Mass Gap Revisited

In a theory with cosmologically varying masses, to some approximation there is no mass gap in the sense that a neutrino mass tends to zero in the early universe, at least from the point of view of a distant observer.

However, the standard model is only about local observations. And so as a matter of principle, given that we do not occupy the early universe, there must be a mass gap, given by the mass of the lightest neutrino, namely $0.00039$ eV.

In lecture 4B (you did watch it, right?) Arkani-Hamed discusses locality in terms of factorization for the coefficients of singularities that arise when internal lines in an old fashioned Feynman diagram go on-shell. This dictates how a proper four particle vertex with helicity labels should decompose into two three particle vertices. There are only two kinds of three particle vertex: (--+) and (-++).

It can be shown that factorization only allows two possible spin values, for a vertex with all legs having the same spin $s$. That is, either $s = 0$ or $s = 2$. Sound familiar? Spin $1$ Yang-Mills theory is OK if we mix spin values on the inputs. In that case, factorization says that the amplitudes must come from Lie group type structures. So the presence of Lie groups is derived from locality. They are not fundamental.