  We present a scattering amplitude formalism to study the tidal heating
effects of nonspinning neutron stars incorporating both worldline effective
field theory and relativistic stellar perturbation theory. In neutron stars,
tidal heating arises from fluid viscosity due to various scattering processes
in the interior. It also serves as a channel for the exchange of energy and
angular momentum between the neutron star and its environment. In the interior
of the neutron star, we first derive two master perturbation equations that
capture fluid perturbations accurate to linear order in frequency. Remarkably,
these equations receive no contribution from bulk viscosity due to a peculiar
adiabatic incompressibility which arises in stellar fluid for non-barotropic
perturbations. In the exterior, the metric perturbations reduce to the
Regge-Wheeler (RW) equation which we solve using the analytical
Mano-Suzuki-Takasugi (MST) method. We compute the amplitude for gravitational
waves scattering off a neutron star, also known as gravitational Raman
scattering. From the amplitude, we obtain expressions for the electric
quadrupolar static Love number and the leading dissipation number to all orders
in compactness. We then compute the leading dissipation number for various
realistic equation-of-state(s) and estimate the change in the number of
gravitational wave cycles due to tidal heating during inspiral in the
LIGO-Virgo-KAGRA (LVK) band.
