Impact Parameter Amplitudes Research Articles

Resonant mode of elastic scattering of protons at ultrahigh energies

An interpretation is given for elastic proton scattering at ultra-high energies with impact-parameter amplitude exceeding the black disk limit. It is shown that this scattering mode can arise due to the contribution of an exceptional intermediate state that unites correlated partons of colliding protons at ultra-high energies into a single coherent system. The behavior of the real part of the amplitude in this mode is discussed.

An Impact Parameter Formalism

A relativistic impact-parameter formalism is introduced without any approximation. It is applicable for all physical values of energy and scattering angle. An impact parameter amplitude is defined from an invariant scattering amplitude by the integral with the Bessel function, in place of the Legendre function in the definition of a partial-wave amplitude. We find that this impact parameter amplitude is an example of a new integral expansion. It is proved that one of the necessary and sufficient conditions that this new expansion is permissible, is that the amplitude should satisfy the Kapteyn equation, which is required in the Webb-Kapteyn theory of the Neumann series expansion of an odd function. The physical interpretation of the amplitude is discussed and cross sections are expressed by simple integrals over the impact parameter without any approximation. The unitarity relation for the amplitude is expressed in the simpler form. Comparison is made with three other relativistic formalisms proposed by Blankenbecler and Goldberger, Ida, and Cottingham and Peiels. In contrast with our formalism, the applications of these three formalisms are limited either to the extremely high energy region or to only the forward angle scattering.

Photoproduction within the two-component Dual Parton Model: Amplitudes and cross sections

In the framework of the Dual Parton Model an approximation scheme to describe high energy photoproduction processes is presented. Based on the distinction between direct, resolved soft, and resolved hard interaction processes we construct effective impact parameter amplitudes. In order to treat low mass diffraction within the eikonal formalism in a consistent way a phenomenological ansatz is proposed. The free parameters of the model are determined by fits to high energy hadro- and photoproduction cross sections. We calculate the partial photoproduction cross sections and discuss predictions of the model at HERA energies. Using hadro- and photoproduction data together, the uncertainties of the model predictions are strongly reduced.

Analytic model for soft and hard hadronic collisions

Starting from a dual analytic model that interpolates soft and hard collisions, we give an explicit expression for its impact parameter amplitude at high energy. The mean effective interaction radius is evaluated and compared with experiment. Constraints on the parameters of the model are discussed.

Fourier-Bessel representation for1r2potentials

An exact Fourier-Bessel impact parameter amplitude S/sub F/(b) is given in the form of a quadrature for the case of nonrelativistic scattering by a 1/r/sup 2/ potential. The only other potential for which S/sub F/(b) is known is the Coulomb potential. An asymptotic expansion of the exact S/sub F/(b) reproduces eikonal expansion results. For small impact parameter, the exact S/sub F/(b) is not correctly represented by the eikonal expansion series due to the 1/r/sup 2/ singularity of the potential. Perturbation theory results are also presented.

On the relation between quantum-mechanical and impact parameter amplitudes in charge exchange

The relation between the impact parameter capture amplitude a( rho ,v) and the quantum-mechanical one f(q,v) in charge transfer is obtained using the Brinkman-Kramers (1930) approximation. Analytical expressions for a( rho ,v) and calculated cross sections are given for some capture reactions using hydrogen-like wavefunctions.

Calculation of Yukawa scattering and impact parameter amplitudes using Legendre Pade approximants

The convergence of a certain sequence of Legendre Pade approximants defined in a previous work (1979) is illustrated numerically for the case of the Yukawa potential, and in particular when this potential is such that the partial wave series converges only slowly. The sequences converge appreciably faster than this series in the physical region, and also converge quite quickly even in regions where the series diverges. The Legendre Pade approximants are used to construct convergent sequences of approximants to the impact parameter amplitude, and numerical results are given in the Yukawa case. The unitarity of the extrapolated partial waves is also studied numerically.

Spin dependence of fermion-fermion scattering in the eikonal limit

The generalized ladder series of Feynman diagrams for scattering of two Dirac particles by neutral-vector-meson exchange is expanded, using functional methods, to obtain a spin-dependent relativistic eikonal approximation and the next term of an expansion about the eikonal limit. The spin dependence of the eikonal scattering amplitude is quite simple. The usual scalar impact-parameter amplitude $\ensuremath{\sim}({e}^{i\ensuremath{\chi}0(b)}\ensuremath{-}1)$ generalizes to $\ensuremath{\sim}{\ensuremath{\gamma}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\gamma}}_{2}({e}^{i\ensuremath{\chi}(b)\ensuremath{-}\ensuremath{\omega}(b)}\ensuremath{-}1) +{\ensuremath{\Sigma}}_{12}{\ensuremath{\Gamma}}_{f}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{b}})$, where ${\ensuremath{\chi}}_{0}(\mathrm{b})$, the eikonal phase shift, is the leading term of an expansion of the complete phase, $\ensuremath{\chi}(b)+i\ensuremath{\omega}(b)$, in powers of inverse momentum. ${\ensuremath{\Sigma}}_{12}={q}^{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}[\frac{{\ensuremath{\gamma}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{k}_{a2}{\ensuremath{\gamma}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}{k}_{a1}}{({k}_{a2}\ifmmode\cdot\else\textperiodcentered\fi{}{k}_{a1})}\ensuremath{-}{\ensuremath{\gamma}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\gamma}}_{2}]$ is a double spin-flip Dirac operator involving the average momenta ${k}_{a1}$ and ${k}_{a2}$ of the two fermions. The double spin-flip amplitude, ${\ensuremath{\Gamma}}_{f}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{b}})$, is approximately given by ${(\stackrel{^}{q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{b})}^{2}{[{{\ensuremath{\chi}}_{0}}^{\ensuremath{'}}(b)]}^{2}$ and is thus entirely determined once the eikonal phase at infinite momentum is known. Additional corrections to the eikonal approximation of relative order ${s}^{\ensuremath{-}1}$ are also determined.

Structure of the Pomeron

At $t = 0$, the Pomeron-pole-plus-cuts series is shown to be a simple power series in the variable $\ensuremath{\upsilon} = \frac{{(\frac{s}{i{s}_{0}})}^{{\ensuremath{\alpha}}_{0}\ensuremath{-}1}}{\mathrm{ln}(\frac{s}{i{s}_{0}})}$, where ${\ensuremath{\alpha}}_{0}$ is the pole intercept. If ${\ensuremath{\alpha}}_{0}>1$, the series moves outside its circle of convergence as $s\ensuremath{\rightarrow}\ensuremath{\infty}$. Appropriate analytic continuation allows the series to saturate the Froissart bound. For $7.5\ensuremath{\lesssim}\sqrt{s}\ensuremath{\lesssim}53$ GeV, $\ensuremath{\upsilon}$ is almost constant, explaining the near-constancy of total cross sections. Excellent fits are obtained to the data for any function of $\ensuremath{\upsilon}$ with the right value and slope. These fits indicate that the dominant contributions come from the $\mathrm{PP}$ and $\mathrm{PPP}$ cuts rather than the pole term. An extension to $t\ensuremath{\ne}0$ yields an intriguing relation between the impact-parameter amplitude and the energy dependence of the forward amplitude.

The overlap slope and multilink interactions in multiperipherallike impact parameter amplitudes

We investigate the overlap function that results from a general multiperipherallike impact parameter structure for multiparticle production. In particular, we generalize the random-walk model in impact parameter to include interactions that extend over more than one link in the multiperipheral chain. It is found that reasonable values for both the magnitude and the shrinkage of the elastic slope can be easily obtained, due to collective effects in impact parameter space.

Analytic properties of impact parameter amplitudes and geometricals-channel models

In the framework of a recently proposed impact parameter representation a formalism is developed for two-body helicity amplitudes. Its relation withs-channel phenomenological models is examined by generalizing the results obtained with a simple dynamical input. Spin properties of the impact parameter amplitudes are also discussed.

Studies of inelastic molecular collisions using impact parameter methods. II. Exact and high-energy approximate calculations

Exact quantum mechanical impact parameter methods have recently been extended to include inelastic collisions through the use of an effective Hamiltonian. In this paper the nature of the impact parameter formulation will be explored through specific examples. First we consider the case of a calculation for He–para-He2 where the exact impact parameter amplitudes have been computed as a Neumann series of transition matrix elements obtained from an effective Hamiltonian close coupling calculation. The purpose of this calculation is to provide a standard of functional behavior for comparison with amplitudes computed by approximate methods. Such approximate calculations were performed for Ar–TlF scattering using both exponential and fractional high-energy forms for the amplitudes. The two approximations are compared in detail, and it is found that the cross sections resulting from the exponential approximation show reasonable agreement with other calculations for the system using conventional (noneffective) close coupling and various sudden approximations. Ranges of validity for the approximations are discussed and general characteristics of the impact parameter amplitudes and the related cross section integrands are summarized. Comparison is also made with model system calculations discussed in a previous publication. A general procedure is presented for smoothing the highly oscillatory cross section integrands that commonly arise in exponential impact parameter treatments.

Studies of inelastic molecular collisions using impact parameter methods. I. Model calculations

In this paper we explore the behavior of an exponential approximation to the impact parameter amplitudes for model systems with potential matrices of simple form. In these matrices the magnitude of the coupling is governed by the difference in the quantum numbers of the coupled states. In the limit of large numbers of states the amplitude functions may be expressed as single integrals, which in many cases can be evaluated in terms of known mathematical functions. The simple structure of this result allows us to analyze the general features of the amplitude functions and the resulting cross section integrands versus impact parameter. Specific examples for potentials with Gaussian and exponential radial dependence are used for illustration. An exploration of the effect of varying the potential parameters for the Gaussian potential is made. In addition, the behavior of functions related to the generalized cross sections for spectral line broadening are considered. Finally, we explore the generalization of the simple model to more elaborate coupling schemes.

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.

Impact parameter methods with effective potentials for inelastic molecular collisions

Previously employed exact quantum impact parameter methods for elastic scattering are extended to inelastic scattering. This is done within the framework of a recently developed effective potential model for the interaction of two molecules. The effective potential Veff and the transition matrix T are represented as integrals over a Bessel function times corresponding impact parameter amplitudes. Three particular impact parameter representations of the effective potential are discussed. It is shown that the form of the equation of motion for the T-matrix amplitude depends on the representation of the potential. This presents no difficulty once a particular choice of potential representation is made. A convenient characteristic wave number κ is introduced into the formulation, and it is shown that the resulting T-matrix is invariant with respect to its magnitude. The choice of κ is then argued on physical grounds. Several approximate equations of motion and their solutions are developed for the T-matrix amplitude. Some specific computational considerations are also discussed in conjunction with this development.

Analytic Properties of the Impact Parameter Amplitude

Analytic properties of the impact parameter amplitude H(s,b) are discussed in the b-plane. It is shown that H(s,b) has algebraic branch points at b=0 and their orders depend on the Regge trajectory, if the amplitde is expressed by the Regge pole. As an example a detailed discussion is given for the Veneziano amplitude. It is also stressed that the s-plane analyticity of H(s,b) is much simpler than that of the partial wave amplitude.

Born Series for Potential Scattering

The Born series for the $T$ matrix is considered with the class of potentials $V(\mathcal{r})={V}_{0}\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dtA}(t)\ifmmode\times\else\texttimes\fi{}{e}^{\ensuremath{-}t{r}^{2}}$. An explicit knowledge of $A(t)$ is not necessary since the resulting scattering amplitude can be expressed in terms of $V$. The exact expressions for the first and second Born approximations are utilized to approximate the general $n\mathrm{th}$-order term in the series. The series is explicitly summed to yield an expression for the scattering amplitude which reduces at high energy to the previous impact-parameter amplitude of Blankenbecler and Goldberger. The continuation of the series beyond its radius of convergence is discussed and exemplified with an exponential potential $V={V}_{0}{e}^{\ensuremath{-}\frac{r}{\ensuremath{\sigma}}}$. A few specific numerical examples are considered to illustrate the behavior of the resulting scattering cross section. Improvement of low-energy total cross sections is noted, particularly for repulsive or weakly attractive potentials.

Variational Estimates of Impact-Parameter Amplitudes

The high-energy impact-parameter amplitude is estimated variationally for a simple potential scattering, using the local-bound property of the reduced zero-energy operator. The result shows that the method may be effectively applied to composite-system scatterings with or without particle exchanges, and also to scattering problems at lower energies.

Impact-Parameter Amplitudes for Large-Angle Scatterings and Particle Correlations

Impact-parameter amplitudes for potential scatterings are calculated in the various approximations, and the results are compared with the exact phase-shift analysis up to the second diffraction minimum. General trends of the approximate cross sections are given which should be taken into account in the study of correlation effect in nucleon-nucleus scattering experiments. An improved form of the amplitude for momentum transfer up to the second diffraction maximum is given. Parametrization of the nuclear correlations in terms of the two-nucleon amplitude is discussed.

Impact Parametrization of the Scattering Amplitude

We study the new impact-parameter representation using ${J}_{0}(kb sin\ensuremath{\vartheta})$ instead of the conventional ${J}_{0}(2\mathrm{kb} sin\frac{1}{2}\ensuremath{\vartheta})$. We give group-theoretic arguments for this representation in the case of unequal-mass spinless scattering. Assuming full Mandelstam analyticity for the scattering amplitude, we describe the analytic properties of the impact parameter amplitude ${A}^{(\ifmmode\pm\else\textpm\fi{})}(k,b)$. For convenience, we have set the masses equal in the discussion of analyticity. The singularity structure of ${A}^{(\ifmmode\pm\else\textpm\fi{})}(k,b)$ governs the analyticity in the $l$-plane of $a_{l}^{}{}_{}{}^{(\ifmmode\pm\else\textpm\fi{})}(k)$, the direct $s$-channel partial-wave amplitude. We study conditions for there to be infinitely many one, or no (moving) Regge poles. Using Serber's optical-potential model, we find infinite families of poles of increasing order in the angular momentum plane. In connection with the Freedman and Wang amplitude, we also state conditions for there to be infinite, one, or no (moving) Lorentz poles. As an example, we quote the Coulomb potential which leads to an infinite number of Lorentz poles. We remark on the actual connection with the eikonal approximation, as well as on earlier work on the impact-parameter representation involving ${J}_{0}(2\mathrm{kb} sin\frac{1}{2}\ensuremath{\vartheta})$.