No particle production in two dimensions: recursion relations and multi-Regge limit

Abstract

We introduce high-energy limits which allow us to derive recursion relations fixing the various couplings of Lagrangians of two-dimensional relativistic quantum field theories with no tree-level particle production in a very straightforward way. The sine-Gordon model, the Bullough-Dodd theory, Toda theories of various kinds and the U(N) non-linear sigma model can all be rediscovered in this way. The results here were the outcome of our explorations at the 2017 Perimeter Institute Winter School.

Holographic four-point functions in Toda field theories in AdS2

“…In flat spacetime, two basic consequences of higher conservation laws are the absence of particle production and factorization of the higher-point S-matrix [1,2,3,4]. Since a Lax pair is in general hard to find, no treelevel particle production is regarded as an important sign of classical integrability, and it is a useful strategy for building classically integrable theories [5,6,2,7,8].…”

From QCD Strings to WZW

“…Before going into details of the inductive procedure, which generalizes the one presented in [39] and makes it possible to fix all g n,k couplings in terms of g 3,1 , let us comment on possible IR divergences (a detailed discussion of these issues in a very close context just appeared in [56]). Generically, theories of massless particles in two dimensions are plagued with IR divergences.…”

“…However, all interactions in (19) are marginal, so a priori one expects to find IR issues in the corresponding on-shell amplitudes. Indeed, these were encountered in the analysis of [39] (see also [56] for a dedicated discussion). There the goal was to construct an integrable model for an U (N ) analogue of (19) with all odd couplings set to zero, g 2J+1,m = 0.…”

“…The discussion in section 3.2 indicates that IR divergencies do not spoil the argument presented in the beginning of the section, and that a pseudofree theory can indeed be constructed. To construct the corresponding action we follow the strategy of [39]. Namely, we fix all the coefficients g n,m by requiring that a sufficiently large subclass of amplitudes vanishes.…”

“…We achieve this goal in section 3. This is done by generalizing an inductive procedure allowing one to build classically integrable actions based on a clever use of multi-Regge limits as presented recently in [39] (see also [40]; some of the early work can be found in [41][42][43]). Imposing that all tree level amplitudes vanish allows one to uniquely fix all higher order terms in the axionic action in terms of the g 3,1 coupling introduced in (5).…”