Approximation of scattering amplitudes using analyticity and experimental data

The representation of scattering amplitudes by analytic functions is studied with particular emphasis on the generality of the method. An application of the Hilbert-space approximation techniques leads to a unified procedure, which properly takes into account the statistical nature of the experimental data.     

Unitary amplitudes with cut-plane analyticity from given scattering data

We continue the study of constructing the scattering amplitude for scalar particles from elastic scattering data at a given energy. Here we require the scattering amplitude, f (z), to be analytic in a suitably cut z-plane; (z is the cosine of the scattering angle). We find that, in the elastic domain, the unitarity constraint is satisfied by a continuum of amplitudes, all corresponding to the same elastic scattering data. This continuum ambiguity in determination of f (z), which was noted earlier in formulations based on a smaller domain on analyticity, is associated with lack of knowledge of the contribution to unitarity from inelastic channels. We discuss the problem of incorporating smooth energy dependence in the determination of the scattering amplitude over some range of energy, and show that, under certain restrictions on the scattering data, there is a continuum of amplitudes having smooth energy dependence.     

Constraints on reggeon amplitudes from analyticity and planar unitarity

Using planar dual amplitudes as a guide, we discuss some features of reggeon amplitudes which are relevant in the context of the topological expansion. We look into the analytic properties and, in particular, discuss the validity of finite-mass sum rules for reggeon-reggeon scattering. We investigate the form taken by planar unitarity when a multiperipheral assumption is added. The integral equations obtained are not of the standard Chew-Goldberger-Low type. We find that pure pole-type solutions (i.e. without Regge cuts) to planar unitarity are possible in a way consistent with the symmetry and factorization properties of reggeon-reggeon amplitudes. The appearance of “good” FMSR in the unitarity integrals follows from a careful treatment of phase space — all possible configurations are counted uniquely — and is crucial in achieving the cut cancellation. Throughout the paper we emphasize various subtle points that have been overlooked in the literature.     

Towards a dynamical interpretation of dual amplitudes with Mandelstam analyticity

The infinite momentum frame loop variable x should be identified with the variable appearing in the integral representation of dual amplitudes with Mandelstam analyticity. Large angle and Regge behavior are discussed under this aspect.     

Geometrical scaling and the quark model in elastic Kp, πp and pp scattering

By combining the idea of geometrical scaling with the naive quark model we arrive at a very simple connection between high energy Kp, πp and pp elastic scattering.     

Geometrical scaling at inelastic threshold

The possibility of geometrical scaling being a low energy phenomenon is investigated using phase shift data. For exotic reactions, in particular in K⁺p scattering, scaling is observed and the high energy (NAL-ISR)GS curve agrees with phase shift inelasticity η down to the inelastic threshold. For reactions with resonances the high energy curves average the phase shift inelasticities. The consequences of these results are discussed and predictions for ππ and Kπ scattering are presented.     

Analytic proton-nucleus scattering amplitudes

Analytic expressions for the elastic nucleon-nucleus amplitudes F(q) and G(q) are derived. They are based on analytic approximations of the nuclear profile functions in the Glauber formulation. Our amplitudes are sums of terms with a familiar structure, black-sphere Bessel functions multiplied by form factors accounting for the diffuseness of the nuclear surface. Their accuracy is tested with the Woods-Saxon density and found to be excellent for momentum transfers q? 5fm^?1.     

Unitarity structure of scattering amplitudes

As a first step towards constructing scattering amplitudes satisfying unitarity, analyticity and crossing symmetry, we derive a linear non-singular integral equation for the total scattering amplitude which is equivalent to the unitarity condition. For this purpose we use the partial-waveN/D representation (with inelasticity) and the convolution theorem for Legendre transforms. We also discuss briefly the choice of two functionsN(s, cos Θ),C(s, cos Θ) which determine the unitary scattering amplitude through the integral equation. These functions may hopefully be chosen so that the analyticity and crossing symmetry requirements are satisfied.

     

Quark-hadron duality and meson-meson scattering amplitudes

We investigate a special concept of quarkhadron duality in meson physics at low and intermediate energies. With a relativistic propagator for confined quarks, we calculated masses and widths of the mesons which areS- andP-wave states ofq¯q (J ^P=0^?, 1^?, 0⁺, 1^+c, 2⁺) as well as meson-meson scattering amplitudes at energies up totrs ≤ 1.3 GeV by including light pseudoscalar and vector mesons into the interaction dynamics. Our investigation shows that the influence of the light meson dynamics resutls in:

1. a contribution to the constituent quark mass of approximately 200 MeV.
2. the appearance of a new soft confinement force barrier responsible for the formation of the highly excited meson states.

     

Dual pairs and representations of scattering amplitudes

A procedure for constructing scattering amplitudes for production processes that is exactly unitary, preserves the bosonic character of the many-particle states, and is invariant with respect to an underlying symmetry group is given. Two simple models, dealing with isospin internal symmetry and the two-dimensional Euclidean space group are presented which illustrate how scattering amplitudes can be represented as matrix elements of groups whose action commutes with the action of the invariance group on the relevant Fock space.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.Work supported in part by the Department of Energy.     

Double-cut of scattering amplitudes and Stokes' Theorem

We show how Stokes' Theorem, in the fashion of the Generalised Cauchy Formula, can be applied for computing double-cut integrals of one-loop amplitudes analytically. It implies the evaluation of phase-space integrals of rational functions in two complex-conjugated variables, which are simply computed by an indefinite integration in a single variable, followed by Cauchy's Residue integration in the conjugated one. The method is suitable for the cut-construction of the coefficients of 2-point functions entering the decomposition of one-loop amplitudes in terms of scalar master integrals.     

Forward pion-nucleus elastic scattering amplitudes

Values for the real part of the forward scattering amplitude have been obtained using dispersion relations from an analysis of pion-nucleus total cross sections for ⁴He, ⁶Li and ¹²C. The results are compared with measured values and those predicted by simple optical models. The dependence of the derived values for the forward scattering amplitudes on uncertainties in the measured cross sections is also discussed.     

Pion-nucleon scattering amplitudes from data and fixedt-analyticity

Summary of recent work of the Karlsruhe group on the determination of N amplitudes at high energies from experimental data and fixed-t analyticity.Presented at the Symposium on Hadron-Hadron Scattering at High Energies, Liblice, Czechoslovakia, June 16–21, 1975. Supported by Bundesministerium für Forschung und Technologie.The author is very grateful toJ. Fischer, J. Piút, O. Dumbrajs and other participants of the symposium for discussions.     

Poles and zeros in ππ scattering amplitudes

We draw attention to the remarkable systematics exhibited in the ππ amplitudes provided by the CERN-Munich data. It is shown that the existence of resonance poles with specific spin and mass values and the behaviours of the amplitude zeros in the Mandelstam plane give strong support to the ideas associated with dual resonance models. It is emphasized that some gauge-like conditions of dual resonance models play an essential role in these considerations.     

Geometrical scaling and the real part of the Pomeron

Consequences of the hyptohesis of geometrical scaling of the inelastic overlap function applied to the Pomeron amplitude are discussed. From analyticity and crossing symmetry some predictions are given for the asymptotic real part of the Pomeron.Presented at the Symposium on Hadron-Hadron Scattering at High Energies, Liblice, Czechoslovakia, June 16–21, 1975.     

Real parts of scattering amplitudes and Regge phenomenology

High energy scattering amplitudes are computed using the Harari model for the imaginary parts and fixed t dispersion relations for the real parts. This procedure induces a corrective term to the real part of the Regge amplitudes whose effects are investigated in π-nucleon and K-nucleon charge exchange reactions.     

Glauber exchange amplitudes for electron-helium scattering

An exact closed form expression is obtained for the Glauber-Bonham-Ochkur exchange amplitudes for the electron helium scattering. The results thus obtained are compared with the results from Ochkur approximation and are used further for obtaining closed form expressions for the differential and total exchange cross-sections for the elastic scattering of electrons from the ground state of helium atom.