Tropical refined curve counting via motivic integration

We propose a geometric interpretation of Block and G"ottsche’s refined tropical curve counting invariants in terms of virtual ${\chi }_{-y}$-specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that this interpretation is correct for linear series of genus 1, and in arbitrary genus after specializing from ${\chi }_{-y}$ to Euler characteristic.