Optical Potential for High-Energy Physics: Theory and Applications

Abstract

A field-theoretic exact two-body optical potential ${V}^{0}$ is defined, corresponding to the exact pseudopotential in nuclear physics. A small-angle, high-energy approximation for scattering amplitudes in the absence of resonances or bound states is suggested, based largely on investigations of Torgerson in a fairly realistic model field theory. This approximation, which corresponds to the eikonal or linearized WKB approximation in a nonrelativistic limit, involves only mass-shell values of ${V}^{0}$ and can be discussed in the framework of dispersion relations and analyticity. The longest range contributions to ${V}^{0}$ are one-pion exchange (when allowed) and the multipheripheral diagrams of Amati, Fubini, and Stanghellini; these contributions are termed "multipheripheral optical potential" (MOP). One possibility for the asymptotic high-energy limit of MOP brings in Regge poles through the multipheripheral diagrams. At energies which are not asymptotic, but are high enough to ensure the usefulness of the eikonal formalism, important non-pole contributions to ${V}^{0}$ are discussed. The difference between $\overline{p}p$ and $\mathrm{pp}$ elastic scattering is explained in such an energy region. As a natural consequence of the picture presented, one obtains distorted-wave Born-approximation (DWBA) correction formulas applicable to any small contributions in ${V}_{0}$; e.g., real part, spin flip, and charge exchange. A special case is the absorptive correction to the $\ensuremath{\rho}$ Regge-pole expression for charge exchange in ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}n$, which has been discussed in a previous paper. The Serber potential which accommodates large-$t$ behavior (although only $\ensuremath{-}\frac{t}{s}\ensuremath{\ll}1$ can be properly described by the eikonal expression) is shown to be a special case of MOP. If MOP is dominated by the Pomeranchuk Regge pole ($P$) in elastic $\mathrm{pp}$ scattering above $10 \frac{\mathrm{BeV}}{c}$, and if we ignore spin, a real part of the forward scattering amplitude for this case is obtained which agrees in sign, with the observed value, but is too small in magnitude; it has an energy dependence ${[\mathrm{ln}(\frac{s}{{s}_{0}})]}^{\ensuremath{-}1}$. The $\overline{p}p$ results should become identical to the $\mathrm{pp}$ ones for energies which are asymptotic for $\overline{p}p$ also. Similar results for the real part hold for all two-body reactions. In $\ensuremath{\pi}p$ scattering, a formalism incorporating spin properly into the eikonal method is presented, and in the asymptotic limit with no "anomalous-moment" terms in the Born approximation (as suggested by a Pomeranchuk pole), a spin-orbit coupling is obtained corresponding to use of the Dirac equation with a 4-vector static central potential. The resulting $\ensuremath{\pi}p$ spin-flip amplitude decreases with increasing energy like ${s}^{\ensuremath{-}\frac{1}{2}}$ relative to nonflip terms, but is presumably dominated by effects of secondary Regge Poles such as ${P}^{\ensuremath{'}}$. To describe multichannel reactions, and to obtain absorptive correction formulas including reactive damping, an exact multichannel optical potential ${{V}_{\mathrm{ij}}}^{0}$ is defined, and a matrix eikonal mass-shell approximation is proposed. Such a method is valid only when certain commutation relations are satisfied for the matrix Born approximation; these are satisfied, for example, if the $t$ dependence of all elements of this matrix is the same, which can be true in many cases if mass differences are ignored. Regge poles and the Byers-Yang model are considered in this context. To include resonances in the $s$ channel, possibilities for extending the eikonal formalism are discussed. A method of formulation utilizing dispersion relations for phase shifts allows an alternative, purely $S$-matrix approach to the eikonal approximation, but is physically more obscure than the field-theoretic and static-potential-theory approaches. Singularities in complex $J$ of the MOP-eikonal approach are explored. It is found that infinite numbers of branch cuts in $J$ correspond to absorptive (DWBA) corrections when Regge poles are used in MOP. An apparent paradox concerning results of Mandelstam on cancellation of cuts is discussed and a possible avenue of resolution, involving treating external particles as Regge poles with signature, is described.