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arXiv:hep-th/0611012 · Journal of Physics A Mathematical and Theoretical

The nestedSU(N) off-shell Bethe ansatz and exact form factors

H Babujian, Angela Foerster, M Karowski

Published Jun 12, 2008• 31 citations
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Abstract

The form factor equations are solved for a SU ( N ) invariant S -matrix under the assumption that the anti-particle is identified with the bound state of N − 1 particles. The solution is obtained explicitly in terms of the nested off-shell Bethe ansatz where the contribution from each level is written in terms of multiple contour integrals. The general solution is illustrated for some operators, such as the energy–momentum tensor, the fields and the current.

DOI
10.1088/1751-8113/41/27/275202

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The form factor equations are solved for a SU ( N ) invariant S -matrix under the assumption that the anti-particle is identified with the bound state of N − 1 particles. The solution is obtained explicitly in terms of the nested off-shell Bethe ansatz where the contribution from each level is written in terms of multiple contour integrals. The general solution is illustrated for some operators, such as the energy–momentum tensor, the fields and the current.
The purpose of the "bootstrap program" for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions.In this article, this program is mainly illustrated in terms of the sinh-Gordon model and the SU (N ) Gross-Neveu model.The nested off-shell Bethe ansatz for an SU (N ) factorizing S-matrix is constructed.We review some previous results on sinh-Gordon form factors and the quantum operator field equation.The problem of how to sum over intermediate states is considered in the short distance limit of the two point Wightman function for the sinh-Gordon model.
The system of SU(N)- and U(N)-matrix difference equations are solved by means of a nested version of a generalized Bethe ansatz, also the called `off shell' Bethe ansatz (Babujian H M 1990 Correlation functions in WZNW model as a Bethe wavefunction for the Gaudin magnets Proc. XXIVth Int. Symp. (Ahrenshoop, Zeuthen)). To solve this new Bethe ansatz in the algebraic language analogous to the conventional case, a new type of monodromy matrices is introduced. They fulfil a new type of Yang - Baxter equations which simplify the proofs. Using a similar approach as for the conventional nested Bethe ansatz the problem is solved iteratively. The vanishing of the `unwanted terms' of the first level ansatz is equivalent to a set of second level difference equations. The solutions are obtained as sums over `off-shell' Bethe vectors. These sums are `Jackson-type integrals'. The highest weight property of the solutions is proved. The solutions are calculated explicitly for several examples of SU(N)- and U(N)-representations.
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.
A system of O(N)-matrix difference equations is solved by means of the off-shell version of the nested algebraic Bethe ansatz. In the nesting process a new object, the ΠΠΠ-matrix, is introduced to overcome the complexities of the O(N) group structure. The proof of the main theorem is presented in detail. In particular, the cancellation of all unwanted terms is shown explicitly. The highest weight property of the solutions is proved.