Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices

Journal of High Energy Physics, Sep 2017

Integrable deformation of SU(2) sigma and lambda models are considered at the classical and quantum levels. These are the Yang-Baxter and XXZ-type anisotropic deformations. The XXZ type deformations are UV safe in one regime, while in another regime, like the Yang-Baxter deformations, they exhibit cyclic RG behaviour. The associ-ated affine quantum group symmetry, realized classically at the Poisson bracket level, has q a complex phase in the UV safe regime and q real in the cyclic RG regime, where q is an RG invariant. Based on the symmetries and RG flow we propose exact factorizable S-matrices to describe the scattering of states in the lambda models, from which the sigma models follow by taking a limit and non-abelian T-duality. In the cyclic RG regimes, the S-matrices are periodic functions of rapidity, at large rapidity, and in the Yang-Baxter case violate parity.

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Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices

HJE Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices Calan Appadu 0 1 Timothy J. Hollowood 0 1 Dafydd Price 0 1 Daniel C. Thompson 0 1 0 Swansea , SA2 8PP , U.K 1 Department of Physics, Swansea University Integrable deformation of SU(2) sigma and lambda models are considered at the classical and quantum levels. These are the Yang-Baxter and XXZ-type anisotropic deformations. The XXZ type deformations are UV safe in one regime, while in another regime, like the Yang-Baxter deformations, they exhibit cyclic RG behaviour. The associated a ne quantum group symmetry, realized classically at the Poisson bracket level, has q a complex phase in the UV safe regime and q real in the cyclic RG regime, where q is an RG invariant. Based on the symmetries and RG S-matrices to describe the scattering of states in the lambda models, from which the sigma models follow by taking a limit and non-abelian T-duality. In the cyclic RG regimes, the S-matrices are periodic functions of rapidity, at large rapidity, and in the Yang-Baxter case violate parity. Integrable Field Theories; Quantum Groups; Sigma Models - Yang 1 Introduction 2 3 2.1 2.2 3.1 3.2 3.3 3.4 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 Classical SU(2) sigma models Lax connection and Poisson brackets Non-local charges and in nite symmetries Classical SU(2) lambda models Target spaces Lax formalism Poisson structure and symmetries Non-local charges and in nite symmetries 4 Renormalization group ow ow The sigma model RG Yang Baxter lambda model RG ow Anisotropic XXZ lambda model RG ow 5 Quantum group S-matrix: q complex phase Quantum group S-matrix: q real The RSOS S-matrix High energy limit The S-matrix proposals 6 Discussion A Lambda spacetimes 1 Introduction Sigma models are fascinating because they are the building blocks of string worldsheet theories but also they share many of the features of QFTs in higher dimensions in a simpler context. And within the space of sigma models, the ones that are integrable have the additional lure of tractability. The key examples are the Principal Chiral Models (PCM), whose target spaces are group manifolds G. There is a G-valued eld f and the action can be written1 S = 2 (1.1) 1We take x = t x and so for vectors A = A0 A1 along with A = 12 (A0 A1). { 1 { outgoing particles as well as the internal quantum numbers i; j; k; l. The PCM can appear as a bosonic sub-sector of a consistent string theory CFT background, e.g. the D1-D5 near horizon geometry, providing a modern holographic motivation for studying this theory. The more prosaic view, which we adopt here, is that PCMs are an exceptionally informative 1 + 1-dimensional QFTs exhibiting asymptotic freedom in the running coupling ( ) and a dynamically generated mass gap. The action given in eq. (1.1) manifests a GL GR global symmetry, f ! U f V . A feature that makes the PCM tractable is that it is classically integrable and the GL GR symmetry is part of a much larger classical Yangian Y (gL) Y (gR) symmetry generated by non-local charges.2 At the quantum level this integrability persists leading to the factorization of its Smatrix [2, 3]. This means that it is completely determined by 2 ! 2 body processes which preserve the individual momenta, as illustrated in gure 1. The states are labelled by their rapidity and by internal quantum numbers i; j; : : :. For example, in the SU(N ) PCM, there are N 1 particle multiplets with mass ma = m sin( a=N ), a =; 1; 2; : : : ; N 1, and each multiplet transforms in the [!a] [!a] representation of the GL GR symmetry, where !a are the highest weight vectors of the ath fundamental representation.3 The 2-body S-matrix has the characteristic product form [4]: S( ) = SGL ( ) SGR ( ) ; (1.2) where = 1 2. The product form re ects the fact that the states transform in a product of representations of GL and GR. The S-matrix building block SG( ) is G-invariant, in fact Yangian invariant, and is built from a rational solution of the Yang-Baxter Equation.4 Since the PCM is asymptotically free and its spectrum is massive and dynamically generated, directly connecting the conjectured quantum S-matrix picture to the Lagrangian description in eq. (1.1) is subtle. Nonetheless, consistency checks can be made by studying the theory in a regime in which perturbation theory can be employed and compared against the factorized S-matrix. The study of the exact solution of the model was initiated in the classic works [4{7]. As a byproduct of the successful comparison of Thermodynamic Bethe 2A concise introduction to this symmetry can be found in [1]. There are also an in nite number of local conserved charges which include and energy and momentum. 3For the groups SO(N ) the representations are actually reducible combinations. 4For the higher rank groups, the product form of the S-matrix must be multiplied by a scalar factor to provide the bound state poles. { 2 { Ansatz and perturbative calculations o (...truncated)


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Calan Appadu, Timothy J. Hollowood, Dafydd Price, Daniel C. Thompson. Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices, Journal of High Energy Physics, 2017, pp. 35, Volume 2017, Issue 9, DOI: 10.1007/JHEP09(2017)035