A review is given of recent developments in compositeness arguments of particles in the framework of Lagrangian field theory and S-matrix theory.

Compositeness Criteria of Particles in Quantum Field Theory and S‐Matrix Theory

The theoretical predictions for the subtraction constants lead to a very accurate dispersive representation of the ππ scattering amplitude below 0.8 GeV. The extension of this representation up to the maximum energy of validity of the Roy equations (1.15 GeV) requires a more precise input at high energies. In this paper we determine the trajectories and residues of the leading Regge contributions to the ππ amplitude (Pomeron, f and ρ), using factorization, phenomenological parametrizations of the πN and NN total cross sections at high energy, and a set of sum rules which connect the high and low energy properties of ππ scattering. We find that nonleading Regge terms are necessary in order to achieve a smooth transition from the partial waves to the Regge representation at or below 2 GeV. We obtain thus a Regge representation consistent both with the experimental information at high energies and the Roy equations for the partial waves with l≤4. The uncertainties in our result for the Regge parameters are sizable, but in the solutions of the Roy equations these only manifest themselves above the $K\hspace {0.03cm}\overline {\rule [0.7em]{0.6em}{0em}}\hspace {-0.8em}K$
 threshold.

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Regge analysis of the ππ scattering amplitude

Colliding high-energy hadrons either produce new particles or scatter elastically with their quantum numbers conserved and no other particles produced. We consider the latter case here. Although inelastic processes dominate at high energies, elastic scattering contributes considerably (18–25%) to the total cross section. Its share first decreases and then increases at higher energies. Small-angle scattering prevails at all energies. Some characteristic features can be seen that provide information on the geometrical structure of the colliding particles and the relevant dynamical mechanisms. The steep Gaussian peak at small angles is followed by the exponential (Orear) regime with some shoulders and dips, and then by a power-law decrease. Results from various theoretical approaches are compared with experimental data. Phenomenological models claiming to describe this process are reviewed. The unitarity condition predicts an exponential fall for the differential cross section with an additional substructure to occur exactly between the low momentum transfer diffraction cone and a power-law, hard parton scattering regime under high momentum transfer. Data on the interference of the Coulomb and nuclear parts of amplitudes at extremely small angles provide the value of the real part of the forward scattering amplitude. The real part of the elastic scattering amplitude and the contribution of inelastic processes to the imaginary part of this amplitude (the so-called overlap function) are also discussed. Problems related to the scaling behavior of the differential cross section are considered. The power-law regime at highest momentum transfer is briefly described.

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Elastic scattering of hadrons

This review describes the development of the physics of hadronic cross sections up to recent LHC results and cosmic ray experiments. We present here a comprehensive review – written with a historical perspective – about total cross sections from medium to the highest energies explored experimentally and studied through a variety of methods and theoretical models for over 60 years. We begin by recalling the analytic properties of the elastic amplitude and the theorems about the asymptotic behavior of the total cross section. A discussion of how proton–proton cross sections are extracted from cosmic rays at higher than accelerator energies and help the study of these asymptotic limits, is presented. This is followed by a description of the advent of particle colliders, through which high energies and unmatched experimental precisions have been attained. Thus the measured hadronic elastic and total cross sections have become crucial instruments to probe the so called soft part of QCD physics, where quarks and gluons are confined, and have led to test and refine Regge behavior and a number of diffractive models. As the c.m. energy increases, the total cross section also probes the transition into hard scattering describable with perturbative QCD, the so-called mini-jet region. Further tests are provided by cross section measurements of $$\gamma p$$
 , $$\gamma ^* p$$
 and $$\gamma ^* \gamma ^*$$
 for models based on vector meson dominance, scaling limits of virtual photons at high $$Q^2$$
 and the BFKL formalism. Models interpolating from virtual to real photons are also tested. It seems to us to be a necessary task to explore bit-by-bit the rigorous consequences of analyticity, unitarity and crossing. Who knows if someday one will not be able to reassemble the pieces of the puzzle. – A. Martin and F. Cheung, based on 1967 A.M. Lectures at Brandeis Summer School and Lectures at SUNY and Stony Brook (Martin and Cheung in Analyticity properties and bounds of the scattering amplitudes. Gordon and Breach Science, New York, 1970).

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Introduction to the physics of the total cross-section at LHC : A Review of Data and Models

The main (old and new) results about analyticity properties of two-body transition amplitudes deducible from general local field theory are reviewed. Most of the proofs are included as well as a short chapter on applications.

Present State of Rigorous Analytic Properties of Scattering Amplitudes

From unitarity alone a lower bound for the derivative of the absorptive part of the forward scattering amplitude with respect to the momentum transfer is obtained, in terms of the elastic and total cross sections. Comparison with high-energy scattering experiments shows that the actual value of this derivative is rather close to the lower bound, which provides some information on the partial-wave distribution. Our result can also be used to obtain consistency requirements on theoretical models. If Regge behavior is assumed for high-energy scattering, namely, $F(s, t)\ensuremath{\simeq}f(t){s}^{\ensuremath{\alpha}(t)}$, then one can show that either ${\ensuremath{\alpha}}^{\ensuremath{'}}(0)g~\ensuremath{\epsilon}g0$ or $\ensuremath{\alpha}(t)\ensuremath{\equiv}\mathrm{const}.$

Unitarity bounds of the scattering amplitude and the diffraction peak

Extending the arguments by Goebel and Sakita, it is shown in a general framework that a pole of the proper vertex function does not lead to a pole in the scattering amplitude. Connections between zeros of the propagator, poles of the proper vertex function, and upper bounds on the coupling constant are discussed in rather general terms as well as in terms of the Zachariasen model. By making use of the analytic continuation of the partial-wave scattering amplitude into the complex angular-momentum plane, a possible physical interpretation of the pole of the vertex function is given.

Poles in the Vertex Function, Zeros of the Propagator, and Bounds on Coupling Constants

By making use of the phase representation, the relation between the high-energy behavior of the symmetric forward scattering amplitude $F(x)$ and the asymptotic properties of the ratio $cot\ensuremath{\delta}=\frac{\mathrm{Re}F}{\mathrm{Im} F}$ is discussed. Starting assumptions are dispersion relations and the Greenberg-Low bound. Lower bounds as well as upper bounds are derived. Under the assumption of the Froissart bound, it is shown that $cot\ensuremath{\delta}$ cannot stay indefinitely greater than an arbitrarily small positive number. Also if the total cross section decreases steadily to a finite limit, but slower than $\frac{\mathrm{const}}{E}$ the real part must tend to $\ensuremath{-}\ensuremath{\infty}$. The results are discussed in connection with those of Khuri and Kinoshita. The unsymmetric case is also treated by the same methods.

S. W. MacDowell

Papers

Unitarity bounds of the scattering amplitude and the diffraction peak

Poles in the Vertex Function, Zeros of the Propagator, and Bounds on Coupling Constants

Phase Representation and High-Energy Behavior of the Forward Scattering Amplitude