  We analyze universal terms that appear in the large system size scaling of
the overlap between the N\'eel state and the ground state of the spin-1/2 XXZ
chain in the antiferromagnetic regime. In a critical theory, the order one term
of the asymptotics of such an overlap may be expressed in terms of $g$-factors,
known in the context of boundary conformal field theory. In particular, for the
XXZ model in its gapless phase, this term provides access to the Luttinger
parameter. In its gapped phase, on the other hand, the order one term simply
reflects the symmetry broken nature of the phase. In order to study the large
system size scaling of this overlap analytically and to compute the order one
term exactly, we use a recently derived finite-size determinant formula and
perform an asymptotic expansion. Our analysis confirms the predictions of
boundary conformal field theory and enables us to determine the exponent of the
leading finite-size correction.
