Angular Momentum Plane Research Articles

Jet-like processes at small angles in QED for arbitrary polarization of the particles

As noticed by Fadin, Kuraev and Lipatov in 1975 incorporation of asymptotic freedom into the BFKL equation splits the QCD pomeron into a series of isolated poles in complex angular momentum plane. Following our earlier work on the proton structure function we explore the phenomenological consequences of the emerging BFKL-Regge factorized expansion for the small-x structure function of the pion F2π(x,Q2). We calculate F2π in a small-x region and find good agreement with the recent H1 determination of F2π(x,Q2). We also present the BFKL-Regge factorization based evaluation of the contribution from hard scattering to the observed rise of the NN, πN and real photo-absorption γN and γπ total cross sections.

Analytic extension of the nuclear algebraic potential.

An analytic extension of the nuclear algebraic potential in the complex energy and angular momentum planes is discussed and an approximation for the algebraic potential in agreement with the known analytic properties of the S-matrix is proposed. The invariance of the energy spectrum of the Coulomb part of the interaction is established. The results are applied to the Regge pole analysis of the $^{12}\mathrm{C}$${+}^{24}$Mg elastic collision at ${\mathit{E}}_{\mathit{l}\mathit{a}\mathit{b}}$=23.0 MeV.

Generalized Lamb waves near mode thresholds of spherical shells.

Pole trajectories as a function of frequency in the complex angular momentum plane for backscattering from empty spherical shells with and without significant fluid loading are presented. These are correlated with Lamb wave threshold features (that depend on Poisson’s ratio) found on a flat plate in vacuum. [R. D. Mindlin, ‘‘Waves and Vibrations in Isotopic, Elastic Plates,’’ in StructuralMechanics (Pergamon, New York, 1960); A. H. Meitzler, J. Acoust. Soc. Am. 38, 835–842 (1965).] The trajectories for the fluid-loaded case are relevant to the high-frequency enhancements and longitudinal (or thickness) resonance of recent investigations [see, e.g., S. G. Kargl and P. L. Marston, J. Acoust. Soc. Am. 89, 2545–2558 (1991)]. The usual poles lie in the first quadrant of the upper half-plane. Near the longitudinal resonance condition a pole associated with a symmetric wave is present in the second and fourth quadrants and is associated with the high-frequency enhancement. Its pole trajectory indicates that the phase and group velocity are oppositely directed over a small range of ka where the enhancement occurs. [Work supported by ONR.]

Nuclear reaction theory of resonances and surface waves

The theory of resonances, regarded as singularities in the complex angular-momentum plane, is here reconsidered taking into account the role played by the so-called «echoes» of resonances. In this way we may describe sequences of resonances and echoes. As a typical example we consider the rotational band produced by the4He-4He elastic scattering. In this theory the poles of theS-function describe families of resonances and are related to the symmetries underlying the dynamics. In order to exhibit this connection we approach the many-body problem by the use of the «grand-angular-momentum tensor». We show, in detail, how various rotational bands emerge in three-body problem. The extension of the analysis to many-body problem is outlined. Furthermore by the use of collective coordinates we briefly illustrate how rotational excitations can be described as hydrodynamical vortices, obtaining a model which fits the complex angular-momentum picture of resonances. At higher energies the resonances smear out towards diffractive effects like the surface waves; this process is described by the motion of poles in the complex angular-momentum plane. Using the ray-tracing method, in the spirit of Keller geometrical diffraction theory, a detailed analysis of surface waves, and of the related phenomena like caustics and rainbows, is developed.

ANALYTIC MULTI-REGGE THEORY AND THE POMERON IN QCD I: THE SUPER-CRITICAL POMERON

The formalism of analytic multi-Regge theory is developed as a basis for the study of abstract critical and super-critical pomeron high-energy behavior and for related studies of the Regge behavior of spontaneously broken gauge theories and the pomeron in QCD. Asymptotic domains of analyticity for multiparticle amplitudes are shown to follow from properties of field theory and S-matrix theory. General asymptotic dispersion relations are then derived for such amplitudes in which the spectral components are described by the graphical formalism of hexagraphs. Further consequences are distinct Sommerfeld-Watson representations for each hexagraph spectral component, together with a complete set of angular momentum plane unitarity equations which control the form of all multi-Regge amplitudes. Because of this constraint of “reggeon unitarity” the critical pomeron solution of the reggeon field theory gives the only known “non-trivial” unitary high-energy S-matrix. By exploiting the full structure of multi-Regge amplitudes as the pomeron becomes super-critical, one can study the simultaneous modification of hadrons and the pomeron. The result is a completely consistent description of the super-critical pomeron appearing in hadron scattering. Reggeon unitarity is satisfied in the super-critical phase by the appearance of a massive “gluon” (Reggeized vector particle) coupling pair-wise to the pomeron.

From resonances to surface waves in pi +-p scattering.

Resonances may be grouped in families and smear out towards diffractionlike effects: surface waves. Classical Breit-Wigner theory, which is local, cannot describe this evolution from resonances to surface waves. We develop a global theory of resonances, by the use of complex angular momentum plane, which is apt to describe this evolution. Phenomenological examples are taken from ${\ensuremath{\pi}}^{+}$-p scattering.

Lamb and creeping waves around submerged spherical shells resonantly excited by sound scattering

Acoustic wave scattering by spherical shells in water in the resonance region is studied. The interaction is studied classically and by the resonance scattering theory (RST). The connection between internal resonances and the Lamb waves excited in the shell is analyzed. Conditions are derived that govern the propagation of Lamb waves in a spherical shell when it is fluid-loaded or in vacuo. These general conditions reduce to the usual conditions for Lamb waves in plates, in the large radii limit. The connection is established between the outer scattering problem and the internal vibrational problem that excites the shell resonances, and it is demonstrated that the creeping-wave series that synthesizes the Franz waves around the shell, can be obtained quite simply without the use of the Watson–Sommerfeld Method (WSM). All that is required is suitable one-term expansions of the denominators of the scattering amplitudes. This eliminates the need for the cumbersome excursions through the complex angular momentum plane ν of the WSM. Calculations are presented for the summed form functions of seven shells of three compositions and thicknesses over quite broad frequency bands. Subtraction of rigid or soft backgrounds permits one to draw many conclusions as to the behavior of the shell response in various spectral regions. The relative phase between the form function and the background is also displayed. This phase is helpful for the identification of actual resonances and for the choice of proper background. It is shown how the poles of the scattering amplitude in the x plane split into two subsets. One set, the Franz set, depends only on shell shape, and is related to the external Franz (creeping) waves in the water. This set does not change with varying shell thickness or composition. The other set, the Lamb set, depends only on material composition, and is related to the Lamb waves in the shell. These poles are displayed in various cases, and it is shown that for thin shells, only one family of Lamb poles (the zeroth-order symmetric family, s0) is dominant, which explains the relatively simple structure of the form function of a thin shell. For thicker shells, many modes interact, and one is forced to go to the usual partial-wave analysis of the RST. A very accurate picture of the scattering process taking place around and inside the shell emerges from this analysis.

The Worden sum rule and perturbative QCD

The amplitude T ++−−( λ 1 λ 2 → λ 1 λ 2) for forward elastic photon-photon scattering is believed to have a J = 0 fixed pole in the complex angular momentum plane. A parton model calculation [1] found that the residue of the fixed pole was given entirely by the “box” diagrams, and the Worden sum rule 2 π ∫ w T ∞ dw w ImT ++−− = − 3 2π ∑ flavours i e i 4 was derived. In this paper the first-order QCD correction to the box diagrams is calculated. The result is not consistent with a J = 0 fixed pole interpretation. The Worden sum rule appears to diverge.

Resonances and surface waves

This paper is devoted to illustrating the differences between resonances and surface waves when both of these phenomena are pictured in the complex angular-momentum plane. After some critical remarks concerning the trajectories connecting several resonances, we discuss in detail the mechanism which produces the surface waves. Using the geometrical theory of diffraction, we develop our model starting from the eikonal and transport equations. Particular attention is devoted to the role played by the caustics. The formulae for large-angle differential cross-sections relative to a completely opaque sphere, and also for an opaque sphere surrounded by a nearly transparent edge, are derived. Finally this surface wave model is discussed in the light of phenomenological examples of angular distributions in α-nuclei and heavy-ion elastic scattering.

On the complex angular momentum theory of scattering

A contribution to the theory of complex angular momentum techniques in the field of atomic and molecular collisions is given. The author derives a new, flexible representation of the scattering amplitude on the basis of realistic assumptions for the behaviour of the S-matrix in the complex angular momentum plane. The representation has the form of a sum of steepest-descent integrals, S-matrix residue terms and a symmetry-type background integral. The flexibility is due to the presence of two integer parameters which may be chosen conveniently so as to make the residue sums sufficiently convergent and to minimise the total number of important terms.

Double logarithmic asymptotics and regge singularities of quark amplitudes with flavour exchange

We derive simple equations in terms of definite signature partial waves for the quark scattering and annihilation amplitudes in the double logarithmic approximation of QCD. We apply the method of isolating the softest virtual particle in the graphs which is based on gauge invariance and unitarity. Besides the particle Regge trajectories and corresponding cuts, there are further singularities in the angular momentum plane generated by perturbative double logarithmic contributions and which may be of phenomenological relevance. We also consider amplitudes with external currents and discuss how to include ultraviolet single logarithms and non-perturbative contributions.

Model for the structure functions of hadrons

Multijet production at extremely high energy, generated by iteration of the standard hard-parton-scattering mechanism, in the framework of quantum chromodynamics, is assumed to be processed by low-p/sub T/ hadron multiplicities satisfying Koba-Nielsen-Olesen scaling. This assumption is argued to lead to a specific structure in the angular-momentum plane, which gives rise to long-range effects in hadron collisions, and in particular to a rising component in the hadron cross section. This component is due to hard parton subcollisions and can be correlated with the hadron structure function. The model is shown to be consistent with ..pi..p, Kp, and pp total-cross-section data, as well as with the ..pi.., K, and p structure-function data.

Double-logarithmic asymptotics of quark scattering amplitudes with flavor exchange

We propose simple equations in terms of the definite-signature partial waves of the quark scattering and annihilation amplitudes with quark-quark and quark-antiquark states in the exchange channel. We discuss the singularities in the complex angular momentum plane generated by the double-logarithmic contributions and point out their relation to the particle Regge trajectories.

Study of quasi-continuum gamma-ray deexcitation of 152–161Dy nuclei populated in α- and 12C-induced reactions

The influence of the reaction mechanism and of the nuclear structure at high angular momentum on the γ-ray deexcitation process is studied for Dy nuclei produced in a (40–80 MeV) and 12C (80–110 MeV) induced reactions on 154, 155, 158, 160Gd and 146, 150Nd targets, respectively. The quasi-continuum (q.c.) γ-ray spectra for selected exit channels are decomposed into collective and statistical parts for which the γ-ray energies, the multiplicities and partly the multi-polarities are determined. Large differences in the constitution of the q.c. γ-ray cascades and in the population strengths of high-spin yrast states in 152Dy and 156Dy are attributed to nuclear structure effects at high angular momentum. This constitution and the centroids of the γ-ray entry regions in the energy versus angular-momentum plane are also evaluated as functions of beam energy and projectile. The γ-ray multiplicities and both coordinates of the entry region centroids increase initially with bombarding energy and then level off or even decrease at the highest energies. The observed increase is reproduced by a statistical-decay model, but at the higher bombarding energies strong deviations occur which are discussed in view of the reaction mechanisms, including incomplete fusion and non-equilibrium neutron emission, particularly absorptive breakup.

The inverse scattering transformation in the angular momentum plane

The inverse scattering transformation (IST) with the angular momentum (λ) as a spectral variable turned out to be a useful method to deal with rotationally invariant problems in field theory at higher spatial dimensions [Refs. (1) and (2)]. We derive the direct and inverse scattering problems for thev-dimensional radial Schrodinger equation for variable λ and fixed energy (negative or positive). We determine the scattering data (SD) in one to one correspondence with the potential and derive the corresponding Gelfand-Levitan equation. A family of exactly solvable potentials for any λ and fixed energy is obtained. The trace identities associated to this IST are derived for both signs of the energy. They relate integrals of local polynomials in the potential and its derivatives timesr2n−1(n≧1) with the SD. The presence of this power ofr makes these relations useful in higher dimensions.

The vacuum channel in non-abelian gauge theories

We discuss the Lipatov equation in the channel with the vacuum quantum numbers, after a continuation in the space-time dimension. We find that the fixed cut in the angular momentum plane, present in the Higgs-regularized solution, is now absent for every value of ϵ.

Two-Reggeon cut in the ordered phase of Reggeon field theory

The ordered phase of Reggeon field theory is analyzed in the framework of the Reggeon quantum spin model with three- and four-point couplings. We work at a particular value of the bare-intercept gap, ${\ensuremath{\Delta}}_{0}={\ensuremath{\Delta}}_{0M}$, for which the spin model is exactly solvable in one transverse dimension. We include the higher single-site states by developing a perturbative expansion for this spin model. The expansion is shown to be characterized by a small dimensionless parameter $\ensuremath{\epsilon}=\frac{16{\ensuremath{\alpha}}_{0}^{\ensuremath{'}}{{\ensuremath{\Delta}}_{0M}}^{3}}{{{r}_{0}}^{4}}$. At each stage of calculation we are able to relate the spin model to the continuum Reggeon field theory by letting the lattice spacing go to zero. The two-point function is then calculated to order $\ensuremath{\epsilon}$. We find that the necessary condition for satisfying Reggeon unitarity, of Regge-Mandelstam singularities appearing in the complex angular momentum plane, is satisfied in the form of the two-Reggeon cut.

Parametrized S matrix models of elastic scattering in chemically reactive systems

The theory of large angle elastic scattering in chemically reactive systems using parametrized S matrix models is investigated Calculations of the large angle scattering are carried out for three new parametrizations, in addition to one that has been used previously. All parametrizations are of the smooth step function type, although their analytic structure in the complex angular momentum plane is completely different. Both meromorphic and non-meromorphic parametrizations are employed. Large angle interference effects are found for all parametrizations provided the opacity function is sufficiently black and the transition from absorbency to transparency is sufficiently rapid.

Integral representations of the Coulomb amplitude

Analytic continuations into the left half of the complex angular momentum plane permit the derivation of some exact integral expressions for the attractive and repulsive Coulomb amplitudes. The problem of background integrals is thus removed and the expressions may be written either in a form which is suitable for expressing the amplitudes as a sum over all the Coulomb poles or as integrals in regions of the complex plane which contain no poles. The latter forms are extremely convenient for the application of the saddle-point method and are used to obtain the semiclassical limits for the amplitudes. All the terms of the Poisson summation formula are correctly taken into account and generalisations to other cases are discussed.

Regge poles in the collective model of the nuclear giant multipole resonances

Giant nuclear resonance states, of higher multipolarity than dipole, have recently been observed in hadron and electron scattering. They may possibly be described by the collective Goldhaber-Teller model as extended to the higher multipolarity case, and as generalized to include spin-isospin vibrations and spin waves. Based on such a model, we show that the multipole resonances, together with the conventional dipole resonances, can be generated by the motion of a few basic Regge poles through the complex angular momentum plane. This viewpoint unifies all the resonances with a given spin-isospin character but arbitrary value of the multipolarity, and suggests the existence of the higher-multipole resonances as a consequence of the existence of the dipole resonance. We also analyze these Regge poles in terms of collective surface waves (“creeping waves”) circumnavigating the nucleus, and we determine their phase velocities and attenuations. Finally, the giant resonances them selves are explained as a resonant reinforcement of phase-matched surface waves.