Assisted baryon number violation from $4k+2$ dimensions
The charge conjugation operator in $4k+2$-dimensional space-time posses the peculiar property of preserving chiralities of the spinor fields its acting upon. This nature of $4k+2-$D fermions and its interactions, on compactifying the extra-dimension, leads to 4-dimensional operators with Kaluza Klein (KK) modes at the lowest order, specifically, in the context of baryon and lepton number violation. With the KK mass degeneracy lifted at 1-loop, these operators generate decays/annihilations of KK-1 fermion modes that violate baryon and lepton number. Moreover, including the fermion bilinear interaction with the broken polarization of the hypercharge gauge boson, namely the ``spineless adjoint scalar", these exotic operators generate proton decay and double proton annihilation. In the simplest scenario, this scalar field, being stable, become the Dark Matter (DM) candidate. Here, we discuss the $\Delta B=1 =\Delta L$ and $\Delta B=2=\Delta L$ operators, in $4k+2-$dimensions, in model independent effective field theory and derive the scale of New Physics using $p \to e^+ + \pi$, $DM+p\to DM+ e^+ + \pi$, $p+p \to e^+ +e^+$, $DM+p \to DM+ \bar{p}+e^++e^+$ and $(DM + e^-) + p \to (DM + e^+) + \bar{p} $ processes. Though the terrestrial experiments are not sensitive enough to Dark Matter influenced $\Delta B=2$, their rate can be enhanced substantially at super dense DM clumps and center of galaxy clusters where the Dark Matter distribution is much denser.