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and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).
In this statement, a Yang–Mills theory is a non-Abelian quantum field theory similar to that underlying the Standard Model of particle physics;
Therefore, the winner must prove that:
Yang–Mills theory exists and satisfies the standard of rigor that characterizes contemporary mathematical physics, in particular constructive quantum field theory
It must have a “mass gap;” namely there must be some constant ∆ > 0 such that every excitation of the vacuum has energy at least ∆.
It must have “quark confinement,” that is, even though the theory is describedintermsofelementaryfields, suchasthequarkfields, thattransform non-trivially under SU(3), the physical particle states—such as the proton, neutron, and pion—are SU(3)-invariant.
It must have “chiral symmetry breaking,” which means that the vacuum is potentially invariant (in the limit, that the quark-bare masses vanish) only under a certain subgroup of the full symmetry group that acts on the quark fields.
