  Searching for deviations in quartic gauge boson couplings (QGCs) is one of
the main goals of the electroweak program at the LHC. We consider positivity
bounds adapted to the Standard Model, and show that a set of positivity
constraints on 18 anomalous QGC couplings can be derived, by requiring that the
vector boson scattering amplitudes of specific channels and polarisations
satisfy the fundamental principles of quantum field theory. We explicitly solve
the positivity inequalities to remove their dependence on the polarisations of
the external particles, and obtain 19 linear inequalities, 3 quadratic
inequalities, and 1 quartic inequality that only involve the QGC parameters and
the weak angle. These inequalities constrain the possible directions in which
deviations from the standard QGC can occur, and can be used to guide future
experimental searches. We study the morphology of the positivity bounds in the
parameter space, and find that the allowed parameter space is carved out by the
intersection of pyramids, prisms, and (approximately) cones. Altogether, they
reduce the volume of the allowed parameter space to only 2.1% of the total. We
also show the bounds for some benchmark cases, where one, two, or three
operators, respectively, are turned on at a time, so as to facilitate a quick
comparison with the experimental results.
