  Primordial non-Gaussianity can lead to a scale-dependent bias in the density
of collapsed halos relative to the underlying matter density. The galaxy power
spectrum already provides constraints on local-type primordial non-Gaussianity
complementary those from the cosmic microwave background (CMB), while the
bispectrum contains additional shape information and has the potential to
outperform CMB constraints in future. We develop the bias model for the halo
density contrast in the presence of local-type primordial non-Gaussianity,
deriving a bivariate expansion up to second order in terms of the local linear
matter density contrast and the local gravitational potential in Lagrangian
coordinates. Non-linear evolution of the matter density introduces a non-local
tidal term in the halo model. Furthermore, the presence of local-type
non-Gaussianity in the Lagrangian frame leads to a novel non-local convective
term in the Eulerian frame, that is proportional to the displacement field when
going beyond the spherical collapse approximation. We use an extended
Press-Schechter approach to evaluate the halo mass function and thus the halo
bispectrum. We show that including these non-local terms in the halo bispectra
can lead to corrections of up to $25\%$ for some configurations, on large
scales or at high redshift.
