Abstract
We investigate the perturbative integrability of massive (1+1)-dimensional bosonic quantum field theories, focusing on the conditions for them to have a purely elastic S-matrix, with no particle production and diagonal scattering, at tree level. For theories satisfying what we call ‘simply-laced scattering conditions’, by which we mean that poles in inelastic 2 to 2 processes cancel in pairs, and poles in allowed processes are only due to one on-shell propagating particle at a time, the requirement that all inelastic amplitudes must vanish is shown to imply the so-called area rule, connecting the 3-point couplings {C}_{abc}^{(3)} to the masses ma, mb, mc of the coupled particles in a universal way. We prove that the constraints we find are universally satisfied by all affine Toda theories, connecting pole cancellations in amplitudes to properties of the underlying root systems, and develop a number of tools that we expect will be relevant for the study of loop amplitudes.
Highlights
We investigate the perturbative integrability of massive (1+1)-dimensional bosonic quantum field theories, focusing on the conditions for them to have a purely elastic S-matrix, with no particle production and diagonal scattering, at tree level
We will use the phrase perturbative integrability to mean the vanishing of all such sums; this might be at tree level, or including all loop diagrams as well
We proved that the theory is completely defined once the mass ratios and the 3-point couplings are known
Summary
A key feature of integrable quantum field theories in 1+1 dimensions is the existence of an infinite tower of higher spin conserved charges. In a generic quantum field theory Mp(nro)d can contain singularities arising from on-shell propagating particles in internal lines; in an integrable model, all such infinities must cancel each other, since otherwise the production amplitude would not vanish.2 This means that if a certain Feynman diagram is singular for particular values of the external momenta, we expect at least one other diagram to become singular for the same choice of the external momenta in such a way that the infinities cancel, making the total Mp(nro)d free of singularities. The purpose of this paper is to make progress in extending these results to any two-dimensional bosonic quantum field theory of the form (1.1) by searching for the constraints on 3- and 4-point couplings necessary for the absence of off-diagonal processes at the tree level We prove that these constraints are satisfied for the affine Toda models, making use of various properties of the associated root systems. Appendix B is devoted to the computation of the residues in 5-point amplitudes, while appendix C reviews some useful properties of Lie algebras needed for the treatment of the affine Toda models
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