• effective field theories
• positivity bounds
• quantum field theory
• S-matrix bootstrap

In this thesis we explore constraints on the possible 2 -> 2 scattering amplitudes in an Effective Field Theory (EFT), which is completable into a ultraviolet theory compatible with the basic principles of Poincaré invariance, causality and unitarity.
In the framework of the S-matrix bootstrap, we exploit dispersive arguments and the positivity of the imaginary part of partial waves to derive "sum rules" that the Wilson coefficients of the EFT have to satisfy.
We first review numerical bounds for an EFT that contains a real massless scalar, both in presence and absence of gravity.
We then generalize the procedure to the case of a complex scalar; we perform the numerics in absence of graviton and photon exchanges and obtain interesting exclusion plots in the space of Wilson coefficients, and we analyze their features. These confirm the expect scaling of the couplings in powers of the cutoff mass, up to order 1 coefficients. We finally compute the sum rules suitable for a numerical work that will look for bounds in presence of both gravity and photons.