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Journal of Physics A Mathematical and Theoretical

Bethe ansatz and exact form factors of the O (6) Gross Neveu-model *

H. Babujian, Angela Foerster, Michael Karowski

Published Jun 29, 2017• 1 citation
Publisher

Abstract

Abstract The isomorphism S U ( 4 ) ≃ O ( 6 ) is used to construct the form factors of the O ( 6 ) Gross–Neveu model as bound state form factors of the S U ( 4 ) chiral Gross–Neveu model. This technique is generalized and is then applied to use the O ( 6 ) as the starting point of the nesting procedure to obtain the O ( N ) form factors for general even N .

DOI
10.1088/1751-8121/aa7c7f

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O the Bethe-Ansatz Construction of the Chiral - Gross-Neveu Model.

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Abstract The isomorphism S U ( 4 ) ≃ O ( 6 ) is used to construct the form factors of the O ( 6 ) Gross–Neveu model as bound state form factors of the S U ( 4 ) chiral Gross–Neveu model. This technique is generalized and is then applied to use the O ( 6 ) as the starting point of the nesting procedure to obtain the O ( N ) form factors for general even N .
The isomorphism SU(4)≃O(6)SU(4) \simeq O(6)SU(4)≃O(6) is used to construct the form factors of the O(6) Gross-Neveu model as bound state form factors of the SU(4) chiral Gross-Neveu model. This technique is generalized and is then applied to use the O(6) as the starting point of the nesting procedure to obtain the O(N) form factors for general even N.
We apply previous results on the O (N) Bethe Ansatz [1-3] to construct a general form factor formula for the O (N) Gross-Neveu model. We examine this formula for several operators, such as the energy momentum, the spin-field and the current. We also compare these results with the 1/N expansion of this model and obtain full agreement. We discuss bound state form factors, in particular for the three particle form factor of the field. In addition for the two particle case we prove a recursion relation for the K-functions of the higher level Bethe Ansatz.
Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.
The process of using the free field representation to construct form factors of two dimensional integrable models is very promising. In this thesis, this procedure is analysed and adapted for application to the chiral Gross-Neveu model. The vertex operators and Zamolodchikov-Faddeev algebra for the particles are presented, with a similar structure producing a representation of the local operators of the theory. Using these techniques, the form factors of the model are then constructed as traces over the space of Zamolodchikov-Faddeev operators, and given in terms of an integral representation. In particular, the two-particle form factors of the current operator are found, and shown to agree with previous results in the literature.