  We perform complex analyses of the gluon propagator at nonzero quark chemical
potential in the long-wavelength limit, using an effective model with a gluon
mass term of the Landau-gauge Yang-Mills theory, which is a Landau-gauge limit
of the Curci-Ferrari model with quantum corrections being included within the
one-loop level. We mainly investigate complex poles of the gluon propagator,
which could be relevant to confinement. Around typical values of the model
parameters, we show that the gluon propagator has one or two pairs of complex
conjugate poles depending on the value of the chemical potential. In addition
to a pair similar to that in the case of zero chemical potential, a new pair
appears near the real axis when the chemical potential is roughly between the
effective quark mass and the effective gluon mass of the model. We discuss
possible interpretations of these poles. Additionally, we prove the uniqueness
of analytic continuation of the Matsubara propagator to a class of functions
that vanish at infinity and are holomorphic except for a finite number of
complex poles and singularities on the real axis.
