  We consider the weakly asymmetric exclusion process with $N=L/2$ particles on
a periodic lattice of $L$ sites, and hopping rates $1$ and $q=1-\mu/\sqrt{L}$
respectively in the forward and in the backward direction. Using Bethe ansatz,
we obtain a systematic perturbative expansion of the spectral gap near $\mu=0$
by solving order by order a simple functional equation. A key point is that
when $\mu\to0$, Bethe roots at a distance $1/\sqrt{L}$ from the edge of the
Fermi sea should not be considered as a continuum, but converge instead at
large $L$ to the complex zeroes of $1+\mathrm{erf}(x)$ after a rescaling by
$\sqrt{L}$.
