Finite-energy sum rules with generalized weight functions are discussed with emphasis on methods for obtaining precise determination of high-energy parameters from low-energy data. Regge parameters for $t=0\mathrm{}$ $\pi N$ charge-exchange amplitudes are deduced as an example from a new sum rule.

A determination is made of the constraints which are imposed upon the low-energy ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ scattering parameters in order that the $S$ matrix may satisfy the known static deuteron properties. A multichannel effective-range representation is used to derive a relationship between the phase parameters ${\delta }_S$, ${\delta }_D$, and ${\epsilon }_1$ at "low" energies. The relationship is violated by all of the present low-energy phenomenological determinations. This indicates a discrepancy between the existing low-energy $\mathrm{np}$ scattering data and the static deuteron properties, and suggests that low-energy energy-dependent phenomenology should manifestly contain the deuteron constraints.

Total cross sections for the production of single $W$ mesons in electron-positron colliding beams and in the electron scattering experiments have been calculated. If the first type of cross sections are estimated by calculating the simplest lowest-order dominant Feynman graph, one finds that they are remarkably large (∼${10}^{-30\mathrm{}}$ ${\mathrm{cm}}^2$). Indeed, a calculation in the literature which includes all lowest-order contributions apparently confirms this result. These estimates are in striking disagreement with calculations for the related process of neutrino-induced production of single $W$ mesons. These suggest a cross section of ∼${10}^{-37\mathrm{}}$ ${\mathrm{cm}}^2$, a result which is to be expected from a naive power-counting argument. We have calculated the cross sections and have found that they are in fact quite small and of the same order of magnitude as the neutrino-induced cross sections. We present a general gauge-invariance argument which illustrates how almost any trivial error in the calculation can easily lead to errors in the result of at least ${\mathrm{}\left[\right(\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n})/(\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{positron}\left)\right]}^2$∼${10}^6$. Basically, gauge invariance demands a critical cancellation in the differential cross section when the mass of the exchanged photon ($K^2$) becomes very small. If such a cancellation does not take place (because of some small error), the region around $K^2=0\mathrm{}$ is enormously enhanced by the smallness of the mass of the positron, and large errors are induced in the final result. An examination of the calculations that lead to a large total cross section does indeed show that each violates the constraint of gauge invariance.

A framework is suggested for interpreting processes of the type $a+b\to a+b+\left(n \mathrm{poins}\right)$ at energies from tens to hundreds of GeV. The model predicts, without any adjustable parameters, the observed average transverse momenta of the pions in the entire range, as well as the existence and observed masses of "fire-balls." The advantages of this model over the multi-Regge models is discussed.

A tractable, relativistic theory for the two-nucleon system is constructed. As a starting point, the Bethe-Salpeter (BS) equation is utilized. In order to reduce the BS equation to three dimensions, the Blankenbecler-Sugar method is generalized to include spin-½ particles. An instantaneous-interaction approximation to the BS equation is also investigated, and results similar to those obtained with the Blankenbecler-Sugar method are derived. Finally, as an application, the generalized potential or interaction kernel is approximated to order $g^2$ and compared to the one-pion-exchange potential (OPEP) used in conjunction with the Schrödinger equation. The singlet states are treated numerically, and results are presented which show that for a lab kinetic energy of 400 MeV, the phase shifts calculated from the relativistic theory differ from those calculated by solving the Lippmann-Schwinger or Schrödinger equation with OPEP by about 20%.

A formula relating the level shifts and widths to the meson-nucleus scattering lengths is examined for a momentum-dependent potential of general form, using a simple argument similar to that of Bethe and deHoffmann and Coulomb effective-range theory.

Expressions are given for the electromagnetic final-state interaction contribution to $D$, the triple-product correlation coefficient, in the nuclear $\beta$-decay transitions ½ → ½, $\frac{3}{2}\to \frac{3}{2}$, ${\frac{3}{2}}^{\pm }\to {\frac{1}{2}}^{\pm }$, ${\frac{5}{2}}^{\pm }\to {\frac{3}{2}}^{\pm }$, and $1^{+}$→$0^{+}$ to all orders of $\frac{Z\alpha }{v_e}$ in the Coulomb interaction.

The reaction ${\pi }^{-}p\to \eta n$ has been observed through the decay mode $\eta \to 2\gamma$ at $T_{{\pi }^{-}}=592, 655, 704, 875, 975, 1117, \mathrm{and} 1300$ MeV. The detection apparatus was a cubic array of six steel-plate optical spark chambers that completely surrounded a liquid-hydrogen target. We identified events attributed to the decay of an $\eta$ by the large c.m. opening angle between the two showers generated in the steel plates by the decay photons. We have calculated the total cross section for $\eta$ production, which is proportional to the number of events under the large-angle peak in the opening-angle distribution. The total cross section rises steeply from threshold to a maximum of about 2.4 mb at 650 MeV, and then falls gradually to about 0.66 mb at 1300 MeV. The differential cross section was obtained by taking the coefficients of a Legendre-polynomial fit to the angular distribution of bisectors of selected two-shower events, and converting them to the coefficients of the $\eta$ c.m. angular distribution. The differential cross section is found to be isotropic at 592 MeV, to require terms through $P_2\left({cos\theta }_{\eta }\right)$ between 655 and 975 MeV, and to have a forward peak fitted by terms through $P_3\left({cos\theta }_{\eta }\right)$ at 1117 MeV and through $P_4\left({cos\theta }_{\eta }\right)$ at 1300 MeV. It is suggested that production at threshold is predominantly through an $S$ state, with some $P_1$ and $D_3$ waves entering by 655 MeV. We suggest that all the absorption in the $S_{11}$ $\pi N$ state can be explained by the $\eta$ production.

A general phenomenological method for studying a two-pion mass spectrum is developed; it is shown that, without assumptions about the production mechanism of the two-pion system, no significant upper limit on the $\omega \to {\pi }^{+}{\pi }^{-}$ branching ratio can be set with present-day experiments. In general, a lower limit may be set if a significant effect is seen. This method is applied to data containing more than $8000\omega \to {\pi }^{+}{\pi }^{-}{\pi }^0$ events. The lower-momentum half of the sample, which shows a significant $\omega \to {\pi }^{+}{\pi }^{-}$ signal, was published previously but is here reanalyzed to set a lower limit on the $\omega \to {\pi }^{+}{\pi }^{-}$ branching ratio. The new data at higher momenta show no significant $\omega \to {\pi }^{+}{\pi }^{-}$ signal. The results from the various momenta are shown to be consistent.