Tree level integrability in 2d quantum field theories and affine Toda models

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 $$ {C}_{abc}^{(3)} $$ 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.

From tree- to loop-simplicity in affine Toda theories I: Landau singularities and their subleading coefficients

“…The different signs, one for each non-zero 3-point coupling, depend on the structure constants of the underlying Lie algebra and respect particular relations that prevent the presence of non-diagonal two to two processes. These relations emerge from the constraints of tree level integrability as we now explain, following the discussion in [5]. Let us consider the following process at tree level…”

“…The rest of this paper is organised as follows. In section 2 we review the mechanism responsible for the cancellation of 4-point tree-level non-elastic processes in perturbation theory with a particular focus on the cancellation of poles in Feynman diagrams connected by flips of type II, according to the convention used in [5,23]. These cancellations are particularly useful to understand the simplification mechanism that manifests itself at one loop.…”

“…The 4-and higher-point couplings can then be found by expanding the potential (2.1) to higher orders. They satisfy various relations which allow them to be fixed in terms of the masses and 3-point couplings [30], relations which turn out also to be necessary conditions for the tree-level integrability of the models [3,5]. The sign, plus or minus, entering (2.2) is not the same for all the 3-point couplings.…”

“…In the review paper [2], the fact that integrability should manifest itself in a priori surprising cancellations between Feynman diagrams contributing to production processes in perturbation theory was emphasised, and a systematic approach to the problem at the tree level was undertaken in [3][4][5][6], where constraints on masses and couplings necessary for the absence of production were given. Though the problem of classifying all the possible bosonic quantum field theories satisfying these constraints remains open, a universal proof of the absence of production for the entire class of affine Toda field theories was found in [5], thereby providing a tree-level proof of their perturbative integrability. In the following, we move to loop level, and consider a subclass of these models, the simply-laced affine Toda theories.…”

“…The S-matrices of these models have been bootstrapped [7][8][9][10][11][12][13][14][15][16][17] and have a beautiful universal structure in terms of the roots and weights of their underlying Lie algebras [15][16][17]. 1 At tree level, this geometrical structure is the reason for the absence of production [5], and all the integrability requirements emerge 1 Results for the S-matrices of more subtle cases such as non-simply laced theories [18][19][20] and supersymmetric models [21,22] have also been found, though a geometrical interpretation of their S-matrices in terms of Dynkin diagrams is less straightforward.…”