Perturbative integrability in 1+1 dimensions and affine Toda theories

Davide Polvara

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Abstract

In this thesis, the perturbative integrability of 1+1 dimensional bosonic massive quantum field theories is investigated. Starting from a theory with a generic polynomial-like potential, the constraints on the masses and Lagrangian couplings emerging by requiring purely elastic amplitudes at the tree level are obtained. It is observed that theories satisfying these constraints are completely determined by their mass ratios and 3-point couplings, while all the higher-order couplings can be obtained recursively in terms of them by imposing the absence of production for higher numbers of external legs. By exploiting different root system properties, it is shown that all the bosonic affine Toda field theories universally satisfy the constraints of purely elasticity at the tree level: a complete proof of their tree-level integrability is therefore provided. Subsequently, the higher-order poles observed in the bootstrapped S-matrices of the ADE series of affine Toda models are studied in perturbation theory. These singular points have been explained in the past in terms of anomalous threshold singularities in certain Feynman diagrams, where multiple propagators go on-shell simultaneously in loop integrations. Networks of Feynman diagrams contributing to these higher-order poles are found and residues at the poles are obtained through perturbation theory, showing agreement with the bootstrapped results. We show that the residues are generated by suitably cutting the loop diagrams into products of tree-level graphs, which will be called ‘atoms’. Most of these atoms simplify between one another and only a small number of them survive matching the bootstrapped results. The simplification mechanism between atoms inside networks is reminiscent of Gauss’s theorem in the space of Feynman diagrams.

Funded in part by the European Union’s Horizon 2020 Framework Program under the Marie Skłodowska-Curie grant agreement No. 764850 (SAGEX).