Search for di-Higgs production and measurement of the Higgs boson self-coupling in the final state with a pair of b quarks and a pair of tau leptons with the ATLAS detector at the LHC, Perspectives on the measurement of the Higgs boson mass and the electron-positron to ZH cross-section at the Future Circular Collider

After the discovery of the Higgs boson in 2012 by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC), precision measurements of the Higgs boson mass and couplings have taken center stage, serving both the validation of the Standard Model (SM) and the exploration for new physics Beyond the Standard Model (BSM). After an introduction to the Standard Model and to the experimental apparatus and tools, the thesis describes the calibration of the $b$-tagging efficiency within ATLAS. The $p_{\text{T}}^{\text{rel}}$ method, which uses muon-in-jets events, is employed here with the DL1r tagger for the first time. In comparing the $p_{\text{T}}^{\text{rel}}$ calibration to the $t\bar{t}$ calibration, the primary method at ATLAS, a clear agreement between the two was observed. The search for double Higgs boson (di-Higgs) production is then the major component of this thesis. This process will give access to the Higgs self-coupling and Higgs potential. This thesis utilizes ATLAS Run 2 data to investigate di-Higgs production through the $HH\rightarrow b\bar{b}\tau^+\tau^-$ decay channels. No significant excess above the expected background from Standard Model processes is observed. For this channel, the observed (expected) upper limit for the di-Higgs production cross-section obtained in our study is $5.9$ ($3.1$) times the SM prediction at a 95% confidence level. The Higgs self-coupling parameter, $\kappa_{\lambda}$, has an observed (expected) 95% confidential interval of $-3.2 < \kappa_{\lambda} < 9.1$ ($-2.5 < \kappa_{\lambda} < 9.2$). Moreover, the 95% confidential interval of di-Higgs to two weak bosons ($HHVV$) coupling parameter, $\kappa_{2V}$, is observed (expected) to be constrained between $-0.4 < \kappa_{2V} < 2.6$ ($-0.2 < \kappa_{2V} < 2.4$). The last part of the thesis introduces the Future Circular Collider, the proposed successor to the LHC. It evaluates the physics feasibility of the FCC-ee, its electron-positron phase, to measure the Higgs mass and the ZH cross-section. By applying the recoil mass method with Z boson decays to muon or electron pairs, the Higgs mass can be measured with a precision of 4.0 MeV, and the ZH cross-section can be determined with an accuracy of $0.69\%$.