Can Fingers of God be Resummed?

Fingers of God (FoGs), Doppler shifts due to the small-scale virial motions of galaxies, are one of the most significant challenges for analytic models of galaxy clustering in redshift surveys. In the effective field theory approach, the characteristic scale of these shifts introduces a new, large length scale, limiting the reach of the theory. This has motivated many phenomenological models of FoGs in the literature where their effect is approximated by a single damping function of the line-of-sight wavenumber λ=kμ, effectively resumming powers of λ due to single-point contractions of small-scale velocities. The goal of this work is to investigate whether such a resummation can be rigorously constructed. We show that common damping models can be understood as the mean-field limit of the single-particle characteristic function (SPCF) of FoG velocities, but that the environmental dependence of the SPCF generates equally large corrections that break the simple damping form. We develop the long-wavelength theory of the SPCF, showing that it admits an expansion in anisotropic bias operators closely related to those derived for selection effects, but with λ-dependent coefficients satisfying selection rules. We estimate these coefficients using the halo model for DESI-like galaxies, showing that they make significant contributions at the level of the 2- and 3-point functions of galaxies. To answer the titular question: potentially, but at the cost of significantly expanding the operator basis and promoting free coefficients to free functions.