Flavour Sphalerons in SO(3)_F Models

• Flavour sphalerons are non-perturbative gauge transitions in a gauged flavour symmetry (e.g., SO(3)_F) that redistribute initial lepton asymmetries between baryon and dark matter sectors.
• They violate specific charge combinations (violating L+X while conserving B and L−X) and, together with electroweak sphalerons, predict precise asymmetry distributions and dark matter mass estimates (e.g., m_DM ≈ 13.4 GeV).
• The mechanism relies on thermal sphaleron rates and tightly constrained flavour symmetry breaking scales, integrating flavour dynamics, anomaly cancellation, and cosmological observations.

Flavour sphalerons are sphaleron transitions associated with a non-abelian gauged flavour or horizontal symmetry rather than with the Standard Model electroweak group. In the explicit $SO(3)_F$ realization, they act on both visible and dark-sector fermions, violate $L+X$, conserve $B$ and $L-X$, and, together with electroweak sphalerons, redistribute a primordial lepton asymmetry into baryon and dark matter asymmetries according to

$$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$

implying

$$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$

in that model (Blennow et al., 19 May 2026). Earlier work formulated closely related mechanisms using broken horizontal symmetries and “dark sphalerons,” while collider and cosmic-ray searches to date have targeted electroweak sphalerons rather than flavour sphalerons (Blennow et al., 2010, Collaboration, 2018, Brooijmans et al., 2016).

1. Definition within sphaleron theory

A sphaleron is a static, finite-energy solution of the classical field equations that is unstable. In variational language, it is a stationary point of the energy functional that is not a minimum. More generally, sphalerons arise when the topology of the configuration space of finite-energy static fields contains non-contractible loops or spheres, so that a min–max construction produces a saddle point of the energy functional. This is the standard field-theoretic meaning of the term, independent of whether the gauge sector is electroweak, flavour, or otherwise (Manton, 2019).

Flavour sphalerons apply this general notion to a gauged flavour symmetry. In the $SO(3)_F$ framework, the non-perturbative gauge configurations of the flavour group generate sphaleron transitions in thermal equilibrium, analogously to electroweak sphalerons of $SU(2)_L$ (Blennow et al., 19 May 2026). A plausible implication is that flavour sphalerons should be understood not as a distinct mathematical category of saddle point, but as the flavour-gauge realization of the general sphaleron mechanism.

The broader topological review literature does not discuss “flavour sphalerons” as a separate particle-physics topic. Likewise, collider analyses that refer simply to “sphalerons” usually mean electroweak vacuum transitions. This distinction is essential, because the anomaly structure, conserved charges, and phenomenological targets differ sharply between the electroweak and flavour cases (Manton, 2019, Collaboration, 2018).

2. Anomaly structure and charge-selection rules

Flavour sphalerons are defined operationally by the global charges they violate in thermal equilibrium. Their significance lies in anomaly-mediated charge transfer between sectors. The explicit $SO(3)_F$ model gives the clearest example: weak sphalerons violate $B+L$ and conserve $L+X$0 and $L+X$1, whereas flavour sphalerons violate $L+X$2, conserve $L+X$3, and conserve $L+X$4. When both are active, the conserved quantity is

$L+X$5

Related models with horizontal or extra non-Standard-Model gauge groups realize analogous, but not identical, selection rules. This places flavour sphalerons within a larger class of non-abelian sphaleron systems that redistribute asymmetries among visible and dark charges.

Gauge sector	Selection rule or conserved combination	Role
$L+X$6 electroweak sphalerons	$L+X$7; conserve $L+X$8	Reprocess visible-sector asymmetry
$L+X$9 flavour sphalerons	Violate $B$0; conserve $B$1 and $B$2	Transfer lepton asymmetry into dark number
$B$3 horizontal “dark sphalerons”	$B$4; with SM sphalerons conserve $B$5	Aidnogenesis via broken horizontal symmetry
$B$6 non-Standard-Model sphalerons	$B$7	Co-generation of baryons and dark matter

The $B$8 case is the one explicitly identified as “flavour sphalerons.” Earlier constructions instead speak of dark sphalerons associated with a horizontal symmetry or with a new non-Standard-Model gauge interaction. This suggests that “flavour sphaleron” is most precise when the non-abelian gauge group is itself a flavour symmetry acting on generations or flavour multiplets (Blennow et al., 19 May 2026, Blennow et al., 2010, Barr et al., 2013).

3. $B$9 flavour sphalerons and aidnogenesis

In the gauged-flavour construction, the Standard Model is extended by

$L-X$0

acting on selected fermion multiplets as triplets. The assignments highlighted for the flavour-sphaleron mechanism are: $L-X$1, $L-X$2, and the dark fermion $L-X$3 as $L-X$4 triplets, together with mirror fermions $L-X$5, $L-X$6, and $L-X$7 as triplets; other Standard Model fermions are singlets. Because the flavour symmetry is chiral with respect to the Standard Model fermion content, anomaly cancellation requires these mirror fermions (Blennow et al., 19 May 2026).

The primordial asymmetry is produced by heavy Majorana neutrinos $L-X$8 with

$L-X$9

which decay out of equilibrium and CP-violatingly, generating an initial lepton asymmetry $$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$0. The model requires $$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$1 to remain unbroken during leptogenesis so that flavour sphalerons are active when the asymmetry is generated. Weak sphalerons then convert part of $$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$2 into baryon number, while flavour sphalerons redistribute asymmetry into the dark sector (Blennow et al., 19 May 2026).

Assuming the relevant interactions, plus weak and flavour sphalerons, are in thermal equilibrium and charge and weak-isospin conservation is imposed, the asymmetry sharing is

$$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$3

Therefore

$$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$4

Using the observed energy density ratio, the dark matter mass is predicted to be

$$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$5

In this framework dark matter arises as baryon-like bound states of a confining $$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$6, so the same construction links leptogenesis, flavour dynamics, and asymmetric dark matter (Blennow et al., 19 May 2026).

4. Flavour symmetry breaking and mass hierarchies

The $$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$7 model embeds flavour sphalerons within a full flavour theory. The symmetry is broken by scalar triplets

$$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$8

with hierarchical vacuum expectation values

$$B=-\frac{16}{61}L_0,\qquad L=\frac{39}{61}L_0,\qquad X=-\frac{6}{61}L_0,$$9

The symmetry-breaking sequence is

$$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$0

with gauge-boson masses approximately

$$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$1

The scalar potential includes quadratic terms, a cubic $$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$2-term, quartic flavon interactions, and a Higgs-flavon portal

$$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$3

which must be suppressed to avoid large Higgs mass corrections (Blennow et al., 19 May 2026).

Fermion masses arise through a seesaw-like structure involving the mirror fermions. Representative terms include

$$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$4

with analogous terms in the down-quark and charged-lepton sectors. After integrating out the heavy mirror fermions, the effective Yukawas satisfy

$$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$5

Thus the lighter generations are associated with larger flavon scales and more suppressed Yukawas (Blennow et al., 19 May 2026).

The group assignments also shape flavour structure. Because $$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$6 and $$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$7 are flavour triplets while $$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$8 are not, the model naturally yields hierarchical quark masses and CKM angles, hierarchical charged-lepton masses, and anarchical PMNS structure. The dark sector follows the same organizing principle: the dark fermions $$\frac{X}{B}=\frac{3}{8},\qquad m_{\rm DM}=13.4\pm0.2~\text{GeV}$$9 are charged under both $SO(3)_F$0 and a confining $SO(3)_F$1, and if a dark-sector seesaw-like suppression is realized, the effective dark Yukawas can be small so that dark baryon masses are dominantly set by confinement. In this sense, flavour sphalerons are not an isolated cosmological ingredient; they are part of a unified flavour-and-dark-sector construction (Blennow et al., 19 May 2026).

5. Thermal conditions, freeze-out, and constraints

For flavour sphalerons to redistribute the primordial asymmetry, they must thermalize before flavour breaking. The rate estimate is

$SO(3)_F$2

and requiring it to exceed the Hubble rate gives roughly

$SO(3)_F$3

This is stated to ensure that flavour sphalerons remain active above the flavour-breaking scale, corresponding to thermalization above roughly

$SO(3)_F$4

The model therefore ties the viability of flavour sphalerons directly to the gauge coupling and the highest flavour-breaking scale (Blennow et al., 19 May 2026).

The flavour-breaking scales are constrained by flavour and electroweak observables. Meson oscillations provide the dominant bounds: kaon mixing gives the strongest bound on $SO(3)_F$5, $SO(3)_F$6 mixing constrains $SO(3)_F$7, and rare LFV decays and electroweak observables constrain $SO(3)_F$8. The benchmark values quoted are

$SO(3)_F$9

These are presented as compatible both with flavour-sphaleron thermalization and with cosmological requirements (Blennow et al., 19 May 2026).

A further cosmological requirement is the removal of the symmetric dark matter component. After $SU(2)_L$0 breaking, direct annihilation through flavour gauge bosons is too suppressed, so the symmetric component forms dark mesons that must decay before BBN. An example decay is

$SU(2)_L$1

with lifetime estimate

$SU(2)_L$2

This leads to an approximate upper bound

$SU(2)_L$3

notably close to the lower bound from flavour physics. The model is therefore constrained from both above and below by the same flavour sector (Blennow et al., 19 May 2026).

Earlier horizontal-symmetry models exhibited the same general interplay. In the $SU(2)_L$4 construction, dark sphalerons were required to satisfy

$SU(2)_L$5

while dark meson decay before BBN required roughly

$SU(2)_L$6

At the same time, the flavour-changing constraint from

$SU(2)_L$7

implied

$SU(2)_L$8

motivating staged breaking

$SU(2)_L$9

followed by lower-scale breaking of the residual $SO(3)_F$0 (Blennow et al., 2010). This earlier result shows that flavour-sphaleron model building is structurally tied to FCNC control and to the fate of the symmetric dark relic.

6. Relation to dark sphalerons, electroweak sphalerons, and searches

Flavour sphalerons are part of a wider family of non-Standard-Model sphaleron mechanisms for asymmetric dark matter. In “Aidnogenesis via Leptogenesis and Dark Sphalerons,” the new sphalerons are associated with a broken horizontal symmetry $SO(3)_F$1, and the explicit anomaly-free model satisfies

$SO(3)_F$2

so that with ordinary electroweak sphalerons the conserved quantity is

$SO(3)_F$3

The chemical-potential analysis gives

$SO(3)_F$4

and predicts

$SO(3)_F$5

A distinct unified scenario based on $SO(3)_F$6 instead uses sphalerons with

$SO(3)_F$7

yielding

$SO(3)_F$8

and, for the minimal model, $SO(3)_F$9 with dark matter mass near $B+L$0 GeV (Blennow et al., 2010, Barr et al., 2013).

These constructions are closely related to flavour sphalerons but are not identical to the explicit $B+L$1 case. The common structure is the coexistence of ordinary electroweak sphalerons with an additional non-abelian sphaleron system that acts on dark-sector charges. What distinguishes flavour sphalerons in the strict sense is that the extra gauge interaction is itself a gauged flavour symmetry and simultaneously organizes flavour hierarchies (Blennow et al., 19 May 2026).

Experimental searches in the cited literature do not directly target flavour sphalerons. The CMS search in proton-proton collisions at $B+L$2 is a search for black holes, string balls, and electroweak sphalerons in high-multiplicity final states using $B+L$3, and sets

$B+L$4

for the nominal electroweak threshold $B+L$5. The paper explicitly states that it does not discuss “flavour sphalerons” as a separate concept; its sphalerons are electroweak sphalerons of the Standard Model (Collaboration, 2018).

The same limitation applies to cosmic-ray proposals. The Pierre Auger study considers electroweak baryon- and lepton-number violating sphaleron processes in ultra-high-energy air showers, with sensitivity around

$B+L$6

from an $B+L$7-based analysis and the possibility of reaching a few microbarns in a dedicated study (Brooijmans et al., 2016). The broader topological review likewise states that it does not discuss flavour sphalerons in the particle-physics sense (Manton, 2019). A common misconception is therefore to identify any sphaleron search with a test of flavour sphalerons; the cited literature shows that the direct searches are presently for electroweak sphalerons, whereas flavour sphalerons remain a model-building and cosmological mechanism tied to gauged flavour dynamics.