Tree-level elastic 2d theories with polynomial interactions remain elastic at one loop once masses are renormalized by bubble diagrams, with a universal one-loop S-matrix in terms of tree-level data.
Exact s-Matrices for the Nonsimply-Laced Affine Toda Theories $a_{2n-1}^{(2)}$
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abstract
We derive the exact, factorized, purely elastic scattering matrices for the $a_{2n-1}^{(2)}$ family of nonsimply-laced affine Toda theories. The derivation takes into account the distortion of the classical mass spectrum by radiative correction, as well as modifications of the usual bootstrap assumptions since for these theories anomalous threshold singularities lead to a displacement of some single particle poles.
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One-loop integrability with shifting masses
Tree-level elastic 2d theories with polynomial interactions remain elastic at one loop once masses are renormalized by bubble diagrams, with a universal one-loop S-matrix in terms of tree-level data.