  We revisit the celebrated Wilemski-Fixman (WF) treatment for the looping time
of a free-draining polymer. The WF theory introduces a sink term into the
Fokker-Planck equation for the $3(N+1)$-dimensional Ornstein-Uhlenbeck process
of the polymer dynamics, which accounts for the appropriate boundary condition
due to the formation of a loop. The assumption for WF theory is considerably
relaxed. A perturbation method approach is developed that justifies and
generalizes the previous results using either a Delta sink or a Heaviside sink.
For both types of sinks, we show that under the condition of a small
dimensionless $\epsilon$, the ratio of capture radius to the Kuhn length, we
are able to systematically produce all known analytical and asymptotic results
obtained by other methods. This includes most notably the transition regime
between the $N^2$ scaling of Doi, and $N\sqrt{N}/\epsilon$ scaling of Szabo,
Schulten, and Schulten. The mathematical issue at play is the non-uniform
convergence of $\epsilon\to 0$ and $N\to\infty$, the latter being an inherent
part of the theory of a Gaussian polymer. Our analysis yields a novel term in
the analytical expression for the looping time with small $\epsilon$, which is
previously unknown. Monte Carlo numerical simulations corroborate the
analytical findings. The systematic method developed here can be applied to
other systems modeled by multi-dimensional Smoluchowski equations.
