  Two-dimensional semi-Dirac fermions are quasiparticles that disperse linearly
in one direction and quadratically in the other. We investigate instabilities
of semi-Dirac fermions towards charge, spin-density wave and superconducting
orders, driven by short-range interactions. We analyze the critical behavior of
the Yukawa theories for the different order parameters using Wilson momentum
shell RG. We generalize to a large number $N_f$ of fermion flavors to achieve
analytic control in 2+1 dimensions and calculate critical exponents at one-loop
order, systematically including $1/N_f$ corrections. The latter depend on the
specific form of the bosonic infrared propagator in 2+1 dimensions, which needs
to be included to regularize divergencies. The $1/N_f$ corrections are
surprisingly small, suggesting that the expansion is well controlled in the
physical dimension. The order-parameter correlations inherit the electronic
anisotropy of the semi-Dirac fermions, leading to correlation lengths that
diverge along the spatial directions with distinct exponents, even at the
mean-field level. We conjecture that the proximity to the critical point may
stabilize novel modulated order phases.
