  Using the "Liouville space" (the space of operators) of the massive Dirac
theory, we define mixed-state form factors of $U(1)$ twist fields. We consider
mixed states with density matrices diagonal in the asymptotic particle basis.
This includes the thermal Gibbs state as well as all generalized Gibbs
ensembles of the Dirac theory. When the mixed state is specialized to a thermal
Gibbs state, using a Riemann-Hilbert problem and low-temperature expansion, we
obtain finite-temperature form factors of $U(1)$ twist fields. We then propose
the expression for form factors of $U(1)$ twist fields in general diagonal
mixed states. We verify that these form factors satisfy a system of nonlinear
functional differential equations, which is derived from the trace definition
of mixed-state form factors. At last, under weak analytic conditions on the
eigenvalues of the density matrix, we write down the large distance form factor
expansions of two-point correlation functions of these twist fields. Using the
relation between the Dirac and Ising models, this provides the large-distance
expansion of the R$\acute{\text{e}}$nyi entropy (for integer
R$\acute{\text{e}}$nyi parameter) in the Ising model in diagonal mixed states.
