  We obtain the pion decay constant and coefficients of fourth derivative terms
in the chiral Lagrangian for massless quarks in the Witten-Sakai-Sugimoto
model. We extract these quantities from the two-pion scattering amplitude,
which we compute directly in the holographic dual through tree-level Witten
diagrams. Identification of the low energy coefficients in the chiral action is
subtle as their values will be shifted when the tower of massive vector bosons
are integrated out. Indeed, by a direct comparison with the existing standard
procedure of constructing the chiral action with radial modes in the gravity
dual, we explicitly show that there are finite 't Hooft coupling corrections
that have been missed. This suggests that past derivations of effective actions
from holographic models may have to be revisited and future derivations more
carefully considered.
