Porter W. Johnson

Determination of the scattering amplitude from the differential cross-section and unitarity

When the differential cross-section for spin-zero elastic scattering is given, the elastic unitarity condition constitutes a nonlinear integral equation for the phase of the scattering amplitude. Existence and uniquences theorems for solutions of the equation were obtained by Newton and Martin. Some improvements of the Newton-Martin results on uniqueness and iterative construction of solutions are obtained. Certain details of rigour in the applications of Schauders theorem by Newton and by Martin are supplied. The case of inelastic spin-zero scattering is treated by adding a term to the unitarity condition to account for absorption. It is shown that in the inelastic region one may have infinitely many different scattering amplitudes with a given differential cross-section. This result is potentially important in phase-shift analysis, since it means that there is a “continuum ambiguity” in the determination of phases and elasticities from scattering data.

Construction of unitary, analytic scattering amplitudes (III). Practical application to αα scattering

We discuss a practical method for exploring continuum ambiguities in phase-shift analysis, and we apply our techniques to αα elastic scattering, above the inelastic threshold. We generate very considerable ambiguity corridors.

QCD asymptotics and kinematic thresholds in deep inelastic scattering

Abstract The Nachtmann moments of deep inelastic scattering structure functions are required by kinematics to contain a so far neglected threshold factor which is dependent on both n and q 2 . Its presence significantly affects the “moment analysis” in the usual QCD phenomenology and it resolves the difficulties connected with improper threshold behavior of the “ξ-scaling” analysis of structure functions.

Construction of analytic, unitary scattering amplitudes from a given differential cross-section : A refined analysis

We extend previous work concerning the construction of unitary scattering amplitudes that correspond to the scattering data at a given energy. The dispersive and absorptive parts are by construction analytic in cosϑ in the small and large Lehmann ellipses, respectively. The dispersive and absorptive parts obtained here, in contrast to those obtained before, are shown to have continuous derivatives on the boundary of their domains of analyticity. The continuum ambiguity in the determination of the scattering amplitude, which is associated with a lack of experimental information on the inelastic contribution to unitarity, is present here as well.

Comparison of asymptotically free theories with high-energy deep inelastic scattering data

Abstract Comparison of the predictionsof asymptotically free gauge theories on ϑ 1n vW2(x, q2)|ϑ 1n q2 in the small x region with recent data on high-energy muon-hadron scattering allows a fairly direct determination of the effective coupling constant. The result is α(q2)≲0.21/[1 + 0.14 1n q2 (GeV2)]. This is much smaller than previous indirect estimates.

Continuum ambiguity in the construction of unitary analytic amplitudes from fixed‐energy‐scattering data

The problem of determining the scattering amplitude for a given fixed‐energy elastic differential cross section is discussed in the spinless case. We show that when the energy is above the inelastic threshold, one may construct an infinte family of unitary scattering amplitudes, by appropriate variation of the elasticity parameters. These amplitudes are analytic in the cosine of the scattering angle throughout the Lehmann ellipse, and all correspond to the same cross section. Hence, even if the cross section is known exactly, there are infinitely many sets of phase shifts. Similar results have been obtained in earlier work, under conditions (on the cross section and elasticities) which seem to be physically unrealistic. In the present paper, the outstanding unrealistic assumptions are avoided. In particular, a finite number of zeros of the dispersive part are now allowed. Each zero reduces the continuum ambiguity by one elasticity parameter, but leaves infinitely many parameters to be varied independently.

Gaussian beam spread in biological tissues.

Flux density distributions were measured in large tissue sections illuminated with 633- and 1064-nm laser radiation delivered by an optical fiber. The results were modeled by solving the 2-D diffusion approximation for an incident Gaussian beam and fitting the data with nonlinear regression. It is shown that the radial average flux density is exponentially attenuated for an arbitrary incident irradiance profile.

Nonconservation of Energy and Loss of Determinism I. Infinitely Many Colliding Balls

An infinite number of elastically colliding balls is considered in a classical, and then in a relativistic setting. Energy and momentum are not necessarily conserved globally, even though each collision does separately conserve them. This result holds in particular when the total mass of all the balls is finite, and even when the spatial extent and temporal duration of the process are also finite. Further, the process is shown to be indeterministic: there is an arbitrary parameter in the general solution that corresponds to the injection of an arbitrary amount of energy (classically), or energy-momentum (relativistically), into the system at the point of accumulation of the locations of the balls. Specific examples are given that illustrate these counter-intuitive results, including one in which all the balls move with the same velocity after every collision has taken place.

Sommerfeld-Watson representation for double-spectral functions: III. Crossing symmetric pion-pion scattering amplitude with Regge poles

We discuss, for the case of pion-pion scattering, a closed system of equations which may be used for a self-consistent calculation of partial-wave amplitudes. It is shown that, for a given sufficiently small input function, the equations have a locally unique solution in a particular Banach space of doubly Hölder continuous partial wave amplitudes. At a fixed point, the scattering amplitude is shown to satisfy both a crossing symmetric unsubtracted Mandelstam representation and the elastic unitarity condition. In this initial study the partial-wave amplitudes are holomorphic in the right half complex angular-momentum plane.

Nonconservation of Energy and Loss of Determinism II. Colliding with an Open Set

An actual infinity of colliding balls can be in a configuration in which the laws of mechanics lead to logical inconsistency. It is argued that one should therefore limit the domain of these laws to a finite, or only a potentially infinite number of elements. With this restriction indeterminism, energy nonconservation and creatio ex nihilo no longer occur. A numerical analysis of finite systems of colliding balls is given, and the asymptotic behaviour that corresponds to the potentially infinite system is inferred.