Determine the σ–η mixing angle at nonzero θ in the two-flavor Schwinger model

Determine the exact θ-dependent mixing angle ω(θ) between the scalar operator σ = \bar{ψ}ψ and the pseudoscalar operator η = -i \bar{ψ} γ^5 ψ that diagonalizes the σ–η sector into mass-eigenstate operators in the two-flavor Schwinger model (1+1-dimensional QED with two degenerate Dirac fermions) at nonzero vacuum angle θ.

Background

In the two-flavor Schwinger model at nonzero vacuum angle θ, the scalar (σ) and pseudoscalar (η) operators mix due to the axial rotation induced by θ. The mass eigenstates are related to the original operators by an additional axial rotation characterized by a mixing angle ω(θ).

While the form of the operator rotation is specified, the precise value of the mixing angle ω(θ) as a function of θ is not analytically known and must be determined numerically. Accurate knowledge of ω(θ) is necessary to construct operators that overlap with the true mass eigenstates and to reliably extract meson spectra at θ ≠ 0.