The principal chiral field in two dimensions on classical lie algebras: The Bethe-ansatz solution and factorized theory of scattering

The exact mass-gaps of the principal chiral models

“…The fact that the theories are integrable means that their scattering matrices are factorizable. Such S-matrices have been conjectured for all the theories corresponding to the classical Lie algebras [1]. Expressions for the complete S-matrices for SU(N ) can be found in [1] and for Sp(2N ) in [2].…”

“…where V a is the a th fundamental representation of G with the standard labelling of the Dynkin diagram [1]. Notice that although the particles are associated to the fundamental representations they are sometimes reducible in the case of SO(N ).…”

“…The S-matrices that have been conjectured to describe the scattering of the states of the model describe r = rank(G) particles which are associated to the fundamental representations of G. The masses of the particles, except for those associated to the spinors of SO(N ), can be described by the universal formula [1] …”

“…Fortunately, we shall not require the expression for the complete S-matrices but only those elements for the particles of the highest weight in each multiplet (so with quantum numbers |ω a , ω a where the ω a 's are the fundamental weights). The S-matrix amongst these states is purely elastic and their expressions can be extracted from [1]:…”

Yang-Baxter equation and reflection equations in integrable models

“…Using the fusion procedure the L-operators in the higher dimensional irreducible representations can be obtained giving rise to the ZF operators for the bound states and to the integrable lattice models of higher spins such as the spin s XXZ-model. The connection of the R-matrices and integrable models with the simple Lie (super-)algebras is reflected in the structure of the Bethe equations: they include r sets of "quasimomenta" , where r is the Lie algebra rank, and the Cartan matrix [37,47].…”