Physicists Rebuild Dark Matter Model Using Symmetries That Cannot Be Inverted

For nearly three decades, neutrinos have been the Standard Model’s most stubborn puzzle. Since the discovery of neutrino oscillations in the 1990s, physicists have known that these ghostly particles must have mass, yet the Standard Model offers no mechanism to give it to them. Any explanation necessarily points to physics beyond the Standard Model, and one of the most elegant proposals on the market is the scotogenic mechanism, which ties the origin of naturally tiny neutrino masses directly to the existence of dark matter. Now, a new theoretical study has rebuilt this celebrated framework on an even more exotic foundation: a type of symmetry that is not a group at all.

The scotogenic model, originally proposed by Ernest Ma in 2006, achieves two goals at once. It generates neutrino masses radiatively, meaning they arise only through quantum loop corrections rather than at tree level, and it supplies a stable dark matter candidate in the same stroke. In its canonical form, the model extends the Standard Model with a second Higgs doublet, usually called eta, and singlet fermions known as right-handed neutrinos. A discrete Z2 symmetry is imposed by hand, forbidding the terms that would otherwise allow neutrino masses at tree level and ensuring the lightest eta component cannot decay, making it a viable dark matter particle. The trouble is that this Z2 symmetry is essentially ad hoc, inserted into the theory without any deeper justification.

Takaaki Nomura of Sichuan University and Oleg Popov of Shenzhen MSU-BIT University have now shown that this ad hoc ingredient can be replaced by something far more fundamental. In a paper published in The European Physical Journal C, they constructed scotogenic models based on non-invertible Z_M symmetries, sometimes called no-group symmetries. Unlike ordinary symmetries, which form groups in which every operation can be undone by an inverse, non-invertible symmetries obey fusion rules in which combining two operations yields a sum of possibilities rather than a single unique result. These exotic structures have surged to prominence in theoretical physics in recent years, appearing naturally in string theory constructions and in the study of generalized symmetries.

The technical machinery behind the new work is subtle but conceptually clean. Instead of assigning ordinary group representations to particles, the authors assign conjugacy classes, denoted [g^k], to each field in the theory. The product of two classes follows the rule [g^k][g^k’] = [g^(k+k’)] + [g^(M-k+k’)], and a term in the Lagrangian is allowed only if the product of the classes of its fields contains the trivial class [g^0]. This selection rule, rooted in constructions from type IIB string theory with magnetized extra dimensions, replaces the familiar group-action invariants of conventional flavor model building. Remarkably, such symmetries can be broken by radiative corrections even when they are exact at tree level, a feature that connects naturally to the radiative character of the scotogenic mechanism itself.

The authors systematically searched for the minimal choice of M and the minimal assignment of classes that could distinguish the three generations of leptons, generate neutrino masses exclusively at one loop, stabilize dark matter, and leave the Standard Model Higgs trivial so that the quark sector remains untouched. The counting is tight: at least three classes are needed to tell the three lepton generations apart, which rules out M below 4, and separating the singlet neutrinos from the Standard Model leptons requires M of at least 8. Smaller choices fail because they inevitably leave dangerous operators, such as the coupling that would let the inert Higgs doublet decay, unfrozen.

Viable models emerge for M equal to 8, 10, and 11. In the M=8 benchmark, the Standard Model leptons carry classes [g^0], [g^4], and [g^2], while two generations of singlet fermions carry [g^3] and [g^1]. This assignment produces a sparse Yukawa coupling matrix connecting the leptons to the singlet fermions through the inert doublet, and it forbids the term that would spoil dark matter stability. An elegant bonus emerges here: the viable models accidentally possess an exact Z2 symmetry at the renormalizable level, with the singlet fermions and the inert doublet each flipping sign, precisely mirroring the symmetry of the original scotogenic model. This accidental Z2 is protected against radiative breaking, so dark matter stability is guaranteed with no new fields beyond the canonical setup.

The payoff for this symmetry engineering is predictivity. Because the class assignments force a specific sparse pattern of Yukawa couplings, the resulting neutrino mass matrix has a one-zero texture, meaning exactly one of its nine entries vanishes. The location of that zero depends on which classes are assigned to which leptons, and each texture imposes analytic relations among the neutrino mixing angles, masses, and CP-violating phases. The authors performed a detailed numerical scan for two benchmark models, fitting the oscillation data from the NuFit 6.0 global analysis and computing a chi-squared measure of the fit quality. In the minimal M=8 benchmark, with only two singlet fermions, the neutrino mass matrix has rank two and only the inverted mass hierarchy can be fitted, with strikingly narrow predictions: the Dirac CP phase is pinned near plus or minus 90 degrees and the Majorana phase alpha_21 near plus or minus 20 degrees.

The richer M=11 benchmark, with three singlet fermions, accommodates both the normal and inverted hierarchies and yields broader but still structured predictions. For the normal hierarchy, the Dirac CP phase can take any value while the Majorana phase alpha_21 prefers the range from 50 to 180 degrees. For the inverted hierarchy, the Dirac phase again clusters around plus or minus 90 degrees. The model also makes concrete statements about observables at the frontier of experiment. The effective mass for neutrinoless double beta decay, which governs whether experiments like KamLAND-Zen can see this hypothetical process, falls in a range where the inverted hierarchy case could be tested in the near future, while normal hierarchy points sit below current bounds. Cosmological limits on the sum of neutrino masses from Planck and the DESI baryon acoustic oscillation measurements further constrain the allowed parameter space.

Charged lepton flavor violation provides another sharp test. The same loop diagrams that generate neutrino masses induce processes such as muon decay to an electron and a photon, muon-to-electron conversion in nuclei, and three-body lepton decays, all of which are bounded by experiments including MEG, Belle II, and earlier searches. The strongest constraint comes from the limit on muon to electron gamma decay, currently about 3.1 times 10 to the minus 13 in branching ratio. Intriguingly, the class assignments dictate which flavor-violating channels are allowed at all: a vanishing (12) element of the neutrino mass matrix permits tau to electron gamma and tau to muon gamma but forbids muon to electron gamma at one loop, while other textures flip the pattern. The authors also checked that quantum corrections do not regenerate the forbidden zero entries, showing that the induced contributions are suppressed by a factor of order 10 to the minus 8 and can be safely neglected.

The broader significance of the work lies in what it demonstrates about the toolkit of modern flavor physics. Discrete flavor symmetries have long been the orthodox route to explaining lepton mixing, modular symmetries have flourished over the past decade, and non-holomorphic modular approaches have offered a more minimal variant. Non-invertible symmetries now join this lineage, and the new study shows they can do real work: replacing an ad hoc stabilizing symmetry with one whose origins trace back to string theory, while simultaneously imposing texture zeros that sharpen the model’s predictions for CP phases, neutrinoless double beta decay, and collider signatures of the inert doublet. As experiments close in on the neutrino mass ordering, the CP phases, and rare lepton decays, models of this kind offer something increasingly precious in beyond-Standard-Model physics: concrete, falsifiable targets.

Subject of Research: A scotogenic model of radiative neutrino masses and dark matter based on non-invertible Z_M symmetries

Article Title: Scotogenic model with non-invertible symmetry

Article References: Nomura, T., & Popov, O. (2026). Scotogenic model with non-invertible symmetry. The European Physical Journal C, 86(9), Article 1070. https://doi.org/10.1140/epjc/s10052-026-16287-7

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16287-7

Keywords: scotogenic model, non-invertible symmetry, neutrino masses, dark matter, flavor symmetry, radiative mass generation, CP violation, neutrinoless double beta decay, charged lepton flavor violation, beyond the Standard Model, string theory, Z_M symmetry