Absence of $O(2)$ symmetry in the Vicsek model

The phase transition in the Vicsek model is widely believed to be associated with spontaneous symmetry breaking of the two-dimensional rotational symmetry $O(2)$. In this paper, we revisit the original Vicsek model introduced in [Phys. Rev. Lett. 75, 1226] and demonstrate that the model lacks $O(2)$ symmetry. As a consequence, we numerically demonstrate that the phase transition reported in the original paper vanishes when the global phase is adaptively chosen.