Ultra-high energy cosmic rays, cosmological constant, and $\theta$-vacua

We propose that the origin of ultra-high energy cosmic rays beyond the GZK cutoff and the origin of small cosmological constant can both be explained by vacuum tunneling effects in a theory with degenerate vacua and fermionic doublets. By considering the possibility of tunneling from a particular winding number state, accompanied by violation of some global quantum number of fermions, the small value of the vacuum dark energy and the production of high energy cosmic rays are shown to be related. We predict that the energy of such cosmic rays should be at least $5\times {10}^{14}$ GeV.