  We study the inner product of two Bethe states, one of which is taken
on-shell, in an inhomogeneous XXX chain in the Sutherland limit, where the
number of magnons is comparable with the length L of the chain and the magnon
rapidities arrange in a small number of macroscopically large Bethe strings.
The leading order in the large L limit is known to be expressed through a
contour integral of a dilogarithm. Here we derive the subleading term. Our
analysis is based on a new contour-integral representation of the inner product
in terms of a Fredholm determinant. We give two derivations of the sub-leading
term. Besides a direct derivation by solving a Riemann-Hilbert problem, we give
a less rigorous, but more intuitive derivation by field-theoretical methods.
For that we represent the Fredholm determinant as an expectation value in a
Fock space of chiral fermions and then bosonize. We construct a collective
field for the bosonized theory, the short wave-length part of which may be
evaluated exactly, while the long wave-length part is amenable to a $1/L$
expansion. Our treatment thus results in a systematic 1/L expansion of
structure factors within the Sutherland limit.
