High-energy Expansion Research Articles

Star interpolation and high-energy expansion for evaluating solid-state Green's functions

Star interpolation and high-energy expansion for evaluating solid-state Green's functions

Solution of the wave equation for the logarithmic potential with application to particle spectroscopy

We present an almost complete solution of the Schrödinger equation for a logarithmic potential. In particular we obtain two pairs of high-energy asymptotic expansions of the boundstate eigenfunctions together with a corresponding expansion of the eigenvalue determined by the secular equation. We also obtain a pair of uniformly convergent solutions and a pair of uniform asymptotic expansions. Various properties of the solutions and eigenvalues are examined, including the scattering problem of the cut-off potential and the behavior of Regge trajectories. Finally the relevance of these investigations to the spectroscopy of heavy quark composites is discussed. In particular we point out that the relevance of the logarithmic potential can be tested only if more than two consecutive energy levels are known. In a separate paper the methods outlined here are applied to quark-confining potentials of the generalized power type.

Quantitative Monitoring of Water Flow Behind And in Wellbore Casing

A nuclear logging technique for measuring water flow in and behind casing was developed and found to determine accurately the linear velocity and volume rate of flow, regardless of flow geometry. This technique detects water channeling caused by faulty cement and allows accurate water production profiling in perforated intervals. production profiling in perforated intervals. Introduction A common problem encountered in production operations is the movement of water in channels within the cement annulus behind casing of oil and gas wells. Such channels provide undesirable paths of communication between formations of different pressures. The channeling of water into oil zones or brine into shallow freshwater formations, which are used to supply potable water for communities, is particularly undesirable.McKinley et al. describe a noise logging technique for locating certain types of fluid movement. This technique primarily measures acoustical noise generated by primarily measures acoustical noise generated by turbulence from high-energy expansion of fluids. Since the noise amplitude generated is a direct function of energy dissipated in the expansion process, this technique does not appear feasible for detecting water channeling when pressure differentials are too low to generate detectable pressure differentials are too low to generate detectable noise amplitudes.Wichmann et al. reported that water flowing outside a 17.78-cm (7-in.) casing was detected by activating the oxygen in the water with 14 million electron volt [MeV] (2243 fJ) neutrons and by detecting this oxygen activation as the water moved past a gamma ray detector. Wichmann et al. concluded that the ability to "tag" any fluid containing oxygen by making it radioactive (without its having been in close contact with the logging tool) is unique and is probably the only way possible to detect the flow of water outside the casing when the water cannot be tagged by conventional tracer techniques. Although the linear flow velocity for a specific interval could be obtained by measuring the time required for the activated water to travel from a point opposite the neutron source to one opposite the detector, they concluded that determination of volume flow rate generally would not be feasible by the activation method. except in very favorable cases.Texaco Inc. has developed a water-flow monitoring system that measures the direction, linear flow velocity, volume flow rate, and radial position of water flowing vertically behind or in wellbore casing. The system is based on a nuclear activation technique in which flowing water is irradiated with neutrons emitted by a logging sonde. These neutrons interact with oxygen nuclei in the water to produce the radioactive isotope nitrogen-16 through the 16 O(n, p) 16 N reaction. 16 N decays with a half-life of 7.13 s, emitting 6.13 and 7.12 MeV (982 to 1141 fJ) gamma radiation. The flow parameters of interest are computed from the energy and intensity response of two gamma ray detectors mounted in the logging sonde.The water-flow monitoring system is similar in some respects to various radioactive tracer techniques, but is unique in the sense that the tracer, 16 N, is "manufactured" in the water. This eliminates the need to perforate the casing and to inject tracer material from an external source. In addition, the technique yields a more quantitative description of the flow parameters than that possible when using conventional tracer techniques. P. 121

Pair production by photons. Screening corrections for intermediate and high energies

The screening correction to the total cross-section for coherent electron pair production by photons is evaluated for intermediate and high energies, by employing the Born-approximation recoil momentum distribution and accurate relativistic form factors. Results are given for all 3≤Z≤92. For each element the results are presented in terms of a seven-parameter formula which is based on a high-energy expansion of the screened cross-section.

Relative density of states and the second virial coefficient of the hard-sphere gas

The first two terms of the high energy expansion of the relative density of states for two hard spheres (needed for the computation of the second virial coefficient at high temperature) are obtained directly from the analytic expression for the phase shifts. This is enough to give the classical second virial and the first quantum correction.

How to calculate the grand partition function in the uncorrelated jet model

We present a new technique for calculating the grand partition function and all quantities of physical interest in the uncorrelated jet model. The method, which is also valid in the large-pT region, consists of the numerical evaluation of the appropriate integral representation in the complex plane. We analyze in detail the difficulties associated with this approach and show how to overcome them. The numerical results are checked with a new high energy expansion for the grand partition function.

Theory of fermion exchange in massive quantum electrodynamics at high energy. V. Twelfth- and higher-order perturbation theory

A study of the high-energy behavior of Compton scattering near the backward direction and of pair annihilation in massive quantum electrodynamics is extended to twelfth order in perturbation theory. These considerations are generalized by introducing a matrix method of calculating higher-order diagrams and the result of the high-energy expansion of an arbitrarily high order of perturbation theory is presented.

High-energy expansion for nuclear multiple scattering

The Watson multiple scattering series is expanded to develop the Glauber approximation plus systematic corrections arising from three effects: (1) deviations from eikonal propagation between scatterings, (2) Fermi motion of struck nucleons, and (3) the kinematic transformation which relates the many-body scattering operators of the Watson series to the physical two-body scattering amplitude. Operators which express effects ignored at the outset to obtain the Glauber approximation are subsequently reintroduced via perturbation expansions. Hence a particular set of approximations is developed which renders the sum of the Watson series to the Glauber form in the center of mass system, and an expansion is carried out to find leading order corrections to that summation. Although their physical origins are quite distinct, the eikonal, Fermi motion, and kinematic corrections produce strikingly similar contributions to the scattering amplitude. It is shown that there is substantial cancellation between their effects and hence the Glauber approximation is more accurate than the individual approximations used in its derivation. It is shown that the leading corrections produce effects of order ${(2k{R}_{c})}^{\ensuremath{-}1}$ relative to the double scattering term in the uncorrected Glauber amplitude, $\ensuremath{\hbar}k$ being momentum and ${R}_{c}$ the nuclear charge radius. The leading order corrections are found to be small enough to validate quantitative analyses of experimental data for many intermediate to high energy cases and for scattering angles not limited to the very forward region. In a Gaussian model, the leading corrections to the Glauber amplitude are given as convenient analytic expressions.NUCLEAR REACTIONS Multiple scattering theory at intermediate to high energy; leading corrections to Glauber approximation due to eikonal, Fermi motion, and kinematic effects.

Correspondence of partial-wave and impact-parameter representations

An alternative but much simpler proof is given that a recently developed high-energy expansion accomplishes the mapping between the partial-wave amplitude and the Fourier-Bessel impact-parameter amplitude. The high-energy expansion is also shown to converge to the exact result for Coulomb scattering at all energies and scattering angles. A Regge pole in the partial-wave amplitude maps to a cut in the Fourier-Bessel impact-parameter amplitude, i.e., the existence of poles in the impact-parameter amplitude is dubious.

S-wave inverse problem and sum rules for potentials with a simple pole at the origin

In a previous paper (F. Calogero and A. Degasperis, J. Math. Phys. 9 (1968), 90) an approach to the inverse problem for the nonrelativistic S-wave potential scattering has been developed with the aim of obtaining explicit and exact expressions of the potential in terms of the phase shift and of the bound states parameters. Since this method, based on the high energy asymptotic expansion, yields the expression of the potential and of all its derivative at the origin only if these quantities are bounded, physically interesting interactions such as the Yukawa potential could not be considered. In this paper this approach has been generalized to potentials with a simple pole at the origin and the expression of the Laurent coefficients of the potential in a neighborhood of the origin has been obtained as an explicit functional of the experimental data. Furthermore the general structure of the high energy expansion of the phase shift has been derived and discussed. The requirement that the potential has only a simple pole at the origin turns out to imply strong constraints on the experimental data. These conditions on the experimental data have been explicitely expressed as a set of infinitely many nonlinear sum rules involving only the phase shift and the energies of the bound states.

High-Energy Expansions of Scattering Amplitudes

The partial-wave series is converted without approximation to a Fourier-Bessel expansion based on a new (infinite series) expansion for the Legendre function. The direct connection between the Fourier-Bessel phase shift and the partial-wave interpolating phase shift is established as an infinite series in powers of ${K}^{\ensuremath{-}2}$ ($K=\mathrm{wave}\mathrm{number}$). The series contains the Glauber eikonal approximation as a leading term and reproduces the results of an eikonal expansion about the Glauber propagator. Corrections to the eikonal approximation are developed and rules are given for an unambiguous interpretation of the eikonal expansion. The relativistic eikonal expansion is discussed for forward and backward scattering without small-angle approximations.

Occurrence of high-energy electrons and surface expansion in laser-heated target plasmas

It is shown that limitations on collisionless electron thermal conduction place a lower limit on the range of energies of electrons in the surface of target plasmas which are heated by incident electromagnetic radiation, and that in cases of contemporary experimental interest these energies can be hundreds of keV. It is then shown that these high-energy electrons can cause a much faster expansion of plasma from the heated surface than would be predicted by a single fluid theory. Numerical simulations demonstrate both effects.

Nuclear radii in the high-energy limit of elastic electron scattering

From the formula for the energy-derivative of the phase shifts a systematic high-energy expansion of the phase shifts for the elastic scattering of electrons from nuclei is deduced. The coefficients of this expansion can be arranged in terms of inverse powers of the energy containing model-independent quantities of the nuclear charge distribution. It is shown that the mean logarithmic radius exp [〈1n r〉] and the inverse square radius 〈1/r 2〉 − 1 2 determine the high-energy behaviour of the phase shifts for small values of the angular momenta. Applications to the elastic electron and positron scattering from nuclei and to dispersion effects are discussed.

First Quantum-Mechanical Correction to the Classical Viscosity Cross Section of Hard Spheres

The classical and first quantum correction terms in a high-energy expansion of the viscosity cross section ${Q}^{(2)}$ for a Boltzmann gas of hard spheres is derived. The first correction is found to be proportional to ${(\frac{1}{k\ensuremath{\sigma}})}^{\frac{4}{3}}$, which is a term nonanalytic in $\ensuremath{\hbar}$ (i.e., ${\ensuremath{\hbar}}^{\frac{4}{3}}$), and results from scattering near the edge of the sphere. A bound is established showing the remainder of the asymptotic series to be of $O[\frac{\mathrm{ln}(k\ensuremath{\sigma})}{{(k\ensuremath{\sigma})}^{2}}]$. This asymptotic formula is compared with calculations based on the exact phase-shift expressions and its range of validity is established. The next correction terms are deduced to be proportional to $\frac{(\mathrm{ln}k\ensuremath{\sigma})}{{(k\ensuremath{\sigma})}^{2}}$ and $\frac{1}{{(k\ensuremath{\sigma})}^{2}}$ which involve ${\ensuremath{\hbar}}^{2}\mathrm{ln}\ensuremath{\hbar}$ and ${\ensuremath{\hbar}}^{2}$, respectively.

Asymptotic form ofn-th-order Spence functions

The constant terms in the large-amplitude expansions ofn-th-order Spence functions are derived. The results are then used to relate the perturbation coefficients in the high-energy expansions of the renormalized Green's functions and the renormalization constants in quantum electrodynamics.

High-Energy Scattering for Yukawa Potentials

A reexamination is made of nonrelativistic scattering by a superposition of Yukawa potentials in the frame of a high-energy perturbation method developed previously by one of the authors. It is found that for Yukawa potentials expandable in ascending powers of r, the S matrix may be written in a remarkably simple form. A high-energy asymptotic expansion is obtained for the phase shift. The residue function at the Regge poles is found to have the same general form as for a Coulomb potential and is formally independent of any high-energy approximation.

Equivalence of the Sudden Approximation to the High-Energy Limit of the First Born Approximation

The ratio of total cross sections for two different inelastic events in a given target as predicted by the sudden approximation is shown to be equal to the high-energy limit of the corresponding first Born approximation to the ratio. This relationship between the two theories is established for the special cases of the ratio of single to double ionization of helium atoms by electrons and by hydrogen atoms. The electron-helium-atom ratio is found to be the same as the hydrogen-atom-helium-atom ratio. The proof supposes that the same approximate wave functions were used in both theories and that the dominant term in the high-energy expansion of the first Born total cross section is independent of energy-conservation requirements. This latter point is established for a general ionization event.

Eigenwerte des Lippmann-Schwinger-Kerns f�r Yukawa- und Nukleon-Nukleon-Potentiale

In several papers we recently obtained simple high-energy asymptotic expansions for the solutions and eigenvalues of wave equations containing generalized superpositions of Yukawa potentials. In the present article we extend these investigations to the calculation of phase shifts and eigenvalues of the Lippmann-Schwinger kernel. We also calculate corresponding Pade approximants and illustrate, by means examples, their usefulness, even in regions of low energies.

High-energy relations for scattering of particles and antiparticles with arbitrary spin following from analyticity and crossing relations

Using analyticity and crossing relations for helicity amplitudes in two-body elementary-particle reactions proceeding via strong interaction, the connection between the high-energy asymptotic expansions of these amplitudes in the direct and the crossed reaction is derived. This connection is simple and generalizes in a straightforward way the corresponding relation for spinless particles. Besides allowing a proof of Pomeranchuk-type theorems for differential cross-sections and polarizations, the result constitutes a very convenient parametrization of the helicity amplitudes, useful in the theoretical analysis of two-body experiments at high energy.

High-energy behaviour of Feynman amplitudes

A method that is more general than existing ones is developed for the determination of the coefficient of the leading term in the high-energy expansions of an arbitrary Feynman integral. Some of the discussion is restricted to the case of heavy bosons having a ψ3-interaction, but it can be generalized to cover other types of Feynman integrals as well.