  The positivity bounds, derived from the axiomatic principles of quantum field
theory (QFT), constrain the signs of Wilson coefficients and their linear
combinations in the Standard Model Effective Field Theory (SMEFT). The precise
determination of these bounds, however, can become increasingly difficult as
more and more SM modes and operators are taken into account. We study two
approaches that aim at obtaining the full set of bounds for a given set of SM
fields: 1) the traditional elastic positivity approach, which exploits the
elastic scattering amplitudes of states with arbitrarily superposed helicities
as well as other quantum numbers, and 2) the newly proposed extremal positivity
approach, which constructs the allowed coefficient space directly by using the
extremal representation of convex cones. Considering the electroweak
gauge-bosons as an example, we demonstrate how the best analytical and
numerical positivity bounds can be obtained in several ways. We further compare
the constraining power and the efficiency of various approaches, as well as
their applicability to more complex problems. While the new extremal approach
is more constraining by construction, we also find that it is analytically
easier to use, numerically much faster than the elastic approach, and much more
applicable when more SM particle states and operators are taken into account.
As a byproduct, we provide the best positivity bounds on the transversal
quartic-gauge-boson couplings, required by the axiomatic principles of QFT, and
show that they exclude $\approx 99.3\%$ of the parameter space currently being
searched at the LHC.
