Analysis note: two-particle correlation in $e^+e^-$ collisions at 91-209 GeV with archived ALEPH data

The first measurement of two-particle angular correlations for charged particles produced in $e^+e^-$ annihilation up to $\sqrt{s}$ = 209 GeV is presented. Hadronic $e^+e^-$ data, archived at center-of-mass energies ranging from 91 to 209 GeV, were collected using the ALEPH detector at LEP between 1992 and 2000. The angular correlation functions have been measured across a wide range of pseudorapidities and the full azimuth in bins of charged particle multiplicity. This is the first such measurement using LEP-II data. With LEP-II data at 91 GeV, neither the beam coordinate analysis nor the thrust coordinate analysis reveals significant long-range correlations, consistent with the finding in the previous measurement with the LEP-I sample. Results for $e^+e^-$ data at energies above 91 GeV, which allow for higher event multiplicities reaching approximately 50, are presented for the first time. A long-range near-side excess in the correlation function has been identified in the thrust axis analysis. Moreover, the two-particle correlation functions were decomposed using a Fourier series, and the resulting Fourier coefficients $v_n$ were compared with event generator outputs. In events with high multiplicity, featuring more than 50 particles, the extracted $v_2$ and $v_3$ magnitudes from the data are higher than those from the Monte Carlo reference.