  We consider a general (beyond $T\bar T$) deformation of the 2D $O(N+1)$
$\sigma$-model by the irrelevant dimension-four operators. The theory deformed
in this most general way is not integrable, and the $S$-matrix loses its
factorization properties. We perform the all-order summation of the leading
infrared logs for the $2 \to 2$ scattering amplitude and provide the exact
result for the $2 \to 2$ $S$-matrix in the leading logarithmic approximation.
These results can provide us with new insights into the properties of the
theories deformed by irrelevant operators more general than the $T\bar T$
deformation.
