  We investigate the competition between d-wave superconductivity and nematic
order in high-$T_c$ superconductor, and examine the role played by gapless
fermionic degrees of freedom. Apart from the competitive interaction with the
superconducting order parameter, the nematic order parameter couples strongly
to gapless nodal quasiparticles. The interplay of these two kinds of
interactions is analyzed by means of renormalization group method. In case the
fermionic degree of freedom are entirely neglected, the competitive interaction
between two bosonic order parameters is strongly relevant, and can lead to
runaway behavior. However, these properties are fundamentally changed once the
dynamics of fermions are taken into account. At the nematic quantum critical
point where an extreme fermion velocity anisotropy occurs, the superconducting
and nematic order parameters are decoupled from each other. Consequently, the
phase transitions are continuous, and d-wave superconductivity can coexist with
nematic order homogeneously. These results indicate that the gapless fermions
can play an important role and should be carefully included in the theoretical
description of competing orders.
