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The positivity bounds, derived from the axiomatic principles of quantum fieldtheory (QFT), constrain the signs of Wilson coefficients and their linearcombinations in the Standard Model Effective Field Theory (SMEFT). The precisedetermination of these bounds, however, can become increasingly difficult asmore and more SM modes and operators are taken into account. We study twoapproaches that aim at obtaining the full set of bounds for a given set of SMfields: 1) the traditional elastic positivity approach, which exploits theelastic scattering amplitudes of states with arbitrarily superposed helicitiesas well as other quantum numbers, and 2) the newly proposed extremal positivityapproach, which constructs the allowed coefficient space directly by using theextremal representation of convex cones. Considering the electroweakgauge-bosons as an example, we demonstrate how the best analytical andnumerical positivity bounds can be obtained in several ways. We further comparethe constraining power and the efficiency of various approaches, as well astheir applicability to more complex problems. While the new extremal approachis more constraining by construction, we also find that it is analyticallyeasier to use, numerically much faster than the elastic approach, and much moreapplicable when more SM particle states and operators are taken into account.As a byproduct, we provide the best positivity bounds on the transversalquartic-gauge-boson couplings, required by the axiomatic principles of QFT, andshow that they exclude $\approx 99.3\%$ of the parameter space currently beingsearched at the LHC.

Elastic positivity vs extremal positivity bounds in SMEFT: a case study in transversal electroweak gauge-boson scatterings