Calogero-Moser Lax pairs with spectral parameter for general Lie algebras

We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite-dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable systems, naturally associated with twisted affine Lie algebras, and construct their Lax operators with spectral parameter (except in the case of G₂).     

Solitons and vertex operators in twisted affine Toda field theories

Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions.     

Neutrino oscillations induced by two-loop radiative mechanism

Two-loop radiative mechanism, when combined with an U(1)_L′ symmetry generated by L_e − L_μ − L_τ (=L′), is shown to provide an estimate of Δm²/Δm²_atm εm_e/m_τ, where ε measures the U(1)_L′-breaking. Since Δm²_atm 3.5×10⁻³ eV², we find that Δm² ε10⁻⁶ eV², which will fall into the allowed region of the LOW solution to the solar neutrino problem for ε 0.1.     

Remarks on stochastic non-linear birth and death models

The first part of this paper is an attempt to formulate and motivate additional work on the important problem of obtaining global bounds applicable to the controlled truncation of the paper relates specifically to the linear birth, quadratic death model. Asymptotic results are given for the first finite difference ΔT_m where T_m is the exactly known mean time to extinction starting from state m (m= 0,1,…). These results are in terms of the environmental carrying capacity n^* taken to be large. For m near zero ΔT_me^n*/(n^*)²; whereas, for m near n^*ΔT_m ≈ (π/2)^1/2/(n^*)^3/2. This indicates the vastly different time scales in those two regions of state space - with considerably slower action near extinction than near n^*.     

Probing the EOS and nn cross section by analyzing the collective flow at high and intermediate energy in the RVUU approach

We have studied the collective flow at high and intermediate energy in a relativistic Vlasov-Uehling-Uhlenbeck (RVUU) approach based on Walecka's QHD-I model, with the aim to probe the nuclear-matter equation of state (EOS) and the in-medium nucleon-nucleon cross section σ. At high energy (1.2 GeV/u), the out-of-plane azimuthal correlation function C(Ψ) is only sensitive to the effective mass m^* and insensitive to the nuclear compressibility K and the effective nucleon-nucleon cross section σ within a reasonable range. We have found that the preferred value of m^* is about 0.85 m. With this value of m^*, from the in-plane mean transverse momentum P_x(Y) which is sensitive to both m^* and σ we have drawn an effective nn cross section σ, namely σ 0.8σ_f where σ_f is the free nucleon-nucleon cross section in Cugnon's parametrization. Taking advantage of the fact that the energy of vanishing flow (EVF) at intermediate energy (around 100 MeV/u) is only sensitive to the nucleon-nucleon cross section σ, we have drawn some information on the nucleon-nucleon cross section σ, namely σ = (1.4±0.2)σ_f.     

The Arnol''d cat: Failure of the correspondence principle

The classical Hamiltonian H = p²/2m + ε(q²/2)Σδ[s-(t/T)] has an integrable mapping of the plane, [q_n+1, p_n+1]= [q_n+1+p_n, q_n+2p_n], as its equations of motion. But then by introducing periodic boundary conditions via (mod 1) applied to both q and p variables, the equations of motion become the Arnol'd cat map, [q_n+1, p_n+1] = [q_n + p_n, q_n + 2p_n], (mod 1), revealing it to be one of the simplest fully chaotic systems which can be derived from a Hamiltonian and analyzed. Consequently, we here quantize the Arnol'd cat and examine its quantum motion for signs of chaos using algorithmic complexity as the litmus. Our analysis reveals that the quantum cat is not chaotic in the deep quantum domain nor does it become chaotic in the classical limit as required by the correspondence principle. We therefore conclude that the correspondence principle, as defined herein, fails for the quantum Arnol'd cat.     

The ΔNγ electromagnetic transition form factors GM(q) and GE(q)

A parameter-free, nonperturbative calculation of the ΔNγ electromagnetic transition amplitudes GM*(q2), GE*(q2), and the resonant multipole ratio REM(q2)≡E1+3/2(q2)/M1+3/2(q2) is performed in terms of the well-known nucleon isovector Sachs form factor GMV. Our methods are fully relativistic with conservation of the electromagnetic current guaranteed. We find that GM*(q2) decreases more rapidly than the nucleon dipole form factor when −q21 GeV2/c2 and that REM(q2) remains small even for very high four-momentum transfer implying that the perturbative QCD prediction REM(q2)→1 is purely asymptotic and is valid only for extremely high |q2|.     

Considerations concerning the radiative corrections to muon decay in the Fermi and Standard theories

The FAC, PMS, and BLM optimization methods are applied to the QED corrections to the muon lifetime in the Fermi V-A theory. The FAC and PMS scales are close to m_e, while the BLM scale nearly coincides with the geometric average √m_em_μ. The optimized expressions are employed to estimate the third order coefficient in the (m_μ) expansion and the theoretical error of the perturbative series. Using arguments based on effective field theory and a simple examination of Feynman diagrams, it is shown that, if contributions of (m_μ²/M_W²) are neglected, the corrections to muon decay in the SM factorize into the QED correction of the Fermi V-A theory and the electroweak amplitude g²/(1 − Δr), both of which are strictly scale-independent. We use the results to clarify how the QED corrections to muon decay and the Fermi constant G_F should be used in the SM, and what is the natural choice of scales if running couplings are employed.     

Magnetic anisotropy in the paramagnetic phase of TbAl2

The magnetic anisotropy of a single crystal of TbAl₂ has been measured by torque magnetometry from below the Curie point up to 170 K, well into the paramagnetic phase. Within a (110) plane the torque can be described by the expression L(θ) = {P sin 2θ} H² + {Q sin 2θ + S sin 4θ} H⁴ + {T sin 4θ} H⁶, where θ is the an gle formed by the magnetization vector with a [001] axis. The first term (in H²) is interpreted as produced by arrays of defects with axial symmetry. The second (in H⁴) and third (in H⁶) terms arise from anisotropic fourth and sixth rank tensor paramagnetic susceptibilities. On the other hand if the anisotropy is described in terms of effective conventional anisotropy constants K₁ and K₂ within the temperature range 90–170 K it is found that both constants change continuously across the Curie temperature and furthermore the [111] direction remains the easy direction in the paramagnetic range. Anisotropy measurements reveal themselves as a sensitive indicator of the level of macroscopic defects in magnetic crystals.     

Generalized Toda Mechanics Associated with Loop Algebras (L)(Cr) and (L)(Dr) and Their Reductions

We construct a class of integrable generalization of Toda mechanics with long-range interactions. These systems are associated with the loop algebras L(C_r) and L(D_r) in the sense that their Lax matrices can be realized in terms of the c=0 representations of the affine Lie algebras C⁽¹⁾_r and D⁽¹⁾_r and the interactions pattern involved bears the typical characters of the corresponding root systems. We present the equations of motion and the Hamiltonian structure. These generalized systems can be identified unambiguously by specifying the underlying loop algebra together with an ordered pair of integers (n,m). It turns out that different systems associated with the same underlying loop algebra but with different pairs of integers (n₁,m₁) and (n₂,m₂) with n₂<n₁ and m₂<m₁ can be related by a nested Hamiltonian reduction procedure. For all nontrivial generalizations, the extra coordinates besides the standard Toda variables are Poisson non-commute, and when either $n$ or m≥3, the Poisson structure for the extra coordinate variables becomes some Lie algebra (i.e. the extra variables appear linearly on the right-hand side of the Poisson brackets). In the quantum case, such generalizations will become systems with noncommutative variables without spoiling the integrability.     

Preferential sputtering and radiation enhanced segregation in palladium-platinum alloys

Five palladium-platinum alloys of different compositions ranging from 10 to 90 at% Pd were sputtered with 1 keV Ar⁺ ions. The surface concentrations were then determined with Auger electron spectroscopy (AES) and low energy ion scattering spectroscopy (ISS). The observed concentrations differed largely from bulk values. This difference is explained in terms of the difference in sputtering yields for platinum and palladium and the radiation enhanced segregation of palladium. The surface was then mechanically cleaned in situ. The surface concentrations measured subsequently corresponded excellently with the bulk values confirming the calibration procedure for the calculation of the surface concentrations. A recently developed model for surface segregation was used to fit the experimental data. Values for the surface segregation energy ΔG and the radiation enhanced diffusion coefficient D were obtained in the range 2000–8000 J/mol and (1.2–8.0)×10⁻²⁰ m²/s, respectively.     

Hyperon-Nucleon bound states and electroproduction of strangeness on He

The A(e, e′K⁺)YX reaction has been investigated in Hall C at Jefferson Lab. Data were taken for Q² ≈ 0.35 and 0.5 GeV² at a beam energy of 3.245 GeV for ³He, ⁴He. The missing mass spectra are fitted with Monte Carlo simulations including Λ, Σ⁰, Σ⁻ hyperon production. Models for quasifree production are compared to the data, excess yields close to threshold are attributed to FSI. Evidence for Λ-hypernuclear bound states is seen for ^3,4He targets.     

On perturbations of the periodic Toda lattice

A class of Hamiltonian systems including perturbations of the periodic Toda lattice and homogeneous cosmological models is studied. Separatrix approximation of oscillation regimes in these systems connected with Coxeter groups is obtained. Hamiltonian systems connected with simple Lie algebras are pointed out, which generalize the system describing periodic Toda lattice and allow theL -A pair representation.     

Final-state hyperon-nucleon interaction in the inclusive K photoproduction from the deuteron

We calculate d(γ, K⁺) inclusive cross sections with the full inclusion of the final ΛN - ΣN interaction. Modern hyperon-nucleon forces and a recently updated production operator for the γ + N → K⁺ + Y process are used. Significant effects of the hyperon-nucleon final-state interaction have been found especially around the K⁺ΣN threshold.     

Polarisation circulaire du rayonnement gamma de Sb, Cs et Ag

The βγ circular polarization correlation of the 3⁻(β⁻ 621 keV)3⁻(γ 1692)2⁺ cascade in ¹²⁴Sb, the 4⁺(β⁻ 662)4⁺ (γ 796)2⁺ (γ 605)0⁺ in ^13Z4Cs and the 6⁺(β⁻529)6⁺ (γ 937)4⁺ (γ 885)2⁺ (γ 658)0⁺ in ^110mAg have been studied by using a Compton effect polarimeter. The measured asymmetry parameters are 0.172±0.004, −0.0702 ±0.0024 and 0.0549±0.0013 respectively.     

The first eigenvalue of the Dirac operator on Kähler manifolds

If M^2m is a closed Kähler spin manifold of positive scalar curvature R, then each eigenvalue λ of type r (r {1, …, [(m + 1)/2]}) of the Dirac operator D satisfies the inequality λ² ≥ rR₀/4r − 2, where R₀ is the minimum of R on M^2m. Hence, if the complex dimension m is odd (even) we have the estimation

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for the first eigenvalue of D. In the paper is also considered the limiting case of the given inequalities. In the limiting case with m = 2r − 1 the manifold M^2m must be Einstein. The manifolds S², S² × S², S² × T², P³( ), F( ), P³( ) × T² and F( ³) × T², where F( ³) denotes the flag manifold and T² the 2-dimensional flat torus, are examples for which the first eigenvalue of the Dirac operator realizes the limiting case of the corresponding inequality. In general, if M^2m is an example of odd complex dimension m, then M^2m × T² is an example of even complex dimension m + 1. The limiting case is characterized by the fact that here appear eigenspinors of D² which are Kählerian twistor-spinors.     

Excitation energy and angular momentum dependence of nuclear level densities and spin cut-off factor in SPA and SPA + RPA approaches

We investigate the excitation energy (E^*) and angular momentum (J) dependence of nuclear level density and spin cut-off factor (σ) within microscopic approaches based on SPA and its extension SPA + RPA representation of the grand partition function for quadrupole-quadrupole interaction model Hamiltonian. For ¹¹⁰Sn, we find that excitation energy dependence of the total level density obtained within these approaches is significantly different. On the other hand, these approaches yield similar behaviour for J-dependence of the level density at fixed values of E^*. Values of σ_{SPA + RPA} at low E^* are found to be slightly smaller than σ_SPA but they tend to become almost the same at higher E^* (> 30 MeV). We also find that Bethe's formula for fixed-J level density based on the spin cut-off approximation can be used to compute (E^*, J) near the yrast line provided one uses an appropriate value of the spin cut-off factor.     

The reactions pp→pΛK and pp→pΣK near their thresholds

The reactions pp→pΛK⁺ and pp→pΣ⁰K⁺ are studied near their thresholds. The strangeness production process is described by the π- and K exchange mechanisms. Effects from the final state interaction in the hyperon-nucleon system are taken into account rigorously. The Λ production turns out to be dominated by K exchange whereas K- as well as π exchange play an important role for the Σ⁰ case. It is shown that the experimentally observed strong suppression of Σ⁰ production compared to Λ production at the same excess energy can be explained by a destructive interference between π and K exchange in the reaction pp→pΣ⁰K⁺. Implications of such an interference on the reaction pp→nΣ⁺K⁺ are pointed out.     

A Class of Integrable Spin Calogero-Moser Systems

 We introduce a class of spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models (called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and gauge transformations. For Lie algebras of A _n -type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided by a general framework which we develop here using dynamical Lie algebroids. Received: 19 October 2001 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research partially supported by NSF grant DMS00-72171.