  Trial wavefunctions like the Moore-Read and Read-Rezayi states which minimize
short range multibody interactions are candidate states for describing the
fractional quantum Hall effects at filling factors $\nu = 1/2$ and $\nu = 2/5$
in the second Landau level. These trial wavefunctions are unique zero energy
states of three body and four body interaction Hamiltonians respectively but
are not close to the ground states of the Coulomb interaction. Previous studies
using extensive parameter scans have found optimal two body interactions that
produce states close to these. Here we focus on short ranged four body
interaction and study two mean field approximations that reduce the four body
interactions to two body interactions by replacing composite operators with
their incompressible ground state expectation values. We present the results
for pseudopotentials of these approximate interactions. Comparison of finite
system spectra of the four body and the approximate interactions at filling
fraction $\nu=3/5$ show that these approximations produce good effective
descriptions of the low energy structure of the four body ineraction
Hamiltonian. The approach also independently reproduces the optimal two body
interaction inferred from parameter scans. We also show that for $n=3$, but not
for $n=4$, the mean field approximations of the $n$-body interaction is
equivalent to particle hole symmetrization of the interaction.
