Lowest Nonvanishing Order Research Articles

On the Gaussian Approximation to Bayesian Posterior Distributions

The present article derives the minimal number N of observations needed to approximate a Bayesian posterior distribution by a Gaussian. The derivation is based on an invariance requirement for the likelihood <img src=image/13422860_01.gif>. This requirement is defined by a Lie group that leaves the <img src=image/13422860_01.gif> unchanged, when applied both to the observation(s) <img src=image/13422860_05.gif> and to the parameter <img src=image/13422860_02.gif> to be estimated. It leads, in turn, to a class of specific priors. In general, the criterion for the Gaussian approximation is found to depend on (i) the Fisher information related to the likelihood <img src=image/13422860_01.gif>, and (ii) on the lowest non-vanishing order in the Taylor expansion of the Kullback-Leibler distance between <img src=image/13422860_01.gif> and <img src=image/13422860_03.gif>, where <img src=image/13422860_04.gif> is the maximum-likelihood estimator of <img src=image/13422860_02.gif>, given by the observations <img src=image/13422860_05.gif>. Two examples are presented, widespread in various statistical analyses. In the first one, a chi-squared distribution, both the observations <img src=image/13422860_05.gif> and the parameter <img src=image/13422860_02.gif> are defined all over the real axis. In the other one, the binomial distribution, the observation is a binary number, while the parameter is defined on a finite interval of the real axis. Analytic expressions for the required minimal N are given in both cases. The necessary N is an order of magnitude larger for the chi-squared model (continuous <img src=image/13422860_05.gif>) than for the binomial model (binary <img src=image/13422860_05.gif>). The difference is traced back to symmetry properties of the likelihood function <img src=image/13422860_01.gif>. We see considerable practical interest in our results since the normal distribution is the basis of parametric methods of applied statistics widely used in diverse areas of research (education, medicine, physics, astronomy etc.). To have an analytical criterion whether the normal distribution is applicable or not, appears relevant for practitioners in these fields.

ON THE ABSENCE OF PENTAQUARK STATES FROM DYNAMICS IN STRONGLY COUPLED LATTICE QCD

We consider an imaginary time functional integral formulation of a two-flavor, 3+1 lattice QCD model with Wilson's action and in the strong coupling regime (with a small hopping parameter, κ &gt; 0, and a much smaller plaquette coupling, [Formula: see text], so that the quarks and glueballs are heavy). The model has local SU (3)c gauge and global SU (2)f flavor symmetries, and incorporates the corresponding part of the eightfold way particles: baryons (mesons) of asymptotic mass ≈-3 ln κ(≈-2 ln κ). We search for pentaquark states as meson–baryon bound states in the energy–momentum spectrum of the model, using a lattice Bethe–Salpeter equation. This equation is solved within a ladder approximation, given by the lowest nonvanishing order in κ and β of the Bethe–Salpeter kernel. It includes order κ2 contributions with a [Formula: see text] exchange potential together with a contribution that is a local-in-space, energy-dependent potential. The attractive or repulsive nature of the exchange interaction depends on the spin of the meson–baryon states. The Bethe–Salpeter equation presents integrable singularities, forcing the couplings to be above a threshold value for the meson and the baryon to bind in a pentaquark. We analyzed all the total isospin sectors, I = 1/2, 3/2, 5/2, for the system. For all I, the net attraction resulting from the two sources of interaction is not strong enough for the meson and the baryon to bind. Thus, within our approximation, these pentaquark states are not present up to near the free meson–baryon energy threshold of ≈-5 ln κ. This result is to be contrasted with the spinless case for which our method detects meson–baryon bound states, as well as for Yukawa effective baryon and meson field models. A physical interpretation of our results emerges from an approximate correspondence between meson–baryon bound states and negative energy states of a one-particle lattice Schrödinger Hamiltonian.

Perturbative Study of Bremsstrahlung and Pair-Production by Spin-½ Particles in the Aharonov–Bohm Potential

In the presence of an external Aharonov–Bohm potential, we investigate the two QED processes of the emission of a bremsstrahlung photon by an electron, and the production of an electron–positron pair by a single photon. Calculations are carried out using the Born approximation within the framework of covariant perturbation theory to lowest non-vanishing order in α. The matrix element for each process is derived, and the corresponding differential cross-section is calculated. In the non-relativistic limit, the resulting angular and spectral distributions and some polarization properties are considered, and compared to results of previous works.

Double-autoionization decay of resonantly excited single-electron states

Perturbation theory in the lowest non-vanishing order in interelectron interaction has been applied to the theoretical investigation of double-ionization decays of resonantly excited single-electron states. The formulae for the transition probabilities were derived in the LS coupling scheme, and the orbital angular momentum and spin selection rules were obtained. In addition to the formulae, which are exact in this order, three approximate expressions, which correspond to illustrative model mechanisms of the transition, were derived as limiting cases of the exact ones. Numerical results were obtained for the decay of the resonantly excited Kr I 3d-15p(1P) state which demonstrated quite clearly the important role of the interelectron interaction in double-ionization processes. On the other hand, the results obtained show that low-energy electrons can appear in the photoelectron spectrum below the ionisation threshold of the 3d shell.

Micro decay at T

The thermal corrections to the decay rate of the muon are calculated to lowest order in \ensuremath{\alpha} and lowest nonvanishing order in the temperature T. An analytical expression is derived for \ensuremath{\Delta}\ensuremath{\Gamma}, the correction to the decay rate, under the assumption that T\ensuremath{\ll}m, where m is the mass of the electron. A rather puzzling feature regarding cancellation of the infrared divergence is discussed at some length. Turning next to mass singularities, in the approximation of T\ensuremath{\ll}m, it would be inconsistent to let m\ensuremath{\rightarrow}0, and, indeed, there are correction terms of the form ln(m/M), where M is the mass of the muon, which do not cancel. In order to consider the m\ensuremath{\rightarrow}0 limit, the proper scheme is m\ensuremath{\ll}T\ensuremath{\ll}M; the additional corrections arising from the thermal bath of ${e}^{+}$${e}^{\mathbf{\ensuremath{-}}}$ pairs, under these conditions, remove the mass singularities.

Multiphoton ionization by intense fields: Corrections to lowest order perturbation theory, with an application to photoionization of H

We consider the perturbation expansion (in powers of the coupling strength of the electron interaction with the radiation field) of the standard time-independent expression for the multiphoton ionization amplitude. We show that corrections beyond lowest nonvanishing order contain divergent diagrams. These divergences cancel in the velocity gauge but not in the length gauge. We discuss the mathematical and physical nature of these divergences, and we present an application to the photoionization of hydrogen.

Three-photon ionisation probability rate of atomic caesium in the spin-orbit coupling scheme and electron spin polarisation effects

Three-photon ionisation probability rates have been calculated for circularly and linearly polarised incident radiation applying time-dependent perturbation theory to the lowest non-vanishing order in the framework of the quantum defect method generalised to the spin-orbit coupling scheme. The infinite summations over the unperturbed atomic states have been performed on the basis of the Green's function formalism. Electron spin polarisation has been evaluated and the possibility of obtaining ejected polarised electrons in the direction of the incident radiation wavevector through three-photon ionisation by circularly polarised radiation is shown.

Quantum effects in the early Universe. IV. Nonlocal effects in particle production in anisotropic models

The dynamical equation governing the evolution of the effective geometry in the presence of the production of conformally invariant scalar particles is solved for a homogeneous model cosmology with small anisotropy and classical radiation. The pair-production probabilities and spectrum are calculated in the one-loop approximation to lowest nonvanishing order in the deviation from exact isotropy.

Coupled electron spin resonance of non-S-state impurities in metals

The dynamical coupling of the individual crystal-field transitions and the conduction electrons is investigated for non$S$-state ions in an arbitrary crystal field. The equations given are the result of a microscopic quantum-statistical treatment. All relevant terms of the transition matrix have been calculated in the lowest nonvanishing order of perturbation theory. The results allow a quantitative description of the experimental ESR spectra. A particular physically realistic example is discussed in detail. A large temperature-dependent $g$ shift and a decrease of the width with increasing concentration is found for the local-moment resonance. At high temperatures the coupling to the excited states plays an important role and in particular affects the conduction-electron resonance shift. The relevance of these effects for actual physical situations is discussed in some detail.

Spectral Representation for Elastic Photon-Photon Scattering

The derivation of spectral forms in source theory has recently been systematically developed for systems of scalar particles. Elastic photon-photon scattering in lowest nonvanishing order is here studied as a simple, but representative, example of the additional considerations necessitated by particles having internal quantum numbers. The amplitude is determined as a double-spectral form augmented by a single-spectral one, the latter being related to the imposition of gauge invariance. Also presented is some discussion on how, in source theory, one obtains at given order all the contributions to the amplitude.

Structure of Dielectric Fluids. II. The Free Energy and the Kerr Effect in Polar Fluids

The Helmholtz free energy of a molecular fluid of rigid dipoles in the absence of any external field is studied. An explicit formal expression for the excess free energy due to the dipolar interaction is found. It is shown that the free energy per particle is independent of the shape of the sample, although shape-dependent long-range correlations are present in the fluid. For the special case of the Onsager model of polar fluids an explicit expression is obtained for the contribution to the free energy due to the dipolar interactions. Next, we study the long-range part of the three-particle correlation function for an arbitrarily shaped sample of the molecular fluid in the absence of an external field. This long-range part is dependent on the shape of the sample. The three-particle correlation function arises in the expression for the Kerr constant, which measures the birefringence of the fluid when a strong electric field is applied. We show explicitly that, in the lowest nonvanishing order in the field, the birefringence is proportional to the square of the resulting local macroscopic electric field with a proportionality constant (the Kerr constant) which is independent of the shape of the sample. A formal expression is given for the Kerr constant in terms of the local properties of the fluid.

Surface-Wave Instability in Helicon Propagation. II: Effect of Collisional Losses

The surface-wave instability analyzed by Baraff and Buchsbaum in the limit of infinite ωcτ is investigated here for ωcτ large but finite. In the previous analysis, the carrier drift velocity required to establish the instability was found to be proportional to (K1−K2), the effective dielectric mismatch across the interface at which the surface wave was excited. Here we find that with ωcτ large but finite, this threshold velocity is increased by an amount proportional to (ωcτ)−1 and inversely proportional to (K1−K2). For a given value of ωcτ there thus exists an optimum choice of (K1−K2) which minimizes the required threshold velocity. These new results can be obtained from heuristic arguments which do not require the calculation of field components beyond lowest nonvanishing order in (ωcτ)−1. The heuristic arguments are here justified rigorously for the physically interesting case of small (K1−K2).

Scattering of massive photons with vector-scalar coupling

The elastic scattering of neutral massive spin-1 particles coupled to scalar nucleons is considered in lowest nonvanishing order of the coupling constant. It is shown that all helicity amplitudes contain terms leading to Kronecker delta singularities in the complex angular-momentum plane. The contribution of these terms to the amplitude in the crossed channel is considered. Some consequences in connection with the Froissart theorem are discussed.

Electromagnetic Effects on Decays ofGEigenstates

from charge independence on boson decay are presented. The theorems are valid when electromagnetism is treated to the lowest nonvanishing order. (L.N.N.)

Momentum-spin correlations in photoelectric effect and in single quantum positron annihilation

The transverse polarization of photoelectrons and the photon emission asymmetry with respect to the polarization plane in single quantum annihilation of transversely polarized positrons is calculated in the lowest non-vanishing order Born approximation. The effects are of the same nature and of the same order of magnitude as the polarization and asymmetry effects in Mott scattering. The momentum spin correlation in quantum theory of radiation is shown to be proportional to the imaginary part of the matrix element integrals. The fact that the correlation only occurs in the second order Born approximation, the first order Born approximation matrix element integrals being real, is closely related to time reversal invariance.