On the Apparent Correlation between X-ray and Neutrino Luminosities of Active Galactic Nuclei

Recent studies have reported a linear correlation between the hard X-ray and high-energy neutrino luminosities of active galactic nuclei (AGN), suggesting a possible physical connection between these two messengers. In this work, we challenge this interpretation by demonstrating that the observed correlation may arise purely from selection effects. We analyze 10 years of IceCube public data for a sample of Seyfert galaxies and blazars from the \textit{Swift} BAT catalog. While our data reproduces the apparent $L_\nu$--$L_X$ correlation for sources with mild (but not significant) neutrino evidence, we show through Monte Carlo simulations that the same correlation appears even when analyzing random sky positions with no astrophysical sources. The key issue is that TS-based source selection effectively restricts the neutrino flux to a narrow range (a factor of several), while the luminosity distance of the sample spans $\sim4$ orders of magnitude. This causes the luminosity $L = 4\pi D_L^2 F$ to be dominated by the distance term rather than intrinsic flux variations, creating an artificial correlation. While a robust flux correlation ($F_\nu$--$F_X$) for high-significance sources may indicate a genuine physical link, our results demonstrate that a luminosity-luminosity correlation alone is insufficient to establish a physical relationship between neutrino and X-ray emission in AGN.