  Bukhvostov and Lipatov have shown that weakly interacting instantons and
anti-instantons in the $O(3)$ non-linear sigma model in two dimensions are
described by an exactly soluble model containing two coupled Dirac fermions. We
propose an exact formula for the vacuum energy of the model for twisted
boundary conditions, expressing it through a special solution of the classical
sinh-Gordon equation. The formula perfectly matches predictions of the standard
renormalized perturbation theory at weak couplings as well as the conformal
perturbation theory at short distances. Our results also agree with the Bethe
ansatz solution of the model. A complete proof the proposed expression for the
vacuum energy based on a combination of the Bethe ansatz techniques and the
classical inverse scattering transform method is presented in the second part
of this work [40].
