Quantum corrections to the classical reflection factor in a2(1) Toda field theory

The O(β²) quantum correction to the classical reflection factor is calculated for one of the integrable boundary conditions of a₂⁽¹⁾ affine Toda field theory. This is found to agree with the conjectured exact reflection factor of the quantum theory. We consider the existence of other exact reflection factors consistent with our perturbative answer and examine the question of how duality transformations might relate theories with different boundary conditions.     

Non-unitarity in quantum affine Toda theory and perturbed conformal field theory

There has been some debate about the validity of quantum affine Toda field theory at imaginary coupling, owing to the non-unitarity of the action, and consequently of its usefulness as a model of perturbed conformal field theory. Drawing on our recent work, we investigate the two simplest affine Toda theories for which this is an issue –a₂⁽¹⁾ and a₂⁽²⁾. By investigating the S-matrices of these theories before RSOS restriction, we show that quantum Toda theory (with or without RSOS restriction) indeed has some fundamental problems, but that these problems are of two different sorts. For a₂⁽¹⁾, the scattering of solitons and breathers is flawed in both classical and quantum theories, and RSOS restriction cannot solve this problem. For a₂⁽²⁾ however, while there are no problems with breather-soliton scattering there are instead difficulties with soliton-excited soliton scattering in the unrestricted theory. After RSOS restriction, the problems with kink-excited kink may be cured or may remain, depending in part on the choice of gradation, as we found earlier [Nucl. Phys. B 489 [FS] (1997) 557]. We comment on the importance of regradations, and also on the survival of R-matrix unitarity and the S-matrix bootstrap in these circumstances.     

Boundary bootstrap principle in two-dimensional integrable quantum field theories

We study the reflection amplitudes of affine Toda field theories with boundary, following the ideas developed by Fring and Koberle [A. Fring, R. Koberle, Nucl. Phys. B 421 (1994) 159; A. Fring, R. Koberle, Nucl. Phys. B 419 (1994) 647; A. Fring, R. Koberle, Int. J. Mod. Phys. A 10 (1995) 739] and focusing our attention on the E_n series elements, because of their interesting structure of higher order poles. We also investigate the corresponding minimal reflection matrices, finding, with respect to the bulk case, a more complicated relation between the spectra of bound states associated to the minimal and to the “dressed” amplitudes.     

On a2(1) reflection matrices and affine Toda theories

We construct new non-diagonal solutions to the boundary Yang-Baxter equation corresponding to a two-dimensional field theory with U_q(a₂⁽¹⁾) quantum affine symmetry on a half-line. The requirements of boundary unitarity and boundary crossing symmetry are then used to find overall scalar factors which lead to consistent reflection matrices. Using the boundary bootstrap equations we also compute the reflection factors for scalar bound states (breathers). These breathers are expected to be identified with the fundamental quantum particles in a₂⁽¹⁾ affine Toda field theory and we therefore obtain a conjecture for the affine Toda reflection factors. We compare these factors with known classical results and discuss their duality properties and their connections with particular boundary conditions.     

Quantum scattering from classical field theory

We show that scattering amplitudes between initial wave packet states and certain coherent final states can be computed in a systematic weak coupling expansion about classical solutions satisfying initial-value conditions. The initial-value conditions are such as to make the solution of the classical field equations amenable to numerical methods. We propose a practical procedure for computing classical solutions which contribute to high energy two-particle scattering amplitudes. We consider in this regard the implications of a recent numerical simulation in classical SU(2) Yang-Mills theory for multiparticle scattering in quantum gauge theories and speculate on its generalization to electroweak theory. We also generalize our results to the case of complex trajectories and discuss the prospects for finding a solution to the resulting complex boundary value problem, which would allow the application of our method to any wave packet to coherent state transition. Finally, we discuss the relevance of these results to the issues of baryon number violation and multiparticle scattering at high energies.     

Wave function renormalization constants and one-particle form factors in Dl(1) Toda field theories

We apply the method of angular quantization to the calculation of the wave function renormalization constants in D₁⁽¹⁾ affine Toda quantum field theories. A general formula for the wave function renormalization constants in ADE Toda field theories is proposed. We also calculate all one-particle form factors and some of the two-particle form factors of an exponential field.     

Integrability of a family of quantum field theories related to sigma models

A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The sinh-Gordon model, the affine sl(3) Toda model, a model called the fermionic sl(2|1) Toda theory, and the N=2 supersymmetric sine-Gordon model. These models are all related to sigma models in various ways. The N=2 supersymmetric sine-Gordon model, in particular, describes the Pohlmeyer reduction of string theory on AdS₂×S², and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space.     

Quantum Group Symmetry in sine-Gordon and Affine Toda Field Theories on the Half-Line

 We consider the sine-Gordon and affine Toda field theories on the half-line with classically integrable boundary conditions, and show that in the quantum theory a remnant survives of the bulk quantized affine algebra symmetry generated by non-local charges. The paper also develops a general framework for obtaining solutions of the reflection equation by solving an intertwining property for representations of certain coideal subalgebras of U _q (ĝ). Received: 10 December 2001 / Accepted: 7 October 2002 Published online: 19 December 2002     

Three-body scattering in configuration space

The asymptotic forms of the wavefunction and Faddeev components in configuration space are shown to determine uniquely the solutions of the Schrödinger or Faddeev differential equations for 2 → (2, 3) and 3 → (2, 3) processes. An antisymmetrized form of the Faddeev differential equation for three equivalent fermions is given and its angular analysis is performed in the general case of local potentials with tensor interaction for neutron-deuteron scattering. We describe a numerical method for solving the corresponding boundary value problem and apply it to scattering and break-up at E_n^1ab = 14.4 MeV in the doublet S state for the four local potentials of Malfliet and Tjon, Reid, de Tourreil and Sprung, and de Tourreil, Rouben and Sprung. For the three realistic potentials, elastic scattering amplitudes differ by 5%, and amplitudes for break-up in the two-neutron state ¹S₀ differ by less than 4%.     

An introduction to on-shell recursion relations

This article provides an introduction to on-shell recursion relations for calculations of tree-level amplitudes. Starting with the basics, such as spinor notations and color decompositions, we expose analytic properties of gauge-boson amplitudes, BCFW-deformations, the large z-behavior of amplitudes, and on-shell recursion relations of gluons. We discuss further developments of on-shell recursion relations, including generalization to other quantum field theories, supersymmetric theories in particular, recursion relations for off-shell currents, recursion relation with nonzero boundary contributions, bonus relations, relations for rational parts of one-loop amplitudes, recursion relations in 3D and a proof of CSW rules. Finally, we present samples of applications, including solutions of split helicity amplitudes and of N = 4 SYM theories, consequences of consistent conditions under recursion relation, Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations for color-ordered gluon tree amplitudes, Kawai-Lewellen-Tye (KLT) relations.     

World-volume action for fractional branes

We study the world-volume action of fractional Dp-branes of type IIA string theory compactified on the orbifold T⁴/Z₂. The geometric relation between these branes and wrapped branes is investigated using conformal techniques. In particular we examine in detail various scattering amplitudes and find that the leading low-energy interactions are consistent with the boundary action derived geometrically.     

Amplitudes of multichannel scattering on the barrier with a constant height in the scattering direction

The immersing method is applied to solve the N-channel scattering problem. In particular, we consider the particle scattering on a multidimensional potential barrier, which is constant in the scattering direction and arbitrary in the lateral direction. For this case the scattering amplitudes t _m and r _m (m = 1, 2, …, N) are determined. Transition from the obtained formulas to the case of thin potential is performed. For this case analytical expressions of transmission amplitudes t _m and reflection amplitudes r _m are obtained. We show that the product of transmission and reflection amplitudes in the channel m does not depend on the scattering channel. It is assumed that the scattering particle falls on the potential with the longitudinal wave vector k _l corresponding to the channel l.     

Reflection matrices for integrable N = 1 supersymmetric theories

We study two-dimensional integrable N = 1 supersymmetric theories (without topological charges) in the presence of a boundary. We find a universal ratio between the reflection amplitudes for particles that are related by supersymmetry and we propose exact reflection matrices for the supersymmetric extensions of the multi-component Yang-Lee models and for the breather multiplets of the supersymmetric sine-Gordon theory. We point out the connection between our reflection matrices and the classical boundary actions for the supersymmetric sine-Gordon theory as constructed by Inami, Odake and Zhang [Phys. Lett. B 359 (1995) 118].     

Quark scattering amplitudes with quasi-elastic unitarity

We consider quark-quark scattering at high energies and fixed momentum transfer. In a model where in the s and u channel intermediate states only gluons with sufficiently small transverse momenta are emitted, the scattering amplitudes are expressed in terms of the S-matrix elements for exactly soluble two-dimensional field theories.     

<Emphasis Type="Italic">W</Emphasis><Subscript>4</Subscript> toda example as hidden Liouville CFT

We construct correlators in the W ₄ Toda 2d conformal field theory for a particular class of representations and demonstrate a relation to a W ₂ (Virasoro) theory with different central charge. The relevance of the classical limits of the constructed 3-point functions and braiding matrices to problems in 4d conformal theories is discussed.     

Universal boundary reflection amplitudes

For all affine Toda field theories we propose a new type of generic boundary bootstrap equations, which can be viewed as a very specific combination of elementary boundary bootstrap equations. These equations allow to construct general solutions for the boundary reflection amplitudes, which are valid for theories related to all simple Lie algebras, that is simply laced and non-simply laced. We provide a detailed study of these solutions for concrete Lie algebras in various representations. The boundary bootstrap equations relating different types of exited boundary states are not automatically solved by our expressions.     

Amplitudes of multichannel scattering on a two-dimensional potential barrier with constant height

The immersing method is applied to solve the N-channel scattering problem for concrete potential. In particular, we consider the particle scattering on a two-dimensional potential barrier, which is constant in the scattering direction and arbitrary in the cross-section direction. For this case the scattering amplitudes t _m and r _m (1,2, m= ...,N) are determined. A transition from the obtained formulas to the case of ?-potential is performed. For this case transmission amplitudes t _m and reflection amplitudes r _m are obtained. It is also shown that the product of transmission and reflection amplitudes along the channel m does not depend on the scattering channel.     

RSOS revisited

We investigate the issues of unitarity and reality of the spectrum for the imaginary coupled affine Toda field theories based on a₁⁽¹⁾ and a₂⁽²⁾ and the perturbed minimal models that arise from their various RSOS restrictions. We show that while all theories based on a₁⁽¹⁾ have real spectra in finite volume, the spectra of a₂⁽²⁾ models is in general complex, with some exceptions. We also correct the S matrices conjectured earlier for the φ₁₅ perturbations of minimal models and give evidence for a conjecture that the RSOS spectra can be obtained as suitable projections of the folded ATFTs in finite volume.     

Multichannel scattering of a particle in a quantum wire of cylindrical symmetry

The Schrödinger equation for a scattering particle in a quantum wire is considered. We discuss two geometrical forms of cross-section of the wire: the rectangular section and the cylindrical one. It is shown that scattering of the particle on an arbitrary potential V = V (x, y, z,), given in the wire, can be considered as a multichannel scattering, where the index of the channel coincides with the index which determines energy levels of the confined transverse motion of the particle. A method for determination of the amplitudes of transmission T _i and reflection R _i in the case of multichannel scattering is proposed. The case of two-channel scattering is considered in detail and a method for determination of the scattering amplitudes T ₁, T ₂ and R ₁, R ₂ is proposed.