Parametrized Love numbers of non-rotating black holes

A set of tidal Love numbers quantifies tidal deformation of compact objects and is a detectable imprint in gravitational waves from inspiralling binary systems. The measurement of black hole Love numbers allows to test strong-field gravity. In this paper, we present a parametrized formalism to compute the Love numbers of static and spherically symmetric black hole backgrounds, connecting the underlying equations of a given theory with detectable quantities in gravitational-wave observations in a theory-agnostic way. With this formalism, we compute the Love numbers in several systems. We further classify black hole Love numbers according to whether they vanish, are nonzero, or are ``running'' (scale-dependent), in theories or backgrounds that deviate perturbatively from the GR values. The construction relies on static linear perturbations and scattering theory. Our analytic and numerical results are in excellent agreement. As a side result, we show how to use Chandrasekhar's relations to relate basis of even parity to odd parity.