The observation of neutrons turning into antineutrons would constitute a discovery of fundamental importance for particle physics and cosmology. Observing the n−n transition would show that baryon number (B) is violated by two units and that matter containing neutrons is unstable. It would provide a clue to how the matter in our universe might have evolved from the B = 0 early universe. If seen at rates observable in foreseeable next-generation experiments, it might well help us understand the observed baryon asymmetry of the universe. A demonstration of the violation of B − L by 2 units would have a profound impact on our understanding of phenomena beyond the Standard Model of particle physics.Slow neutrons have kinetic energies of a few meV. By exploiting new slow neutron sources and optics technology developed for materials research, an optimized search for oscillations using free neutrons from a slow neutron moderator could improve existing limits on the free oscillation probability by at least three orders of magnitude. Such an experiment would deliver a slow neutron beam through a magnetically-shielded vacuum chamber to a thin annihilation target surrounded by a low-background antineutron annihilation detector. Antineutron annihilation in a target downstream of a free neutron beam is such a spectacular experimental signature that an essentially background-free search is possible. An authentic positive signal can be extinguished by a very small change in the ambient magnetic field in such an experiment. It is also possible to improve the sensitivity of neutron oscillation searches in nuclei using large underground detectors built mainly to search for proton decay and detect neutrinos. This paper summarizes the relevant theoretical developments, outlines some ideas to improve experimental searches for free neutron oscillations, and suggests avenues both for theoretical investigation and for future improvement in the experimental sensitivity.

This report, prepared for the Community Planning Study -Snowmass 2013 -summarizes the theoretical motivations and the experimental efforts to search for baryon number violation, focussing on nucleon decay and neutron-antineutron oscillations. Present and future nucleon decay search experiments using large underground detectors, as well as planned neutron-antineutron oscillation search experiments with free neutron beams are highlighted. OverviewBaryon Number, B, is observed to be an extremely good symmetry of Nature. The stability of ordinary matter is attributed to the conservation of baryon number. The proton and the neutron are assigned B = +1, while their antiparticles have B = −1, and the leptons and antileptons all have B = 0. The proton, being the lightest of particles carrying a non-zero B, would then be absolutely stable if B is an exactly conserved quantum number. Hermann Weyl formulated the principle of conservation of baryon number in 1929 primarily to explain the stability of matter [1]. Weyl's suggestion was further elaborated by Stueckelberg [2] and Wigner [3] over the course of the next two decades. The absolute stability of matter, and the exact conservation of B, however, have been questioned both on theoretical and experimental grounds. Unlike the stability of the electron which is on a firm footing as a result of electric charge conservation

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