S-matrix Bootstrap Research Articles

A Critical Look at 50 Years Particle Theory from the Perspective of the Crossing Property

The crossing property is perhaps the most subtle aspect of the particle-field relation. Although it is not difficult to state its content in terms of certain analytic properties relating different matrixelements of the S-matrix or formfactors, its relation to the localization- and positive energy spectral principles requires a level of insight into the inner workings of QFT which goes beyond anything which can be found in typical textbooks on QFT. This paper presents a recent account based on new ideas derived from "modular localization" including a mathematic appendix on this subject. Its main novel achievement is the proof of the crossing property of formfactors from a two-algebra generalization of the KMS condition. The main content of this article is the presentation of the derailments of particle theory during more than 4 decades: the S-matrix bootstrap, the dual model and its string theoretic extension. Rather than being related to crossing, string theory is the (only known) realization of a dynamic infinite component one-particle wave function space and its associated infinite component field. Here "dynamic" means that, unlike a mere collection of infinitely many irreducible unitary Poincar\'e group representation or free fields, the formalism contains also operators which communicate between the different irreducible Poincar\'e represenations (the levels of the "infinite tower") and set the mass/spin spectrum. Wheras in pre-string times there were unsuccessful attempts to achieve this in analogy to the O(4,2) hydrogen spectrum by the use of higher noncompact groups, the superstring in d=9+1, which uses instead (bosonic/fermionic) oscillators obtained from multicomponent chiral currents is the only known unitary positive energy solution of the dynamical infinite component pointlike localized field project.

Structure functions of the 2d [formula omitted] non-linear sigma models

We investigate structure functions in the 2-dimensional (asymptotically free) non-linear O ( n ) σ-models using the non-perturbative S-matrix bootstrap program. In particular, the exact small (Bjorken) x behavior is derived. Structure functions in the special case of the n = 3 model are accurately computed over the whole x range for − q 2 / M 2 < 10 5 , and some moments are compared with results from renormalized perturbation theory. Some results concerning the structure functions in the 1 / n approximation are also presented.

Structure functions of 2d integrable asymptotically free models

We investigate structure functions in the 2-dimensional (asymptotically free) non-linear O(n) σ-models using the non-perturbative S-matrix bootstrap program. In particular the exact small (Bjorken) x behavior is exhibited; the structure is rather universal and is probably the same in a wide class of (integrable) asymptotically free models. Structurally similar universal formulae may also hold for the small x behavior of QCD in 4-dimensions. Structure functions in the special case of the n=3 model are accurately computed over the whole x range for −q2/M2<105, and some moments are compared with results from renormalized perturbation theory. Some remarks concerning the structure functions in the 1/n approximation are also made.

On the universal representation of the scattering matrix of affine Toda field theory

By exploiting the properties of q-deformed Coxeter elements, the scattering matrices of affine Toda field theories with real coupling constant related to any dual pair of simple Lie algebras may be expressed in a completely generic way. We discuss the governing equations for the existence of bound states, i.e. the fusing rules, in terms of q-deformed Coxeter elements, twisted q-deformed Coxeter elements and undeformed Coxeter elements. We establish the precise relation between these different formulations and study their solutions. The generalized S-matrix bootstrap equations are shown to be equivalent to the fusing rules. The relation between different versions of fusing rules and quantum conserved quantities, which result as nullvectors of a doubly q-deformed Cartan like matrix, is presented. The properties of this matrix together with the so-called combined bootstrap equations are utilised in order to derive generic integral representations for the scattering matrix in terms of quantities of either of the two dual algebras. We present extensive case-by-case data, in particular on the orbits generated by the various Coxeter elements.

Non-unitarity in quantum affine Toda theory and perturbed conformal field theory

There has been some debate about the validity of quantum affine Toda field theory at imaginary coupling, owing to the non-unitarity of the action, and consequently of its usefulness as a model of perturbed conformal field theory. Drawing on our recent work, we investigate the two simplest affine Toda theories for which this is an issue – a 2 (1) and a 2 (2). By investigating the S-matrices of these theories before RSOS restriction, we show that quantum Toda theory (with or without RSOS restriction) indeed has some fundamental problems, but that these problems are of two different sorts. For a 2 (1), the scattering of solitons and breathers is flawed in both classical and quantum theories, and RSOS restriction cannot solve this problem. For a 2 (2) however, while there are no problems with breather-soliton scattering there are instead difficulties with soliton-excited soliton scattering in the unrestricted theory. After RSOS restriction, the problems with kink-excited kink may be cured or may remain, depending in part on the choice of gradation, as we found earlier [Nucl. Phys. B 489 [FS] (1997) 557]. We comment on the importance of regradations, and also on the survival of R-matrix unitarity and the S-matrix bootstrap in these circumstances.

Questionable arguments for the correctness of perturbation theory in non-Abelian models

We analyze the arguments put forward recently by Niedermayer et al in favor of the correctness of conventional perturbation theory in non-Abelian models and supposedly showing that our super-instanton counterexample was sick. We point out that within their own set of assumptions, the proof of Niedermayer et al regarding the correctness of perturbation theory is incorrect and provide a correct proof under more restrictive assumptions. We reply also to their claim that the S-matrix bootstrap approach of Balog et al supports the existence of asymptotic freedom in the O(3) model.