Probing sphalerons: energy scale, cross-section, and event rate predictions*

Sphalerons are unstable, static, finite-energy solutions of classical field equations that facilitate baryon and lepton number-violating transitions, making them essential in the context of many aspects of electroweak theory, symmetry breaking, and baryogenesis. These non-perturbative topological field configurations could provide critical insights into matter-antimatter asymmetry in the universe and offer direct phenomenological evidence for Beyond Standard Model physics. Current estimates place the sphaleron energy barrier height between 9 and 20 TeV, but there remains uncertainty due to higher-order electroweak corrections, influence of the Higgs potential, and thermal effects. Studies in the last decade have revealed that periodic sphaleron potential may lead to pass-bands in the energy barrier, and to exotic high-multiplicity lepton, jet and/or vector boson signatures that violate both lepton number and baryon number, with experimentally accessible cross sections starting at the LHC center of mass energy of 13 TeV and above. This research refines these theoretical estimates, develops collider-search strategies, and assesses the feasibility of detecting sphalerons at high-energy colliders such as the Large Hadron Collider, High Luminosity LHC, and Future Circular Collider. It integrates advanced Monte Carlo simulations, Bayesian inference methods, LHC Run 2-3 datasets to probe potential sphaleron-induced signatures, setting upper limits on their production cross-section. It involves event reconstruction techniques using BaryoGEN, HERBVI, HERWIG, Pythia to differentiate sphaleron-like events from Standard Model backgrounds. Further, sensitivity projections for next-generation colliders are generated, evaluating the required energy thresholds and luminosity constraints for sphaleron discovery. In the absence of positive detection, results from this research would still yield crucial constraints on sphalerons, shaping future experimental and theoretical investigations in high-energy physics. By combining collider phenomenology, computational modeling, early-universe constraints, it offers a path for sphaleron detection.