Virial coefficients of trapped and un-trapped three-component fermions with three-body forces in arbitrary spatial dimensions

Using a coarse temporal lattice approximation, we calculate the first few terms of the virial expansion of a three-species fermion system with a three-body contact interaction in $d$ spatial dimensions, both in homogeneous space as well as in a harmonic trapping potential of frequency $\omega$. Using the three-body problem to renormalize, we report analytic results for the change in the fourth- and fifth-order virial coefficients $\Delta b_4$ and $\Delta b_5$ as functions of $\Delta b_3$. Additionally, we argue that in the $\omega \to 0$ limit the relationship $b_n^\text{T} = n^{-d/2} b_n$ holds between the trapped (T) and homogeneous coefficients for arbitrary temperature and coupling strength (not merely in scale-invariant regimes). Finally, we point out an exact, universal (coupling- and frequency-independent) relationship between $\Delta b_3^\text{T}$ in 1D with three-body forces and $\Delta b_2^\text{T}$ in 2D with two-body forces.