Extend the conserved-quantity and filtering analysis to higher codimension

Identify a conserved quantity for codimension n>2, such as the total angular momentum on the transverse sphere, and determine whether an analogous level-dependent filtering mechanism persists when radial and angular motion are coupled.

Background

The analysis in the paper depends on the flat transverse metric dr2+\gamma d\theta2 available only for codimension two. For n>2, the transverse space is a curved sphere, causing radial and angular sectors to couple and destroying separate conservation of the individual angular velocities. It remains unresolved whether another conserved quantity can support a corresponding mass-generation and localization-filtering mechanism.

References

Several directions remain open. First, the zero mode $l=0$ requires dedicated treatment, since $A_l$ is singular in this limit and the small-$r$ expansion of the effective potential must be revisited directly in terms of $\dot\theta\to0$. Second, it would be worthwhile to obtain the full localization profile analytically -- determining $r_{min}(l)$ and its rate of divergence as a function of $\gamma$ and the fermionic condensate $\xi_k\psi_kr(0)$ -- turning the qualitative filtering picture above into a quantitative localization law for the observable brane spectrum. Third, it would be worthwhile to identify a conserved quantity that survives for $n>2$, such as the total angular momentum on the transverse sphere, and to investigate whether an analogous -- even if not level-uniform -- filtering mechanism persists.

— Mass for Particles from Extra Dimensions  (2609.08697 - Souza et al., 8 Sep 2026) in Section 'Conclusions and Perspectives'