Bazhanov, Vladimir V.Lukyanov, Sergei L.Runov, Boris A.2016-11-082016-11-080550-3213http://hdl.handle.net/1885/110177Bukhvostov 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 [42] .The research of SL is supported by the NSF under grant number NSF-PHY-1404056.© 2016 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.http://creativecommons.org/licenses/by/3.0/2016-09-1910.1016/j.nuclphysb.2016.08.0312016-11-07