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A set of tidal Love numbers quantifies tidal deformation of compact objects and is a detectable imprint in gravitational waves from inspiraling binary systems. The measurement of black hole Love numbers allows us 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 general relativity 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.
Katagiri et al. (Tue,) studied this question.