Hierarchies without Symmetries from Extra Dimensions

TL;DR

This work shows that small fermion Yukawas and proton stability can arise from geometric separation of Standard Model fermions in a thick extra-dimensional wall, rather than from high-energy flavor or baryon-number symmetries. By localizing fermions at different points along the fifth dimension while allowing gauge fields to propagate through the wall, the resulting 4D couplings are suppressed by the overlaps of Gaussian wave functions, e.g. $e^{-rac{1}{2}\mu^{2} r^{2}}$ for Yukawas and $e^{-rac{3}{4}\mu^{2} r^{2}}$ for proton-decay operators. A model-independent prediction is that Standard Model fermions acquire non-universal couplings to Kaluza-Klein gauge excitations, which in turn enables cartography of the fermion locations at colliders if the wall thickness is near the TeV scale. The framework naturally accounts for Yukawa hierarchies and proton stability without invoking flavor symmetries, and its collider implications—such as non-universal KK couplings and potential atomic parity-violation signals—provide a practical path to testing the geometry-driven mechanism.

Abstract

It is commonly thought that small couplings in a low-energy theory, such as those needed for the fermion mass hierarchy or proton stability, must originate from symmetries in a high-energy theory. We show that this expectation is violated in theories where the Standard Model fields are confined to a thick wall in extra dimensions, with the fermions "stuck" at different points in the wall. Couplings between them are then suppressed due to the exponentially small overlaps of their wave functions. This provides a framework for understanding both the fermion mass hierarchy and proton stability without imposing symmetries, but rather in terms of higher dimensional geography. A model independent prediction of this scenario is non-universal couplings of the Standard Model fermions to the ``Kaluza-Klein'' excitations of the gauge fields. This allows a measurement of the fermion locations in the extra dimensions at the LHC or NLC if the wall thickness is close to the TeV scale.

Figures (8)

• Figure 1: Profile of Standard Model fermion wave functions (vertical axis) in the extra dimensions (horizontal axis). The fermions freely propagate in 3+1 dimensions (not shown) and are "stuck" at different locations in the extra dimensions. The gauge and Higgs fields' wave functions occupy the whole width of the thick wall. Direct couplings between the fermions are then suppressed by the exponentially small overlap of their wave functions. If -- as shown here -- quarks and leptons live on opposite ends of the wall profile protons become essentially stable. The hierarchy of Yukawa couplings arises from order one (in units of the fermion wave function width) distances between left and right handed components of the fermions.
• Figure 2: Profile of the scalar domain wall field $\Phi$ in the $x_5$ dimension. A chiral zero mode fermion is localized at the zero of $\Phi$.
• Figure 3: Yukawa coupling: the Gaussian wave functions of the fermions $l$ and $e^c$ overlap only in an exponentially small region, suppressing the effective Yukawa coupling exponentially.
• Figure 4: Tree-level proton decay diagram, drawn in position space for the fifth dimension. The quarks stuck at one end of the wall and the lepton stuck at the other end propagate to some interior point s where they interact via the higher-dimensional QQQL operator. The "free" propagator to go to a point in the bulk is given by the value of the (Gaussian) zero-mode wave function at that point.
• Figure 5: General proton decay diagram including higher order effects. The blobs on propagators denotes the all-order propagators, and the blob on the vertex denotes the corrected vertex. The corrected propagator is nothing other than the corrected zero-mode wave function. The corrected vertex is still local on scales larger than the width of the fermion wave function. This diagram is therefore well-approximated by the overlap between the corrected wave functions of the quark and lepton zero modes, which gives the enormous suppression of proton decay.