Jan Fischer

General laws of hadron scattering at very high energies

Abstract We review the general properties of particle scattering at extremely high energy and fixed momentum transfer, concentrating on features which can be derived from analyticity, crossing symmetry, unitarity and other general principles. The progress since 1973 is reported; it is pointed out that the rigorous approach yields fairly tight high-energy correlations between the phase and the modulus of the crossing-even as well as the crossing-odd forward scattering amplitude, without assuming any particular model of the interaction mechanism. The approach becomes particularly predictive when the formalism is supplemented by the phenomenological assumption that the total cross section rises unboundedly at high energies. Relation of the results to existing hadron-hadron scattering data and to future experiments is discussed.

On the validity and practical applicability of derivative analyticity relations

We examine derivative analyticity relations (DAR), which were originally proposed by Bronzan as an alternative to dispersion relations and in which the dispersion integral is replaced by a tangent series of derivatives. We characterize the class of functions satisfying DAR, and show that outside this class the dispersion integral represents a Borel‐like sum of tangent series. We point out difficulties connected with the application of DAR.

A new expression for the high‐energy upper bound on the scattering amplitude

The two Froissart–Martin high‐energy upper bounds for forward and nonforward scattering are combined into one formula under the additional assumption that the scattering amplitude is polynomially bounded in energy for all scattering angles inside the Lehmann–Martin ellipse. The method used presents a modification of that of Kinoshita, Loeffel, and Martin. The analogous bound for the scattering of particles with spin is obtained as well. Using the same method, a bound for the case of complex scattering angles is also derived and ways leading to its improvement by using the solution of the Dirichlet problem are suggested.

On an asymptotic estimate of the n-loop correction in perturbative QCD.

A recently proposed method of estimating the asymptotic behaviour of QCD perturbation theory coefficients is critically reviewed and shown to contain numerous invalid mathematical operations and unsubstantiated assumptions. We discuss in detail why this procedure, based solely on renormalization group (RG) considerations and analyticity constraints, cannot lead to such estimates. We stress the importance of correct renormalization scheme (RS) dependence of any meaningful asymptotic estimate and argue that the unambiguous summation of QCD perturbation expansions for physical quantities requires information from outside of perturbation theory itself.

Odderon classes - general criteria

Abstract Predictions based on axiomatic field theory are obtained from the assumption that the observed difference between the pp and the p p elastic scattering is caused by an “odderon-like” term in the scattering amplitude. A wide class of amplitudes violating the Pomeranchuk relation is considered and consequences for the behaviour of ϱ(pp) and ϱ( p p ) are drawn.

Improvement of the Froissart-Martin bound for complex scattering angles

Extension of the Froissart-Martin bound for complex scattering angles is improved using the solution of the Dirichlet boundary value problem for doubly connected domain. The Froissart-Martin bound for physical scattering angles is used as input value on one of the two boundaries. The obtained bound is valid in an ellipse smaller than the Lehmann-Martin one. Possiblities of further improvements and applications are discussed. Details will be published elsewhere (Chaichian M., Fischer J., Nelipa N. F.,Vrkoč I. Preprint of the University of Helsinki, HU-TFT-81-22, to be published).

Extension of the froissart bound to unphysical scattering angles

High-energy upper bound on elastic scattering amplitude at fixed complex angle is improved by making better estimates during derivation. A further improvement is reached by solving the Dirichlet boundary value problem in which the fixed-angle Froissart bound is used as input value on a part of the boundary. Further development of the method is discussed.

Asymptotic correlations between total cross-sections and forward scattering amplitudes

Recent experimental results on hadron-hadron total cross-sections and on the real parts of forward scattering amplitudes are discussed on the basis of rigorous model-independent theory. Some consequences of the asymptotic theorems obtained recently by the authors are presented.