  The discovery of new, flavor-dependent neutrino interactions would provide
compelling evidence of physics beyond the Standard Model. We focus on
interactions generated by the anomaly-free, gauged, abelian lepton-number
symmetries, specifically $L_e-L_\mu$, $L_e-L_\tau$, and $L_\mu-L_\tau$, that
introduce a new matter potential sourced by electrons and neutrons, potentially
impacting neutrino flavor oscillations. We revisit, revamp, and improve the
constraints on these interactions that can be placed via the flavor composition
of the diffuse flux of high-energy astrophysical neutrinos, with TeV-PeV
energies, i.e., the proportion of $\nu_e$, $\nu_\mu$, and $\nu_\tau$ in the
flux. Because we consider mediators of these new interactions to be
ultra-light, lighter than $10^{-10}$ eV, the interaction range is ultra-long,
from km to Gpc, allowing vast numbers of electrons and neutrons in celestial
bodies and the cosmological matter distribution to contribute to this new
potential. We leverage the present-day and future sensitivity of high-energy
neutrino telescopes and of oscillation experiments to estimate the constraints
that could be placed on the coupling strength of these interactions. We find
that, already today, the IceCube neutrino telescope demonstrates potential to
constrain flavor-dependent long-range interactions significantly better than
existing constraints, motivating further analysis. We also estimate the
improvement in the sensitivity due to the next-generation neutrino telescopes
such as IceCube-Gen2, Baikal-GVD, KM3NeT, P-ONE, and TAMBO.

High Energy Physics - Phenomenology;High Energy Astrophysical Phenomena;High Energy Physics - Experiment;Instrumentation and Detectors