Non-integrable quantum field theories as perturbations of certain integrable models

Abstract

We approach the study of non-integrable models of two-dimensional quantum field theory as perturbations of the integrable ones. By exploiting the knowledge of the exact S-matrix and Form Factors of the integrable field theories we obtain the first order corrections to the mass ratios, the vacuum energy density and the S-matrix of the non-integrable theories. As interesting applications of the formalism, we study the scaling region of the Ising model in an external magnetic field at T ∼ T c and the scaling region around the minimal model M 2,7 . For these models, a remarkable agreement is observed between the theoretical predictions and the data extracted by a numerical diagonalization of their Hamiltonian.

Matrix elements of the operator in integrable quantum field theory

“…For the case we discussed in this paper this means adding to the scaling action (2.10) the contribution of the irrelevant operators, TT among them. Since the scaling theory is integrable one can think of computing perturbatively the corrections to the S-matrix [26]. In particular, the first order correction coming from TT to the amplitude of a process with n particles in the initial state and m particles in the final state will be proportional to the form factor F TT m+n on a (m + n)-particle state with vanishing total energy and momentum.…”

“…It can be checked that this conclusion is unchanged if the limit in (4.25) is not taken symmetrically on the two rapidities. The symmetric limit gives the model independent finite result [26] …”

Confinement in the q-state Potts field theory

“…4). In this section we discuss the evolution of the mass spectrum in d = 2 starting from the case of small magnetic field for which perturbation theory around integrable quantum field theories [10] can be used.…”

“…for η ± → +∞) the corrections to the mass spectrum are determined by the matrix elements of the magnetic operator σ q (x) on the asymptotic states of the unperturbed (η ± = +∞) theory. At leading order, the correction to the mass matrix is [10] (Fig. 5a)…”

“…The possibility of a quantitative study in weak field comes from integrability of the q-state Potts field theory in zero-field [9]. Hence, form factor perturbation theory [10] allows to express mass corrections in terms of the matrix elements of the magnetic operator computed in [11].…”