  We revisit the $DD^\ast$ interactions in chiral effective field theory up to
the third order for the first time. We deal with the pion-exchanged
interactions via local momentum-space regularization, in which we focus on
their long-range behaviors through demanding their contributions vanish at the
origin in the coordinate space. The short-range contact interactions and
subleading pion-charmed meson couplings are estimated with the phenomenological
resonance saturation model. The subleading pion-charmed meson couplings are
much weaker than those in the pion-nucleon system, thus the $DD^\ast$ binding
mechanism is very different with that of the $NN$ system. We also obtain the
analytic structure of the two-pion exchange interactions in the coordinate
space, and we find that its asymptotic behavior at long distance is similar to
but slightly different with the $NN$ interactions. We get the same asymptotic
behavior of the two-pion exchange interaction with that from HAL QCD method but
appearing in the longer distance rather than $1 <r<2\text{ fm}$. The binding
solution only exists in the isoscalar channel. Our calculation supports the
molecular interpretation of $T_{cc}^+$.
