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Form-factor bootstrap and the operator content of perturbed minimal models

Koubek, A, Cambridge Univ. (United Kingdom). Dept. of Applied Mathematics and Theoretical Physics

Published Jan 1, 1994• 31 citations
OpenAlex

Abstract

The form-factor bootstrap approach is applied to the perturbed minimal models $M_{2,2n+3}$ in the direction of the primary field $\\phi_{1,3}$. These theories are integrable and contain $n$ massive scalar particles, whose $S$--matrix is purely elastic. The form-factor equations do not refer to a specific operator. We use this fact to classify the operator content of these models. We show that the perturbed models contain the same number of primary fields as the conformal ones. Explicit solutions are constructed and conjectured to correspond to the off-critical primary fields $\\phi_{1,k}$.

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•Oct 2, 2025•0 citations
The form-factor bootstrap approach is applied to the perturbed minimal models M2,2n+3M_{2,2n+3}M2,2n+3​ in the direction of the primary field phi1,3\\phi_{1,3}phi1,3​. These theories are integrable and contain nnn massive scalar particles, whose SSS--matrix is purely elastic. The form-factor equations do not refer to a specific operator. We use this fact to classify the operator content of these models. We show that the perturbed models contain the same number of primary fields as the conformal ones. Explicit solutions are constructed and conjectured to correspond to the off-critical primary fields phi1,k\\phi_{1,k}phi1,k​.
The form factors of the descendant operators in the massive Lee–Yang model are determined up to level 7. This is first done by exploiting the conserved quantities of the integrable theory to generate the solutions for the descendants starting from the lowest non-trivial solutions in each operator family. We then show that the operator space generated in this way, which is isomorphic to the conformal one, coincides, level by level, with that implied by the S -matrix through the form factor bootstrap. The solutions that we determine satisfy asymptotic conditions carrying the information about the level that we conjecture to hold for all the operators of the model.
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Completing the Bootstrap Program for TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ-Deformed Massive Integrable Quantum Field Theories
In recent years a considerable amount of attention has been devoted to the investigation of 2D quantum field theories perturbed by certain types of irrelevant operators. These are the composite field TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ - constructed out of the components of the stress-energy tensor - and its generalisations - built from higher-spin conserved currents. The effect of such perturbations on the infrared and ultraviolet properties of the theory has been extensively investigated. In the context of integrable quantum field theories, a fruitful perspective is that of factorised scattering theory. In fact, the above perturbations were shown to preserve integrability. The resulting deformed scattering matrices - extensively analysed with the thermodynamic Bethe ansatz - provide the first step in the development of a complete bootstrap program. In this letter we present a systematic approach to computing matrix elements of operators in generalised TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ-perturbed models, based on employing the standard form factor program. Our approach is very general and can be applied to all theories with diagonal scattering. We show that the deformed form factors, just as happens for the SSS-matrix, factorise into the product of the undeformed ones and of a perturbation- and theory-dependent term. From these solutions, correlation functions can be obtained and their asymptotic properties studied. Our results set the foundations of a new research program for massive integrable quantum field theory perturbed by irrelevant operators.
In this paper, we present a method to compute the minimal form factors (MFFs) of diagonal integrable field theories perturbed by generalized T T ¯ perturbations. Building on existing results by the same authors, these MFFs are constructed in such a way as not to allow for any free parameters, an issue that plagued previous solutions. The MFFs are derived from a generalization of the standard integral representation which has been used for UV-complete theories since the birth of the form factor bootstrap program. By UV-complete we mean theories whose short-distance/high-energy limit is a local conformal field theory. Their asymptotics is characterized by exponential decay at large rapidities. By computing higher particle form factors, we find that any natural higher-particle solutions involve the cancellation of parts of the newly found MFF. We conclude that the assumption that the form factor equations, particularly the kinematic residue equation, remain unchanged in the presence of T T ¯ perturbations, is too strong. There is a trade-off between having MFFs satisfying desirable analyticity and asymptotic properties and finding analytic solutions to the form factor equations, which is likely solved by nontrivial changes to the form factor equations, especially those where locality or semilocality of fields are essential assumptions.
The purpose of the "bootstrap program" is to construct integrable quantum field theories in 1+1 dimensions in terms of their Wightman functions explicitly. As an input the integrability and general assumptions of local quantum field theories are used. The object is to be achieved in tree steps: 1) The S-matrix is obtained using a qualitative knowledge of the particle spectrum and the Yang-Baxter equations. 2) Matrix elements of local operators are calculated by means of the "form factor program" using the S-matrix as an input. 3) The Wightman functions are calculated by taking sums over intermediate states. The first step has been performed for a large number of models and also the second one for several models. The third step is unsolved up to now. Here the program is illustrated in terms of the sine-Gordon model alias the massive Thirring model. Exploiting the "off-shell" Bethe Ansatz we propose general formulae for form factors. For example the n-particle matrix element for all higher currents are given and in particular all eigenvalues of the higher conserved charges are calculated. Furthermore quantum operator equations are obtained in terms of their matrix elements, in particular the quantum sine-Gordon field equation. Exact expressions for the finite wave function and mass renormalization constants are calculated.
In this paper, we present a method to compute the minimal form factors (MFFs) of diagonal integrable field theories perturbed by generalized TTbar perturbations. Building on existing results by the same authors, these MFFs are constructed in such a way as not to allow for any free parameters, an issue that plagued previous solutions. The MFFs are derived from a generalization of the standard integral representation which has been used for UV-complete theories since the birth of the form factor bootstrap program. By UV-complete we mean theories whose short-distance/high-energy limit is a local conformal field theory. Their asymptotics is characterized by exponential decay at large rapidities. By computing higher particle form factors, we find that any natural higher-particle solutions involve the cancellation of parts of the newly found MFF. We conclude that the assumption that the form factor equations, particularly the kinematic residue equation, remain unchanged in the presence of TTbar perturbations, is too strong. There is a trade-off between having MFFs satisfying desirable analyticity and asymptotic properties and finding analytic solutions to the form factor equations, which is likely solved by nontrivial changes to the form factor equations, especially those where locality or semilocality of fields are essential assumptions.