  Moments of the leptonic angular distribution in the Drell-Yan process have
recently been shown to be sensitive probes of a specific class of dimension-8,
four-fermion operators in the Standard Model Effective Field Theory, involving
a pair of quarks and leptons. The same operators are also subject to positivity
bounds, when requiring the associated (unknown) UV completion to obey basic
principles of quantum field theory. We perform a phenomenological study to
quantify the sensitivity of the high-luminosity LHC to this set of operators
and, by extension, the positivity bounds. We further extend the angular basis
of moments and consider double differential information to improve the ability
to disentangle the different operators, leading to a sensitivity to new physics
scales up to 3 TeV. We use this information to explore the violation of
positivity at the LHC as a way to test the underlying principles of quantum
field theory. Finally, we present a case study which combines our results with
information from other (current and prospective) experiments, as well as the
positivity cone to infer the properties of possible tree-level UV completions.
The data lead to robust, model-independent lower bounds on the $M/\sqrt{g}$
combination of the particle mass and coupling, for states that couple to
right-handed leptons and/or up quarks.
