On bosonic Thirring model in Minkowski signature

We present the way to continue the bosonic Thirring model or $\beta \gamma$-system with quartic interaction to Minkowski signature, based on the symmetries of this model. It is shown that the considered Minkowski version of the model is one-loop renormalizable. Based on this, we find the amplitudes of the scattering of the excitations corresponding to the $\gamma$ and $\bar{\gamma }$ fields up to the one-loop order. In particular, it was computed that the $2\to 2$ amplitudes of these excitations possess the property of reflectionless scattering and thus the corresponding $S$-matrix of such excitations satisfies the Yang-Baxter equation. The obtained $S$-matrix elements for $\gamma$ and $\bar{\gamma }$ are shown to coincide with the corresponding $S$-matrix elements of the solitons in the complex sine-Gordon model proposed by Dorey and Hollowood.