An exact version of Pagels's sum rule for neutral-pion decay is derived for massless pions. It is inconsistent, however, in the weak-coupling limit. If we modify this limit by including the $\sigma$ meson, then the sum rule is consistent and also yields the physically reasonable relation ${\kappa }_p=-{\kappa }_n$.

The contribution of high-mass states in the dispersion relation for decay amplitudes is considered. A finite dispersion relation (FDR) implemented by duality and finite-energy sum rules (FESR), is suggested as a useful tool for such calculations, and an application to the decay $\omega \to \pi \pi \gamma$ is done in some detail.

The qualitative behavior of large-angle scattering by a rotationally invariant optical potential is investigated using the WKB approximation. It is shown that the strong $s$ dependence exhibited by elastic proton-proton scattering requires an energy-dependent potential. The source of the inaccuracy of the eikonal approximation, at large angles, is found by comparing eikonal phase shifts with those given by the WKB approximation.

The decay modes ${K_1}^0\to {\mu }^{+}{\mu }^{-}$ and ${K_2}^0\to {\mu }^{+}{\mu }^{-}$ are discussed within the framework of first-order weak combined with fourth-order electromagnetic interactions. The possibility of obtaining lower bounds to the corresponding decay rates is examined, and estimates of the lower bounds are given in a perturbation-theory model. Possible enhancement effects due to strong interactions have been estimated through a simple model of final-state interactions. The assumption of $\mathrm{CP}$ invariance is made throughout.

It is shown that the ordinary Born approximation for $\mathrm{pn}$ and $p\overline{p}$ charge-exchange scattering correctly accounts for (1) the shape of the forward peak for $0\le \mathrm{}(-t)\mathrm{}\le \frac{{\mu }^2}{2}$ at $P_L=8\mathrm{} \frac{\mathrm{GeV}}{c}$, and (2) the energy dependence of the cross sections at $t=0\mathrm{}$ in the energy range $P_L=2-8\mathrm{} \frac{\mathrm{GeV}}{c}$. This result is analogous to the well-known success of the electric Born approximation in ${\pi }^{+}$ photoproduction. It is then shown that the simplest interpretation of this surprising result within the framework of Regge-pole theory is in terms of the fixed poles which are allowed by unitarity in hadronic amplitudes at certain nonsense points of right signature. Finally, it is shown how such a fixed pole at a nonsense point of one helicity amplitude affects another amplitude for which the corresponding point is sense.

The new invariant amplitudes of Bardeen and Tung for nucleon Compton scattering, which are free of both kinematic singularities and zeros, and zeros, are examined. The forward scattering amplitude, and the continuous-dispersion sum rules derived therefrom, are obtained. Using the data of a recent calculation by Damashek and Gilman, tests of these sum rules are shown to be quite satisfactory, indicating the validity of the dispersion relation, the good parametrization of the forward proton Compton scattering amplitude, and the presence of a $J=0\mathrm{}$ fixed pole within the accuracy of present experiment.

The inelastic channels $K^{*}\pi$ and $K$ are included in a three-channel model of the $I=\frac{1}{2} p$-wave $K\pi$ amplitude. Feynman diagrams for vector-meson exchange are used as input to multichannel $\frac{N}{D}$ equations, which are solved to obtain the scattering amplitude. Coupling constants which cannot be measured experimentally or calculated using $\mathrm{SU}\left(3\right)\mathrm{}$ are obtained by employing $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$ in the rest system of one of the particles at a given vertex. The $K^{*}\pi$ channel, neglected in previous calculations, is observed to influence the amplitude strongly, but the width of the $K^{*}\mathrm{}\left(890\right)\mathrm{}$ is calculated to be 210 MeV or about four times the experimental value.

The predictions of the Amati-Bertocchi-Fubini-Stanghellini-Tonin multiperipheral model based on pion pole dominance are calculated quantitatively and compared with experimental data, by using the Veneziano representation for the $\pi -\pi$ amplitude, with a cutoff at the mass of the $g$ resonance for the input kernel, and neglecting off-shell effects. We find ${\alpha }_P\mathrm{}\left(0\right)=0.30\mathrm{}$, ${\alpha }_{\rho }\mathrm{}\left(0\right)=0.16\mathrm{}$, an average multiplicity growing as 0.74 $\ln s$, an elasticity factor of 0.69, an average pion-pair transverse momentum equal to 0.43 $\frac{\mathrm{GeV}}{c}$, and an average invariant mass squared equal to 0.84 Ge${\mathrm{V}}^2$. The only serious discrepancies with experiment are the trajectory heights, which correspond to a kernel strength $2\frac{1}{2}-5$ times too weak. The inclusion of the high-subenergy contribution and the inclusion of off-shell effects by several methods of off-shell continuation are considered, but none is found to be satisfactory. The importance of interference terms, due to different ways of arranging the final-state particles along the multiperipheral chain, is discussed.

We elaborate on the application of the parton concepts to inelastic lepton-nucleon scattering. We propose a series of sum rules connecting the structure function $\nu W_2(\nu ,Q^2)$ and the properties of the partons. In particular, a sum rule is derived relating the structure function of the neutron and the proton which depends only on the charge distribution of the partons. We then attempt to generalize the quark-parton model of Bjorken and Paschos (BP) so as to fit the data on inelastic $e-p$ scattering which are presently available. We find that we are unable to fit the data with this model. A new model is therefore developed in which the partons are not identified with quarks, and the minimum number of partons is four (whereas in the BP model it is three). It is found that this model gives an excellent fit to the data in the approximation that the scale invariance is true. It is pointed out that the minimum number of partons plays an important role in parton models. Our fits also show that the mean square charge per parton is very small, which rules out any reasonable possibility of associating individual quarks with partons.

We present a treatment of $K_{l3\mathrm{}}$ decays based on current commutators and using a covariant sum-rule formulation. Working in a general kinematical configuration and dispersing on a line (in the plane of the squared pion mass and of the momentum transfer) which belongs to a family of curves all parallel to the "steepest" (kaon rest frame) parabola, we write down definite expressions for the form factors $f_{+}$ and $f_{-}$. The relation of our procedure to other recently reported approaches is discussed.

The distribution over the number of prongs in inelastic hadronic collisions can be calculated by assuming (i) that the multiplicities of produced pions form a Poisson distribution, and (ii) that all the isospin final states are equally probable. The set of distributions for ${\pi }^{-}p$ inelastic collisions have been calculated, and agree well with experiments for all energies.

Recent experiments reporting the $2\pi$ decay mode of the $\omega$ meson and its interference with that mode of the $\rho$ meson are analyzed in terms of the propagator method for mixing of particle states. Both vector mixing and mass mixing are included, and the energy dependence of the $\rho$ width is explicitly taken into account. It is shown that the effect as observed in the reaction $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ can be accounted for in terms of available parameters. The consequent determination of the parameters makes possible an analysis of strong-interaction phenomena of the same type, giving information on the relative phases of $\omega$ and $\rho$ production amplitudes. This is applied to the $\rho -\omega$ interference observed in ${\pi }^{+}p\to {\pi }^{+}{\pi }^{-}{\Delta }^{++}$. Other experiments are suggested.

Problems associated with the passage of high-energy charged-particle pairs through matter (the Chudakoff effect) are investigated. It is shown that such an effect could not produce lightly ionized tracks which would be mistaken for quarks in cosmic-ray experiments. Treatments of polarization of the medium and Landau straggling are given.

A phenomenological study of the strong reaction ${\overline{\Omega }}^{+}+p\to K^{+}+K^{+}+K^0$ with low-energy ${\overline{\Omega }}^{+}\text{'}\mathrm{s}$ may determine the parity of ${\overline{\Omega }}^{+}$. Assuming $S$-wave interaction in the initial state, the presence of the final-state configuration in which the $K^0$ is at rest in the c.m. system would indicate that the parity of ${\overline{\Omega }}^{+}$ is odd. Other significant predictions based on features in the Dalitz plot are also presented.

An impulse model is presented for the reactions $K^{-}+{\mathrm{He}}^4\to Y+\pi +{\mathrm{H}}^3$. The model incorporates the low-energy $\overline{K}$-nucleon parametrization of Kim. The calculation is done for capture at rest from both $s$ and $p$ atomic orbital states and also for kaon interactions in flight. The results are compared with experiment for the final states $\Lambda {\pi }^{-}{\mathrm{He}}^3$ and ${\Sigma }^{+}{\pi }^{-}{\mathrm{H}}^3$ from at-rest kaon interactions.

The $K^{\pm }-p$ elastic scattering was measured in a small angular region near 180° ($-0.995\mathrm{}\ge cos{\theta }^{*}\ge -1.0\mathrm{}$) at six momenta from 0.41 to 0.72 $\frac{\mathrm{GeV}}{c}$. The forward-scattered protons were detected with scintillation counters and identified by time-of-flight measurements. The $K^{+}-p$ elastic differential cross section at 180° is found to be essentially constant with a value near 0.9 mb/sr. The sum of the $P$-wave phase shifts ($2{\delta }_{\frac{3}{2}}+{\delta }_{\frac{1}{2}}$) is found to be negative in this momentum region. The $K^{-}$ elastic differential cross section is found to dip to a minimum (0±0.2 mb/sr) at 0.66 $\frac{\mathrm{GeV}}{c}$.

A connection between an idea of Dashen's concerning $\mathrm{SU}\left(3\right)\mathrm{}$ mass splitting and a similar treatment based upon the algebraic approach of Cabibbo, Horwitz, and Ne'eman is pointed out. Comment is also made on results obtained recently by Buccella et al. concerning hadron electromagnetic mass splittings.

It is shown that the assumption of vanishing commutators between space components of currents is incompatible with the conjecture connecting the observed $\mathrm{SU}\left(6\right)\mathrm{}$-like hadron spectrum with the algebra of the currents and its contraction.

An $O(4, 2)$ model for the electromagnetic transitions of the $I=\frac{1}{2}$ nucleon resonances is inserted into a finite-energy sum rule for forward off-shell Compton scattering in order to estimate the neutron-proton mass splitting. The result is encouraging. Comments are made on the importance of the nonpole terms and of the high-momentum-transfer behavior of the form factor in the case of the pion mass difference.