  In some quantum many particle systems, the fermions could form Cooper pairs
by exchanging intermediate bosons. This then drives a superconducting phase
transition or a superfluid transition. Such transitions should be theoretically
investigated by using proper non-perturbative methods. Here we take the neutron
superfluid transition as an example and study the Cooper pairing of neutrons
mediated by neutral $\pi$-mesons in the low density region of a neutron matter.
We perform a non-perturbative analysis of the neutron-meson coupling and
compute the pairing gap $\Delta$, the critical density $\rho_{c}$, and the
critical temperature $T_c$ by solving the Dyson-Schwinger equation of the
neutron propagator. We first carry out calculations under the widely used bare
vertex approximation and then incorporate the contribution of the lowest-order
vertex correction. This vertex correction is not negligible even at low
densities and its importance is further enhanced as the density increases. The
transition critical line on density-temperature plane obtained under the bare
vertex approximation is substantially changed after including the vertex
correction. These results indicate that the vertex corrections play a
significant role and need to be seriously taken into account.
