  Using unitarity, analyticity and crossing symmetry, we derive universal sum
rules for scattering amplitudes in theories invariant under an arbitrary
symmetry group. The sum rules relate the coefficients of the energy expansion
of the scattering amplitudes in the IR to total cross sections integrated all
the way up to the UV. Exploiting the group structure of the symmetry, we
systematically determine all the independent sum rules and positivity
conditions on the expansion coefficients. For effective field theories the
amplitudes in the IR are calculable and hence the sum rules set constraints on
the parameters of the effective Lagrangian. We clarify the impact of gauging on
the sum rules for Goldstone bosons in spontaneously broken gauge theories. We
discuss explicit examples that are relevant for WW-scattering, composite Higgs
models, and chiral perturbation theory. Certain sum rules based on custodial
symmetry and its extensions provide constraints on the Higgs boson coupling to
the electroweak gauge bosons.
