  In this work, we present an effective field theory to describe a
two-component Fermi gas near a $d$-wave interaction resonance. The effective
field theory is renormalizable by matching with the low energy $d$-wave
scattering phase shift. Based on the effective field theory, we derive
universal properties of the Fermi gas by the operator product expansion method.
We find that beyond the contacts defined by adiabatic theorems, the asymptotic
expressions of the momentum distribution and the Raman spectroscopy involve two
extra contacts which provide additional information of correlations of the
system. Our formalism sets the stage for further explorations of many-body
effects in a $d$-wave resonant Fermi gas. Finally we generalise our effective
field theory for interaction resonances of arbitrary higher partial waves.
