  The first moment the chirality-odd twist-3 parton distribution $e(x)$ is
related to the pion-nucleon $\sigma$-term which is important for phenomenology.
However, the possible existence of a singular contribution proportional to
$\delta(x)$ in the distribution prevents from the determination of the
$\sigma$-term with $e(x)$ from experiment. There are two approaches to show the
existence. The first one is based on an operator identity. The second one is
based on a perturbative calculation of a single quark state with finite quark
mass. We show that all contributions proportional to $\delta (x)$ in the first
approach are in fact cancelled. To the second approach we find that $e(x)$ of a
multi-parton state with a massless quark has no contribution with $\delta (x)$.
Considering that a proton is essentially a multi-parton state, the effect of
the contribution with $\delta(x)$ is expected to be suppressed by light quark
masses with arguments from perturbation theory. A detailed discussion about the
difference between cut- and uncut diagrams of $e(x)$ is provided.
