Form Factors Of Descendant Operators Research Articles

The complex sinh-Gordon model: form factors of descendant operators and current-current perturbations

We study quasilocal operators in the quantum complex sinh-Gordon theory in the form factor approach. The free field procedure for descendant operators is developed by introducing the algebra of screening currents and related algebraic objects. We work out null vector equations in the space of operators. Further we apply the proposed algebraic structures to constructing form factors of the conserved currents Tk and Θk. We propose also form factors of current-current operators of the form TkT−l. Explicit computations of the four-particle form factors allow us to verify the recent conjecture of Smirnov and Zamolodchikov about the structure of the exact scattering matrix of an integrable theory perturbed by a combination of irrelevant operators. Our calculations confirm that such perturbations of the complex sinh-Gordon model and of the ℤN symmetric Ising models result in extra CDD factors in the S matrix.

Algebraic approach to form factors in the complex sinh-Gordon theory

We study form factors of the quantum complex sinh-Gordon theory in the algebraic approach. In the case of exponential fields the form factors can be obtained from the known form factors of the $Z_N$-symmetric Ising model. The algebraic construction also provides an Ansatz for form factors of descendant operators. We obtain generating functions of such form factors and establish their main properties: the cluster factorization and reflection equations.

Form factors of descendant operators: reduction to perturbed M (2, 2s + 1) models

In the framework of the algebraic approach to form factors in two-dimensional integrable models of quantum field theory we consider the reduction of the sine-Gordon model to the $\Phi_{13}$-perturbation of minimal conformal models of the $M(2,2s+1)$ series. We find in an algebraic form the condition of compatibility of local operators with the reduction. We propose a construction that make it possible to obtain reduction compatible local operators in terms of screening currents. As an application we obtain exact multiparticle form factors for the compatible with the reduction conserved currents $T_{\pm2k}$, $\Theta_{\pm(2k-2)}$, which correspond to the spin $\pm(2k-1)$ integrals of motion, for any positive integer~$k$. Furthermore, we obtain all form factors of the operators $T_{2k}T_{-2l}$, which generalize the famous $T\bar T$ operator. The construction is analytic in the $s$ parameter and, therefore, makes sense in the sine-Gordon theory.

Form factors of descendant operators: resonance identities in the sinh-Gordon model

We study the space of local operators in the sinh-Gordon model in the framework of the bootstrap form factor approach. Our final goal is to identify the operators obtained by solving bootstrap equations with those defined in terms of the Lagrangian field. Here we try to identify operators at some very particular points, where the phenomenon of operator resonance takes place. The operator resonance phenomenon being perturbative, nevertheless, results in exact identities between some local operators. By applying an algebraic approach developed earlier for form factors we derive an infinite set of identities between particular descendant and exponential operators in the sinh-Gordon theory, which generalize the quantum equation of motion. We identify the corresponding descendant operators by comparing them with the result of perturbation theory.

Form factors in sinh- and sine-Gordon models, deformed Virasoro algebra, Macdonald polynomials and resonance identities

We continue the study of form factors of descendant operators in the sinh- and sine-Gordon models in the framework of the algebraic construction proposed in [1]. We find the algebraic construction to be related to a particular limit of the tensor product of the deformed Virasoro algebra and a suitably chosen Heisenberg algebra. To analyze the space of local operators in the framework of the form factor formalism we introduce screening operators and construct singular and cosingular vectors in the Fock spaces related to the free field realization of the obtained algebra. We show that the singular vectors are expressed in terms of the degenerate Macdonald polynomials with rectangular partitions. We study the matrix elements that contain a singular vector in one chirality and a cosingular vector in the other chirality and find them to lead to the resonance identities already known in the conformal perturbation theory. Besides, we give a new derivation of the equation of motion in the sinh-Gordon theory, and a new representation for conserved currents.

Form factors of descendant operators in the Bullough-Dodd model

We propose a free field representation for the form factors of descendant operators in the Bullough-Dodd model. This construction is a particular modification of Lukyanov’s technique for solving the form factors axioms. We prove that the number of proposed solutions in each level subspace of the chiral sectors coincide with the number of the corresponding descendant operators in the Lagrangian formalism. We check that these form factors possess the cluster factorization property. Besides, we propose an alternative free field representation which allows us to study analytic properties of the form factors effectively. In particular, we prove that the form factors satisfy non trivial identities known as the “reflection relations”. We show the existence of the reflection invariant basis in the level subspaces for a generic values of the parameters.

Form factors of descendant operators: free field construction and reflection relations

The free field representation for form factors in the sinh-Gordon model and the sine-Gordon model in the breather sector are modified to describe the form factors of descendant operators, which are obtained from the exponential ones, eiαφ, by means of the action of the Heisenberg algebra associated with the field φ(x). As a check of the validity of the construction, we count the numbers of operators defined by the form factors at each level in each chiral sector. Another check is related to the so-called reflection relations, which identify in the breather sector the descendants of the exponential fields eiαφ and for generic values of α. We prove the operators defined by the obtained families of form factors to satisfy such reflection relations. A generalization of the construction for form factors to the kink sector is also proposed.

Form factors of descendant operators in the massive Lee–Yang model

The form factors of the descendant operators in the massive Lee–Yang model are determinedup to level 7. This is first done by exploiting the conserved quantities of the integrable theoryto generate the solutions for the descendants starting from the lowest non-trivial solutionsin each operator family. We then show that the operator space generated in this way, whichis isomorphic to the conformal one, coincides, level by level, with that implied by theS-matrix through the form factor bootstrap. The solutions that we determine satisfyasymptotic conditions carrying the information about the level that we conjecture to holdfor all the operators of the model.

FACTORIZED CHARACTERS AND FORM FACTORS OF DESCENDANT OPERATORS IN PERTURBED CONFORMAL SYSTEMS

We determine which characters of minimal models factorize into an infinite product like the one of the spin operator in the Ising model. In particular, all characters in the Ising model and in the nonunitary minimal Lee-Yang model factorize. This information is used to determine form factors of descendant operators, especially in the scaling Lee-Yang model.