Revisiting Neutrino Self-Interaction Constraints from $Z$ and $\tau$
  decays
  Given the elusive nature of neutrinos, their self-interaction is particularly
difficult to probe. Nevertheless, upper limits on the strength of such an
interaction can be set by using data from terrestrial experiments. In this work
we focus on additional contributions to the invisible decay width of $Z$ boson
as well as the leptonic $\tau$ decay width in the presence of a neutrino
coupling to a relatively light scalar. For invisible $Z$ decays we derive a
complete set of constraints by considering both three-body bremsstrahlung as
well as the loop correction to two-body decays. While the latter is usually
regarded to give rather weak limits we find that through the interference with
the Standard Model diagram it actually yields a competitive constraint. As far
as leptonic decays of $\tau$ are concerned, we derive a first limit on neutrino
self-interactions that is valid across the whole mass range of a light scalar
mediator. Our bounds on the neutrino self-interaction are leading for $m_\phi
\gtrsim 300$ MeV and interactions that prefer $\nu_\tau$. Bounds on such
$\nu$-philic scalar are particularly relevant in light of the recently proposed
alleviation of the Hubble tension in the presence of such couplings.
