Constraints on the Higgs self-coupling at the LHC with $\sqrt{s}=13$ TeV and Long-Lived Particles searches with a future lepton collider

The Standard Model (SM) of particle physics provides a comprehensive framework that has been experimentally verified to an impressive degree, with the Higgs boson as a pivotal element. Investigations into di-Higgs interactions offer vital insights into the self-coupling behavior of the Higgs boson and potentially elucidate the mechanisms underlying electroweak symmetry breaking. Moreover, the Higgs boson acts as a prospective avenue for exploring a multitude of theories beyond the Standard Model, with Long-Lived Particles (LLPs) representing just one of several intriguing models yet to be fully explored. In the dissertation's first analysis, attention is centered on the production of SM Higgs boson pairs ($HH$) decaying to multilepton final states, especially 3-lepton. Utilizing $140~\text{fb}^{-1}$ of proton-proton collision data at $\sqrt{s} = 13~\text{TeV}$ from the ATLAS detector's Run 2, the study predominantly explores $HH$ production via gluon-gluon fusion, with Vector Boson Fusion contributing additional signal yield. Critical to this work is the mitigation of the primary $WZ$ background, addressed through reweighting to account for mismodeling at high jet multiplicities, and the secondary fake-lepton background, estimated via the Template Fit method. A Gradient Boosted Decision Tree is employed for optimal signal-background discrimination, further refined through a 3-fold training technique and exhaustive hyperparameter tuning. Remarkably, this 3-lepton channel achieves an upper limit of $28.09^{+12.81}_{-7.86}$ on the $HH$ cross-section over SM , constituting a 9.4-fold enhancement over previous analyses. When considering all multilepton channels, the expected upper limit stands at $9.74^{+13.91}_{-7.02}$, making the 3-lepton channel result as one of the best limits among pure leptonic decay channels. The second key facet of this dissertation revolves around the $ HH \rightarrow b\bar{b}\tau\tau $ analysis, building on 140 fb$^{-1}$ of Full Run 2 ATLAS data. This work distinguishes itself by the targeted refinement of methodological approaches, particularly concerning the $ \kappa_\lambda $ and $ \kappa_{2V} $ coupling modifiers. Employing advanced Multivariate Analysis (MVA) techniques, the analysis optimizes for both gluon-gluon fusion and Vector Boson Fusion production modes. Specifically, event categorization is influenced by the invariant mass ($ m_{\text{HH}} $) of the Higgs pair in the ggF region, with an additional VBF category introduced to enhance sensitivity to $ \kappa_{2V} $. Utilizing MVA outputs as the ultimate discriminants has engendered significant improvements over legacy data: a 17\% boost in the systematic-adjusted baseline for $ \mu_{\text{HH}} $, and 11.9\% and 19.8\% enhancements in the 95\% confidence intervals for $ \kappa_\lambda $ and $ \kappa_{2V} $, respectively. The HH+H combination analysis concludes the dissertation's SM di-Higgs investigations. This synthesis incorporates the primary di-Higgs channels ( \bbbb, \bbyy, and \bbtt ) with single-Higgs observables to impose stringent constraints on key coupling modifiers, specifically $ \kappa_\lambda $ and $ \kappa_{2V} $. In a comprehensive fit, where $ \kappa_t $, $ \kappa_b $, $ \kappa_{\tau} $, and $ \kappa_V $ are allowed to vary freely, the analysis attains a 95\% confidence interval of $ -1.4 < \kappa_{\lambda} < 6.1 $, which closely approximates the expected range of $ -2.2 < \kappa_{\lambda} < 7.7 $. The stability of these results is corroborated by a minimal sensitivity to $ \kappa_{2V} $ fluctuations, affecting the observed $ \kappa_\lambda $ constraints by less than 5\%. This substantiates the concordance of all analyzed coupling modifiers with the SM, within the limits of the associated uncertainties. In the later part of the dissertation, the investigation transitions to scrutinizing decay modes of the Higgs boson in the realm of BSM physics, specifically targeting LLPs. To this end, state-of-the-art machine learning algorithms, such as Convolutional Neural Networks (CNNs) and Graph Neural Networks (GNNs), are deployed for direct analysis on raw detector output, which significantly enhances the expected signal selection efficiency. For example, in the case of a 50~GeV LLP with a lifetime of 1~ns, the expected signal efficiency reaches 99\%. This research achieves an expected upper limit of $4 \times 10^{-6}$ for the branching ratio of Higgs decaying to LLPs under the assumption of $10^6$ Higgs, which significantly outperforms the upper limit of $1 \times 10^{-3}$ currently established by ATLAS and CMS, For LLPs with lifetimes greater than 1~ns, the analysis yields an expected upper limit that is one order of magnitude better than the expected results from the International Linear Collider. Last but not least, the appendix section of the dissertation describes a preliminary study on dark photon searches via a proposed fixed target experiment with 8 GeV electrons recoiling from a High-Z metal target.