Observation of flow vector fluctuations in p$-$Pb collisions at $\mathbf{\sqrt{\textit{s}_{_{\bf NN}}}}=$ 5.02 TeV

Measurements of transverse momentum ($p_{\rm T}$) and pseudorapidity ($\eta$) dependent flow vector fluctuations in p$-$Pb collisions at $\sqrt{s_{_{\rm NN}}} = 5.02$ TeV at the CERN Large Hadron Collider are presented. By studying long-range two-particle correlations with a template fit method, potential biases from non-flow effects such as jets and resonance decays are effectively suppressed. Significant $p_{\rm T}$- and $\eta$-dependent fluctuations of the second-harmonic flow vector are observed with more than 5$\sigma$ confidence in p$-$Pb collisions, similar to the observations in Pb$-$Pb collisions. The influence of residual non-flow effects has been evaluated and cannot account for the observed fluctuations, thereby confirming the observation of flow vector fluctuations in small collision systems at the LHC. Comparisons to model calculations from 3DGlauber+MUSIC+UrQMD and the parton transport model from AMPT are also presented. The measurements provide constraints on the theoretical modelling of the three-dimensional initial geometry and its event-by-event fluctuations, offering critical insights into the origin of collective flow in small collision systems at the LHC.