H. J. de Vega

Method for calculating finite size corrections in Bethe ansatz systems: Heisenberg chain and six-vertex model

Abstract A systematic procedure for computing large but finite size corrections in integrable theories is presented. It is based on an exact contour integral representation for the physical magnitudes. The saddle-point method applied to it provides the finite size corrections. We deal with the XXZ Heisenberg chain and the six-vertex model. The extension to any model solvable by the Bethe ansatz is straightforward provided the mass gap is not zero. The leading large size corrections for the two lowest lying states of the XXZ chain are derived in closed form. Leading finite size corrections to the excitations are obtained also.

Classical and quantum algebras of nonlocal charges in

We investigate the algebras of the non-local charges and their generating functionals (the monodromy matrices) in classical and quantum non-linear σ models. In the case of the classical chiral σ models it turns out that there exists no definition of the Poisson bracket of two monodromy matrices satisfying antisymmetry and the Jacobi identity. Thus, the classical non-local charges do not generate a Lie algebra. In the case of the quantum O(N) non-linear σ model, we explicitly determine the conserved quantum monodromy operator from a factorization principle together withP,T, and O(N) invariance. We give closed expressions for its matrix elements between asymptotic states in terms of the known two-particleS-matrix. The quantumR-matrix of the model is found. The quantum non-local charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.

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We analyze the phenomenon of preheating, i.e., explosive particle production due to parametric amplification of quantum fluctuations in the unbroken symmetry case, or spinodal instabilities in the broken symmetry phase, using the Minkowski space O({ital N}) vector model in the large {ital N} limit to study the nonperturbative issues involved. We give analytic results for weak couplings and times short compared to the time at which the fluctuations become of the same order as the tree level terms, as well as numerical results including the full back reaction. In the case where the symmetry is unbroken, the analytical results agree spectacularly well with the numerical ones in their common domain of validity. In the broken symmetry case, interesting situations, corresponding to slow roll initial conditions from the unstable minimum at the origin, give rise to a new and unexpected phenomenon: the dynamical relaxation of the vacuum energy. That is, particles are abundantly produced at the expense of the quantum vacuum energy while the zero mode comes back to almost its initial value. In both cases we obtain analytically and numerically the equation of state which in both cases can be written in terms of an effective polytropic index that interpolatesmorexa0» between vacuum and radiationlike domination. We find that simplified analyses based on the harmonic behavior of the zero mode, giving rise to a Mathieu equation for the nonzero modes, miss important physics. Furthermore, such analyses that do not include the full back reaction and do not conserve energy result in unbound particle production. Our results rule out the possibility of symmetry restoration by nonequilibrium fluctuations in the cases relevant for new inflationary scenarios. Estimates of the reheating temperature and the inconsistency of a kinetic approach to thermalization when a nonperturbatively large number of particles are created are discussed. {copyright} {ital 1996 The American Physical Society.}«xa0less

models

Abstract We give a generalization of the XXZ model where the spin takes n + 1 values, and the hamiltonian is not parity invariant. The diagonalization problem is solved exactly by the use of the n nested Bethe ansatz. We compute the ground-state energy in both the ferromagnetic and the antiferromagnetic case.

Analytic and numerical study of preheating dynamics.

We present a model-independent analysis of dark matter (DM) decoupling both ultrarelativistically (UR) and non-relativistically (NR) based on the DM phase-space density D = ρ DM /σ 3 DM . We derive explicit formulae for the DM particle mass m and for the number of ultrarelativistic degrees of freedom g d at decoupling. We find that for DM particles decoupling UR both at local thermal equilibrium (LTE) and out of LTE, m turns out to be in the keV scale. For example, for DM Majorana fermions decoupling at LTE the resulting mass is m ≃ 0.85 keV. For DM particles decoupling NR, √mT d results in the keV scale (T d is the decoupling temperature) and the value of m is consistent with the keV scale. In all cases, DM turns out to be cold DM (CDM). In addition, lower and upper bounds on the DM annihilation cross-section for NR decoupling are derived. We evaluate the free-streaming (Jeans) wavelength and Jeans mass: they are independent of the type of DM except for the DM self-gravity dynamics. The free-streaming wavelength today turns to be in the kpc range. These results are based on our theoretical analysis, on astronomical observations of dwarf spheroidal satellite galaxies in the Milky Way and on N-body numerical simulations. We analyse and discuss the results for D from analytic approximate formulae for both linear fluctuations and the (non-linear) spherical model and from N-body simulations results. In this way we obtain upper bounds for the DM particle mass, which are all below the 100-keV range.

Exact solution of the Zn+1 × Zn+1 symmetric generalization of the XXZ model

At energies of the order or larger than the Planck mass, the curved space-time geometry created by the particles dominates their collision process. The so-called Aichelburg-Sexl (AS) metric is relevant in this problem. We find the exact solution to the quantization and scattering of a closed bosonic string in D-dimensional AS geometry. The mass spectrm and critical dimension are the same as in flat space-time but there is non-trivial elastic and inelastic scattering by the shock wave. We find the exact (non-linear) and the Bogoliubov (linear) transformations relating the ingoing and outgoing string mode operators. The transition amplitudes between the internal modes and the total pair-creation rate are computed (this pair mode creation is a genuine string effect which does not exist in the quantum point particle theory). We find the deflection angle and the quantum string corrections to the classical cross section. The ground-state scattering amplitude is analyzed. Comparison with the black-hole results and recent investigations of string collisions in flat space-time are discussed.

Model-independent analysis of dark matter points to a particle mass at the keV scale

Abstract We apply the quantum inverse scattering method to a discretization of the generalized Toda field theory. In this way we get a family of solutions of the factorization equations. These solutions provide (i) a solvable many particle system with different types of particles interacting via non-relativistic complex potentials; (ii) a presumably solvable generalization of the six-vertex model in a planar lattice.

Quantum string scattering in the Aichelburg-Sexl geometry

We investigate the Baxter equation for the Heisenberg spin model corresponding to a generalized BFKL equation describing composite states of n Reggeized gluons in the multi-color limit of QCD. The Sklyanin approach is used to find a unitary transformation from the impact parameter representation to the representation in which the wave function factorizes as a product of Baxter functions and a pseudovacuum state. We show that the solution of the Baxter equation is a meromorphic function with poles

Solutions of the factorization equations from Toda field theory

(ensuremath{lambda}ensuremath{-}{ir)}^{ensuremath{-}(nensuremath{-}1)}