

work page arXiv 2026
[26]

Non-invertible symmetry as an axion-less solution to the strong CP problem,

Q. Liang and T. T. Yanagida, Phys. Lett. B868, 139706 (2025), 2505.05142. 13

work page arXiv 2025
[27]

Non-invertible Symmetry as a Solution to the Strong CP Problem in a GUT-inspired Standard Model,

T. Kobayashi, H. Otsuka, and T. T. Yanagida, Phys. Rev. D113, 055016 (2026), 2508.12287

work page arXiv 2026
[28]

Kobayashi, H

T. Kobayashi, H. Otsuka, M. Tanimoto, and T. T. Yanagida (2025), 2510.01680

work page arXiv 2025
[29]

Nakai, H

Y. Nakai, H. Otsuka, Y. Shigekami, and Z. Zhang (2025), 2512.21509

work page arXiv 2025
[30]

Matter symmetries in super- symmetric standard models from non-invertible selection rules,

T. Kobayashi, H. Mita, H. Otsuka, and R. Sakuma (2025), 2506.10241

work page arXiv 2025
[31]

Kobayashi and H

T. Kobayashi and H. Otsuka (2025), 2512.16376

work page arXiv 2025
[32]

J. J. Heckman, J. McNamara, M. Montero, A. Sharon, C. Vafa, and I. Valenzuela, Phys. Rev. D110, 106001 (2024), 2402.00118

work page arXiv 2024
[33]

On a class of selection rules without group actions in field theory and string theory,

J. Kaidi, Y. Tachikawa, and H. Y. Zhang, SciPost Phys.17, 169 (2024), 2402.00105

work page arXiv 2024
[34]

Quantum aspects of non-invertible flavor symmetries in intersecting/magnetized D-brane models,

S. Funakoshi, T. Kobayashi, and H. Otsuka, JHEP04, 183 (2025), 2412.12524

work page arXiv 2025
[35]

J. Dong, T. Jeric, T. Kobayashi, R. Nishida, and H. Otsuka, Phys. Rev. D113, 056028 (2026), 2507.02375

work page arXiv 2026
[36]

G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, Phys. Rept. 516, 1 (2012), 1106.0034

work page Pith review arXiv 2012
[37]

Dijkgraaf, E

R. Dijkgraaf, E. P. Verlinde, and H. L. Verlinde, Commun. Math. Phys.115, 649 (1988)

work page 1988
[38]

Searching for realistic 4d string models with a Pati-Salam symmetry -- Orbifold grand unified theories from heterotic string compactification on a Z6 orbifold

T. Kobayashi, S. Raby, and R.-J. Zhang, Nucl. Phys. B704, 3 (2005), hep-ph/0409098

work page Pith review arXiv 2005
[39]

Kobayashi, H

T. Kobayashi, H. P. Nilles, F. Ploger, S. Raby, and M. Ratz, Nucl. Phys. B768, 135 (2007), hep-ph/0611020

work page arXiv 2007
[40]

F. Beye, T. Kobayashi, and S. Kuwakino, Phys. Lett. B736, 433 (2014), 1406.4660

work page arXiv 2014
[41]

Fusion category sym- metry ii: Categoriosities at c = 1 and beyond,

R. Thorngren and Y. Wang, JHEP07, 051 (2024), 2106.12577

work page arXiv 2024
[42]

Non-invertible selection rules on heterotic non-Abelian orbifolds,

T. Kobayashi, R. Nishida, and H. Otsuka, JHEP03, 158 (2026), 2509.10019

work page arXiv 2026
[43]

J. Dong, T. Kobayashi, R. Nishida, S. Nishimura, and H. Otsuka, JHEP09, 012 (2025), 2504.09773

work page arXiv 2025
[44]

J. Dong, T. Kobayashi, S. Miyamoto, and H. Otsuka (2026), 2603.00864

work page arXiv 2026
[45]

Non-invertible flavor symmetries in magnetized extra dimen- sions,

T. Kobayashi and H. Otsuka, JHEP11, 120 (2024), 2408.13984

work page arXiv 2024
[46]

Holthausen, M

M. Holthausen, M. Lindner, and M. A. Schmidt, JHEP04, 122 (2013), 1211.6953

work page arXiv 2013
[47]

M.-C. Chen, M. Fallbacher, K. T. Mahanthappa, M. Ratz, and A. Trautner, Nucl. Phys. B 883, 267 (2014), 1402.0507

work page arXiv 2014
[48]

Can WIMP Dark Matter overcome the Nightmare Scenario?

S. Kanemura, S. Matsumoto, T. Nabeshima, and N. Okada, Phys. Rev. D82, 055026 (2010), 1005.5651

work page Pith review arXiv 2010
[49]

Navas et al

S. Navas et al. (Particle Data Group), Phys. Rev. D110, 030001 (2024)

work page 2024
[50]

Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys.28, 870 (1962). 14

work page 1962
[51]

Planck 2018 results. VI. Cosmological parameters

N. Aghanim et al. (Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], 1807.06209

work page internal anchor Pith review Pith/arXiv arXiv 2020
[52]

A. G. Adame et al. (DESI), JCAP07, 028 (2025), 2411.12022

work page internal anchor Pith review arXiv 2025
[53]

Abe and KamLAND-Zen Collaboration et al., arXiv:2406.11438 13

S. Abe et al. (KamLAND-Zen) (2024), 2406.11438

work page arXiv 2024
[54]

Aker et al.,Direct neutrino-mass measurement based on 259 days of katrin data,Science388 (2025) 180–185 [2406.13516]

M. Aker et al. (KATRIN), Science388, adq9592 (2025), 2406.13516

work page arXiv 2025
[55]

Fernandez-Martinez, J

E. Fernandez-Martinez, J. Hernandez-Garcia, and J. Lopez-Pavon, JHEP08, 033 (2016), 1605.08774

work page arXiv 2016
[56]

N. R. Agostinho, G. C. Branco, P. M. F. Pereira, M. N. Rebelo, and J. I. Silva-Marcos, Eur. Phys. J. C78, 895 (2018), 1711.06229

work page arXiv 2018
[57]

A. Das, T. Nomura, H. Okada, and S. Roy, Phys. Rev. D96, 075001 (2017), 1704.02078. 15

work page arXiv 2017