Positivity constraints on aQGC: carving out the physical parameter space

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.