  We show that to understand the orthogonality catastrophe in the half-filled
lattice model of spinless fermions with repulsive nearest neighbor interaction
and a local impurity in its Luttinger liquid phase one has to take into account
(i) the impurity scaling, (ii) unusual finite size $L$ corrections of the form
$\ln(L)/L$, as well as (iii) the renormalization group flow of the umklapp
scattering. The latter defines a length scale $L_u$ which becomes exceedingly
large the closer the system is to its transition into the charge-density wave
phase. Beyond this transition umklapp scattering is relevant in the
renormalization group sense. Field theory can only be employed for length
scales larger than $L_u$. For small to intermediate two-particle interactions,
for which the regime $L > L_u$ can be accessed, and taking into account the
finite size corrections resulting from (i) and (ii) we provide strong evidence
that the impurity backscattering contribution to the orthogonality exponent is
asymptotically given by $1/16$. While further increasing the two-particle
interaction leads to a faster renormalization group flow of the impurity
towards the cut chain fixed point, the increased bare amplitude of the umklapp
scattering renders it virtually impossible to confirm the expected asymptotic
value of $1/16$ given the accessible system sizes. We employ the density matrix
renormalization group.
