Mapping out EFTs with analytic S-matrix

The aim of this thesis is to apply bootstrap methods based on scattering amplitudes in the context of low-energy effective field theories (EFTs), with a focus on leading deviations from the pure Standard Model (SM) of particle physics in the form of higher dimensional operators. It consists of a general introduction and two parts. General Introduction provides a quick review of the fundamental concepts such as the SM Lagrangian and its interpretation as an EFT; then the scattering matrix (S-matrix) and its general properties derived from compatibility with quantum mechanics and special relativity. Part I is dedicated to the physics of lepton flavor number violation (LFV) focusing on the processes: $\mu \to \gamma e$, $\mu \to eee$ and $\mu N \to e N$. We motivate why they are among the best probes of beyond the Standard Model (BSM) physics at and beyond TeV scales, given that the sensitivity on these observables will improve by up-to-four orders of magnitude in the next decade. LFV searches put stringent lower bounds on the energy scale of possible flavor violating new physics effects, rendering important understanding renormalization group running effects in the intepretation of null results. We perform a complete analysis of operator mixing chains up to two-loops in perturbation theory, powered by on-shell amplitude techniques. In Part II, we turn to nonperturbative physics of the scattering amplitude in the framework of S-matrix bootstrap in four dimensions. We demonstrate that nonlinear unitarity can bound the amplitude and its derivatives evaluated at low energies, even when they lack a dispersive representation. The overarching aim is to inject the low-energy EFT structure into the bootstrap algorithm and scan over the space of consistent EFTs, focusing on non-dispersive observables. We study the singlet and $O(n)$ flavored amplitudes, and provide bounds on the Higgs self-couplings in the custodial symmetric limit of the SM. We also establish contact with the positivity bounds on dispersive observables whenever we can.