  We consider constraints on the S-matrix of any gapped, Lorentz invariant
quantum field theory in 1 + 1 dimensions due to crossing symmetry and
unitarity. In this way we establish rigorous bounds on the cubic couplings of a
given theory with a fixed mass spectrum. In special cases we identify
interesting integrable theories saturating these bounds. Our analytic bounds
match precisely with numerical bounds obtained in a companion paper where we
consider massive QFT in an AdS box and study boundary correlators using the
technology of the conformal bootstrap.
