  A weakly interacting boson-fermion mixture model was investigated using
Wisonian renormalization group analysis. This model includes one boson-boson
interaction term and one boson-fermion interaction term. The scaling dimensions
of the two interaction coupling constants were calculated as 2-D at tree level
and the Gell-Mann-Low equations were derived at one-loop level. We find that in
the Gell-Mann-Low equations the contributions from the fermion loops go to zero
as the length scale approaches infinity. After ignoring the fermion loop
contributions two fixed points were found in 3 dimensional case. One is the
Gaussian fixed point and the other one is Wilson-Fisher fixed point. We find
that the boson-fermion interaction decouples at the Wilson-Fisher fixed point.
We also observe that under RG transformation the boson-fermion interaction
coupling constant runs to negative infinity with a small negative initial
value, which indicates a boson-fermion pairing instability. Furthermore, the
possibility of emergent supersymmetry in this model was discussed.
