2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/145172We revisit the Littlest Higgs model with $T$-parity, and discover a $Z_2$ symmetry on collective symmetry. It is dubbed collective parity ($\EuScript{C}$-parity). We demonstrate that $T$-parity is consistent with $\EuScript{C}$-parity. We further investigate the origin of the collective symmetry in the context of composite Higgs and find a new path to the UV complete theory of the model. In addition, we demonstrate that $T$-parity violating processes naturally take place.14 pages, 2 figuresHigh Energy Physics - Phenomenologytext