Regge Pole Theory Research Articles

Spin and Polarization in High-Energy Hadron-Hadron and Lepton-Hadron Scattering

The role of spin degrees of freedom in high-energy hadron-hadron and lepton-hadron scattering is reviewed with emphasis on the dominant role of soft, diffractive, non-perturbative effects. Explicit models based on analyticity and Regge-pole theory, including the pomeron trajectory (gluon exchange in the t channel) are discussed. We argue that there is a single, universal pomeron in Nature, manifest as relatively “soft” or “hard”, depending on the kinematics considered. Both the pomeron and the non-leading (secondary) Regge trajectories, made of quarks are non-linear, complex functions. They are populated by a finite number of resonances: known baryons and mesons in case of the reggeons and hypothetical glueballs in case of the pomeron (“oddballs” on the odderon trajectory). Explicit models and fits are presented that may be used in recovering generalized parton distributions from deeply virtual Compton scattering and electoproduction of vector mesons.

New physics from TOTEM’s recent measurements of elastic and total cross sections

We analyze the recently discovered phenomena in elastic proton–proton scattering at the LHC, challenging the standard Regge-pole theory: the low- ‘break’ (departure from the exponential behavior of the diffraction cone), the accelerating rise with energy of the forward slope B(s, t = 0), the absence of secondary dips and bumps on the cone and the role of the odderon in the forward phase of the amplitude, ρ(13 TeV) = 0.1 ± 0.01, and especially its contribution at the dip region, measured recently by TOTEM. Relative contributions from different components to the scattering amplitude are evaluated from the fitted model.

Recent LHC/TOTEM data challenging the standard Regge pole theory

We analyse the recently discovered phenomena in elastic protonproton scattering at the LHC: the low-|t| “break” (departure from the exponential behaviour of the diffraction cone), the accelerating rise with energy of the forward slope B(s), the absence of secondary dips and bumps on the cone and the role of the odderon in the forward phase of the amplitude, ρ(13 TeV) = 0:09 ± 0:01, especially its contribution at the dip region, measured recently by TOTEM. The dip at 13 TeV seems to become more shallow with respect to lower energies, and we attribute this important new phenomena to the odderon contribution. These findings call for a revision of the standeard Regge-pole theory.

Novel theoretical approaches to small-angle electron scattering

The difficulties of obtaining reliable measurements of the electron differential cross sections (DCSs) for atomic, ionic and molecular transitions at and near zero scattering angles are well documented. Hence, the need for reliable theoretical calculations. Recently, three theoretical approaches have been derived to investigate and guide measurements of small-angle, including zero, electron DCSs in atoms, ions and molecules. The first method, the momentum dispersion method (MDM), based on Regge Pole theory, uses the analytical continuation of the generalized oscillator strength (GOS) function to obtain the smaller angle, including zero, data from the more reliably measured larger angular data. The second method, the forward scattering function (FSF), represents a unique path of the GOS function to the OOS. It is therefore useful for normalizing the measured relative electron DCSs through the GOSs. Very recently, a singular behavior has been found in the electron–atom scattering DCS at small momentum transfer, K coming from second-order terms and a new generalized Lassettre expansion has been derived. At forward scattering, it is expected to yield the unique long sought after curve that normalizes the measured relative electron DCSs to the OOSs. The utility of the methodologies is demonstrated using atomic, ionic and molecular transitions.

Recent theoretical advances in small-angle electron scattering

Two recently developed methodologies for small-angle, including zero, electron scattering are presented briefly and applied to various atomic and molecular transitions. The first is the momentum dispersion method of Haffad et al. (1996) which is based on Regge pole theory; it uses the analytical continuation of the generalized oscillator strength (GOS) function to obtain the small angle, including zero, data from their more reliably determined larger angular measurements. The second method is the forward scattering function of Avdonina et al. (1997) which employs only the optical oscillator strength as input; it describes a unique path to reach the Lassettre limit without traversing the nonphysical region and provides a stringent test of both theory and measurement at zero scattering angles. Combined, these methods are also effective in normalizing the measured relative electron differential cross-sections (DCSs). Electron impact excitation DCSs for H, He, Na, Xe, F 2, SF 6 and N 2 are investigated through the GOSs to demonstrate the power and versatility of the methods to analyze small-angle electron impact data and identify spurious behavior. The results cover the energy range from near threshold to the first Born Approximation limit; in particular, for He 1 1S–2 1P 0 the most recent measurements are compatible with the forward scattering function.

Hadronic diffractive cross sections and the rise of the total cross section.

A model for high-energy hadronic cross sections is proposed. It is based on Regge theory and perturbative QCD, and includes soft and hard mechanisms as well as diffractive processes. The validity range of Regge-pole theory in the description of total, elastic, single-, and double-diffractive cross sections is investigated and inconsistencies found already at CERN LHC and/or SSC energies. Examining unitarity constraints, modifications of the cross section formulas are proposed which allow a continued use of formulas in the Regge spirit to describe elastic and diffractive events. The nondiffractive cross section is allowed to rise at a rate consistent with unitarization of multiple parton-parton scatterings. However, in our picture, the rise of the total cross section with increasing energy is only partly due to the minijet cross sections; the diffractive topologies rise as well. Fully differential distributions are given and convenient parametrizations derived for the integrated rates of elastic and diffractive events. Predictions for the various partial cross sections at Fermilab Tevatron, LHC, and SSC energies are given and compared to other estimates.

On the non-radial oscillations of a star. IV An application of the theory of Regge poles

It is shown how the flow of energy in the form of gravitational radiation, through a star in non-radial oscillations, can be determined by a suitable adaptation of Regge’s theory of potential scattering in the quantum theory. While the resonant scattering of axial gravitational waves can be treated by Regge’s theory, essentially, in its original form, the treatment of the resonant scattering of polar gravitational waves requires a generalization of the theory beyond its standard range and the incorporation of a ‘flux integral’ derived from independent considerations. Illustrative examples of the applications of the theory are provided.

Arbitrary-order three-turning-point phase-integral formula for the S matrix in Regge-pole theory.

Arbitrary-order three-turning-point phase-integral formula for the S matrix in Regge-pole theory.

Accurate evaluation of quantal saddles and Regge pole positions and residues for calculation of elastic differential cross section by the Prüfer phase function method

The Prüfer phase function method for calculation of differential cross section by Regge pole theory is described and tested. The approach provides a procedure for exact quantal calculation of Regge pole positions and residues, quantum saddles and other quantities required for evaluation of the differential cross section to virtually any desired accuracy and with realistic error control. Applications to a complex optical potential and to rainbow scattering are presented.

Green's Functions for Composite Particles in Quantum Chromodynamics

We discuss the bound state problem of quantum chromodynamics. We give the reduc­ tion formula for nonlocal gauge-invariant composite operators which describe composite particles using the method of Haag, Nishijima and Zimmermann. We develop the method of auxiliary fields for nonlocal gauge-invariant composite operators. § 1. Introduction Different models of strong interactions have been proposed corresponding to different energy regions considered; that is, current algebra and chiral dynamics for the low energy region; the dual resonance model and the dual string model for the Regge energy region; and the parton model for the deep inelastic region. Recently, quantum chromodynamics (QCD) has attracted much interest of particle physicists as a model which may combine properties of the models of three energy regions. Quarks are probably confined; at least, they have not appeared in the experi­ ments up to the present. Therefore the ordinary perturbative expansion with quarks and gluons as intermediate states may not reveal the real physical world, because the ordinary perturbation theory restricts asymptotic conditions. The ex­ pansions must probably be carried out using bound states of quarks and gluons as intermediate states. Moreover, the Regge pole theory and the dual model indicate that not only stable hadrons but also unstable resonances should be included in the intermediate states, while the dual string model suggests that the intermediate states have semiclassical string configurations. Further, only the states which can be described by color singlet states have been observed and no massless particles participating in strong interactions seem to exist. These facts suggest that the color gauge invariance remains unbroken even by the state vectors and the asymptotic states are gauge-invarianel, 2J (see also Ref.

Low-energy nucleon-nucleon potential from Regge-pole theory

The results from a potential model for the low-energy $\mathrm{NN}$ interaction based on Regge-pole theory are presented. The forces are due to the dominant parts of the $\ensuremath{\pi}$, $\ensuremath{\eta}$, ${\ensuremath{\eta}}^{\ensuremath{'}}$, $\ensuremath{\rho}$, $\ensuremath{\omega}$, $\ensuremath{\varphi}$, $\ensuremath{\delta}$, $\ensuremath{\epsilon}$, ${S}^{*}$ trajectories in the complex $J$ plane, which are the well-known one-boson-exchange forces. Novel features are the dominant $J=0$ parts of the Pomeron, $f$ ${f}^{\ensuremath{'}}$ and ${A}_{2}$ trajectories. At the Reggeon vertices we use exponential form factors, as suggested by high-energy fits. The Pomeron, $f$, and ${f}^{\ensuremath{'}}$ trajectories lead essentially to repulsive central Gaussian potentials. This soft-core, partially nonlocal potential model fits the $\mathrm{NN}$ data with $\frac{{\ensuremath{\chi}}^{2}}{\mathrm{data}}=2.09$, which is lower than any other model we know of. The $\mathrm{NN}$ coupling constants have reasonable values, and the contributions of the Pomeron and tensor trajectories agree with estimates from high-energy fits.

Why has Regge pole theory survived?

In spite of a very mixed record of success and failure, Regge theory, in which the notion of complex angular momentum is introduced into quantum mechanics still lives on. What has dissuaded physicists from discarding it, and what is our present assessment of the theory?

Interference effects in large angle elastic scattering of chemically reactive systems

A computational study is made of a Regge pole theory of large angle elastic scattering in chemically reactive systems. When the transition from absorbancy to transparency is rapid in the S matrix, summation of the partial wave expansion shows the Regge theory accurately predicts oscillations and a backward glory in the large angle scattering. When the transition is less rapid a new oscillatory effect is predicted. It arises from the interference of a surface wave with the direct elastic scattering.

Upper bound on the slope parameter

We find the axiomatic upper bound on the slope parameter of the scattering amplitude, which is logarithmically divergent with respect to energies in the region 0 < t < 4 m 2 π . This upper bound is realized in Regge-pole theory. We also obtain explicitly the numerical upper bound on the slope parameter of the linear Regge trajectory.

Theory of Regge poles for 1/r2 potentials. II. An exactly solvable example at zero energy

We give the exact solution of the Schrödinger equation at zero energy and derive an expression in closed form for the Regge poles for a particular 1/r2 potential with the behavior r2V (r) = −V0 at r = 0 and r2V (r) = −V2 at r = ∞. We give detailed results on the properties and distribution of the Regge poles in the λ plane and find them to be in agreement with the predictions of a previous paper in this series.

Theory of Regge poles for 1/r2 potentials. I

In the following series of papers we give a detailed study of the theory of Regge poles for 1/r2 potentials with the behavior r2 V (r) = −V0 at r = 0 and r2 V (r) = −V2 at r = ∞. We give a complete description of the distribution of the Regge poles in the λ plane, which is cut from −V01/2 to V01/2 and study the behavior of the pole trajectories. We find that the high energy limit of the Regge poles is controlled by the parameter p = (λ2−V0)1/2 while at low energies the relevant parameter is q = (λ2−V2)1/2. This means that the point λ = 0 for the case of Yukawa potentials corresponds here to the point q = 0. We also find that the Regge trajectories λ (E) may have branch points of the square root type at finite, in general complex, values of E at which points the pole passes the origin λ = 0. We further find that the kinematic singularity of the S matrix at k = 0 is more complicated than it is for Yukawa potentials and is here characterized by the Floquet parameter ν (λ,k) associated with the Schrödinger equation. We illustrate these and other results with some

Theory of Regge poles for 1/r2 potentials. III. An exact solution of Schrödinger’s equation for arbitrary l and E

We give the exact solution of the radial Schrödinger equation and derive the S matrix for arbitrary energy and angular momentum for a particular 1/r2 potential with the behavior r2V (r) = −V0 at r = 0 and r2V (r) = −V2 at r = ∞. We obtain an expression for the Regge poles and study their properties and distribution at low energy. The present results are in agreement with those obtained in a previous paper in this series.

Elastic and inelastic scattering of high-energy particles on the deuteron

The scattering amplitudes on the deuteron at small momentum transfers are described, taking into account the shadow corrections. In the framework of Regge pole theory, the inelastic shadow corrections are expressed in terms of the amplitudes of inelastic processes. The validity limit of the formulae thus obtained is discussed.

Electronic structure of transition metals. III.d-band resonance and Regge-pole theory

This is the third paper in our current series on the electronic structure and properties of the transition metals in the pseudopotential approximation. Here we review, reformulate, and generalize the resonance model of $s\ensuremath{-}d$ hybridization in one-electron energy bands so as to be applicable to scattering processes at arbitrary energies different from the eigenvalues of the one-electron wave equation in a crystal. This is achieved by formulating the resonance model in two ways, first in the framework of the one-electron energy-band secular equation given by the Korringa-Kohn-Rostoker (KKR) method in Ziman's "plane-wave representation," and second as a scattering problem in terms of standard partial-wave phase-shift analysis. The former leads to a real resonance energy, say ${E}_{d}$, while the latter leads to a complex resonance energy, say ${E}_{d}+\frac{1}{2}\mathrm{iW}$ where $W$ is a measure of $s\ensuremath{-}d$ interaction which also determines the $d$ bandwidth. To relate ${E}_{d}$ and $W$, and hence determine $W$ in the framework of the KKR or any other valid transition-metal pseudopotential form factor, the scattering problem is reformulated in complex angular momentum representation using the Regge-pole theory in nonrelativistic potential scattering; the result, $W\ensuremath{\propto}{(\frac{\mathrm{dE}}{\mathrm{dl}})}_{l=2}$, is then interpreted in the context of the transition-metal-model-potential form factor discussed in the first two papers of this series and the quantitative predictions are compared with the results of augmented-plane-wave one-electron energy-band calculation and those of the renormalized-atom method, and with recent photoemission data. The agreement between the various results is systematically good for transition metals of the $3d$, $4d$, and $5d$ series, as far as the results are known.

Minimal Regge model for meson-baryon scattering: Duality, SU(3), and phase-modified absorptive cuts

A model is presented which incorporates economically all of the modifications to simple SU(3)-symmetric dual Regge-pole theory which are required by existing data on ${0}^{\ensuremath{-}}$${\mathrm{\textonehalf{}}}^{+}$${\ensuremath{\rightarrow}0}^{\ensuremath{-}}$${\mathrm{\textonehalf{}}}^{+}$ processes. The basic assumptions are no-exotics duality, minimally broken SU(3) symmetry, and absorptive Regge cuts phase-modified by the Ringland prescription. We first describe qualitatively how these assumptions suffice for the description of all measured reactions, and then present the results of a detailed fit to 1987 data points for 18 different reactions.