  We derive non-linear differential equations for correlation functions of U(1)
twist fields in the two-dimensional massive Dirac theory. Primary U(1) twist
fields correspond to exponential fields in the sine-Gordon model at the
free-fermion point, and it is well-known that their vacuum two-point functions
are determined by integrable differential equations. We extend part of this
result to more general quantum states (pure or mixed) and to certain
descendents, showing that some two-point functions are determined by the
sinh-Gordon differential equations whenever there is translation and parity
invariance, and the density matrix is the exponential of a bilinear expression
in fermions. We use methods involving Ward identities associated to the
copy-rotation symmetry in a model with two independent, anti-commuting copies.
Such methods were used in the context of the thermally perturbed Ising quantum
field theory model. We show that they are applicable to the Dirac theory as
well, and we suggest that they are likely to have a much wider applicability to
free fermion models in general. Finally, we note that our form-factor study of
descendents twist fields combined with a CFT analysis provides a new way of
evaluating vacuum expectation values of primary U(1) twist fields: by deriving
and solving a recursion relation.
