Compactification Without Orientation, or a Topological Scenario for CP Violation

TL;DR

This paper investigates a simple yet nontrivial non-orientable compactification by placing a free 6D Dirac fermion on ${\mathbb R}^{3,1} \times K_2$ with pin$^+$ and pin$^-$ boundary conditions. The Klein bottle induces translation breaking and two parity walls that host localized, symmetry-sensitive effects; depending on boundary choices ${\sf R}_4^+$ or ${\sf R}_4^-$, parity, charge conjugation, and CP are broken or preserved in distinct patterns, with fermion bilinears acting as order parameters peaked near the walls. Remarkably, CP violation can emerge in 3+1D from the bulk via wall-localized vevs, and a small-volume CP-violation scenario is proposed through KK-mode mixing and a bulk-to-brane coupling. The analysis also quantifies the Casimir-energy structure and wall tension, showing localized energy near parity walls and massless- versus massive-field contributions that differ between ${\sf R}_4^+$ and ${\sf R}_4^-$ cases. Overall, the work suggests that non-orientable compactifications offer a topological route to CP violation and potential baryogenesis, motivating further exploration in phenomenology and cosmology.

Abstract

In higher dimensional theories, we often assume that the extra dimensions form an orientable space, perhaps with singularities. However, many physical theories are well-defined on non-orientable spaces, and many spaces are not orientable, so it is reasonable to explore what happens if the assumption of orientability is relaxed. Here we consider the simplest example of free 6D theories compactified on a flat Klein bottle. We focus on a Dirac fermion in 6D, with boundary conditions that define ${\rm pin}^+$ and ${\rm pin}^-$ structures. Translation invariance is broken by the boundary conditions, which leads to sharp features localized near the parity walls (fixed points of the reflection used to construct the Klein bottle). For a scalar field, there is a position-dependent energy density, peaked near the parity walls. A Dirac fermion can lead to breaking of parity, charge conjugation and CP in 3+1 dimensions. Order parameters for this breaking are provided by the vevs of certain fermion bilinears, again peaked near the parity walls. As one potential application, these results suggest mechanisms for CP violation and baryogenesis, enabled by compactification on a Klein bottle.

Figures (7)

• Figure 1: Left: a Klein bottle of size $(2 \pi R_4, 2 \pi R_5)$ built from a single fundamental tile in the $(x^4,x^5)$ plane. The sides are identified as shown. The flip axis at $x^4 = 0$ is indicated by a vertical dashed line. The identified left and right sides, at $x^4 = \pm \pi R_4$, form a second flip axis. Right: the Klein bottle can be lifted to tile the entire plane. A fundamental tile, shaded in the lower left corner, and some of its images are shown. We have put symbols L, R on the Klein bottle to help keep track of orientation. Note that, as an intermediate step, the Klein bottle can be lifted to a covering torus of size $(2 \pi R_4, 4 \pi R_5)$.
• Figure 2: Top panel: the operations that perform ${\sf R}_x {\sf R}_\theta = {\sf R}_x {\sf D}(\theta) {\sf R}_x {\sf D}(-\theta)$ are equivalent to a rotation ${\sf D}(-2\theta)$. Bottom panel: the operations that perform ${\sf R}_\theta {\sf R}_x = {\sf D}(\theta) {\sf R}_x {\sf D}(-\theta) {\sf R}_x$ are equivalent to a rotation ${\sf D}(2\theta)$.
• Figure 3: The wall function $W^+$ in (\ref{['W+']}) for a massless field, shown as a function of $x^4/R_4$. Left: $R_4 = 4$ and $R_5 = 1$. Right: $R_4 = 10$ and $R_5 = 1$. When $R_4 \gg R_5$, the peaks near the origin are well-separated from those near the $x^4=\pm \pi R_4$ edges, and $W^+$ is small in the region between the peaks.
• Figure 4: In red, the no-winding term $w_4 = w_5 = 0$ in the sum (\ref{['eq:wall6D']}) for $W^+(x^4)$. Compare to the blue line, which sums over $w_4 = {-1,0,1}$ with $w_5 = 0$. (The plot is for $R_4 = 10$, $R_5 = 1$). A second double-bumped wall has appeared around the identified axes $x_4=\pm {\pi}R_4$.
• Figure 5: A plot of $W^-(x^4)$ vs. $x^4/R_4$, for a massless field with $R_4 = 3$ and $R_5 = 1$.